diff --git a/-9E1T4oBgHgl3EQfUwOX/content/tmp_files/2301.03094v1.pdf.txt b/-9E1T4oBgHgl3EQfUwOX/content/tmp_files/2301.03094v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d36c8ed4f06196a03ff6de5fb6cb43eea76b09c --- /dev/null +++ b/-9E1T4oBgHgl3EQfUwOX/content/tmp_files/2301.03094v1.pdf.txt @@ -0,0 +1,666 @@ +A Divide-Align-Conquer Strategy for Program +Synthesis +Jonas Witt +Department of General Psychology and Methodology +University of Bamberg +Stef Rasing +Department of Computer Science +TU Delft +Sebastijan Dumanˇci´c +Department of Computer Science +TU Delft +Tias Guns +Department of Computer Science +KU Leuven +Claus-Christian Carbon +Department of General Psychology and Methodology +University of Bamberg +Abstract +A major bottleneck in search-based program synthesis is the exponentially growing +search space which makes learning large programs intractable. Humans mitigate +this problem by leveraging the compositional nature of the real world: In structured +domains, a logical specification can often be decomposed into smaller, complemen- +tary solution programs. We show that compositional segmentation can be applied +in the programming by examples setting to divide the search for large programs +across multiple smaller program synthesis problems. For each example, we search +for a decomposition into smaller units which maximizes the reconstruction ac- +curacy in the output under a latent task program. A structural alignment of the +constituent parts in the input and output leads to pairwise correspondences used to +guide the program synthesis search. In order to align the input/output structures, +we make use of the Structure-Mapping Theory (SMT), a formal model of human +analogical reasoning which originated in the cognitive sciences. We show that +decomposition-driven program synthesis with structural alignment outperforms +Inductive Logic Programming (ILP) baselines on string transformation tasks even +with minimal knowledge priors. Unlike existing methods, the predictive accuracy +of our agent monotonically increases for additional examples and achieves an +average time complexity of O(m) in the number m of partial programs for highly +structured domains such as strings. We extend this method to the complex setting +of visual reasoning in the Abstraction and Reasoning Corpus (ARC) for which ILP +methods were previously infeasible. +1 +Introduction +Learning large programs remains an open challenge in program synthesis [Alur et al., 2018]. The +reason for that is the combinatorial nature of program synthesis, which is typically cast as a search +over a space of programs defined by a formal grammar. Therefore, the larger the solution program +(i.e. the more lines of code that need to be synthesized), the larger the search space that needs to be +traversed. +Preprint. Under review. +arXiv:2301.03094v1 [cs.AI] 8 Jan 2023 + +(a) Example ARC task +(b) Example real world string transformation task +Figure 1: Programming by examples tasks from the Abstraction and Reasoning Corpus (ARC) +[Chollet, 2019] and the real world string transformation data set [Cropper and Dumancic, 2020]. +Agents must search for a program that transforms inputs to outputs. In Figure 1a: "Replace every +orange square with a hole by a green cross of the same size". +An often employed strategy to scale combinatorial problem solving is divide-and-conquer. The +existing divide-and-conquer strategies in program synthesis divide the problem by splitting the +examples of a task into subsets [Alur et al., 2017; Cropper, 2022]. They conquer each subset +independently (i.e. synthesise a program solving the subset of examples) and combine independent +solutions into a global solution program. These approaches assume that the solution for one example +is easy to find, which is not always true. +In this work, we introduce a divide-align-conquer strategy that exploits the inherent structure of +examples to scale program synthesis. That is, it divides the individual examples into parts that can +be solved independently, rather then dividing (whole) examples into groups. To illustrate our point, +consider the tasks in Fig. 1. Both tasks have inner structure that can be exploited: the images contain +various objects, while strings are composed of words. The synthesis problem becomes substantially +easier once we make explicit use of the compositional domain structure: it is easier to transform +individual objects and words into their target form rather than entire images and strings. Following +this idea, we split a single synthesis task into multiple synthesis subtasks (i.e. over individual objects) +which are solved independently. +The segmentation makes repeating transformations explicit: every word in the string task (Figure +1b) is manipulated in the same way (i.e. take the first character) and groups of objects in an image +undergo the same transformation (i.e. replace by a green cross). This means that a unique subprogram +discovered once can be useful for other parts of an example. Working with subprograms also +improves the generalizability of the global program as it flexibly applies to a variety of different task +configurations (e.g. spatial configurations in a visual scene). +A major challenge that arises when exploiting compositionality is how to identify meaningful +correspondences between the many input and output objects that in turn form the input to the +synthesis step. One could explore every possible object correspondence but that is likely to diminish +the benefit of problem decomposition as the number of synthesis steps (needed to discover that all +but one correspondence was meaningless) will be large. The problem is further complicated by the +fact that not every input object needs to be present in the output and, thus, does not need to have a +corresponding object. +We argue that this problem can be circumvented by structurally aligning the input and output examples +via analogical reasoning. Analogical reasoning helps to efficiently identify meaningful object +correspondences, which guide the individual synthesis steps. We can rank possible correspondences +and explore them iteratively until the entire output is solved. In order to rank correspondences, we +use the Structure-Mapping Theory (SMT) [Gentner, 1983], a formal account of analogical reasoning +in humans, and the Structure-Mapping-Engine (SME) [Falkenhainer et al., 1989], a computational +model implementing SMT. +In order to demonstrate the benefit of a divide-align-conquer strategy, we implement our approach in +an agent called BEN and evaluate on two domains: the established setting of string transformation +tasks and a new challenging domain of visual reasoning tasks called Abstraction and Reasoning +Corpus (ARC) [Chollet, 2019]. ARC is a collection of heterogeneous visual reasoning data sets +and an interesting benchmark for two reasons: First, visual reasoning programs tend to be large (in +current program synthesis context) and are out of reach for existing synthesis techniques. Second, +visual domains offer an intuitive notion of analogy between objects based on their visual properties +and relational context (i.e. spatial positioning) [Johnson et al., 2021; Wagemans et al., 2012]. +2 + +11 +02Inputs +Outputs +Principles Of Programming Languages +POPL +International Semantic Web Conference +ISWC +Very Large Data Bases +VLDB +Algorithmic Learning Theory +ALT2 +Related work +Our work lies in search-based program synthesis, especially the programming by examples (PBE) +systems which learn programs that are consistent with a semantic specification demonstrated by +representative input/output examples. Learning is formulated as search through a language space of +hypotheses. Synthesizing large programs is challenging as the search space grows exponentially (b|p|) +in the size of programs p, and the average branching factor b. Different strategies have been proposed +to guide the search towards more promising candidates. Ellis et al. [2019], Cropper and Dumancic +[2020], and Nye et al. [2021] leverage the intermediate states of programs and evaluate their distance +to the target solution. Other approaches use re-representation techniques to condense programs and, +thus, make it is easier to search them: compression-driven rewriting or functional abstraction (i.e. +predicate invention) [Henderson and Muggleton, 2014; Cropper et al., 2020; Dumancic et al., 2021]. +The approach in this paper is compatible with all the techniques above. Instead of focusing on search +strategies used during synthesis itself, we exploit the innate structure of input/output examples to +split the learning problem into multiple smaller synthesis tasks. This trades some of the exponential +complexity in a deep program search with a linear number of additional synthesis tasks. +Our work on using analogies to guide the decomposition builds on computational models of analogical +reasoning [Evans, 1964; Mitchell, 1993]. In the past, these models were applied to psychometric +tests of intelligence which often feature analogy problems [Snow et al., 1984], i.e. in number series +completion, string transformations, verbal analogies, and Raven’s matrices. For instance, Lovett and +Forbus [2017] use analogical reasoning to compute ’patterns of variance’ (descriptive statements of +how objects change) across subsequent scenes within each row of a Raven’s matrix. Ín contrast, we +explicitly learn actionable transformation programs which are capable of generating new outputs. +A Kaggle competition that popularized the Abstraction and Reasoning Corpus (ARC) [Chollet, 2019] +has seen all top-ranked agents perform a brute force ’generate and test’ approach using elaborately +handcrafted domain specific languages (DSL) [Wind, 2020; de Miquel et al., 2020]. Instead, we are +investigating a more systematic program synthesis approach based on segmentation and structural +alignment. +3 +Problem Definition +We solve a standard PBE task: given a set of (positive) examples Q, where each example ql is an +input/output pair ql = (Il, Rl), find a program p such that ∀(I, R) ∈ Q, +p(I) = R. +Our proposed approach refines this monolithic program synthesis problem into three, mutually +dependent, divide-align-conquer stages: decomposition, structural alignment, and program synthesis. +We assume all inputs Il and Rl are of a shared type T, e.g. images or strings. The decomposition +part requires finding a function decompose : T → {oi}m +i=1, which segments the input Il or output +Rl of an example ql into a set of objects. +This decomposition function is not given, but has to be learned, i.e. from a parameterized family of +decomposition functions D (section 4.1). +Next, to structurally align input and output objects o resulting from the decomposition, the goal +is to identify a set of object correspondences corr : decompose(I) → decompose(R), a partial +map connecting (a subset of) input objects (from decompose(I)) to the ones in the output (from +decompose(R)). +On each of those correspondences, a (smaller) program synthesis can be preformed, to find a +transformation program τ, such that τ(oi) = or for some (oi, or) ∈ corr. We then require a function +compose : {or}n +r=1 → T which composes a set of objects into an output of type T. This functions is +typically bound to the domain (i.e. concatenate substrings, overlay images) and does not have to be +learnt on each task. +The precise problem we solve is: given a set of input/output examples Q = {(Il, Rl)}t +l=1, find: (1) a +decomposition function, (2) a structural alignment between the resulting objects, (3) a set of programs +T = {τ1, τ2, ...} and (4) a composition function, such that +3 + +Figure 2: Scene segmentation decomposes examples into constituent parts (i.e. visual objects in +a scene). Program synthesis then finds a transformation for an individual pair of constituents in +the input and output. Constituent pairs are considered in the order in which they contribute to the +structural alignment of the input/output example (analogy). For each unique resulting transformation, +we learn a concept in object space which specifies to which input objects a transformation is applied. +A task program is the collection of these transformations and their associated concepts. Program +synthesis can trigger a new structural alignment if unable to find a set of transformations that fully +reconstructs the output. Analogical matching can in turn re-trigger new scene segmentations. +• for every object or in the output, or ∈ decompose(Rl), there exists an object oi in the input +oi ∈ decompose(Il), with corr(or) = oi, and a transformation program τk transforming oi +into or : τk(oi) = or +• when applying all transformation programs τ ∈ T to the decomposed input objects oi ∈ +decompose(Ol), then the composition of all resulting output objects {ot} yields the correct +output Ol, compose({ot}) = Ol. +Every transformation program τ takes the form of a rule ’if [concept definition] then [transformation +function]’, where a concept definition is a set of properties that an object must exhibit for the rule +to activate. On all objects in the input that satisfy these properties, the transformation function is +executed. Transformation functions are synthesized from a set of language primitives (i.e. ’replace’, +’change color’, ’capitalize’). +The divide-align-conquer stages depend on each other: Synthesis is performed on object correspon- +dences derived from structural alignment which in turn works on objects discovered by the initial +decomposition. The final composition depends on the objects reconstructed by programs τk. +These dependencies can be exploited to arrive at useful programs fast. For example, in the next part +we will show how analogical matching can prioritize structural alignments so that meaningful object +correspondences are found early on, minimizing the need to learn transformation programs for the +complete space defined by the Cartesian product of input/output objects. +4 +Architectural overview +Figure 2 shows an overview of the architecture of our proposed approach, visualized on an example +task from the ARC data set. Note how the program synthesis consists of two parts, one part searches +for a transformation function on a given input/output pair, while another part searches for a generic +concept definition across all input objects making use of that transformation function. Together, +the two can find transformation programs of the form ’if [concept definition] then [transformation +function]’ +We now detail the algorithmic approach to each of the divide-align-conquer parts. +4.1 +Divide - Decomposition +First, we decompose the given input/output examples into individual objects. +In case of visual reasoning, the decomposition outputs a set of objects in an image, together with +their symbolic description in terms of features (i.e. width, height, bounding box, color) and relational +4 + +Divide +Align +Conguer +trigger new structural alignment +Io 1 +trigger re-segmentation +Program synthesis +Structural +Transformation +alignment +(口，+）（-，-) +programs +(+,+) .. +To: 口→+ +Decomposition ++←+: +→ Tb +0。 01 +T2: + . → ta +Tb +■ +replace(color(0) +Transformation search +Concept learningplacement on the canvas. For ARC, we use 15 features that are defined a priori and listed in the +appendix (Table S2). In the case of string transformations, strings are described using 14 different +features (Table S4). +We learn a decompose function by synthesizing a segmentor using a context-free grammar of possible +segmentation rules. +In case of images, the matching rules specify conditions under which two pixels are joined into an +object (i.e. equal color, neighboring). This approach works well on ARC scenes as they are noiseless +and have a coarse resolution (30x30 pixels). In case of strings, we consider all non alphanumerical +characters (i.e. comma, whitespace) as possible segmentation points within a string and define +segmentors over subsets of non alphanumerical characters. In addition, we also consider numeric +characters as segmentation candidates and include a segmentor that can split strings into all individual +characters as is required for some of the tasks. +4.2 +Align - Structural alignment via analogy +Having extracted the objects from the examples, we proceed by structurally aligning sets of objects +in the input with sets of objects in the output, which form the subtasks for the synthesis. This step +is combinatorially expensive. In the worst case, we would have to inspect every possible pair of +input/output objects in order to find a meaningful match. +To prevent such a combinatorial explosion, our method leverages the Structure-Mapping Theory (SMT +[Gentner, 1983] from cognitive science which accounts for human capabilities of analogical reasoning. +SMT argues that humans solve the combinatorial explosion of possible mappings between a base +and target domain through structural alignment. The probability of mapping (parts of) the base onto +(parts of) the target is proportional to the agreement of matched relational structure (’systematicity +principle’) and the feature similarity between matched objects. The Structure-Mapping-Engine (SME) +[Falkenhainer et al., 1989], a computational model implementing SMT, finds this proportion by +identifying the largest isomorphic subgraphs between the input and output scene in order to extract +object pairs with similar relational context and features (pseudo-code in the appendix, Section S0.5). +We use the propositional description of each object identified during segmentation to perform +analogical matching of objects with SME. This propositional description can contain both features of +objects and information on their relational positioning (i.e. neighboring objects, preceding substring): +(above +(object (ID o0) (color orange) (bbox [[111],[101],[111]]) (width 3) ...) +(object (ID o1) (color orange) (bbox [[010],[111],[010]]) (width 3) ...) +) ... +In the example input/output images of Figure 1a, input objects match with their intuitive counterparts +at the same location in the output. These are found and receive very high scores by SME. Matches +between i.e. the orange and green crosses are also found, due to surface similarities. However, their +score, and hence probability of being mapped onto each other, is much lower as they can only explain +small portions of the overall relational structure in the two scenes. +4.3 +Conquer - Program synthesis +Having structurally aligned pairs of objects through analogical reasoning, we synthesize a program +that transforms the input object of the correspondence into the one in the output (pseudocode in +the appendix, Alg. S1). We repeatedly perform greedy set covering over output objects: picking +the highest-ranking object correspondence, synthesizing the transformation program for that pair, +and deciding to which other objects the individual transformation program should be applied; the +procedure is repeated until all output objects are covered. +4.3.1 +Searching for a transformation +Each input/output pair of objects identified via analogical matching forms one task for the program +synthesis step. To synthesise the programs, we follow an enumeration approach, exploring all +programs up to a predefined depth (d=3). Focusing on the enumerative approach allows us to better +quantify the impact of structural alignment as it eliminates the impact of smart search. The programs +in both domains are constructed from 11 low-level primitives. In the visual reasoning domain, the +primitives include basic image operations such as coloring, scaling, translating, and rotating objects +5 + +(Table S3), while the string transformation tasks use primitives to manipulate character sequences +such as capitalizing, dropping leading characters, and adding special characters (Table S5). +For example, to solve Figure 1a, we find among others the following transformation functions: +o.identity() and o.replace_by( ).color(green). As this example shows, learning transformations on +pairs of objects directly (in this case, an orange square in the input and a green cross in the output) +lets us efficiently induce complex arguments to the transformation primitives: The +-shape used +in the replacement operation does not need to be searched for in the entire image, but is directly +available through the corresponding output object. +4.3.2 +Learning a concept definition +In the previous step, we synthesised transformation programs but do not yet know to which input +objects a transformation should be applied. We, therefore, learn the description of objects to which +each transformation program should be applied. The concept definition takes the form of a Disjunctive +Normal Form (DNF) over object features. +More precisely, for each synthesized transformation program f, we partition the set of input objects +into three groups: 1) the set of positive objects P such that applying the transformation function on +an object in this set reconstructs an object in the output ∀o ∈ P : f(o) ∈ decompose(R). 2) the set +of negative objects N, that generate false pixels in the output if f is applied to an object from this set +, and 3) a set of neutral objects N, which don’t generate false pixels but also don’t reconstruct any +object in the output (i.e. partial object reconstructions or out-of-bounds transformations). +The goal is to learn a function that covers all examples in P and none of the examples in N. This +corresponds to a standard noise-free concept learning problem [Valiant, 1985], for which we learn a +DNF in which each conjunct associates a value to one object feature. +In order to learn a DNF over the variety of object features, we first hash any non-numeric object +features (i.e. color, pixel array) and double one-hot encode all resulting attributes, such that there is a +Boolean for each attribute-value combination as well as its negation (i.e. ’color == blue’ and ’color +!= blue’; ’pixels == [[1,0],[0,1]]’ and ’pixels != [[1,0],[0,1]]’ etc). +In the running example (Figure 1a), the learnt DNFs contain just a single Boolean attribute set to true: +if o.pixels == [[1,1,1],[1,0,1],[1,1,1]] then o.replace_by([[0,1,0],[1,1,1],[0,1,0]]).color(green) +if o.pixels != [[1,1,1],[1,0,1],[1,1,1]] then o.identity() +To bring everything together, this learned DNF is combined with the transformation function to create +a transformation rule ’if [concept definition] then [transformation function]’ where the DNF is the +concept definition and the transformation function is f. +4.3.3 +Covering remaining objects +The previous two subsections describe how a program is synthesised for a single object correspon- +dence. To synthesise programs for any remaining output objects, we remove all currently matched +output objects from consideration and continue with the next best scoring analogical matching. +Note that incorrectly matched pairs of objects will never cause the approach to miss meaningful +transformations, because the analogies only influence the order in which object pairs are considered +for synthesis. In the worst case, it will check all possible object correspondences. +5 +Experiments +We claim that the use of segmentation and analogical matching for structured domains enables agents +to learn complex programs in less time. To this end, we evaluate our agent BEN across two domains: +string transformation tasks (Section 5.1) which are a well established experimentation domain within +program synthesis and abstract visual reasoning (Section 5.2). We select these domains because +they have inherent structure that can be exploited and provide an intuitive notion of analogy. Our +experiments seek to test three hypotheses: +H1 BEN achieves higher predictive accuracy than state-of-the-art ILP methods. Its performance +monotonically increases with the number of input examples. +6 + +1 +1 +1 +1 +1 +- +1 +1 +1 +- +1 +1(a) Predictive accuracy: percentage of solved test cases +(b) Mean learning time +Figure 3: Performance real world string transformations +H2 BEN performs significantly below its worst case time complexity of O(n2) (n segmented +objects) on real-world synthesis tasks which hold rich structural analogies to efficiently +guide its search. +H3 Solution times for BEN linearly scale with the number of unique transformation programs +in a task solution program. +We compare BEN to Brute [Cropper and Dumancic, 2020] and Metagol [Muggleton et al., 2015; +Cropper and Muggleton, 2016] as two state-of-the-art ILP systems for learning recursive programs. +In addition, we evaluate two ablated versions of BEN as baselines: one using randomized object pairs +instead of analogically matched object pairs and the other without segmentation at all performing +synthesis on the input examples directly. +Brute performs best-first-search which is guided by an example-dependent loss function and was +specifically designed for the synthesis of large programs. We also evaluate Brute with a traditional +entailment-based loss function. Metagol works with user-specified meta-rules which serve as a +declarative bias on the type of clauses that are considered as hypotheses, thus, limiting the search space. +We supply Metagol with the identity, inverse, precon, postcon, chain meta-rules as recommended +for learning dyadic programs by Cropper and Muggleton [2016]. In addition to being able to learn +recursive programs, both Brute and Metagol can also invent new predicates on the fly. BEN in its +current state can neither learn recursive programs nor perform predicate invention. +All experiments were run on a desktop with a single Intel Xeon E5 CPU and 64GB of RAM. We +use OR-Tools and a Clojure-based implementation of SME [van der Meer, 2010]. Our code and an +overview of the tasks solved by BEN are available in the supplementary materials. +5.1 +Experiment 1: String Transformations +First, we consider the typical setting of string transformation tasks and vary the number of examples +presented to agents. +Materials +We use a publicly available data set of 130 real world string transformation tasks from +Cropper and Dumancic [2020]. The initial subset of tasks was curated by Gulwani [2011] from online +Microsoft Excel forums, later expanded by Lin et al. [2014] with additional handcrafted spreadsheet +manipulations and has been repeatedly used as a benchmark in program synthesis. +BEN consistently outperforms both Metagol and Brute on tasks with 3 or more examples (Figure +3a). If the training set consists of only a single example, BEN cannot determine how well a +transformation generalizes to other training examples. As a consequence, it tends to learn overly +specific transformations (i.e. using a replace operation). In much the same way, the concept learning +in BEN requires a minimum amount of 2-3 examples in order to learn an informative selector with +high generalization power to held-out test instances. BEN’s performance monotonically increases +with the number of input examples and ends up surpassing that of both Metagol and Brute by over +20 percentage points which supports hypothesis H1. Notably, learning times in BEN saturate as the +number of examples increases unlike for established ILP methods (Figure 3b). This has to do with the +fact that BEN always discovers transformations on a single pair of substrings within a single example +and then immediately applies it to all other substrings (across examples) which can make use of them. +7 + +BEN + Brute + Brute_uniform → Metagol +100 +Predictive accuracy [%] +75 +50 +25 +0 +2 +3 +4 +5 +6 +8 +9 +No. examplesBEN + +Brute + +Brute_uniform + Metagol +Learning time [seconds] +30 +20 +2 +3 +4 +5 +6 +8 +9 +No. examplesSolution times in BEN are theoretically bounded by a worst case time complexity of O(n2) in the +number n of substrings in the input and output (dashed graph in Figure 4a). In practice, however, +useful transformations are already recovered much faster after only a few object correspondences +which supports hypothesis H2. Solutions times in BEN scale linearly O(m) with the number m of +unique transformations τ in a successful program T. Unlike the search space in Brute and Metagol +which exponentially grows with the size of T, the search space in BEN remains constant and is +traversed once for every unique transformation τ. This is the case even as the size of the overall +program grows which supports hypothesis H3 (Figure 4b). +5.2 +Experiment 2: Abstract Visual Reasoning +We apply BEN to abstract visual reasoning tasks in order to demonstrate the impact of segmentation +and analogical reasoning on program synthesis in high-dimensional domains which typically lead +to combinatorially exploding program spaces for traditional synthesis methods. In this setting, we +evaluate ablated baselines of BEN (Section 5.2.1) and compare to the state-of-the-art agents outside +of ILP (Section 5.2.2). +Materials +We use the training part of the Abstraction and Reasoning Corpus (ARC) [Chollet, 2020] +(Apache 2.0 license), which consists of 400 data sets that each contains 2-10 examples comprising +an input and an output where the outputs are generated by an unspecified program that we wish to +synthesize. The search space of programs is determined by developers and agents themselves; the +benchmark does not specify a language over programs. +5.2.1 +Ablation analysis +The results in Figure 5a show that analogical reasoning allows BEN (blue bars) to solve more tasks in +less time. In the restrictive case of a 1 min time allowance per task, BEN solves 8 times more tasks +than the random object-pair baseline. The performance lead is still at 30% even for a 20 min time +budget which allows the random search to eventually cover a larger fraction of the total search space. +Without segmentation, repeating transformations are less likely to be recovered during the synthesis +which leads to less than 25% of tasks being solved within 20 min. We verified that each of the tasks +solved by either of the two baselines is also solved by BEN. +BEN has an average runtime of 135 seconds and a standard deviation of 120 seconds which is +significantly lower than that of the random baseline, with a mean of 206 seconds and a standard +deviation of 163 seconds, evaluated using a Wilcoxon signed-ranks test on paired samples, Z=855, +p<.001 (Figure 4a). The runtime includes the computation of the structural alignment and the time to +identify the correct object pairs, showing that the additional computational effort of SME is limited +and easily compensated for in the overall algorithm. +The baseline using random object pairs poorly scales to larger scenes (β = 0.62, p<.001) which is +consistent with hypothesis H2. The solution times in BEN remain low even for many object pairs +in crowded scenes (β = 0.18, p>0.1). The use of analogical structural alignment and the number of +transformations in a successful program are both significant predictors of task solution times in BEN, +F(7,152)=26.26,p<.001, with an R2 of 56%. +(a) Solution times per No. of segmented substrings +(b) Solution times per program size +Figure 4: Time complexity of BEN on string transformations +8 + +BEN +Program size +10 +20 +●30 +60 +Solution time [seconds] +40 +20 +0 +10 +15 +20 +25 +5 +No. segmented sub stringsNo. segmented sub strings +5 +10 +15 +20 +60 +Solution time [seconds] +40 +20 +0 +10 +20 +30 +40 +Program sizeDetailed ablation results across different parameters are listed in the appendix. +5.2.2 +Limitations +After evaluating the effect of analogies on complex program synthesis and the importance of the +different components in BEN, we take a closer look at the tasks that BEN did not solve. +The winning Kaggle agent solves 45.5% of tasks compared to 25% by BEN. This difference in +performance is directly related to the amount of prior knowledge given to those agents: the Kaggle +agent uses 46 specialized primitives compared to only 11 generic geometric transformations in BEN. +In other words, its performance is in large parts due to hand-crafted primitives rather than better +search. To make the comparison fairer, we conservatively reduce the agent’s primitives to the ones that +semantically match BEN’s (leaving 30 primitives, Table S7). In this case, its performance drops to +25%. That is, BEN solves the same amount of tasks with only a third of the primitives. For the Kaggle +agent, the computational challenge of synthesizing large programs was overcome by hand-crafted +subprograms which outsourced the cognitive effort to the developer rather than the program synthesis. +We instead use ARC to demonstrate progress on the synthesis of complex programs from low-level +primitives. BEN uniquely solves 9% of the data set (36 tasks) on which the Kaggle winner fails. +BEN fails tasks for one of three main reasons: the scene segmentation does not yield meaningful +objects, the transformation search fails to transform an otherwise correct object pair (due to missing +transformation knowledge), or the concept learning cannot distinguish between positive and negative +objects (due to missing object features). +6 +Discussion and future work +Humans remain far better than machines at learning complex programs. Even in a few-shot learning +setting, as in ARC or for strings transformations, humans effortlessly induce large programs which +generalize well to previously unseen test cases. +Our work suggests that program synthesis is able to exploit the compositional nature of structured +domains to guide the search for well-generalizing programs in vast language spaces. That is done by +decomposing the problem of learning a single nested program into a three-stage process: First, we +find a segmentation of inputs into their smallest compositional units. Second, we align input/output +units into corresponding pairs and, third, perform multiple program synthesis tasks on constituent +parts. Crucially, we separate the problem of searching for object-specific transformations from +the task of learning the contexts in which they apply (the concept definition) which increases the +generalizability of programs. +Although BEN solves more string transformation tasks than state-of-the-art ILP baselines and a fair +share of highly heterogeneous visual reasoning tasks, it also deviates from human task solvers in +important ways. Humans seem to make efficient use of context switching, encoding only those object +features and relational structures which appear important to the task at hand. This reduces the load +of having to process many different encodings all at once as well as limits the search space, and +(a) Ablated baselines +(b) Solution times per number of object pairs +Figure 5: Abstract visual reasoning performance on ARC +9 + +BEN +Random object pairs +800 +Solution time [seconds] +600 +400 +200 +0 +0 +5 +10 +15 +No. of object pairsBEN +Random object pairs +wlo scene segmentation +104 +101 +100 +90 +No. of solved tasks +80 +75 +70 +52 +50 +24 +25 +23 +23 +17 +3 +1 +0 +1 min +5 min +10 min +20 min +Time allowance per taskneurally-guided search could be a computational means to this, which would require an additional +learning component. +Finally, the assumption of independent object transformations is a simplification. Humans are readily +able to find a suitable sequential order of object transformations for example to make use of over- +painting. The methods proposed in this paper do not rely on independent object transformations and, +hence, in future work, we will investigate ways of composing programs with complex dependencies +between their transformations. +References +Rajeev Alur, Arjun Radhakrishna, and Abhishek Udupa. Scaling Enumerative Program Synthesis +via Divide and Conquer. In Tools and Algorithms for the Construction and Analysis of Systems +(TACAS), March 2017. +Rajeev Alur, Rishabh Singh, Dana Fisman, and Armando Solar-Lezama. Search-Based Program +Synthesis. Commun. ACM, 61(12):84–93, November 2018. +François Chollet. On the Measure of Intelligence. arXiv, 1911.01547, 2019. +François Chollet. The Abstraction and Reasoning Corpus (ARC), 2020. +Andrew Cropper and Sebastijan Dumancic. Learning Large Logic Programs By Going Beyond +Entailment. 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AAAI Press, 2022. +Alejandro de Miquel, Roderic Guigo Corominas, and Yuji Ariyasu. 2nd place solution ARC Kaggle +competition, 2020. +Sebastijan Dumancic, Tias Guns, and Andrew Cropper. Knowledge Refactoring for Inductive +Program Synthesis. In Thirty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Thirty- +Third Conference on Innovative Applications of Artificial Intelligence, IAAI 2021, The Eleventh +Symposium on Educational Advances in Artificial Intelligence, EAAI 2021, Virtual Event, February +2-9, 2021, pages 7271–7278. AAAI Press, 2021. +Kevin Ellis, Maxwell I. Nye, Yewen Pu, Felix Sosa, Josh Tenenbaum, and Armando Solar-Lezama. +Write, Execute, Assess: Program Synthesis with a REPL. In Hanna M. Wallach, Hugo Larochelle, +Alina Beygelzimer, Florence d’Alché Buc, Emily B. Fox, and Roman Garnett, editors, Advances in +Neural Information Processing Systems 32: Annual Conference on Neural Information Processing +Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pages 9165–9174, +2019. +Thomas G. Evans. A Heuristic Program to Solve Geometric-Analogy Problems. In Proceedings of +the April 21-23, 1964, Spring Joint Computer Conference, AFIPS ’64, pages 327–338, New York, +NY, USA, 1964. Association for Computing Machinery. +Brian Falkenhainer, Kenneth D. Forbus, and Dedre Gentner. The Structure-Mapping Engine: Algo- +rithm and Examples. Artificial Intelligence, 41:1–63, 1989. +Dedre Gentner. Structure-Mapping: A Theoretical Framework for Analogy. Cognitive Science, +7:155–170, 1983. +10 + +Sumit Gulwani. Automating String Processing in Spreadsheets using Input-Output Examples. In +PoPL’11, January 26-28, 2011, Austin, Texas, USA, January 2011. +Robert Henderson and Stephen Muggleton. Automatic Invention of Functional Abstractions. 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Tenenbaum, and Armando +Solar-Lezama. Representing Partial Programs with Blended Abstract Semantics. In 9th Interna- +tional Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. +OpenReview.net, 2021. +Richard E. Snow, Patrick C. Kyllonen, and Brachia Marshalek. The topography of ability and learning +correlations. In R. J. Sternberg, editor, Advances in the psychology of human intelligence, pages +47–103. Erlbaum, Hillsdale, NJ, 1984. +L. G. Valiant. Learning Disjunction of Conjunctions. In Proceedings of the 9th International Joint +Conference on Artificial Intelligence - Volume 1, IJCAI’85, pages 560–566, San Francisco, CA, +USA, 1985. Morgan Kaufmann Publishers Inc. +Stefan van der Meer. Making meaningful movements. Master’s thesis, Radboud University, Nijmegen, +Netherlands, 2010. +Johan Wagemans, James Elder, Michael Kubovy, Stephen Palmer, Mary Peterson, Manish Singh, and +Rüdiger Heydt. A Century of Gestalt Psychology in Visual Perception: I. Perceptual Grouping and +Figure-Ground Organization. Psychological Bulletin, 138:1172–217, 2012. +Johan Sokrates Wind. 1st place solution ARC Kaggle competition, 2020. +11 + diff --git a/-9E1T4oBgHgl3EQfUwOX/content/tmp_files/load_file.txt b/-9E1T4oBgHgl3EQfUwOX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d5ac502909c22a78104251f9c17766b1496d3d0 --- /dev/null +++ b/-9E1T4oBgHgl3EQfUwOX/content/tmp_files/load_file.txt @@ -0,0 +1,460 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf,len=459 +page_content='A Divide-Align-Conquer Strategy for Program ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='Synthesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='Jonas Witt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='Department of General Psychology and Methodology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='University of Bamberg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='Stef Rasing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='Department of Computer Science ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='TU Delft ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='Sebastijan Dumanˇci´c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='Department of Computer Science ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='TU Delft ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='Tias Guns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='Department of Computer Science ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='KU Leuven ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='Claus-Christian Carbon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='Department of General Psychology and Methodology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='University of Bamberg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='Abstract ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='A major bottleneck in search-based program synthesis is the exponentially growing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='search space which makes learning large programs intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Humans mitigate this problem by leveraging the compositional nature of the real world: In structured domains, a logical specification can often be decomposed into smaller, complemen- tary solution programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We show that compositional segmentation can be applied in the programming by examples setting to divide the search for large programs across multiple smaller program synthesis problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' For each example, we search for a decomposition into smaller units which maximizes the reconstruction ac- curacy in the output under a latent task program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' A structural alignment of the constituent parts in the input and output leads to pairwise correspondences used to guide the program synthesis search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In order to align the input/output structures, we make use of the Structure-Mapping Theory (SMT), a formal model of human analogical reasoning which originated in the cognitive sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We show that decomposition-driven program synthesis with structural alignment outperforms Inductive Logic Programming (ILP) baselines on string transformation tasks even with minimal knowledge priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Unlike existing methods, the predictive accuracy of our agent monotonically increases for additional examples and achieves an average time complexity of O(m) in the number m of partial programs for highly structured domains such as strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We extend this method to the complex setting of visual reasoning in the Abstraction and Reasoning Corpus (ARC) for which ILP methods were previously infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 1 Introduction Learning large programs remains an open challenge in program synthesis [Alur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The reason for that is the combinatorial nature of program synthesis, which is typically cast as a search over a space of programs defined by a formal grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Therefore, the larger the solution program (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' the more lines of code that need to be synthesized), the larger the search space that needs to be traversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='03094v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='AI] 8 Jan 2023 (a) Example ARC task (b) Example real world string transformation task Figure 1: Programming by examples tasks from the Abstraction and Reasoning Corpus (ARC) [Chollet, 2019] and the real world string transformation data set [Cropper and Dumancic, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Agents must search for a program that transforms inputs to outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In Figure 1a: "Replace every orange square with a hole by a green cross of the same size".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' An often employed strategy to scale combinatorial problem solving is divide-and-conquer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The existing divide-and-conquer strategies in program synthesis divide the problem by splitting the examples of a task into subsets [Alur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Cropper, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' They conquer each subset independently (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' synthesise a program solving the subset of examples) and combine independent solutions into a global solution program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' These approaches assume that the solution for one example is easy to find, which is not always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In this work, we introduce a divide-align-conquer strategy that exploits the inherent structure of examples to scale program synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' That is, it divides the individual examples into parts that can be solved independently, rather then dividing (whole) examples into groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' To illustrate our point, consider the tasks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Both tasks have inner structure that can be exploited: the images contain various objects, while strings are composed of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The synthesis problem becomes substantially easier once we make explicit use of the compositional domain structure: it is easier to transform individual objects and words into their target form rather than entire images and strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Following this idea, we split a single synthesis task into multiple synthesis subtasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' over individual objects) which are solved independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The segmentation makes repeating transformations explicit: every word in the string task (Figure 1b) is manipulated in the same way (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' take the first character) and groups of objects in an image undergo the same transformation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' replace by a green cross).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' This means that a unique subprogram discovered once can be useful for other parts of an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Working with subprograms also improves the generalizability of the global program as it flexibly applies to a variety of different task configurations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' spatial configurations in a visual scene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' A major challenge that arises when exploiting compositionality is how to identify meaningful correspondences between the many input and output objects that in turn form the input to the synthesis step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' One could explore every possible object correspondence but that is likely to diminish the benefit of problem decomposition as the number of synthesis steps (needed to discover that all but one correspondence was meaningless) will be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The problem is further complicated by the fact that not every input object needs to be present in the output and, thus, does not need to have a corresponding object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We argue that this problem can be circumvented by structurally aligning the input and output examples via analogical reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Analogical reasoning helps to efficiently identify meaningful object correspondences, which guide the individual synthesis steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We can rank possible correspondences and explore them iteratively until the entire output is solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In order to rank correspondences, we use the Structure-Mapping Theory (SMT) [Gentner, 1983], a formal account of analogical reasoning in humans, and the Structure-Mapping-Engine (SME) [Falkenhainer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=', 1989], a computational model implementing SMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In order to demonstrate the benefit of a divide-align-conquer strategy, we implement our approach in an agent called BEN and evaluate on two domains: the established setting of string transformation tasks and a new challenging domain of visual reasoning tasks called Abstraction and Reasoning Corpus (ARC) [Chollet, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' ARC is a collection of heterogeneous visual reasoning data sets and an interesting benchmark for two reasons: First, visual reasoning programs tend to be large (in current program synthesis context) and are out of reach for existing synthesis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Second, visual domains offer an intuitive notion of analogy between objects based on their visual properties and relational context (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' spatial positioning) [Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Wagemans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=', 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 2 11 02Inputs Outputs Principles Of Programming Languages POPL International Semantic Web Conference ISWC Very Large Data Bases VLDB Algorithmic Learning Theory ALT2 Related work Our work lies in search-based program synthesis, especially the programming by examples (PBE) systems which learn programs that are consistent with a semantic specification demonstrated by representative input/output examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Learning is formulated as search through a language space of hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Synthesizing large programs is challenging as the search space grows exponentially (b|p|) in the size of programs p, and the average branching factor b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Different strategies have been proposed to guide the search towards more promising candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Ellis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' [2019], Cropper and Dumancic [2020], and Nye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' [2021] leverage the intermediate states of programs and evaluate their distance to the target solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Other approaches use re-representation techniques to condense programs and, thus, make it is easier to search them: compression-driven rewriting or functional abstraction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' predicate invention) [Henderson and Muggleton, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Cropper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Dumancic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The approach in this paper is compatible with all the techniques above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Instead of focusing on search strategies used during synthesis itself, we exploit the innate structure of input/output examples to split the learning problem into multiple smaller synthesis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' This trades some of the exponential complexity in a deep program search with a linear number of additional synthesis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Our work on using analogies to guide the decomposition builds on computational models of analogical reasoning [Evans, 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Mitchell, 1993].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In the past, these models were applied to psychometric tests of intelligence which often feature analogy problems [Snow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=', 1984], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' in number series completion, string transformations, verbal analogies, and Raven’s matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' For instance, Lovett and Forbus [2017] use analogical reasoning to compute ’patterns of variance’ (descriptive statements of how objects change) across subsequent scenes within each row of a Raven’s matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Ín contrast, we explicitly learn actionable transformation programs which are capable of generating new outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' A Kaggle competition that popularized the Abstraction and Reasoning Corpus (ARC) [Chollet, 2019] has seen all top-ranked agents perform a brute force ’generate and test’ approach using elaborately handcrafted domain specific languages (DSL) [Wind, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' de Miquel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Instead, we are investigating a more systematic program synthesis approach based on segmentation and structural alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 3 Problem Definition We solve a standard PBE task: given a set of (positive) examples Q, where each example ql is an input/output pair ql = (Il, Rl), find a program p such that ∀(I, R) ∈ Q, p(I) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Our proposed approach refines this monolithic program synthesis problem into three, mutually dependent, divide-align-conquer stages: decomposition, structural alignment, and program synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We assume all inputs Il and Rl are of a shared type T, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' images or strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The decomposition part requires finding a function decompose : T → {oi}m i=1, which segments the input Il or output Rl of an example ql into a set of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' This decomposition function is not given, but has to be learned, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' from a parameterized family of decomposition functions D (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Next, to structurally align input and output objects o resulting from the decomposition, the goal is to identify a set of object correspondences corr : decompose(I) → decompose(R), a partial map connecting (a subset of) input objects (from decompose(I)) to the ones in the output (from decompose(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' On each of those correspondences, a (smaller) program synthesis can be preformed, to find a transformation program τ, such that τ(oi) = or for some (oi, or) ∈ corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We then require a function compose : {or}n r=1 → T which composes a set of objects into an output of type T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' This functions is typically bound to the domain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' concatenate substrings, overlay images) and does not have to be learnt on each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The precise problem we solve is: given a set of input/output examples Q = {(Il, Rl)}t l=1, find: (1) a decomposition function, (2) a structural alignment between the resulting objects, (3) a set of programs T = {τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='} and (4) a composition function, such that 3 Figure 2: Scene segmentation decomposes examples into constituent parts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' visual objects in a scene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Program synthesis then finds a transformation for an individual pair of constituents in the input and output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Constituent pairs are considered in the order in which they contribute to the structural alignment of the input/output example (analogy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' For each unique resulting transformation, we learn a concept in object space which specifies to which input objects a transformation is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' A task program is the collection of these transformations and their associated concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Program synthesis can trigger a new structural alignment if unable to find a set of transformations that fully reconstructs the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Analogical matching can in turn re-trigger new scene segmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' for every object or in the output, or ∈ decompose(Rl), there exists an object oi in the input oi ∈ decompose(Il), with corr(or) = oi, and a transformation program τk transforming oi into or : τk(oi) = or when applying all transformation programs τ ∈ T to the decomposed input objects oi ∈ decompose(Ol), then the composition of all resulting output objects {ot} yields the correct output Ol, compose({ot}) = Ol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Every transformation program τ takes the form of a rule ’if [concept definition] then [transformation function]’, where a concept definition is a set of properties that an object must exhibit for the rule to activate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' On all objects in the input that satisfy these properties, the transformation function is executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Transformation functions are synthesized from a set of language primitives (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' ’replace’, ’change color’, ’capitalize’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The divide-align-conquer stages depend on each other: Synthesis is performed on object correspon- dences derived from structural alignment which in turn works on objects discovered by the initial decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The final composition depends on the objects reconstructed by programs τk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' These dependencies can be exploited to arrive at useful programs fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' For example, in the next part we will show how analogical matching can prioritize structural alignments so that meaningful object correspondences are found early on, minimizing the need to learn transformation programs for the complete space defined by the Cartesian product of input/output objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 4 Architectural overview Figure 2 shows an overview of the architecture of our proposed approach, visualized on an example task from the ARC data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Note how the program synthesis consists of two parts, one part searches for a transformation function on a given input/output pair, while another part searches for a generic concept definition across all input objects making use of that transformation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Together, the two can find transformation programs of the form ’if [concept definition] then [transformation function]’ We now detail the algorithmic approach to each of the divide-align-conquer parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='1 Divide - Decomposition First, we decompose the given input/output examples into individual objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In case of visual reasoning, the decomposition outputs a set of objects in an image, together with their symbolic description in terms of features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' width, height, bounding box, color) and relational 4 Divide Align Conguer trigger new structural alignment Io 1 trigger re-segmentation Program synthesis Structural Transformation alignment (口，+）（-，-) programs (+,+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='. To: 口→+ Decomposition +←+: → Tb 0。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 01 T2: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' → ta Tb ■ replace(color(0) Transformation search Concept learningplacement on the canvas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' For ARC, we use 15 features that are defined a priori and listed in the appendix (Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In the case of string transformations, strings are described using 14 different features (Table S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We learn a decompose function by synthesizing a segmentor using a context-free grammar of possible segmentation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In case of images, the matching rules specify conditions under which two pixels are joined into an object (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' equal color, neighboring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' This approach works well on ARC scenes as they are noiseless and have a coarse resolution (30x30 pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In case of strings, we consider all non alphanumerical characters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' comma, whitespace) as possible segmentation points within a string and define segmentors over subsets of non alphanumerical characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In addition, we also consider numeric characters as segmentation candidates and include a segmentor that can split strings into all individual characters as is required for some of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='2 Align - Structural alignment via analogy Having extracted the objects from the examples, we proceed by structurally aligning sets of objects in the input with sets of objects in the output, which form the subtasks for the synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' This step is combinatorially expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In the worst case, we would have to inspect every possible pair of input/output objects in order to find a meaningful match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' To prevent such a combinatorial explosion, our method leverages the Structure-Mapping Theory (SMT [Gentner, 1983] from cognitive science which accounts for human capabilities of analogical reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' SMT argues that humans solve the combinatorial explosion of possible mappings between a base and target domain through structural alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The probability of mapping (parts of) the base onto (parts of) the target is proportional to the agreement of matched relational structure (’systematicity principle’) and the feature similarity between matched objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The Structure-Mapping-Engine (SME) [Falkenhainer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=', 1989], a computational model implementing SMT, finds this proportion by identifying the largest isomorphic subgraphs between the input and output scene in order to extract object pairs with similar relational context and features (pseudo-code in the appendix, Section S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We use the propositional description of each object identified during segmentation to perform analogical matching of objects with SME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' This propositional description can contain both features of objects and information on their relational positioning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' neighboring objects, preceding substring): (above (object (ID o0) (color orange) (bbox [[111],[101],[111]]) (width 3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=') (object (ID o1) (color orange) (bbox [[010],[111],[010]]) (width 3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=') ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In the example input/output images of Figure 1a, input objects match with their intuitive counterparts at the same location in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' These are found and receive very high scores by SME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Matches between i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' the orange and green crosses are also found, due to surface similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' However, their score, and hence probability of being mapped onto each other, is much lower as they can only explain small portions of the overall relational structure in the two scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='3 Conquer - Program synthesis Having structurally aligned pairs of objects through analogical reasoning, we synthesize a program that transforms the input object of the correspondence into the one in the output (pseudocode in the appendix, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We repeatedly perform greedy set covering over output objects: picking the highest-ranking object correspondence, synthesizing the transformation program for that pair, and deciding to which other objects the individual transformation program should be applied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' the procedure is repeated until all output objects are covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='1 Searching for a transformation Each input/output pair of objects identified via analogical matching forms one task for the program synthesis step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' To synthesise the programs, we follow an enumeration approach, exploring all programs up to a predefined depth (d=3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Focusing on the enumerative approach allows us to better quantify the impact of structural alignment as it eliminates the impact of smart search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The programs in both domains are constructed from 11 low-level primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In the visual reasoning domain, the primitives include basic image operations such as coloring, scaling, translating, and rotating objects 5 (Table S3), while the string transformation tasks use primitives to manipulate character sequences such as capitalizing, dropping leading characters, and adding special characters (Table S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' For example, to solve Figure 1a, we find among others the following transformation functions: o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='identity() and o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='replace_by( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='color(green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' As this example shows, learning transformations on pairs of objects directly (in this case, an orange square in the input and a green cross in the output) lets us efficiently induce complex arguments to the transformation primitives: The shape used in the replacement operation does not need to be searched for in the entire image, but is directly available through the corresponding output object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='2 Learning a concept definition In the previous step, we synthesised transformation programs but do not yet know to which input objects a transformation should be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We, therefore, learn the description of objects to which each transformation program should be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The concept definition takes the form of a Disjunctive Normal Form (DNF) over object features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' More precisely, for each synthesized transformation program f, we partition the set of input objects into three groups: 1) the set of positive objects P such that applying the transformation function on an object in this set reconstructs an object in the output ∀o ∈ P : f(o) ∈ decompose(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 2) the set of negative objects N, that generate false pixels in the output if f is applied to an object from this set , and 3) a set of neutral objects N, which don’t generate false pixels but also don’t reconstruct any object in the output (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' partial object reconstructions or out-of-bounds transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The goal is to learn a function that covers all examples in P and none of the examples in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' This corresponds to a standard noise-free concept learning problem [Valiant, 1985], for which we learn a DNF in which each conjunct associates a value to one object feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In order to learn a DNF over the variety of object features, we first hash any non-numeric object features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' color, pixel array) and double one-hot encode all resulting attributes, such that there is a Boolean for each attribute-value combination as well as its negation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' ’color == blue’ and ’color !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='= blue’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' ’pixels == [[1,0],[0,1]]’ and ’pixels !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='= [[1,0],[0,1]]’ etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In the running example (Figure 1a), the learnt DNFs contain just a single Boolean attribute set to true: if o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='pixels == [[1,1,1],[1,0,1],[1,1,1]] then o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='replace_by([[0,1,0],[1,1,1],[0,1,0]]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='color(green) if o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='pixels !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='= [[1,1,1],[1,0,1],[1,1,1]] then o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='identity() To bring everything together, this learned DNF is combined with the transformation function to create a transformation rule ’if [concept definition] then [transformation function]’ where the DNF is the concept definition and the transformation function is f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='3 Covering remaining objects The previous two subsections describe how a program is synthesised for a single object correspon- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' To synthesise programs for any remaining output objects, we remove all currently matched output objects from consideration and continue with the next best scoring analogical matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Note that incorrectly matched pairs of objects will never cause the approach to miss meaningful transformations, because the analogies only influence the order in which object pairs are considered for synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In the worst case, it will check all possible object correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 5 Experiments We claim that the use of segmentation and analogical matching for structured domains enables agents to learn complex programs in less time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' To this end, we evaluate our agent BEN across two domains: string transformation tasks (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='1) which are a well established experimentation domain within program synthesis and abstract visual reasoning (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We select these domains because they have inherent structure that can be exploited and provide an intuitive notion of analogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Our experiments seek to test three hypotheses: H1 BEN achieves higher predictive accuracy than state-of-the-art ILP methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Its performance monotonically increases with the number of input examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 6 1 1 1 1 1 1 1 1 1 1(a) Predictive accuracy: percentage of solved test cases (b) Mean learning time Figure 3: Performance real world string transformations H2 BEN performs significantly below its worst case time complexity of O(n2) (n segmented objects) on real-world synthesis tasks which hold rich structural analogies to efficiently guide its search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' H3 Solution times for BEN linearly scale with the number of unique transformation programs in a task solution program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We compare BEN to Brute [Cropper and Dumancic, 2020] and Metagol [Muggleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Cropper and Muggleton, 2016] as two state-of-the-art ILP systems for learning recursive programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In addition, we evaluate two ablated versions of BEN as baselines: one using randomized object pairs instead of analogically matched object pairs and the other without segmentation at all performing synthesis on the input examples directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Brute performs best-first-search which is guided by an example-dependent loss function and was specifically designed for the synthesis of large programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We also evaluate Brute with a traditional entailment-based loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Metagol works with user-specified meta-rules which serve as a declarative bias on the type of clauses that are considered as hypotheses, thus, limiting the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We supply Metagol with the identity, inverse, precon, postcon, chain meta-rules as recommended for learning dyadic programs by Cropper and Muggleton [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In addition to being able to learn recursive programs, both Brute and Metagol can also invent new predicates on the fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' BEN in its current state can neither learn recursive programs nor perform predicate invention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' All experiments were run on a desktop with a single Intel Xeon E5 CPU and 64GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We use OR-Tools and a Clojure-based implementation of SME [van der Meer, 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Our code and an overview of the tasks solved by BEN are available in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='1 Experiment 1: String Transformations First, we consider the typical setting of string transformation tasks and vary the number of examples presented to agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Materials We use a publicly available data set of 130 real world string transformation tasks from Cropper and Dumancic [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The initial subset of tasks was curated by Gulwani [2011] from online Microsoft Excel forums, later expanded by Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' [2014] with additional handcrafted spreadsheet manipulations and has been repeatedly used as a benchmark in program synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' BEN consistently outperforms both Metagol and Brute on tasks with 3 or more examples (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' If the training set consists of only a single example, BEN cannot determine how well a transformation generalizes to other training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' As a consequence, it tends to learn overly specific transformations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' using a replace operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In much the same way, the concept learning in BEN requires a minimum amount of 2-3 examples in order to learn an informative selector with high generalization power to held-out test instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' BEN’s performance monotonically increases with the number of input examples and ends up surpassing that of both Metagol and Brute by over 20 percentage points which supports hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Notably, learning times in BEN saturate as the number of examples increases unlike for established ILP methods (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' This has to do with the fact that BEN always discovers transformations on a single pair of substrings within a single example and then immediately applies it to all other substrings (across examples) which can make use of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 7 BEN + Brute + Brute_uniform → Metagol 100 Predictive accuracy [%] 75 50 25 0 2 3 4 5 6 8 9 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' examplesBEN + Brute + Brute_uniform + Metagol Learning time [seconds] 30 20 2 3 4 5 6 8 9 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' examplesSolution times in BEN are theoretically bounded by a worst case time complexity of O(n2) in the number n of substrings in the input and output (dashed graph in Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In practice, however, useful transformations are already recovered much faster after only a few object correspondences which supports hypothesis H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Solutions times in BEN scale linearly O(m) with the number m of unique transformations τ in a successful program T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Unlike the search space in Brute and Metagol which exponentially grows with the size of T, the search space in BEN remains constant and is traversed once for every unique transformation τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' This is the case even as the size of the overall program grows which supports hypothesis H3 (Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='2 Experiment 2: Abstract Visual Reasoning We apply BEN to abstract visual reasoning tasks in order to demonstrate the impact of segmentation and analogical reasoning on program synthesis in high-dimensional domains which typically lead to combinatorially exploding program spaces for traditional synthesis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In this setting, we evaluate ablated baselines of BEN (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='1) and compare to the state-of-the-art agents outside of ILP (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Materials We use the training part of the Abstraction and Reasoning Corpus (ARC) [Chollet, 2020] (Apache 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='0 license), which consists of 400 data sets that each contains 2-10 examples comprising an input and an output where the outputs are generated by an unspecified program that we wish to synthesize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The search space of programs is determined by developers and agents themselves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' the benchmark does not specify a language over programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='1 Ablation analysis The results in Figure 5a show that analogical reasoning allows BEN (blue bars) to solve more tasks in less time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In the restrictive case of a 1 min time allowance per task, BEN solves 8 times more tasks than the random object-pair baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The performance lead is still at 30% even for a 20 min time budget which allows the random search to eventually cover a larger fraction of the total search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Without segmentation, repeating transformations are less likely to be recovered during the synthesis which leads to less than 25% of tasks being solved within 20 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We verified that each of the tasks solved by either of the two baselines is also solved by BEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' BEN has an average runtime of 135 seconds and a standard deviation of 120 seconds which is significantly lower than that of the random baseline, with a mean of 206 seconds and a standard deviation of 163 seconds, evaluated using a Wilcoxon signed-ranks test on paired samples, Z=855, p<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='001 (Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The runtime includes the computation of the structural alignment and the time to identify the correct object pairs, showing that the additional computational effort of SME is limited and easily compensated for in the overall algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The baseline using random object pairs poorly scales to larger scenes (β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='62, p<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='001) which is consistent with hypothesis H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The solution times in BEN remain low even for many object pairs in crowded scenes (β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='18, p>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The use of analogical structural alignment and the number of transformations in a successful program are both significant predictors of task solution times in BEN, F(7,152)=26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='26,p<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='001, with an R2 of 56%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' (a) Solution times per No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' of segmented substrings (b) Solution times per program size Figure 4: Time complexity of BEN on string transformations 8 BEN Program size 10 20 30 60 Solution time [seconds] 40 20 0 10 15 20 25 5 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' segmented sub stringsNo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' segmented sub strings 5 10 15 20 60 Solution time [seconds] 40 20 0 10 20 30 40 Program sizeDetailed ablation results across different parameters are listed in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='2 Limitations After evaluating the effect of analogies on complex program synthesis and the importance of the different components in BEN, we take a closer look at the tasks that BEN did not solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The winning Kaggle agent solves 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content='5% of tasks compared to 25% by BEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' This difference in performance is directly related to the amount of prior knowledge given to those agents: the Kaggle agent uses 46 specialized primitives compared to only 11 generic geometric transformations in BEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In other words, its performance is in large parts due to hand-crafted primitives rather than better search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' To make the comparison fairer, we conservatively reduce the agent’s primitives to the ones that semantically match BEN’s (leaving 30 primitives, Table S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' In this case, its performance drops to 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' That is, BEN solves the same amount of tasks with only a third of the primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' For the Kaggle agent, the computational challenge of synthesizing large programs was overcome by hand-crafted subprograms which outsourced the cognitive effort to the developer rather than the program synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' We instead use ARC to demonstrate progress on the synthesis of complex programs from low-level primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' BEN uniquely solves 9% of the data set (36 tasks) on which the Kaggle winner fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' BEN fails tasks for one of three main reasons: the scene segmentation does not yield meaningful objects, the transformation search fails to transform an otherwise correct object pair (due to missing transformation knowledge), or the concept learning cannot distinguish between positive and negative objects (due to missing object features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 6 Discussion and future work Humans remain far better than machines at learning complex programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Even in a few-shot learning setting, as in ARC or for strings transformations, humans effortlessly induce large programs which generalize well to previously unseen test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Our work suggests that program synthesis is able to exploit the compositional nature of structured domains to guide the search for well-generalizing programs in vast language spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' That is done by decomposing the problem of learning a single nested program into a three-stage process: First, we find a segmentation of inputs into their smallest compositional units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Second, we align input/output units into corresponding pairs and, third, perform multiple program synthesis tasks on constituent parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Crucially, we separate the problem of searching for object-specific transformations from the task of learning the contexts in which they apply (the concept definition) which increases the generalizability of programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Although BEN solves more string transformation tasks than state-of-the-art ILP baselines and a fair share of highly heterogeneous visual reasoning tasks, it also deviates from human task solvers in important ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Humans seem to make efficient use of context switching, encoding only those object features and relational structures which appear important to the task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' This reduces the load of having to process many different encodings all at once as well as limits the search space, and (a) Ablated baselines (b) Solution times per number of object pairs Figure 5: Abstract visual reasoning performance on ARC 9 BEN Random object pairs 800 Solution time [seconds] 600 400 200 0 0 5 10 15 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' of object pairsBEN Random object pairs wlo scene segmentation 104 101 100 90 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' of solved tasks 80 75 70 52 50 24 25 23 23 17 3 1 0 1 min 5 min 10 min 20 min Time allowance per taskneurally-guided search could be a computational means to this, which would require an additional learning component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Finally, the assumption of independent object transformations is a simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Humans are readily able to find a suitable sequential order of object transformations for example to make use of over- painting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' The methods proposed in this paper do not rely on independent object transformations and, hence, in future work, we will investigate ways of composing programs with complex dependencies between their transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Johan Wagemans, James Elder, Michael Kubovy, Stephen Palmer, Mary Peterson, Manish Singh, and Rüdiger Heydt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' A Century of Gestalt Psychology in Visual Perception: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Perceptual Grouping and Figure-Ground Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Psychological Bulletin, 138:1172–217, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' Johan Sokrates Wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 1st place solution ARC Kaggle competition, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} +page_content=' 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfUwOX/content/2301.03094v1.pdf'} diff --git a/.gitattributes b/.gitattributes index 5515d81f8e8e5a2ef2e27566f65e10c5433cb73b..8b511d2a693092acf7efe36a592ce0f120680d6b 100644 --- a/.gitattributes +++ b/.gitattributes @@ -132,3 +132,7 @@ YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf filter=lfs diff=lfs merge=lfs -tex 29E2T4oBgHgl3EQfjQdC/content/2301.03966v1.pdf filter=lfs diff=lfs merge=lfs -text 4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf filter=lfs diff=lfs merge=lfs -text aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf filter=lfs diff=lfs merge=lfs -text +89FST4oBgHgl3EQfajh_/vector_store/index.faiss 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To further improve the performance, +recent works mainly focus on designing more complex network +structures and exploiting extra supervised information, e.g., +semantic segmentation. These methods optimize the models by +exploiting the reconstructed relationship between the target +and reference images in varying degrees. However, previous +methods prove that this image reconstruction optimization is +prone to get trapped in local minima. In this paper, our +core idea is to guide the optimization with prior knowledge +from pretrained Flow-Net. And we show that the bottleneck +of unsupervised monocular depth estimation can be broken +with our simple but effective framework named FG-Depth. In +particular, we propose (i) a flow distillation loss to replace the +typical photometric loss that limits the capacity of the model +and (ii) a prior flow based mask to remove invalid pixels +that bring the noise in training loss. Extensive experiments +demonstrate the effectiveness of each component, and our +approach achieves state-of-the-art results on both KITTI and +NYU-Depth-v2 datasets. +I. INTRODUCTION +Accurate depth estimation is critical for many applications +in computer vision, such as robotic perception [1], [2], +augmented reality [3], and 3D modeling [4]. Monocular depth +estimation has become a challenging and promising field, +attracting the attention of many researchers. Recently, deep +learning-based monocular depth estimation [5], [6], [7], [8], +[9] has been able to achieve high accuracy by narrowing +the gap between predicted depth and ground truth. However, +these methods are limited by the expensive annotation cost. +The emergence of self-supervised approaches [10], [11], [12] +addresses problems requiring depth annotations, typically +trained using the photometric loss to reconstruct and warp +images between target frames and source frames from monoc- +ular videos, stereo pairs, or stereo videos. The photometric +loss widely used in self-supervised depth estimation is based +on implicit assumptions [10] that 1) the scene is static; 2) no +occlusion occurs between target frames and source frames; +3) the surface is Lambertian. However, these assumptions +are so hard to be met on real data that the optimization of +photometric loss is prone to be trapped in local minima [13] +and the performance of the model is limited. +Regarding the above problems, Waston et al. [13] used +SGM algorithm results as depth hints to guide the model +to reach better minima. Zhan et al. [15] and Shu et al. [16] +1Junyu Zhu, Lina Liu and Yong Liu are with the Institute of Cyber- +Systems and Control, Zhejiang University, Hangzhou 310027, P. R. China. +(∗Yong Liu is the corresponding author, email: yongliu@iipc.zju.edu.cn) +2Wanlong Li, Feng Wen and Hongbo Zhang are with Huawei Noah’s Ark +Lab, Beijing, China. +Input +(a) +(b) +(c) +(d) +Fig. 1. +Comparison with state-of-the-art methods. (a)MonoDepth2 [11], +(b)DepthHints [13], (c)EPCDepth [14], (d)Ours. +proposed the feature reconstruction loss to make the training +loss more sensitive to low-texture regions and more robust to +illumination change. [10] uses stereo pair based photometric +loss to avoid the influence of dynamic objects in image +warping. Other methods ignore the defects of photometric +loss and try to improve depth estimation performance by +introducing additional semantic segmentation constraints, such +as [17]. They use semantic segmentation constraints to further +the depth quality near object boundaries. However, annotating +semantic segmentation in real data is expensive. Although +low-cost semantic segmentation labels can be easily obtained +from synthetic data, existing semantic segmentation models +trained on synthetic data cannot generalize well to real data +due to the domain shift [18]. +Despite many improvements, almost all existing unsuper- +vised methods rely heavily on photometric loss with non- +negligible defects, and performance has reached a bottleneck. +Therefore, we believe that it is hard to make significant +progress if the training process still relies on typical photo- +metric loss. +In order to break the bottleneck of unsupervised monocular +depth estimation, inspired by [19], in this paper, we design +a new loss to replace the widely used photometric loss. +Similar to [19], our Depth-Net also learns monocular depth +by distilling prior knowledge of an optical flow estimation +network that has strong generalization and can still generalize +well to real data when trained on low-cost synthetic data. But +there are three main differences between our method and [19]. +Besides the intuitive depth level, our proposed loss further +restricts the Depth-Net from the color level. And we propose +a mask to filter out out-of-range pixels at distillation time to +accelerate convergence. Also, our network architectures are +different from theirs. +Our framework is trained based on stereo pairs to avoid +influence from moving objects. Firstly, based on the fact that +the depth pseudo labels can be generated from prior flow +for stereo pairs and the warping procedure in computing +arXiv:2301.08414v1 [cs.CV] 20 Jan 2023 + +Bundesverfas +sungsgericht +Richtung +Waldstadtphotometric loss is based on flow which can be synthesized +from depth estimation or directly predicted by pretrained +optical flow estimation network, we propose a flow distillation +loss to restrict model from depth and color levels. Secondly, +Depth-Net is usually only effective within a certain range, +and pixels outside the estimated range also inhibit training +because they always fail to match the corresponding pixels +in the warping procedure. In the previous method [11], [20], +there are some pixels beyond the above range that cannot be +removed by the previous methods during the training. To this +end, taking full advantage of the prior flow, we propose a +mask to remove pixels outside the estimation range. Fig. 1 +shows that our approach (Fig. 1(d)) can break the bottleneck +of self-supervised monocular depth estimation compared to +other methods(Fig. 1(a) to (c)). +To summarize, our main contributions are listed below +threefold: +• We introduce a flow distillation loss for restricting the +model from depth and color levels to replace the typical +photometric loss. +• We propose a prior flow based mask for pixels out +of the estimation range at distillation time to improve +performance. +• The proposed model achieves state-of-the-art perfor- +mance on the KITTI and NYU-Depth-v2 datasets. +II. RELATED WORKS +A. Supervised Monocular Depth Estimation +For the task of supervised monocular depth estimation, +ground truth depth labels are used to supervise the training +of the model with RGB monocular images as input. The +ground truth depth labels are usually captured with lidars or +RGBD cameras which have the disadvantage of high cost +or limited depth range and usage scenario. Most supervised +methods regard monocular depth estimation as a regression +task. Eigen et al. [21] is the first to employ CNN in supervised +monocular depth estimation. Xu et al. [22] applied CRF +to optimally combine multi-scale information derived from +the inner layers of CNN to improve the performance of a +CNN depth estimator. Fu et al. [5] found that a performance +improvement can be achieved when the depth estimation is +regarded as a classification task. With more complex and +well-designed CNN architectures, [6], [7] refreshed previous +records. And in recent years, thanks to the development +of ViT [23], several models [24], [8], [9], [25] have been +proposed to help the accuracy reach new heights. +B. Unsupervised Monocular Depth Estimation +To avoid the expensive cost of depth annotation, unsuper- +vised methods usually use photometric loss between adjacent +frames to train the model. As the earliest works in the +self-supervised depth estimation field, [10] uses stereo pairs +to train the Depth-Net and [12] is trained by monocular +video with an extra Pose-Net to predict the relative pose +between adjacent frames. [12] introduced a mask predicted +by the network to reduce the influence of occlusions and +moving objects. Godard et al. [11] made a noticeable +improvement by proposing a minimum photometric loss to +handle occlusions, an auto mask to ignore pixels that violate +camera motion assumptions, and a full-resolution multi-scale +sampling method to make the prediction more accurate. +Noting that the model can struggle during the training to +find the global optimum when minimizing the photometric +loss because of low-texture regions and illumination change, +[13] introduced SGM algorithm results as extra constraints in +training loss and [15], [16] took reconstruction on feature level +into consideration. To further improve the performance, some +works [26], [17], [18] brought extra semantic segmentation +constraints into the training loss but extra semantic labels +on real data actually increase the burden of annotation. +And recently, Peng et al. [14] introduce an effective data +augmentation method for stereo-based models. +C. Optical Flow Estimation +Optical flow estimation is the task of estimating per-pixel +displacement between two frames. Recently, many state- +of-the-art deep learning based approaches [27], [28], [29], +[30], [31] have been proposed for optical flow estimation. In +supervised optical flow estimation tasks, models are usually +trained on synthetic data that has dense accurate optical flow +labels. After the training on the synthetic data, they can +usually generalize well on the real data. As an exceptional +case of optical flow estimation, stereo matching has additional +constraints that the displacement is always negative along +the horizontal direction and always zero along the vertical +direction. Also, many stereo matching models trained on +synthetic data have superior generalization ability on real +data. +III. METHOD +A. Method Overview +In this paper, we propose a flow distillation loss to replace +the typical photometric loss (introduced in Sec. III-B), a prior +flow based mask to remove pixels out of the estimation range +for self-supervised depth estimation networks. Our goal is to +train depth networks using stereo pairs and constrain them +with our proposed flow distillation loss and prior flow based +mask. The pipeline of our framework is shown in Fig. 2. +In the following subsections, we first introduce the typical +photometric loss and automatic mask in Sec. III-B, then +describe our flow distillation loss in Sec III-C. and prior flow +based mask in Sec III-D. +B. Photometric Loss and Automatic Mask +Previous stereo-based self-supervised works typically use +photometric loss Lp to train the model, assuming that the +surface is Lambertian and has no occlusion [12]. Lp is +between target frame It and synthesized frame Is +t which +is interpolated on source frame Is using predicted depth and +relative camera pose, and is defined as: +Lp = pe(It, Is +t ) += α +2 (1 − SSIM(It, Is +t )) + (1 − α)∥It − Is +t ∥1 +1 +(1) +1SSIM [32] is computed over a 3 × 3 pixel window, and α = 0.85. + +Flow-Net +Depth-Net +concat +������ +������ +������ +������→��� +��������� +������ +������ +������ +��� +������ +��� +��������� +��������������������� +��������������������� +��������� +��� +Fig. 2. +Framework illustration. Given a real stereo pair (It, Is) that is refactored from training data by data grafting [14], the Flow-Net pretrained +on synthetic data infers the prior flow ˆFt→s that can then be converted to the pesudo depth lable ˆDt and the mask Mf that removes those pixels out +of estimation range. Multi-scale depth maps are estimated by the Depth-Net from It and here we only draw the maximum scale depth map Dt as an +example. Is +t and ˆIs +t are synthesized from Is using Dt and ˆDt respectively by inverse warping. Depth regression loss Ldr is computed between Dt and +ˆDt. Flow-guided photometric loss Lfp is computed between Is +t and ˆIs +t . Finally, total training loss LM +fd is calculated using Ldr, Lfp and Mf. +Also widely used is the automatic mask Mp for occlusion +proposed in [11], which is formulated as: +Mp = [pe(It, Is +t ) < pe(It, Is)] 2 +(2) +It is still difficult to minimize the Lp in real data. This is +because there are multiple local minima with similar magni- +tudes, especially in regions of low texture and not fulfilling the +assumption of color consistency [15]. And the Lp is disturbed +by occluded pixels. But it’s hard to remove occluded pixels +completely by Mp based on a simple comparison of geometry +relationships. Also, removing occluded pixels means less +supervision information. Furthermore, pixels out of estimation +range can inhibit training but are usually neglected by previous +methods. To avoid the above problems, we propose a flow +distillation loss and a prior flow based mask. Our framework +can achieve better performance due to an easier optimized +loss function and more reliable supervision information. +C. Flow Distillation Loss +The flow distillation loss Lfd consists of depth regression +loss Ldr and flow-guided photometric loss Lfp: +Lfd = Ldr + Lfp +(3) +The Ldr is given by +Ldr = log(|Dt − ˆDt| + 1) +(4) +The Dt is the depth estimation and the pesudo depth lable +ˆDt is computed from +ˆDt = +fxb +| ˆFt→s| +(5) +where fx is the focal length of the camera and b is the +baseline of the stereo. +2[] here is the Iverson bracket. +And the Lfp is adopted to express the discrepancy between +the reconstructions from Is respectively using Dt and ˆDt: +� +� +� +Lfp = |Is +t − ˆIs +t | +Is +t = fw(Is, Dt) +ˆIs +t = fw(Is, ˆDt) +(6) +where fw is the differentiable inverse warping operation. +The inspiration behind flow-guided photometric loss Lfp +is that the warping procedure in computing photometric +loss is based on the flow which can be synthesized from +depth estimation or directly predicted by pretrained optical +flow estimation network. Optimization of Lfp can be easier +to reach better global minima because when Lfp reaches +minima, synthesized flow is closed to the predicted flow. +And pretrained optical flow estimation network can predict +enough accurate flow, so depth estimation can be more closed +to ground truth after optimization of Lfp. By contrast, typical +Lp is harder to be optimized because of illumination changes. +The intuitive display is shown in Fig. 3. The red pixels +marked in Fig. 3(a) are the pixel pairs matched by the stereo +pair. Fig. 3(b) shows the loss curve for optimizing the depth +of this matching point with depth regression loss Ldr, flow- +guided photometric loss Lfp, flow distillation loss Lfd and +photometric loss Lp, respectively. There is the same global +minimum for Ldr, Lfp, and Lfd, which is almost identical to +the ground truth. And the Lfd curve is steeper than Ldr and +Lfp, so it makes optimization of Lfd easier. For Lp, there +are multiple minimum points, where the optimal point fails +to learn the correct depth value. Therefore, the Lfd is easier +for optimization and has a more accurate global minimum, +when compared with the Lp. +D. Prior Flow based Mask +We use a prior flow based mask Mf to remove those +pixels out of range by checking the length of prior rigid flow. + +Bundesverfas +sungsgericht +RichtungBundesverfa +sungsgeritr +FichtungBlBundesrerfa +sungsgerrr +FithtungBundesierfa +sungsgerier +RithtungBundesverfas +sungsgericht +Richtung +WaldstaatBundesverfa +sungsgerith +ichtung +Waldstadt(a) +(b) +Fig. 3. +Loss visualization. (a)A pair of matching pixels on the left image +and the right image. (b)The relationship between the loss and the depth of +matching points in the left subplot (a). The flow distillation loss is easier +for optimization and has a more accurate global minimum when compared +with photometric loss. +(a) +(b) +(c) +Fig. 4. +Masks visualization. (a)Input target frame and source frame. +(b)Auto-mask Mp proposed in [11]. (c)Prior flow based mask. +The mask value Mf(pi) of the pixel at position pi can be +formulated as: +� +� +� +� +� +� +� +Mf(pi) = +� 1 +, pi ∈ V +0 +, else +V = +� +pt +����| ˆFt→s(pi)| > fxb +δ +� +(7) +where ˆFt→s denotes the prior flow from target frame to source +frame and δ are set to 80. +In depth estimation, out-of-range depths (greater than +80m in KITTI) drop dramatically in accuracy. In previous +works, masks do not remove all those pixels out-of-range. +So, noise is brought in photometric loss because those out-of- +range pixels always fail to match corresponding pixels in the +warping procedure. Visualization results in Fig. 4 intuitively +show that compared with automatic mask Mp proposed +in [11], our mask Mf can filter out out-of-range pixels more +completely, making it more stable and less susceptible to +noise interference. +E. Final Training Loss +We combine the flow distillation loss and prior flow based +mask as: +LM +fd = 1 +T +� +i +Mf(pi)Lfd(pi) +(8) +where T denotes the number of pixels reserved by the mask, +and average over each scale. +F. Network Architecture +We implement the Flow-Net with RAFT-Stereo [33] which +is based on GRU [34] and has excellent accuracy and good +generalization ability. For simplicity, we directly use the +official model3 that is pretrained on Scene Flow dataset [35]. +For the Depth-Net, we use the same architecture as [14] +which uses ResNet18 as backbone and RSU block as +the bridge between different scale features and disparity +prediction blocks to output full-scale predictions. The outputs +σ of the prediction blocks are further constrained between +0.1 and 80 units with D = 1/(aσ + b). +IV. EXPERIMENTS +In this section, we evaluate our proposed model on the +KITTI dataset [36] to verify its state-of-art performance +and we validate the generalization ability of our model on +the NYU-Depth-v2 dataset [37]. Furthermore, we conduct +an ablation study to demonstrate the effectiveness of our +contributions. +A. Datasets +a) KITTI: KITTI dataset was captured by a driving +vehicle with cameras and depth sensors around the mid-size +city of Karlsruhe, in rural areas, and on highways. It is widely +used for outdoor monocular depth estimation and we use the +Eigen split [21] that consists of 22600 stereo image pairs for +training and 697 images for testing. The train set is from 32 +scenes and the test set is from other 29 scenes. +b) NYU-Depth-v2: NYU-Depth-v2 dataset was collected +with a Microsoft Kinect sensor in total of 582 indoor scenes. +To validate the generalization ability of our model, we use +the official test set that consists of 654 images with depth +GTs. +B. Inplementation Details +Our work is implemented in PyTorch on one Nvidia Tesla +V100 GPU. For training, we use the Adam optimizer [38](β1 += 0.9, β2 = 0.999). The total number of epochs is set to +20 with a batch size of 12 and an input/output resolution of +192×640 unless otherwise specified. The initial learning rate +is 1 × 10−4 and decays after the 10th epoch with a factor of +0.1. For evaluation, we resize the estimated depth map to the +ground-truth depth resolution using bilinear interpolation. +With a 50% chance, we flip the input images horizontally, +apply data grafting [14] with the same setting as [14] +and add color augmentations where we perform random +brightness, contrast, saturation, and hue jitter by sampling +uniform distributions in ranges of [0.8,1.2], [0.8,1.2], [0.8,1.2], +[0.9,1.1] respectively. The color augmentations are applied +to the images that are fed to the Depth-Net rather than those +fed to the Flow-Net and the loss function. +3https://github.com/princeton-vl/RAFT-Stereo + +1.5 +Ldr +Lfp +1.0 +Lfd +: +! +Lp +loss +1 +GT depth +0.5 +Ldr minimum +Lfp minimum +0.0 +Lfd minimum +11.34 +11.34 +Lp minimum +11.22 +i.11.34 +14.80 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +depthestimation(metersTABLE I +QUANTITATIVE RESULTS ON THE KITTI DATASET USING EIGEN SPLIT [21]. FOR RED METRICS, LOWER IS BETTER. AND HIGHER IS BETTER FOR +BLUE METRICS. PP REPRESENTS POST-PROCESSING [10]. IN DATA COLUMN, D REFERS TO THE METHODS SUPERVISED BY THE GROUND TRUTH DEPTH, +M MEANS THAT THE SUPERVISION IS FROM MONOCULAR VIDEO, S MEANS THAT THE SUPERVISION IS FROM STEREO PAIRS, C† MEANS USING +PREDICTED SEGMENTATION LABLES AND S∗ MEANS USING EXTRA SYNTHETIC SCENEFLOW DATASET. THE BEST RESULTS ARE IN BOLD. +Method +PP Data +Resolution +Abs Rel Sq Rel RMSE RMSElog +δ1 +δ2 +δ3 +DORN [5] (ResNet101) +D +385 × 513 crop +0.099 +0.593 +3.714 +0.161 +0.897 0.966 0.986 +BTS [6] (DenseNet-161) +D +352 × 704 crop +0.091 +0.555 +4.033 +0.174 +0.904 0.967 0.984 +AdaBins [8] (EfficientNet-B5) +D +352 × 704 crop +0.086 +0.475 +3.621 +0.167 +0.918 0.970 0.985 +NeWCRFs [25] (swin-ViT) +D +352 × 1120 crop +0.080 +0.426 +3.400 +0.158 +0.930 0.973 0.986 +MonoDepth2 [11] + +MS +192 × 640 +0.104 +0.786 +4.687 +0.194 +0.876 0.958 0.980 +DepthHints [13] + +MS +192 × 640 +0.105 +0.769 +4.627 +0.189 +0.875 0.959 0.982 +HR-Depth [20] +MS +192 × 640 +0.107 +0.785 +4.612 +0.185 +0.887 0.962 0.982 +CADepth-Net [39] (ResNet50) +MS +192 × 640 +0.102 +0.752 +4.502 +0.181 +0.894 0.964 0.983 +Guo et al. [19] w/o Fintuned (VGG-16) + +SS∗ +384 × 1280 +0.109 +0.822 +4.656 +0.192 +0.868 0.958 0.981 +Guo et al. [19] Fintuned (VGG-16) + +SS∗ +384 × 1280 +0.099 +0.745 +4.424 +0.182 +0.884 0.963 0.983 +MonoDepth2 [11] + +S +192 × 640 +0.108 +0.842 +4.891 +0.207 +0.866 0.949 0.976 +DepthHints [13] + +S +192 × 640 +0.106 +0.780 +4.695 +0.193 +0.875 0.958 0.980 +EPCDepth [14] + +S +192 × 640 +0.099 +0.754 +4.490 +0.183 +0.888 0.963 0.982 +Ours + +SS∗ +192 × 640 +0.093 +0.634 +4.123 +0.174 +0.900 0.967 0.984 +MonoDepth2 [11] + +MS +320 × 1024 +0.104 +0.775 +4.562 +0.191 +0.878 0.959 0.981 +DepthHints [13] + +MS +320 × 1024 +0.098 +0.702 +4.398 +0.183 +0.887 0.963 0.983 +HR-Depth [20] +MS +320 × 1024 +0.101 +0.716 +4.395 +0.179 +0.899 0.966 0.983 +Feat-Depth [16] (ResNet50) +MS +320 × 1024 +0.099 +0.697 +4.427 +0.184 +0.889 0.963 0.982 +CADepth-Net [39] (ResNet50) +MS +320 × 1024 +0.096 +0.694 +4.264 +0.173 +0.908 0.968 0.984 +MonoDepth2 [11] + +S +320 × 1024 +0.105 +0.822 +4.692 +0.199 +0.874 0.954 0.977 +DepthHints [13] + +S +320 × 1024 +0.099 +0.723 +4.445 +0.187 +0.886 0.962 0.981 +EdgeDepth [17] + +SC† +320 × 1024 +0.097 +0.675 +4.350 +0.180 +0.890 0.965 0.983 +EPCDepth [14] + +S +320 × 1024 +0.093 +0.671 +4.297 +0.178 +0.899 0.965 0.983 +Ours + +SS∗ +320 × 1024 +0.088 +0.583 +3.924 +0.168 +0.909 0.970 0.985 +DepthHints [13] (ResNet50) + +S +320 × 1024 +0.096 +0.710 +4.393 +0.185 +0.890 0.962 0.983 +EdgeDepth [17] (ResNet50) + +SC† +320 × 1024 +0.091 +0.646 +4.244 +0.177 +0.898 0.966 0.983 +EPCDepth [14] (ResNet50) + +S +320 × 1024 +0.091 +0.646 +4.207 +0.176 +0.901 0.966 0.983 +Ours (ResNet50) + +SS∗ +320 × 1024 +0.086 +0.575 +3.873 +0.166 +0.910 0.971 0.985 +Input +Monodepth2 [11] +DepthHints [13] +EPCDepth [14] +Ours(FG-Depth) +Fig. 5. +Qualitative results on the KITTI dataset using Eigen split. Our model, BTB-Depth, produces the sharpest results even in low-texture regions +and on thin structures. +C. Depth Estimation Performance +Firstly, we verify the performance of our model on the +KITTI dataset. For a fair comparison, we use the metrics +proposed in [21] with Garg’s crop [40] and a standard +distance cap of 80 meters. The same as other comparative +self-supervised methods, we use the same post-processing +steps as theirs [10]. The quantitative results are summarized +in Tab. I and the qualitative results are shown in Fig. 5. +The quantitative results show that our model, FG-Depth, +comprehensively exceeds all existing unsupervised methods +that are even trained with stereo video(MS). Compared with +[17] which uses extra expensive semantic segmentation labels, + +TABLE II +QUANTITATIVE RESULTS ON THE NYU-DEPTH-V2 DATASET. +Method +Abs Rel +Sq Rel +RMSE +RMSElog +δ1 +δ2 +δ3 +Monodepth2 [11] +0.362 +0.718 +1.312 +0.384 +0.477 +0.758 +0.898 +EPCDepth [14] +0.281 +0.341 +0.912 +0.319 +0.554 +0.833 +0.943 +Ours(FG-Depth) +0.269 +0.318 +0.888 +0.312 +0.560 +0.840 +0.947 +Input +Monodepth2 [11] EPCDepth [14] Ours(FG-Depth) +Ground truth +Fig. 6. +Qualitative results on the NYUV2 dataset. +our framework uses additional low-cost synthetic optical flow +dataset and gets better performance. Compared with [19] +which also distills knowledge from optical flow network +pretrianed on sceneflow dataset, FG-Depth performs better +event at low resolution. Despite lack of the supervision from +ground truth depth maps, the high-resolution performance of +FG-Depth is even close to AdaBins [8], a recent state-of-the- +art supervised method, and FG-Depth has fewer parameters +meanwhile. Besides, the qualitative results show that FG- +Depth can produce sharper results even in some low-texture +regions and on some thin structures. +Then, we validate the performance on the NYU-Depth- +v2 dataset using our model trained on the KITTI just as +EPCDepth [17] did. The quantitative results in Tab. II and +the qualitative results in Fig. 6 verify the strong generalization +ability of our model. +D. Ablation studies +To better understand the effect of each component of our +proposed model, we perform an ablation study and present the +results in Tab. III. The results show that all our components +can lead to significant performance when combined together. +a) Flow distillation loss: Although Lp is common in +previous self-supervised works, we show that it actually limits +the capacity of models. As shown in Tab. III, in all cases, +being trained with Lfd can outperform those being trained +with Lp. +b) Prior flow based mask: Tab. III also shows that prior +flow based mask Mf significantly improves performance and +its improvement is more significant than Mp proposed in [11] +even though Mf dosen’t remove occlusion for Lp while Lfd +isn’t disturbed by occlusion. +c) Loss function combinations: Tab. IV lists perfor- +mance of different combinations of loss function. The results +show that using Ldr can already get impressive performance +and combining Ldr with Lfp can achieve state-of-the-art +performance which is consistent with the analysis in Sec. III- +C. +d) Pipeline: For a fair comparison with [19], we give +results under different pipelines in Tab. V. The results on the +first row and on the third row show that our networks have +better performance even though at a smaller resolution. And +the results in the last row show that our contributions can +significantly improve the accuracy and even outperform [19] +finetuned with the supervised method. +TABLE III +ABLATION STUDIES. Base REFERS TO THE NETWORK ARCHITECTURE, +Lp REFERS TO THE PHOTOMETRIC LOSS, Lfd REFERS TO THE FLOW +DISTILLATION LOSS, Mp REFERS TO THE AUTO MASK PROPOSED IN [11] +AND Mf REFERS TO OUR PRIOR FLOW BASED MASK. +Method +Abs Rel +Sq Rel +RMSE +RMSElog +δ1 +δ2 +δ3 +Base+LP +0.106 +1.300 +5.850 +0.201 +0.872 +0.953 +0.977 +Base+LP +Mp +0.104 +0.919 +5.176 +0.202 +0.873 +0.953 +0.976 +Base+LP +Mf +0.100 +0.730 +4.499 +0.195 +0.878 +0.956 +0.979 +Base+Lfd +0.099 +1.102 +5.230 +0.180 +0.894 +0.965 +0.983 +Base+Lfd+Mp +0.097 +0.970 +5.182 +0.180 +0.896 +0.965 +0.983 +Base+Lfd+Mf +0.093 +0.634 +4.123 +0.174 +0.900 +0.967 +0.984 +TABLE IV +ABLATION STUDIES ON LOSS FUNCTION COMBINATIONS. Lp REFERS +TO THE PHOTOMETRIC LOSS, Ldr REFERS TO THE DEPTH REGRESSION +LOSS AND Lfp REFERS TO THE FLOW-GUIDED PHOTOMETRIC LOSS. +Loss +Abs Rel +Sq Rel +RMSE +RMSElog +δ1 +δ2 +δ3 +Ldr +0.094 +0.643 +4.139 +0.175 +0.896 +0.965 +0.985 +Lfp +0.098 +0.718 +4.230 +0.177 +0.892 +0.966 +0.984 +Ldr+Lfp +0.093 +0.634 +4.123 +0.174 +0.900 +0.967 +0.984 +TABLE V +ABLATION STUDIES ON PIPLINES. unsupFt AND supFt RESPECTIVELY +REFERS TO FITUNING THE FLOW-NET USING UNSUPERVISED AND +SUPERVISED METHOD ON REAL DATA. disp REFERS TO USING DISPARITY +TO SUPERVISE THE DEPTH-NET FOR ALL PIXELS WITH PREDICTION OF +FLOW-NET WITHOUT FITUNING AS THE PIPLINE ON THE FIRST ROW DID. +pipline +resolution +Abs Rel +Sq Rel +RMSE +RMSElog +δ1 +δ2 +δ3 +Guo [19] w/oF t +384 × 1280 +0.109 +0.822 +4.656 +0.192 +0.868 +0.958 +0.981 +Guo [19] unsupF t +384 × 1280 +0.099 +0.745 +4.424 +0.182 +0.884 +0.963 +0.983 +Guo [19] supF t +384 × 1280 +0.097 +0.653 +4.170 +0.170 +0.889 +0.967 +0.986 +ours(disp) +192 × 640 +0.103 +1.353 +5.768 +0.185 +0.891 +0.964 +0.982 +ours(Ldr+Lfp+Mf ) +192 × 640 +0.093 +0.634 +4.123 +0.174 +0.900 +0.967 +0.984 +V. 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Reid, “Unsupervised cnn for +single view depth estimation: Geometry to the rescue,” in European +Conference on Computer Vision, pp. 740–756, Springer, 2016. + diff --git a/19FAT4oBgHgl3EQfDBzX/content/tmp_files/load_file.txt b/19FAT4oBgHgl3EQfDBzX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7ffc2f96aa1f3330175088850bc792c36a0b50c0 --- /dev/null +++ b/19FAT4oBgHgl3EQfDBzX/content/tmp_files/load_file.txt @@ -0,0 +1,856 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf,len=855 +page_content='FG-Depth: Flow-Guided Unsupervised Monocular Depth Estimation Junyu Zhu1, Lina Liu1, Yong Liu1∗, Wanlong Li2, Feng Wen2 and Hongbo Zhang2 Abstract— The great potential of unsupervised monocular depth estimation has been demonstrated by many works due to low annotation cost and impressive accuracy comparable to supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' To further improve the performance, recent works mainly focus on designing more complex network structures and exploiting extra supervised information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=', semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' These methods optimize the models by exploiting the reconstructed relationship between the target and reference images in varying degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' However, previous methods prove that this image reconstruction optimization is prone to get trapped in local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' In this paper, our core idea is to guide the optimization with prior knowledge from pretrained Flow-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' And we show that the bottleneck of unsupervised monocular depth estimation can be broken with our simple but effective framework named FG-Depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' In particular, we propose (i) a flow distillation loss to replace the typical photometric loss that limits the capacity of the model and (ii) a prior flow based mask to remove invalid pixels that bring the noise in training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Extensive experiments demonstrate the effectiveness of each component, and our approach achieves state-of-the-art results on both KITTI and NYU-Depth-v2 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' INTRODUCTION Accurate depth estimation is critical for many applications in computer vision, such as robotic perception [1], [2], augmented reality [3], and 3D modeling [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Monocular depth estimation has become a challenging and promising field, attracting the attention of many researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Recently, deep learning-based monocular depth estimation [5], [6], [7], [8], [9] has been able to achieve high accuracy by narrowing the gap between predicted depth and ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' However, these methods are limited by the expensive annotation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The emergence of self-supervised approaches [10], [11], [12] addresses problems requiring depth annotations, typically trained using the photometric loss to reconstruct and warp images between target frames and source frames from monoc- ular videos, stereo pairs, or stereo videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The photometric loss widely used in self-supervised depth estimation is based on implicit assumptions [10] that 1) the scene is static;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 2) no occlusion occurs between target frames and source frames;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 3) the surface is Lambertian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' However, these assumptions are so hard to be met on real data that the optimization of photometric loss is prone to be trapped in local minima [13] and the performance of the model is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Regarding the above problems, Waston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' [13] used SGM algorithm results as depth hints to guide the model to reach better minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Zhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' [15] and Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' [16] 1Junyu Zhu, Lina Liu and Yong Liu are with the Institute of Cyber- Systems and Control, Zhejiang University, Hangzhou 310027, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' (∗Yong Liu is the corresponding author, email: yongliu@iipc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='cn) 2Wanlong Li, Feng Wen and Hongbo Zhang are with Huawei Noah’s Ark Lab, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Input (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Comparison with state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' (a)MonoDepth2 [11], (b)DepthHints [13], (c)EPCDepth [14], (d)Ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' proposed the feature reconstruction loss to make the training loss more sensitive to low-texture regions and more robust to illumination change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' [10] uses stereo pair based photometric loss to avoid the influence of dynamic objects in image warping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Other methods ignore the defects of photometric loss and try to improve depth estimation performance by introducing additional semantic segmentation constraints, such as [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' They use semantic segmentation constraints to further the depth quality near object boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' However, annotating semantic segmentation in real data is expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Although low-cost semantic segmentation labels can be easily obtained from synthetic data, existing semantic segmentation models trained on synthetic data cannot generalize well to real data due to the domain shift [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Despite many improvements, almost all existing unsuper- vised methods rely heavily on photometric loss with non- negligible defects, and performance has reached a bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Therefore, we believe that it is hard to make significant progress if the training process still relies on typical photo- metric loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' In order to break the bottleneck of unsupervised monocular depth estimation, inspired by [19], in this paper, we design a new loss to replace the widely used photometric loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Similar to [19], our Depth-Net also learns monocular depth by distilling prior knowledge of an optical flow estimation network that has strong generalization and can still generalize well to real data when trained on low-cost synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' But there are three main differences between our method and [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Besides the intuitive depth level, our proposed loss further restricts the Depth-Net from the color level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' And we propose a mask to filter out out-of-range pixels at distillation time to accelerate convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Also, our network architectures are different from theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Our framework is trained based on stereo pairs to avoid influence from moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Firstly, based on the fact that the depth pseudo labels can be generated from prior flow for stereo pairs and the warping procedure in computing arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='08414v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='CV] 20 Jan 2023 Bundesverfas sungsgericht Richtung Waldstadtphotometric loss is based on flow which can be synthesized from depth estimation or directly predicted by pretrained optical flow estimation network, we propose a flow distillation loss to restrict model from depth and color levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Secondly, Depth-Net is usually only effective within a certain range, and pixels outside the estimated range also inhibit training because they always fail to match the corresponding pixels in the warping procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' In the previous method [11], [20], there are some pixels beyond the above range that cannot be removed by the previous methods during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' To this end, taking full advantage of the prior flow, we propose a mask to remove pixels outside the estimation range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 1 shows that our approach (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 1(d)) can break the bottleneck of self-supervised monocular depth estimation compared to other methods(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 1(a) to (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' To summarize, our main contributions are listed below threefold: We introduce a flow distillation loss for restricting the model from depth and color levels to replace the typical photometric loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' We propose a prior flow based mask for pixels out of the estimation range at distillation time to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The proposed model achieves state-of-the-art perfor- mance on the KITTI and NYU-Depth-v2 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' RELATED WORKS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Supervised Monocular Depth Estimation For the task of supervised monocular depth estimation, ground truth depth labels are used to supervise the training of the model with RGB monocular images as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The ground truth depth labels are usually captured with lidars or RGBD cameras which have the disadvantage of high cost or limited depth range and usage scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Most supervised methods regard monocular depth estimation as a regression task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Eigen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' [21] is the first to employ CNN in supervised monocular depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' [22] applied CRF to optimally combine multi-scale information derived from the inner layers of CNN to improve the performance of a CNN depth estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' [5] found that a performance improvement can be achieved when the depth estimation is regarded as a classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' With more complex and well-designed CNN architectures, [6], [7] refreshed previous records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' And in recent years, thanks to the development of ViT [23], several models [24], [8], [9], [25] have been proposed to help the accuracy reach new heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Unsupervised Monocular Depth Estimation To avoid the expensive cost of depth annotation, unsuper- vised methods usually use photometric loss between adjacent frames to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' As the earliest works in the self-supervised depth estimation field, [10] uses stereo pairs to train the Depth-Net and [12] is trained by monocular video with an extra Pose-Net to predict the relative pose between adjacent frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' [12] introduced a mask predicted by the network to reduce the influence of occlusions and moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Godard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' [11] made a noticeable improvement by proposing a minimum photometric loss to handle occlusions, an auto mask to ignore pixels that violate camera motion assumptions, and a full-resolution multi-scale sampling method to make the prediction more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Noting that the model can struggle during the training to find the global optimum when minimizing the photometric loss because of low-texture regions and illumination change, [13] introduced SGM algorithm results as extra constraints in training loss and [15], [16] took reconstruction on feature level into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' To further improve the performance, some works [26], [17], [18] brought extra semantic segmentation constraints into the training loss but extra semantic labels on real data actually increase the burden of annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' And recently, Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' [14] introduce an effective data augmentation method for stereo-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Optical Flow Estimation Optical flow estimation is the task of estimating per-pixel displacement between two frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Recently, many state- of-the-art deep learning based approaches [27], [28], [29], [30], [31] have been proposed for optical flow estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' In supervised optical flow estimation tasks, models are usually trained on synthetic data that has dense accurate optical flow labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' After the training on the synthetic data, they can usually generalize well on the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' As an exceptional case of optical flow estimation, stereo matching has additional constraints that the displacement is always negative along the horizontal direction and always zero along the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Also, many stereo matching models trained on synthetic data have superior generalization ability on real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Method Overview In this paper, we propose a flow distillation loss to replace the typical photometric loss (introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' III-B), a prior flow based mask to remove pixels out of the estimation range for self-supervised depth estimation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Our goal is to train depth networks using stereo pairs and constrain them with our proposed flow distillation loss and prior flow based mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The pipeline of our framework is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' In the following subsections, we first introduce the typical photometric loss and automatic mask in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' III-B, then describe our flow distillation loss in Sec III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' and prior flow based mask in Sec III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Photometric Loss and Automatic Mask Previous stereo-based self-supervised works typically use photometric loss Lp to train the model, assuming that the surface is Lambertian and has no occlusion [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Lp is between target frame It and synthesized frame Is t which is interpolated on source frame Is using predicted depth and relative camera pose, and is defined as: Lp = pe(It, Is t ) = α 2 (1 − SSIM(It, Is t )) + (1 − α)∥It − Is t ∥1 1 (1) 1SSIM [32] is computed over a 3 × 3 pixel window, and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Flow-Net Depth-Net concat ������ ������ ������ ������→��� ��������� ������ ������ ������ ��� ������ ��� ��������� ��������������������� ��������������������� ��������� ��� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Framework illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Given a real stereo pair (It, Is) that is refactored from training data by data grafting [14], the Flow-Net pretrained on synthetic data infers the prior flow ˆFt→s that can then be converted to the pesudo depth lable ˆDt and the mask Mf that removes those pixels out of estimation range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Multi-scale depth maps are estimated by the Depth-Net from It and here we only draw the maximum scale depth map Dt as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Is t and ˆIs t are synthesized from Is using Dt and ˆDt respectively by inverse warping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Depth regression loss Ldr is computed between Dt and ˆDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Flow-guided photometric loss Lfp is computed between Is t and ˆIs t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Finally, total training loss LM fd is calculated using Ldr, Lfp and Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Also widely used is the automatic mask Mp for occlusion proposed in [11], which is formulated as: Mp = [pe(It, Is t ) < pe(It, Is)] 2 (2) It is still difficult to minimize the Lp in real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' This is because there are multiple local minima with similar magni- tudes, especially in regions of low texture and not fulfilling the assumption of color consistency [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' And the Lp is disturbed by occluded pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' But it’s hard to remove occluded pixels completely by Mp based on a simple comparison of geometry relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Also, removing occluded pixels means less supervision information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Furthermore, pixels out of estimation range can inhibit training but are usually neglected by previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' To avoid the above problems, we propose a flow distillation loss and a prior flow based mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Our framework can achieve better performance due to an easier optimized loss function and more reliable supervision information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Flow Distillation Loss The flow distillation loss Lfd consists of depth regression loss Ldr and flow-guided photometric loss Lfp: Lfd = Ldr + Lfp (3) The Ldr is given by Ldr = log(|Dt − ˆDt| + 1) (4) The Dt is the depth estimation and the pesudo depth lable ˆDt is computed from ˆDt = fxb | ˆFt→s| (5) where fx is the focal length of the camera and b is the baseline of the stereo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 2[] here is the Iverson bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' And the Lfp is adopted to express the discrepancy between the reconstructions from Is respectively using Dt and ˆDt: � � � Lfp = |Is t − ˆIs t | Is t = fw(Is, Dt) ˆIs t = fw(Is, ˆDt) (6) where fw is the differentiable inverse warping operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The inspiration behind flow-guided photometric loss Lfp is that the warping procedure in computing photometric loss is based on the flow which can be synthesized from depth estimation or directly predicted by pretrained optical flow estimation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Optimization of Lfp can be easier to reach better global minima because when Lfp reaches minima, synthesized flow is closed to the predicted flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' And pretrained optical flow estimation network can predict enough accurate flow, so depth estimation can be more closed to ground truth after optimization of Lfp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' By contrast, typical Lp is harder to be optimized because of illumination changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The intuitive display is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The red pixels marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 3(a) are the pixel pairs matched by the stereo pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 3(b) shows the loss curve for optimizing the depth of this matching point with depth regression loss Ldr, flow- guided photometric loss Lfp, flow distillation loss Lfd and photometric loss Lp, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' There is the same global minimum for Ldr, Lfp, and Lfd, which is almost identical to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' And the Lfd curve is steeper than Ldr and Lfp, so it makes optimization of Lfd easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' For Lp, there are multiple minimum points, where the optimal point fails to learn the correct depth value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Therefore, the Lfd is easier for optimization and has a more accurate global minimum, when compared with the Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Prior Flow based Mask We use a prior flow based mask Mf to remove those pixels out of range by checking the length of prior rigid flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Bundesverfas sungsgericht RichtungBundesverfa sungsgeritr FichtungBlBundesrerfa sungsgerrr FithtungBundesierfa sungsgerier RithtungBundesverfas sungsgericht Richtung WaldstaatBundesverfa sungsgerith ichtung Waldstadt(a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Loss visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' (a)A pair of matching pixels on the left image and the right image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' (b)The relationship between the loss and the depth of matching points in the left subplot (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The flow distillation loss is easier for optimization and has a more accurate global minimum when compared with photometric loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Masks visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' (a)Input target frame and source frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' (b)Auto-mask Mp proposed in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' (c)Prior flow based mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The mask value Mf(pi) of the pixel at position pi can be formulated as: � � � � � � � Mf(pi) = � 1 , pi ∈ V 0 , else V = � pt ����| ˆFt→s(pi)| > fxb δ � (7) where ˆFt→s denotes the prior flow from target frame to source frame and δ are set to 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' In depth estimation, out-of-range depths (greater than 80m in KITTI) drop dramatically in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' In previous works, masks do not remove all those pixels out-of-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' So, noise is brought in photometric loss because those out-of- range pixels always fail to match corresponding pixels in the warping procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Visualization results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 4 intuitively show that compared with automatic mask Mp proposed in [11], our mask Mf can filter out out-of-range pixels more completely, making it more stable and less susceptible to noise interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Final Training Loss We combine the flow distillation loss and prior flow based mask as: LM fd = 1 T � i Mf(pi)Lfd(pi) (8) where T denotes the number of pixels reserved by the mask, and average over each scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Network Architecture We implement the Flow-Net with RAFT-Stereo [33] which is based on GRU [34] and has excellent accuracy and good generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' For simplicity, we directly use the official model3 that is pretrained on Scene Flow dataset [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' For the Depth-Net, we use the same architecture as [14] which uses ResNet18 as backbone and RSU block as the bridge between different scale features and disparity prediction blocks to output full-scale predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The outputs σ of the prediction blocks are further constrained between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='1 and 80 units with D = 1/(aσ + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' EXPERIMENTS In this section, we evaluate our proposed model on the KITTI dataset [36] to verify its state-of-art performance and we validate the generalization ability of our model on the NYU-Depth-v2 dataset [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Furthermore, we conduct an ablation study to demonstrate the effectiveness of our contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Datasets a) KITTI: KITTI dataset was captured by a driving vehicle with cameras and depth sensors around the mid-size city of Karlsruhe, in rural areas, and on highways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' It is widely used for outdoor monocular depth estimation and we use the Eigen split [21] that consists of 22600 stereo image pairs for training and 697 images for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The train set is from 32 scenes and the test set is from other 29 scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' b) NYU-Depth-v2: NYU-Depth-v2 dataset was collected with a Microsoft Kinect sensor in total of 582 indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' To validate the generalization ability of our model, we use the official test set that consists of 654 images with depth GTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Inplementation Details Our work is implemented in PyTorch on one Nvidia Tesla V100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' For training, we use the Adam optimizer [38](β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The total number of epochs is set to 20 with a batch size of 12 and an input/output resolution of 192×640 unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The initial learning rate is 1 × 10−4 and decays after the 10th epoch with a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' For evaluation, we resize the estimated depth map to the ground-truth depth resolution using bilinear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' With a 50% chance, we flip the input images horizontally, apply data grafting [14] with the same setting as [14] and add color augmentations where we perform random brightness, contrast, saturation, and hue jitter by sampling uniform distributions in ranges of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='8,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='2], [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='8,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='2], [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='8,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='2], [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='9,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='1] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The color augmentations are applied to the images that are fed to the Depth-Net rather than those fed to the Flow-Net and the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='com/princeton-vl/RAFT-Stereo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='5 Ldr Lfp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='0 Lfd : !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Lp loss 1 GT depth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='5 Ldr minimum Lfp minimum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='0 Lfd minimum 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='34 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='34 Lp minimum 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='22 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='34 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='80 8 9 10 11 12 13 14 15 16 17 depthestimation(metersTABLE I QUANTITATIVE RESULTS ON THE KITTI DATASET USING EIGEN SPLIT [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' FOR RED METRICS, LOWER IS BETTER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' AND HIGHER IS BETTER FOR BLUE METRICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' PP REPRESENTS POST-PROCESSING [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' IN DATA COLUMN, D REFERS TO THE METHODS SUPERVISED BY THE GROUND TRUTH DEPTH, M MEANS THAT THE SUPERVISION IS FROM MONOCULAR VIDEO, S MEANS THAT THE SUPERVISION IS FROM STEREO PAIRS, C† MEANS USING PREDICTED SEGMENTATION LABLES AND S∗ MEANS USING EXTRA SYNTHETIC SCENEFLOW DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' THE BEST RESULTS ARE IN BOLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Method PP Data Resolution Abs Rel Sq Rel RMSE RMSElog δ1 δ2 δ3 DORN [5] (ResNet101) D 385 × 513 crop 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='593 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='897 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='986 BTS [6] (DenseNet-161) D 352 × 704 crop 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='555 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='984 AdaBins [8] (EfficientNet-B5) D 352 × 704 crop 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='475 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='621 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='167 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='985 NeWCRFs [25] (swin-ViT) D 352 × 1120 crop 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='426 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='930 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='986 MonoDepth2 [11] \x13 MS 192 × 640 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='786 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='687 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='876 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='982 CADepth-Net [39] (ResNet50) MS 192 × 640 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='752 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Our model, BTB-Depth, produces the sharpest results even in low-texture regions and on thin structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Depth Estimation Performance Firstly, we verify the performance of our model on the KITTI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' For a fair comparison, we use the metrics proposed in [21] with Garg’s crop [40] and a standard distance cap of 80 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The same as other comparative self-supervised methods, we use the same post-processing steps as theirs [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The quantitative results are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' I and the qualitative results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The quantitative results show that our model, FG-Depth, comprehensively exceeds all existing unsupervised methods that are even trained with stereo video(MS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Compared with [17] which uses extra expensive semantic segmentation labels, TABLE II QUANTITATIVE RESULTS ON THE NYU-DEPTH-V2 DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Method Abs Rel Sq Rel RMSE RMSElog δ1 δ2 δ3 Monodepth2 [11] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='362 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='718 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='384 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='758 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='898 EPCDepth [14] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='281 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='341 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='833 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='943 Ours(FG-Depth) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='560 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='947 Input Monodepth2 [11] EPCDepth [14] Ours(FG-Depth) Ground truth Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Qualitative results on the NYUV2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' our framework uses additional low-cost synthetic optical flow dataset and gets better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Compared with [19] which also distills knowledge from optical flow network pretrianed on sceneflow dataset, FG-Depth performs better event at low resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Despite lack of the supervision from ground truth depth maps, the high-resolution performance of FG-Depth is even close to AdaBins [8], a recent state-of-the- art supervised method, and FG-Depth has fewer parameters meanwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Besides, the qualitative results show that FG- Depth can produce sharper results even in some low-texture regions and on some thin structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Then, we validate the performance on the NYU-Depth- v2 dataset using our model trained on the KITTI just as EPCDepth [17] did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The quantitative results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' II and the qualitative results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 6 verify the strong generalization ability of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Ablation studies To better understand the effect of each component of our proposed model, we perform an ablation study and present the results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The results show that all our components can lead to significant performance when combined together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' a) Flow distillation loss: Although Lp is common in previous self-supervised works, we show that it actually limits the capacity of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' III, in all cases, being trained with Lfd can outperform those being trained with Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' b) Prior flow based mask: Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' III also shows that prior flow based mask Mf significantly improves performance and its improvement is more significant than Mp proposed in [11] even though Mf dosen’t remove occlusion for Lp while Lfd isn’t disturbed by occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' c) Loss function combinations: Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' IV lists perfor- mance of different combinations of loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The results show that using Ldr can already get impressive performance and combining Ldr with Lfp can achieve state-of-the-art performance which is consistent with the analysis in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' III- C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' d) Pipeline: For a fair comparison with [19], we give results under different pipelines in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' The results on the first row and on the third row show that our networks have better performance even though at a smaller resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' And the results in the last row show that our contributions can significantly improve the accuracy and even outperform [19] finetuned with the supervised method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' TABLE III ABLATION STUDIES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Base REFERS TO THE NETWORK ARCHITECTURE, Lp REFERS TO THE PHOTOMETRIC LOSS, Lfd REFERS TO THE FLOW DISTILLATION LOSS, Mp REFERS TO THE AUTO MASK PROPOSED IN [11] AND Mf REFERS TO OUR PRIOR FLOW BASED MASK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Method Abs Rel Sq Rel RMSE RMSElog δ1 δ2 δ3 Base+LP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='300 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='872 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='977 Base+LP +Mp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='919 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='976 Base+LP +Mf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='730 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='956 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='979 Base+Lfd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='099 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='102 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='983 Base+Lfd+Mp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='970 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='983 Base+Lfd+Mf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='634 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='984 TABLE IV ABLATION STUDIES ON LOSS FUNCTION COMBINATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Lp REFERS TO THE PHOTOMETRIC LOSS, Ldr REFERS TO THE DEPTH REGRESSION LOSS AND Lfp REFERS TO THE FLOW-GUIDED PHOTOMETRIC LOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Loss Abs Rel Sq Rel RMSE RMSElog δ1 δ2 δ3 Ldr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='643 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='985 Lfp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='718 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='984 Ldr+Lfp 0.' metadata={'source': 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+page_content='103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='353 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='768 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='891 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='964 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='982 ours(Ldr+Lfp+Mf ) 192 × 640 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='634 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content='984 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' CONCLUSION In this paper, to break the bottleneck of unsupervised monocular depth estimation, noting that optical flow esti- mation models have strong generalization ability and the typical photometric loss is defective, we propose a flow distillation loss and a prior flow based mask to improve the performance of the unsupervised monocular depth estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' And the experiments demonstrate that our model, FG-Depth, can achieve state-of-the-art performance on the KITTI dataset using Eigen split and NYU-Depth-v2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' In future work, to further improve the performance, we will explore more methods to make full use of prior optical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Also, we will try to apply our contributions to other categories that use monocular video(M) or stereo video(MS) as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' REFERENCES [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Hadsell, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Sermanet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Ben, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' Reid, “Unsupervised cnn for single view depth estimation: Geometry to the rescue,” in European Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} +page_content=' 740–756, Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FAT4oBgHgl3EQfDBzX/content/2301.08414v1.pdf'} diff --git a/3NE4T4oBgHgl3EQfbAwV/content/tmp_files/2301.05068v1.pdf.txt b/3NE4T4oBgHgl3EQfbAwV/content/tmp_files/2301.05068v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a4225a424dda2f44e623f9e8de464100fbdbfe18 --- /dev/null +++ b/3NE4T4oBgHgl3EQfbAwV/content/tmp_files/2301.05068v1.pdf.txt @@ -0,0 +1,2149 @@ +Observability and Identifiability Analyses of Models +for Agricultural Anaerobic Digestion Plants +Simon Hellmann1, Arne-Jens Hempel2, Stefan Streif3, Sören Weinrich1 +1 DBFZ Deutsches Biomasseforschungszentrum gGmbH, Leipzig, Germany, +{simon.hellmann, soeren.weinrich}@dbfz.de +2 Saxon University of Cooperative Education, Glauchau, Germany, +arne-jens.hempel@ba-sachsen.de +3 Chemnitz University of Technology, Chemnitz, Germany, +stefan.streif@etit.tu-chemnitz.de +This is the extended version of a paper submitted to the 24th International +Conference on Process Control on January 13, 2023. It is available under +a CC BY-NC-ND 4.0 license. +Abstract +Dynamic operation of anaerobic digestion plants requires advanced process monitoring and +control. Different simplifications of the Anaerobic Digestion Model No. 1 (ADM1) have been +proposed recently, which appear promising for model-based process automation and state +estimation. As a fundamental requirement, observability and identifiability of these models are +analysed in this work, which was pursued through algebraic and geometric analysis. Manual +algebraic assessment was successfull for small models such as the ADM1-R4 and simplified +versions of the ADM1-R3, which were derived in this context. However, for larger model +classes the algebraic approach showed to be insufficient. By contrast, the geometric approach, +implemented in the STRIKE_GOLDD toolbox, allowed to show observability for more complex +models (including ADM1-R4 and ADM1-R3), employing two independent algorithms. The +present study lays the groundwork for state observer design, parameter estimation and advanced +control resting upon ADM1-based models. +Key words: +ADM1, model simplification, Biogas technology, parameter estimation, +STRIKE_GOLDD +1 +arXiv:2301.05068v1 [eess.SY] 12 Jan 2023 + +1 Introduction +Anaerobic digestion (AD) allows to convert numerous organic feedstocks into biogas, which can +either be upgraded and injected into the natural gas grid or combusted to produce renewable +electricity and heat. +The AD process naturally shows strongly nonlinear behavior and is sensitive to process inhibition +[1]. Moreover, the process is prone to instability, especially during dynamic feeding [2]. To +avoid instable process behavior, monitoring and control schemes are required. +Many investigations focus on automated operation of domestic or industrial wastewater +treatment plants [3–6]. +Additionally, advanced control for efficient and demand-oriented +biogas production of agricultural AD plants are frequently examined [2, 7, 8]. +A bottleneck in full-scale application of AD is the lack of online measurements, especially +regarding reliable stability indicators, such as volatile fatty acids (VFA) and alkalinity [9]. A +remedy to overcome this shortage is to apply soft sensors (or state observers), which use readily +available external measurements and a mathematical model of the process to estimate internal, +non-measurable process states [10]. +A prominent process model was presented by Bernard et al. [11], which has successfully been +used in multiple monitoring and control applications [3, 4, 12]. +However, nonlinear aspects of the AD process are described in more detail within the established +Anaerobic Digestion Model No. 1 (ADM1) [13]. While the model proposed by Bernard et +al. only includes pH inhibition, the ADM1 covers process inhibition through pH, nitrogen +limitation and ammonia. Still, successful applications of the ADM1 in full-scale monitoring +and control applications have not yet been reported, mostly because of its complexity and vast +number of parameters [14]. +Yet, in an agricultural setting, which is an important application of AD, both the orginal +ADM1 and the model by Bernard et al. cannot directly be used due to their underlying +reference unit (chemical oxygen demand, COD) [15]. Thus, Weinrich and Nelles have recently +proposed mass-based simplifications of the ADM1 [15, 16]. These models represent a suitable +alternative for application in monitoring and control of agricultural AD plants [7]. Individual +model variations differ significantly in their number of differential equations, state variables +and required parameters (Fig. 1). Simplified models (such as the ADM1-R4) combine nutrient +degradation and biogas formation based on first-order sum reactions, whereas more detailed +models (such as the ADM1-R3) depict specific degradation pathways and ammonia or pH +inhibition during acetoclastic methanogensis. +2 + +Observability is a model property which indicates whether internal states can be inferred based +on input-output measurements [17]. Likewise, identifiability implies that model parameters can +be calibrated based on input-output measurements. Assessing observability and identifiability +is therefore a fundamental requirement before implementing state and parameter estimation. +Numerous approaches for assessing observability and identifiability have been proposed [18–20]. +However, especially for complex models they are seldom analyzed a priori because of the +computational complexity of the symbolic calculations involved [20]. +This contribution analyses observability and identifiability of different ADM1 simplifications +proposed by Weinrich and Nelles [15]. For this purpose, two different approaches are pursued: +the differential algebraic and differential geometric approach.1 +Typical measurements at +full-scale AD plants are assumed to be available to ensure feasibility of future process control +schemes based on these analyses. +2 Methods +In this work, we consider systems of ordinary differential equations of the form: +M : +� +� +� +� +� +� +� +� +� +˙x += f (x(t), u(t), θ) +y += h (x(t), θ) +x(t0) = x0 +(1) +with state variables x ∈ Rn, initial state x0, manipulated variables u ∈ Rp, measurement +variables y ∈ Rq, and model parameters θ ∈ Rm. Generally, model parameters (such as +microbial growth or decay rates) can be time-variant. However, they vary at a slow rate of +change and their dynamics does not follow a defined differential equation. Therefore, their +implicit time dependence is suppressed in the notation. +2.1 Modelling of the Anaerobic Digestion Process +Based on available model simplifications in Fig. 1, observability and identifiability were assessed +for the ADM1-R4, ADM1-R3 and ADM1-R2 [15]. Mass concentration of ash was integrated to +compute volatile solids (VS) measurements. +1The two approaches are simply referred to as algebraic and geometric in the following. +3 + +Figure 1: Characteristics of different ADM1 simplifications [15]. +2.1.1 Model Simplification +Both ADM1-R4 and ADM1-R3 were further simplified throughout the investigation. For this +purpose, individual model components were isolated and omitted or incorporated systematically +to assess their influence on observability. Fig. 2 illustrates the full ADM1-R4 and its model +parts qualiltatively. These model parts are decay of microbial biomass and its stoichiometric +feedback as macro nutrients (part A, in green), and gas solubility of methane and carbon +dioxide (part B, in orange). +The ADM1-R3 allows to isolate more model parts as shown in Fig. 3. Part A and B were left +identical as for the ADM1-R4 (in green and orange, respectively). Part C (in purple) describes +inhibition through nitrogen limitation. Part D (in blue) covers inhibition through pH and +ammonia, as well as dissociation of ammonium/ammonia. Lastly, part E (in red) contains the +computation of pH, which includes the charge balance of available anions and cations. +Carbohydrates +Proteins +Polymers +Lipids +First-order +kine�cs +Biomass decay +A +Liquid- as +g +transfer +Methane +Carbon dioxide +Biogas +Microorganisms +B +Figure 2: Model components of the ADM1-R4. +4 + +Polymers +Monomers +Acetic acid +Organic acids +Biogas +ADM1 +ch I pr I li +su|aa|fa +va |bulpro +CH4 ICO2 / H2 +Hydrogen +Polymers +Monomers +Organic acids +Acetic acid +Biogas +ADM1-R1 +ch |pr|li +su|aa|fa +va | bu |pro +CH4 1 CO2 +ac +Polymers +Organic acids +Acetic acid +Biogas +ADM1-R2 +ch | pr |li +va|bu|pro +CH4 I CO2 +ac +Polymers +Acetic acid +Biogas +ADM1-R3 +ch I pr |li +CH4 I CO2 +ac +Polymers +Biogas +ADM1-R4 +> +ch |pr |li +CH4 1 CO2The core elements of the models ADM1-R4 and ADM1-R3 without any additional model +parts are denoted as BMR4 and BMR3 (base model, BM). Augmenting them with additional +model parts results in e.g. BMR4+A or BMR4+B. The same notation applies for individual +ADM1-R3 model variations. The investigated models are summarized in Tab. 2. A full set of +the corresponding model equations is given in the appendix. +2.1.2 Data Availability +For observability and identifiabilty analyses, measurement signals in Tab. 1 were considered. +Online measurements were generally restricted to pH and gas composition, expressed as partial +pressures of methane and carbon dioxide [21]. For detailed assessment of observability using the +algebraic approach, acetic acid was also considered as available online. For lab and pilot-scale +settings, this is a reasonable assumption [22]. Generally however, online measurements of acetic +acid were not considered, which represents the more realistic scenario in a full-scale agricultural +setting [21]. +Offline measurements of total solids (TS), volatile solids (VS) and inroganic nitrogen (IN) were +assumed to be slowly time-variant in between samples [23]. To obtain time-continuous signals, +sample-and-hold behavior was supposed. +2.2 Observability Analyses +Definition. A state variable xi(t) is observable if its initial state xi(0) can be reconstructed +from measurements of inputs u(τ) and outputs y(τ) over a finite time τ ∈ [0, t]. A system M +is fully observable if all of its n states are observable. If at least one state is not observable, the +system is not fully observable [24]. +pH calcula�on +Liquid- as +g +transfer +Methane +Carbon dioxide +Biogas +Microorganisms +Ace�c acid +Organic acids +Carbohydrates +Proteins +Polymers +Lipids +E +pH inhibi�on +D +N limita�on +C +Dissocia�on +NH inhibi�on +3 +D +D +B +Monod +kine�cs +First-order +kine�cs +Biomass decay +A +Figure 3: Model components of the ADM1-R3. +5 + +In this work, two pathways to assess observability were pursued: the algebraic and the geometric +approach. In the former, a system of equations of output variables and their time derivatives is +established and solved for individual model states. The latter assesses the observability rank +condition, which relies on an observability matrix. +2.2.1 Algebraic Approach +In the algebraic approach, a system of equations of output variables y and their time derivatives +y, ˙y, ¨y, . . . is established and solved for the state variables xi, i ∈ 1, . . . , n. The system of +equations can be summarized as +Y = Y(x, u, θ), where Y = (y, ˙y, ¨y, . . .)T . +(2) +Time derivatives of the outputs are obtained by iteratively computing the Lie derivatives with +respect to f. We assumed constant input signals during individual feeding events ( ˙u = 0) and +constant parameters ( ˙θ = 0), resulting in: +y += +h(x, θ) +(3) +˙y += +Lfh(x, θ) = ∂h(x, θ) +∂x +f(x, u, θ), +(4) +¨y += +L2 +fh(x, θ) = ∂Lfh(x, θ) +∂x +f(x, u, θ), +(5) +... +y(k) += +L(k) +f h(x, θ) = +∂L(k−1) +f +h(x, θ) +∂x +f(x, u, θ) +(6) +Observability is given if the system of equations (2) can be solved algebraically for all states +xi, i = 1 . . . n and has at least one solution [17]. This requires n equations, which can either be +obtained by incorporating all of the q elements of each y, ˙y etc.; or by building higher-order +time derivatives y(k) and incorporating their elements. If the system of equations can be solved +uniquely, global observability is shown. Multiple solutions indicate local observability [17]. +Table 1: Measurement signals considered for individual model classes +Model class +Online +Offlinea +ADM1-R4 +gas composition (CH4, CO2) +TS, VS, IN +ADM1-R3b +gas composition (CH4, CO2), pH +TS, VS, IN +aTotal solids (TS), volatile solids (VS) and inorganic nitrogen (IN) +bDuring algebraic oberservability analyses acetic acid was also +considered as an optional online measurement. +6 + +Implementation +The algebraic approach was implemented in Mathematica (version 13.0, +Wolfram Research, Inc.). Equations which could be solved for state variables manually were +excluded from the system of equations to minimize computational demand. Complexity of +the system of equations was further reduced by seeking to incorporate terms with minimal +orders of time derivatives first. This is because generally complexity of symbolic derivatives y(k) +increases rapidly with the order of derivatives k [20]. Selection of terms among the q elements +of y(k) followed a heuristic procedure. +Since the algebraic approach turned out to be inconclusive for the ADM1-R3, the models +ADM1-R4 and ADM1-R3 were systematically simplified (see Sec. 2.1). The maximum allowable +model complexity for the algebraic approach was determined by systematically adding model +components, starting from the base models (BMR4 and BMR3), and terminating when the +system of equations could no longer be solved. +In the algebraic approach, all parameters and influent concentrations were assumed to be known +and time-invariant. Moreover, for the ADM1-R3 model classes, measurements of acetic acid +were assumed to be available online in order to achieve conclusive statements on observability, +see Sec. 3.1. +2.2.2 Geometric Approach +In the differential geometric approach observability is investigated by computing the rank of +an observability matrix. Two different algorithms were considered in this context, which differ +in the way the observability matrix is built: Full Input-State-Parameter Observabilty (FISPO) +and Observability Rank Condition with Direct Feedthrough (ORC-DF). Both algorithms rely +on Lie derivatives of the output. For better understanding, computation of the observability +matrix is explained by means of the FISPO algorithm. In a second step, disparities of the +ORC-DF algorithm are illustrated. +In FISPO, the observability matrix O(x) is built by taking the Lie derivatives of the output +symbolically and computing their partial derivatives with respect to the states x: +O(x) = +� +� +� +� +� +� +� +� +∂h(x) +∂x +∂ +∂x (Lfh(x)) +... +∂ +∂x +� +Ln−1 +f +h(x) +� +� +� +� +� +� +� +� +� +. +(7) +A model is locally observable in a neighborhood N(x0) of a point x0 if it holds that [18] +rank (O(x0)) = n. +(8) +7 + +The ORC-DF algorithm pursues a different approach in computing the observability matrix +and is only applicable for input-affine systems [19], which can be described as: +˙x += +f(x) + +p +� +j=1 +gj(x)uj, +(9) +y += +h0(x) + +p +� +j=1 +hj(x)uj. +(10) +f, gj ∈ Rn and h0, hj ∈ Rp are vectorial functions, which describe the output y uniquely. They +are aggregated in a column vector Ω. The observability matrix in ORC-DF is built repeatedly +using symbolic computation. Initially, it is computed as partial derivatives of Ω with respect to +the states x. If the observability rank condition can be satisfied, the system is locally observable. +If not, the vector Ω is extended by the Lie derivatives of the previous version of Ω with respect +to f and gj. This procedure continues until the rank condition is satisfied or a maximum +number of iterations is reached. Details can be found in [19]. +Implementation +Both geometric approaches were implemented in the Matlab toolbox +STRIKE_GOLDD 4.0 [25]. The toolbox allows to assess local observability (and structural +local identifiability) with the FISPO and ORC-DF algorithms. +To ensure that statements on observability of the models are independent from the numeric +value of the initial state x0, no previously known initial states x0 were assumed for both +procedures. This results in symbolic computations of O(x) and its rank. +In the geometric approach, full model classes ADM1-R4 and ADM1-R3 as well as all submodels +were analyzed. Inlet concentrations were assumed to be known and time-invariant. +Computations of both algebraic and geometric approaches were conducted on a standard +personal computer.2 +2.3 Identifiability Analyses +Definition. A model parameter θi, i = 1, . . . , m is structurally identifiable (s.i.) if for almost +all true parameter values θ∗ the parameter estimate ˆθ can be determined from input-output +behavior of the model M [26]: +M +�x(t), u(t), ˆθ +� = M +�x(t), u(t), θ∗� +(11a) +⇒ ˆθ = θ∗. +(11b) +2Intel Core i5 processor (1.7 GHz), 32 GB RAM and Windows 10 operating system. +8 + +If all model parameters θi are s.i., the model is s.i.. A parameter θi is locally structural +identifiability (l.s.i.) if (11) holds in a neighborhood N(θ∗). A full model is l.s.i. if all its +parameters θi are l.s.i.. If at least one of them is not, the model is not l.s.i.. +Identifiability was analyzed as part of the differential geometric approach and considered as +augmented observability [18, 19]. For this purpose, the state vector is augmented by the model +parameters, for which trivial dynamics is assumed: +˙˜x = +� +˙x +˙θ +� += +� +f(x, u, θ) +0 +� +. +(12) +Consequently, for FISPO, computation of the extended observability matrix is augmented to: +O(˜x) = +� +� +� +� +� +� +� +� +∂h(˜x) +∂˜x +∂ +∂˜x (Lfh(˜x)) +... +∂ +∂˜x +� +Ln+m−1 +f +h(˜x) +� +� +� +� +� +� +� +� +� +. +(13) +The system is considered observable and identifiable if the rank conditions satisfies [18] +rank (O(˜x0)) = n + m. +(14) +For ORC-DF the state differential equations (9) are augmented by a zero vector of dimension +m × 1, accounting for the trivial dynamics of the parameters. The output equations remain +unchanged. +For the ADM1-R4, the rate constants of hydrolysis and decay were assumed as time-variant +and analyzed for identifiability. For the ADM1-R3, the following parameters were investigated: +maximum growth rate and half-saturation constant of acetoclastic methanogens as well as the +inhibition constant for non-competitive ammonia inhibition. +3 Results and Discussion +3.1 Algebraic Approach +The full ADM1-R4 (BMR4+AB) could be shown to be globally observable by using the +measurements given in Tab. 2. The underlying system of equations involves online measurements +of the gas composition (CH4 and CO2). Furthermore, to guarantee observability, measurements +of IN, TS and VS need to be available. This follows directly from the model equations: all +three states only appear in their corresponding differential equation. Therefore, if they were not +available as measurements, they could not be observable because they would not be introduced +9 + +Table 2: Measurements and their time derivatives required for algebraic observability of model +simplifications +Nominal measurementsa,b +1st derivative +2nd derivative +3rd deriv.e +Model name nc +t[s]d +CH4 +CO2 +IN +TS VS +Ac +CH4 +CO2 +IN TS +CH4 +CO2 +IN TS +CH4 +CO2 +BMR4 +9 +9 +x +x +x +x +x +x +x +x +x +BMR4+A +9 +3 +x +x +x +x +x +x +x +x +x +BMR4+B +11 +7 +x +x +x +x +x +x +x +x +x +x +x +BMR4+AB +11 +10 +x +x +x +x +x +x +x +x +x +x +x +BMR3+AC +f +11 +1061 +x +x +x +x +x +x +x +x +x +x +x +1038 +x +x +x +x +x +x +x +x +x +x +x +BMR3+BC +f +13 +5352 +x +x +x +x +x +x +x +x +x +x +x +x +x +6353 +x +x +x +x +x +x +x +x +x +x +x +x +x +BMR3+ABC +f +13 +3861 +x +x +x +x +x +x +x +x +x +x +x +x +x +8730 +x +x +x +x +x +x +x +x +x +x +x +x +x +aCH4, CO2 - partial pressures of methane and carbon dioxide; IN - inorganic nitrogen; TS, VS - total and volatile +solids; Ac - acetic acid +bCH4 and CO2 were assumed as online measurements, as well as Ac for ADM1-R3 +variants. IN, TS and VS were assumed as offline measurements. +cNumber of states +dComputation time +ederivative +fOnly local observability could be shown because two solutions to the equation systems were found. +into the system of equations via other measurements, regardless of the degree of time derivatives. +Model simplifications were applied to the ADM1-R4, resulting in the submodels BMR4, +BMR4+A and BMR4+B. This was done to explore the effect of individual model parts on +the complexity of the system of equations, indicated by the computation time required to +solve the system of equations. However, reduction in computational complexity could not +clearly be attributed to omitted model parts, as summarized in Tab. 2: The simplest model +variant (BMR4) showed the second highest computation time among the ADM1-R4 models, +whereas the fastest computation time could be achieved with the less reduced variant BMR4+A. +Nevertheless, all ADM1-R4 submodels could be shown to be globally observable and required +only short computation times. +For the full ADM1-R3 (BMR3+ABCDE), the solver of Mathematica failed to deliver conclusive +statements on observability, likely due to the complexity of the resulting terms in the system +of equations. Neither a set of solutions nor an empty set could be returned. Instead, the +kernel systematically died in the process of solving the system of equations. Hence, with the +algebraic approach, observability of the full ADM1-R3 could not be determined. Therefore, +model complexity of the ADM1-R3 was reduced by isolating and omitting individual model +parts (see Sec. 2.1). +Up to this point, measurements of CH4, CO2, IN, TS and VS were assumed. Additionally, +for all ADM1-R3 submodels online measurements of acetic acid had to be taken into account +to show local observability, see Tab. 2. The obtained results are therefore strictly valid only +10 + +for lab- and pilot-scale settings [22]. To find solutions of the systems of equations, first and +second-order derivatives of the offline measurements IN and TS are required, see Sec. 2.1.2. +With given nominal values of VS, it was preferred to include time derivatives of TS over VS, +because both terms added the same information to the system of equations but TS delivered +less complex terms. +For both BMR3+AC and BMR3+BC, local observability could be shown for two combinations +of equations, each only differing by one equation: the second derivative of IN or of TS, as +shown in Tab. 2. Both respective combinations were solved in similar computation times. +BMR3+ABC is the most complex model variant which still delivered a solution of the resulting +system of equations. +Two solutions of the systems of equations each could be found by +either incorporating the first derivative of IN or TS. The latter resulted in more than double +computation times (Tab. 2). Based on specific model equations, this was not expected because +the first derivatives of both IN and TS have the same algebraic structure and are linearly +dependent except for their convection terms. +By contrast, for BMR3+AC and BMR3+BC computation times of the two respective solutions +did not differ much. This indicates that computational complexity of the terms resulting from +the second derivatives of IN and TS does not change as much as for their first derivatives +(observed in BMR3+ABC). Omitting model part B (cf. BMR3+AC) resulted in a significant +reduction of computation times, which is in line with the findings for ADM1-R4 model variants. +Omitting part A instead does not allow to draw such clear conclusions (cf. BMR3+BC), since +computation times are in between the two observable combinations of BMR3+ABC. +When restricting measurements to gas composition, IN, TS and VS, none of the ADM1-R3 +submodels could be shown to be observable using the algebraic approach. For higher-order +variants than BMR3+ABC, inhibition through pH and ammonia need to be considered, as well +as computation of pH. This entails a higher number of state variables, which leads to symbolic +systems of equations too complex to be solved in the presented framework. +Lower-order model variants of the ADM1-R3 were not considered because (i) no clear tendency +of reduction in computation time could be observed by omitting model parts, and (ii) for +practical applications, lower-order models were already given through the ADM1-R4 and +corresponding submodels. +3.2 Geometric Approach +Based on the implementation in the STRIKE_GOLDD toolbox, the geometric approach proved +to be successful in showing both local observability and structural local identifiability of the +11 + +Table 3: Model properties and computation times of FISPO and ORC-DF algorithms for +ADM1-R4 and ADM1-R3. +Model class +number of +computation time in s +states +parameters +FISPO +ORC-DF +ADM1-R4 +11 +4 +3 +7 +ADM1-R3 +17 +7 +11959 +811 +full models ADM1-R4 and ADM1-R3. Moreover, this was achieved in (i) considerably shorter +computation times, (ii) without assuming acetic acid as an online measurement, and (iii) by +employing two independent algorithms (FISPO and ORC-DF). This is summarized in Tab. 3. +As noted by [27], computational efficiency of FISPO and ORC-DF can differ vastly depending +on the model structure, which is apparent for the larger ADM1-R3. +Furthermore, computational effort decreases significantly when neglecting model parts of the +full ADM1-R3: computation times of the submodel BMR3-ABC, for instance, could be reduced +to 12 and 5 s for the FISPO and ORC-DF algorithms, respectively. This is also considerably +faster than those computation times achieved with the algebraic approach, cf. Tab. 2. +Higher-order model classes such as the ADM1-R2 are of similar structure as the ADM1-R3, +but involve significantly more time-variant parameters and contain a more detailled acid +spectrum (acetic to valeric acid instead of only acetic acid) [15]. In an agricultural setting, +these acid measurements are not available online. However, even when assuming them to be +available as online measurements, both algorithms of the geometric approach fail to deliver +conclusive statements. This is likely due to the aggravated complexity of the involved symbolic +expressions. +A practical application of the ADM1-R2 is thus not anticipated for monitoring and control +schemes due to the lack of online measurements for the individual VFA. It was therefore not +further simplified into submodels and no further observability and identifiability analyses were +pursued. +4 Conclusion +The models ADM1-R4 and ADM1-R3 were analysed for observability and identifiability using +differential algebraic and geometric approaches. With the former, observability of the ADM1-R4 +was successfully shown. However, the algebraic approach failed for the larger ADM1-R3 if only +typical measurements at agricultural biogas plants were considered. In this scenario however, +the geometric approach succeeded to show observability and identifiability for both ADM1-R3 +and ADM1-R4, and exhibited a higher computational efficiency. This emphasizes that the +12 + +ADM1 simplifications of Weinrich and Nelles [15] are indeed promising for state and parameter +estimation as well as process automation for agricultural AD plants. +Acknowledgment +The authors are thankful for funding from German Federal Ministry of Food and Agriculture of +the junior research group on simulation, monitoring and control of anaerobic digestion plants +(grant 2219NR333). S. H. thanks Terrance Wilms for his encouragement and advice. +Appendix +The ADM1 simplifications of Weinrich and Nelles [15] have been transformed into standard +control notation. This section summarizes the model equations and parameters of all successfully +investigated models, starting with the simplest model structures. The following notation was +applied: +• xi - states (mass concentrations of the species involved), +• yi - measurements +• u - control variable (feed volume flow) +• ξi - time-variant, uncertain parameters (inlet mass concentration) +• θi - time-variant parameters +• ci - (aggregated) time-invariant parameters +• aij - time-invariant stoichiometric coefficients +ADM1-R4 Models +Aside from the ADM1-R4, the following models were analyzed: BMR4+B, BMR4+A, and +BMR4. +ADM1-R4 +State vector: +x = [Sch4, SIC, SIN, Sh2o, Xch, Xpr, Xli, Xbac, Xash, Sch4,gas, Sco2,gas]T +(15) +13 + +Differential equations: +˙x1 = c1 (ξ1 − x1) u + a11θ1x5 + a12θ2x6 + a13θ3x7 − c2x1 + c3x10 +(16a) +˙x2 = c1 (ξ2 − x2) u + a21θ1x5 + a22θ2x6 + a23θ3x7 − c2x2 + c4x11 +(16b) +˙x3 = c1 (ξ3 − x3) u − a31θ1x5 + a32θ2x6 − a33θ3x7 +(16c) +˙x4 = c1 (ξ4 − x4) u − a41θ1x5 − a42θ2x6 − a43θ3x7 +(16d) +˙x5 = c1 (ξ5 − x5) u − θ1x5 + a54θ4x8 +(16e) +˙x6 = c1 (ξ6 − x6) u − θ2x6 + a64θ4x8 +(16f) +˙x7 = c1 (ξ7 − x7) u − θ3x7 + a74θ4x8 +(16g) +˙x8 = c1 (ξ8 − x8) u + a81θ1x5 + a82θ2x6 + a83θ3x7 − θ4x8 +(16h) +˙x9 = c1 (ξ9 − x9) u +(16i) +˙x10 = c15x3 +10 + c16x2 +10x11 + c17x10x2 +11 + c18x2 +10 + c19x10x11 + c20x10 + c5x1 +(16j) +˙x11 = c17x3 +11 + c16x10x2 +11 + c15x2 +10x11 + c19x2 +11 + c18x10x11 + c21x11 + c6x2 +(16k) +aij are the absolute values of the entries of the petersen matrix given in Tab. 5. i denotes the +column (component) and j the row (process). For brevity, only those entries with an absolute +value ̸= 1 or 0 were denoted specifically. +Measurements: +y1 = ˙Vg = c6x2 +10 + c7x10x11 + c8x2 +11 + c9x10 + c10x11 + c11 +(17a) +y2 = pch4 = c12x10 +(17b) +y3 = pco2 = c13x11 +(17c) +y4 = SIN = x3 +(17d) +y5 = TS = 1 − 1 +c14 +x4 +(17e) +y6 = V S = 1 − +1 +c14 − x4 +x9 +(17f) +Tab. 4 summarizes the aggregated parameters ci and the time-variant parameters θi used for +ADM1-R4 and its submodels. +BMR4+B +The state vector remains as in (15). Differential equations: +˙x1 = c1 (ξ1 − x1) u + a11θ1x5 + a12θ2x6 + a13θ3x7 + a14θ4x8 − c2x1 + c3x10 +(18a) +˙x2 = c1 (ξ2 − x2) u + a21θ1x5 + a22θ2x6 + a23θ3x7 + a24θ4x8 − c2x2 + c4x11 +(18b) +˙x3 = c1 (ξ3 − x3) u − a31θ1x5 + a32θ2x6 − a33θ3x7 + a34θ4x8 +(18c) +˙x4 = c1 (ξ4 − x4) u − a41θ1x5 − a42θ2x6 − a43θ3x7 − a44θ4x8 +(18d) +14 + +˙x5 = c1 (ξ5 − x5) u − θ1x5 +(18e) +˙x6 = c1 (ξ6 − x6) u − θ2x6 +(18f) +˙x7 = c1 (ξ7 − x7) u − θ3x7 +(18g) +˙x8 = c1 (ξ8 − x8) u + a81θ1x5 + a82θ2x6 + a83θ3x7 − a84θ4x8 +(18h) +˙x9 = c1 (ξ9 − x9) u +(18i) +˙x10 = c15x3 +10 + c16x2 +10x11 + c17x10x2 +11 + c18x2 +10 + c19x10x11 + c20x10 + c5x1 +(18j) +˙x11 = c17x3 +11 + c16x10x2 +11 + c15x2 +10x11 + c19x2 +11 + c18x10x11 + c21x11 + c6x2. +(18k) +Note that omitting model part A entails modified stoichiometric constants aij as given in Tab. 6. +The other parameters remain the same and are summarized in Tab.4. The measurements +remain as in (17). +BMR4+A +State vector: +x = [SIN, Sh2o, Xch, Xpr, Xli, Xbac, Xash, Sch4,gas, Sco2,gas]T +(19) +Differential equations, where aij are given in Tab. 5 and the remaining parameters in Tab.4: +˙x1 = c1 (ξ1 − x1) u − a31θ1x3 + a32θ2x4 − a33θ3x5 +(20a) +˙x2 = c1 (ξ2 − x2) u − a41θ1x3 − a42θ2x4 − a43θ3x5 +(20b) +˙x3 = c1 (ξ3 − x3) u − θ1x3 + a54θ4x6 +(20c) +˙x4 = c1 (ξ4 − x4) u − θ2x4 + a64θ4x6 +(20d) +˙x5 = c1 (ξ5 − x5) u − θ3x5 + a74θ4x6 +(20e) +˙x6 = c1 (ξ6 − x6) u + a81θ1x3 + a82θ2x4 + a83θ3x5 − θ4x6 +(20f) +˙x7 = c1 (ξ7 − x7) u +(20g) +˙x8 = c22a11θ1x3 + c22a12θ2x4 + c22a13θ3x5 + c15x3 +8 + c16x2 +8x9 + c17x8x2 +9 + c18x2 +8+ ++ c19x8x9 + c23x8 +(20h) +˙x9 = c22a21θ1x3 + c22a22θ2x4 + c22a23θ3x5 + c17x3 +9 + c16x8x2 +9 + c15x2 +8x9 + c19x2 +9+ ++ c18x8x9 + c23x9 +(20i) +Measurements: +y1 = ˙Vg = c6x2 +8 + c7x8x9 + c8x2 +9 + c9x8 + c10x9 + c11, +(21a) +y2 = pch4 = c12x8, +(21b) +y3 = pco2 = c13x9, +(21c) +y4 = SIN = x1, +(21d) +y5 = TS = 1 − 1 +c14 +x2, +(21e) +15 + +y6 = V S = 1 − +1 +c14 − x2 +x7. +(21f) +BMR4 +The state vector remains as in (19). Differential equations: +˙x1 = c1 (ξ1 − x1) u − a31θ1x3 + a32θ2x4 − a33θ3x5 + a34θ4x6 +(22a) +˙x2 = c1 (ξ2 − x2) u − a41θ1x3 − a42θ2x4 − a43θ3x5 − a44θ4x6 +(22b) +˙x3 = c1 (ξ3 − x3) u − θ1x3 +(22c) +˙x4 = c1 (ξ4 − x4) u − θ2x4 +(22d) +˙x5 = c1 (ξ5 − x5) u − θ3x5 +(22e) +˙x6 = c1 (ξ6 − x6) u + a81θ1x3 + a82θ2x4 + a83θ3x5 − a84θ4x6 +(22f) +˙x7 = c1 (ξ7 − x7) u +(22g) +˙x8 = c22a11θ1x3 + c22a12θ2x4 + c22a13θ3x5 + c22a14θ4x6 + c15x3 +8 + c16x2 +8x9+ ++ c17x8x2 +9 + c18x2 +8 + c19x8x9 + c23x8 +(22h) +˙x9 = c22a21θ1x3 + c22a22θ2x4 + c22a23θ3x5 + c22a24θ4x6 + c17x3 +9 + c16x8x2 +9+ ++ c15x2 +8x9 + c19x2 +9 + c18x8x9 + c23x9 +(22i) +aij are given in Tab. 6 and the remaining parameters in Tab.4. The measurements remain as +in (21). +16 + +Table 4: Aggregated and time-variant parameters of ADM1-R4 and its submodels. +Aggregated notation +Notation by Weinrich and Nelles [15] +u +˙Vf +θ1 +kch +θ2 +kpr +θ3 +kli +θ4 +kdec +c1 +V −1 +l +c2 +kLa +c3 +kLaKH,ch4 ¯RT +c4 +kLaKH,co2 ¯RT +c5 +kLaVlV −1 +g +c6 +kpp−1 +0 +� ¯RT ¯ +M−1 +ch4 +�2 +c7 +2kpp−1 +0 +� ¯RT +�2 ¯ +M−1 +ch4 ¯ +M−1 +co2 +c8 +kpp−1 +0 +� ¯RT ¯ +M−1 +co2 +�2 +c9 +kpp−1 +0 +¯RT ¯ +M−1 +ch4 (2ph2o − p0) +c10 +kpp−1 +0 +¯RT ¯ +M−1 +co2 (2ph2o − p0) +c11 +kpp−1 +0 +(ph2o − p0) ph2o +c12 +¯RT ¯ +M−1 +ch4 +c13 +¯RT ¯ +M−1 +co2 +c14 +ρl +c15 +−kpp−1 +0 V −1 +g +� ¯RT ¯ +M−1 +ch4 +�2 +c16 +−2kpp−1 +0 V −1 +g +� ¯RT +�2 ¯ +M−1 +ch4 ¯ +M−1 +co2 +c17 +−kpp−1 +0 V −1 +g +� ¯RT ¯ +M−1 +co2 +�2 +c18 +−kpp−1 +0 V −1 +g +¯RT ¯ +M−1 +ch4 (2ph2o − p0) +c19 +−kpp−1 +0 V −1 +g +¯RT ¯ +M−1 +co2 (2ph2o − p0) +c20 +−kLaVlV −1 +g +KH,ch4 ¯RT − kpp−1 +0 V −1 +g +(ph2o − p0) ph2o +c21 +−kLaVlV −1 +g +KH,co2 ¯RT − kpp−1 +0 V −1 +g +(ph2o − p0) ph2o +c22 +VlV −1 +g +c23 +−kpp−1 +0 V −1 +g +(ph2o − p0) ph2o +17 + +Table 5: Petersen matrix of ADM1-R4, derived from [28]. +Component i → +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +j Process ↓ +Sch4 +SIC +SIN +Sh2o +Xch +Xpr +Xli +Xbac +Xash +Sch4,gas +Sco2,gas Process rate rj +1 Fermentation Xch +0.2482 +0.6809 +-0.0207 +-0.0456 +-1 +0.1372 +θ1 x5 +2 Fermentation Xpr +0.3221 +0.7954 +0.1689 +-0.4588 +-1 +0.1723 +θ2 x6 +3 Fermentation Xli +0.6393 +0.5817 +-0.0344 +-0.4152 +-1 +0.2286 +θ3 x7 +4 Decay Xbac +0.18 +0.77 +0.05 +-1 +θ4 x8 +5 Phase transition Sch4 +-1 +c22 +c2x1 − c3x10 +6 Phase transition SIC +-1 +c22 +c2x2 − c4x11 +Table 6: Petersen matrix of BMR4+B, derived from [28]. +Component i → +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +j Process ↓ +Sch4 +SIC +SIN +Sh2o +Xch +Xpr +Xli +Xbac +Xash +Sch4,gas +Sco2,gas Process rate rj +1 Fermentation Xch +0.2482 +0.6809 +-0.0207 +-0.0456 +-1 +0.1372 +θ1 x5 +2 Fermentation Xpr +0.3221 +0.7954 +0.1689 +-0.4588 +-1 +0.1723 +θ2 x6 +3 Fermentation Xli +0.6393 +0.5817 +-0.0344 +-0.4152 +-1 +0.2286 +θ3 x7 +4 Decay Xbac +0.3247 +0.7641 +0.1246 +-0.3822 +-0.8312 +θ4 x8 +5 Phase transition Sch4 +-1 +c22 +c2x1 − c3x10 +6 Phase transition SIC +-1 +c22 +c2x2 − c4x11 +18 + +ADM1-R3 Models +Aside from ADM1-R3, the following models were analyzed: BMR3+AC, BMR3+BC, and +BMR3+ABC. +ADM1-R3 +State vector: +x = +�Sac, Sch4, SIC, SIN, Sh2o, Xch, Xpr, Xli, Xbac, Xac, Xash, ... +Sion, Sac−, Shco3−, Snh3, Sch4,gas, Sco2,gas +�T +(23) +Differential equations: +˙x1 = c1 (ξ1 − x1) u + a11θ1x6 + a12θ2x7 + a13θ3x8 − a14θ5 +x1 x10 +θ6 + x1 +Iac +(24a) +˙x2 = c1 (ξ2 − x2) u + a21θ1x6 + a22θ2x7 + a23θ3x8 − c5x2 + c6x16 + a24θ5 +x1 x10 +θ6 + x1 +Iac +(24b) +˙x3 = c1 (ξ3 − x3) u + a31θ1x6 + a32θ2x7 − a33θ3x8 − c5x3 + c5x14 + c7x17 + a34θ5 +x1 x10 +θ6 + x1 +Iac +(24c) +˙x4 = c1 (ξ4 − x4) u − a41θ1x6 + a42θ2x7 − a43θ3x8 − a44θ5 +x1 x10 +θ6 + x1 +Iac +(24d) +˙x5 = c1 (ξ5 − x5) u − a51θ1x6 − a52θ2x7 − a53θ3x8 + a54θ5 +x1 x10 +θ6 + x1 +Iac +(24e) +˙x6 = c1 (ξ6 − x6) u − θ1x6 + a65θ4x9 + a66θ4x10 +(24f) +˙x7 = c1 (ξ7 − x7) u − θ2x7 + a75θ4x9 + a76θ4x10 +(24g) +˙x8 = c1 (ξ8 − x8) u − θ3x8 + a85θ4x9 + a86θ4x10 +(24h) +˙x9 = c1 (ξ9 − x9) u + a91θ1x6 + a92θ2x7 + a93θ3x8 − θ4x9 +(24i) +˙x10 = c1 (ξ10 − x10) u − θ4x10 + θ5 +x1 x10 +θ6 + x1 +Iac +(24j) +˙x11 = c1 (ξ11 − x11) u +(24k) +˙x12 = c1 (ξ12 − x12) u +(24l) +˙x13 = c29 (x1 − x13) − c9x13SH+ +(24m) +˙x14 = c30 (x3 − x14) − c10x14SH+ +(24n) +˙x15 = c31 (x4 − x15) − c11x15SH+ +(24o) +˙x16 = c22x3 +16 + c23x2 +16x17 + c24x16x2 +17 + c25x2 +16 + c26x16x17 + c12x2 + c27x16 +(24p) +˙x17 = c24x3 +17 + c23x16x2 +17 + c22x2 +16x17 + c26x2 +17 + c25x16x17 + c12x3 − c12x14 + c28x17 +(24q) +Iac and SH+ are defined as +Iac = +c3 +c3 + Sc2 +H+ +x4 +x4 + c8 +θ7 +θ7 + x15 +(25) +SH+ = −Φ +2 + 1 +2 +� +Φ2 + c4 , where +(26) +19 + +Φ = x12 + x4 − x15 +17 +− x14 +44 − x13 +60 . +(27) +The absolute values of the stoichiometric coefficients aij are given in the petersen matrix of +ADM1-R3, Tab. 8. Note that for all ADM1-R3 models, definition and indexing of parameters +and stoichiometric constants is independent from all ADM1-R4 models. +Measurements: +y1 = ˙Vg = c13x2 +16 + c14x16x17 + c15x2 +17 + c16x16 + c17x17 + c18 +(28a) +y2 = pch4 = c19x16 +(28b) +y3 = pco2 = c20x17 +(28c) +y4 = pH = − log10 SH+ +(28d) +y5 = SIN = x4 +(28e) +y6 = TS = 1 − 1 +c21 +x5, +(28f) +y7 = V S = 1 − +1 +c21 − x5 +x11 +(28g) +y8 = Sac = x1 +(28h) +Tab. 7 summarizes the aggregated parameters ci and the time-variant parameters θi used for +ADM1-R3 and all of its submodels. +BMR3+ABC +State vector: +x = [Sac, Sch4, SIC, SIN, Sh2o, Xch, Xpr, Xli, Xbac, Xac, Xash, Sch4,gas, Sco2,gas]T +(29) +Differential equations: +˙x1 = c1 (ξ1 − x1) u + a11θ1x6 + a12θ2x7 + a13θ3x8 − a14θ5 +x1 x10 +θ6 + x1 +x4 +x4 + c8 +(30a) +˙x2 = c1 (ξ2 − x2) u + a21θ1x6 + a22θ2x7 + a23θ3x8 − c5x2 + c6x12 + a24θ5 +x1 x10 +θ6 + x1 +x4 +x4 + c8 +(30b) +˙x3 = c1 (ξ3 − x3) u + a31θ1x6 + a32θ2x7 − a33θ3x8 − c5x3 + c7x13 + a34θ5 +x1 x10 +θ6 + x1 +x4 +x4 + c8 +(30c) +˙x4 = c1 (ξ4 − x4) u − a41θ1x6 + a42θ2x7 − a43θ3x8 − a44θ5 +x1 x10 +θ6 + x1 +x4 +x4 + c8 +(30d) +˙x5 = c1 (ξ5 − x5) u − a51θ1x6 − a52θ2x7 − a53θ3x8 + a54θ5 +x1 x10 +θ6 + x1 +x4 +x4 + c8 +(30e) +˙x6 = c1 (ξ6 − x6) u − θ1x6 + a65θ4x9 + a66θ4x10 +(30f) +˙x7 = c1 (ξ7 − x7) u − θ2x7 + a75θ4x9 + a76θ4x10 +(30g) +˙x8 = c1 (ξ8 − x8) u − θ3x8 + a85θ4x9 + a86θ4x10 +(30h) +˙x9 = c1 (ξ9 − x9) u + a91θ1x6 + a92θ2x7 + a93θ3x8 − θ4x9 +(30i) +20 + +˙x10 = c1 (ξ10 − x10) u − θ4x10 + θ5 +x1 x10 +θ6 + x1 +x4 +x4 + c8 +(30j) +˙x11 = c1 (ξ11 − x11) u +(30k) +˙x12 = c22x3 +12 + c23x2 +12x13 + c24x12x2 +13 + c25x2 +12 + c26x12x13 + c12x2 + c27x12 +(30l) +˙x13 = c24x3 +13 + c23x12x2 +13 + c22x2 +12x13 + c26x2 +13 + c25x12x13 + c12x3 + c28x13 +(30m) +aij have to be taken from Tab. 8. The other involved parameters are given in Tab. 7. +Measurements: +y1 = ˙Vg = c13x2 +12 + c14x12x13 + c15x2 +13 + c16x12 + c17x13 + c18 +(31a) +y2 = pch4 = c19x12 +(31b) +y3 = pco2 = c20x13 +(31c) +y4 = SIN = x4 +(31d) +y5 = TS = 1 − 1 +c21 +x5 +(31e) +y6 = V S = 1 − +1 +c21 − x5 +x11 +(31f) +y7 = Sac = x1 +(31g) +BMR3+BC +The state vector remains as in (29). Differential equations: +˙x1 = c1 (ξ1 − x1) u + a11θ1x6 + a12θ2x7 + a13θ3x8 − a14θ5 +x1 x10 +θ6 + x1 +x4 +x4 + c8 ++ a15θ4x9+ ++ a16θ4x10 +(32a) +˙x2 = c1 (ξ2 − x2) u + a21θ1x6 + a22θ2x7 + a23θ3x8 − c5x2 + c6x12 + a24θ5 +x1 x10 +θ6 + x1 +x4 +x4 + c8 ++ ++ a25θ4x9 + a26θ4x10 +(32b) +˙x3 = c1 (ξ3 − x3) u + a31θ1x6 + a32θ2x7 − a33θ3x8 − c5x3 + c7x13 + a34θ5 +x1 x10 +θ6 + x1 +x4 +x4 + c8 ++ ++ a35θ4x9 + a36θ4x10 +(32c) +˙x4 = c1 (ξ4 − x4) u − a41θ1x6 + a42θ2x7 − a43θ3x8 − a44θ5 +x1 x10 +θ6 + x1 +x4 +x4 + c8 ++ a45θ4x9+ ++ a46θ4x10 +(32d) +˙x5 = c1 (ξ5 − x5) u − a51θ1x6 − a52θ2x7 − a53θ3x8 + a54θ5 +x1 x10 +θ6 + x1 +x4 +x4 + c8 +− a55θ4x9+ +− a56θ4x10 +(32e) +˙x6 = c1 (ξ6 − x6) u − θ1x6 +(32f) +˙x7 = c1 (ξ7 − x7) u − θ2x7 +(32g) +˙x8 = c1 (ξ8 − x8) u − θ3x8 +(32h) +˙x9 = c1 (ξ9 − x9) u + a91θ1x6 + a92θ2x7 + a93θ3x8 − a95θ4x9 + a96θ4x10 +(32i) +˙x10 = c1 (ξ10 − x10) u − θ4x10 + θ5 +x1 x10 +θ6 + x1 +x4 +x4 + c8 +(32j) +21 + +˙x11 = c1 (ξ11 − x11) u +(32k) +˙x12 = c22x3 +12 + c23x2 +12x13 + c24x12x2 +13 + c25x2 +12 + c26x12x13 + c12x2 + c27x12 +(32l) +˙x13 = c24x3 +13 + c23x12x2 +13 + c22x2 +12x13 + c26x2 +13 + c25x12x13 + c12x3 + c28x13 +(32m) +Note that omitting model part A entails modified stoichiometric constants aij as given in +Tab. 9. The other involved parameters remain the same. They are given in Tab. 7. +Measurements: +y1 = ˙Vg = c13x2 +12 + c14x12x13 + c15x2 +13 + c16x12 + c17x13 + c18 +(33a) +y2 = pch4 = c19x12 +(33b) +y3 = pco2 = c20x13 +(33c) +y4 = SIN = x4 +(33d) +y5 = TS = 1 − 1 +c21 +x5 +(33e) +y6 = V S = 1 − +1 +c21 − x5 +x11 +(33f) +y7 = Sac = x1 +(33g) +BMR3+AC +State vector: +x = [Sac, SIN, Sh2o, Xch, Xpr, Xli, Xbac, Xac, Xash, Sch4,gas, Sco2,gas]T +(34) +Differential equations: +˙x1 = c1 (ξ1 − x1) u + a11θ1x4 + a12θ2x5 + a13θ3x6 − a14θ5 +x1 x8 +θ6 + x1 +x2 +x2 + c8 +(35a) +˙x2 = c1 (ξ2 − x2) u − a41θ1x4 + a42θ2x5 − a43θ3x6 − a44θ5 +x1 x8 +θ6 + x1 +x2 +x2 + c8 +(35b) +˙x3 = c1 (ξ3 − x3) u − a51θ1x4 − a52θ2x5 − a53θ3x6 + a54θ5 +x1 x8 +θ6 + x1 +x2 +x2 + c8 +(35c) +˙x4 = c1 (ξ4 − x4) u − θ1x4 + a65θ4x7 + a66θ4x8 +(35d) +˙x5 = c1 (ξ5 − x5) u − θ2x5 + a75θ4x7 + a76θ4x8 +(35e) +˙x6 = c1 (ξ6 − x6) u − θ3x6 + a85θ4x7 + a86θ4x8 +(35f) +˙x7 = c1 (ξ7 − x7) u + a91θ1x4 + a92θ2x5 + a93θ3x6 − θ4x7 +(35g) +˙x8 = c1 (ξ8 − x8) u + θ5 +x1 x8 +θ6 + x1 +x2 +x2 + c8 +− θ4x8 +(35h) +˙x9 = c1 (ξ9 − x9) u +(35i) +˙x10 = c32a21θ1x4 + c32a22θ2x5 + c32a23θ3x6 + c32a24θ5 +x1 x8 +θ6 + x1 +x2 +x2 + c8 ++ +c22x3 +10 + c23x2 +10x11 + c24x10x2 +11 + c25x2 +10 + c26x10x11 + c33x10 +(35j) +22 + +˙x11 = c32a31θ1x4 + c32a32θ2x5 + c32a33θ3x6 + c32a34θ5 +x1 x8 +θ6 + x1 +x2 +x2 + c8 ++ +c24x3 +11 + c23x10x2 +11 + c22x2 +10x11 + c26x2 +11 + c25x10x11 + c33x11 +(35k) +aij have to be taken from Tab. 8. The involved parameters remain the same. They are given +in Tab. 7. Measurements: +y1 = ˙Vg = c13x2 +10 + c14x10x11 + c15x2 +11 + c16x10 + c17x11 + c18 +(36a) +y2 = pch4 = c19x10 +(36b) +y3 = pco2 = c20x11 +(36c) +y4 = SIN = x2 +(36d) +y5 = TS = 1 − 1 +c21 +x3 +(36e) +y6 = V S = 1 − +1 +c21 − x3 +x9 +(36f) +y7 = Sac = x1 +(36g) +23 + +Table 7: Aggregated and time-variant parameters of ADM1-R3 and its submodels. +Aggregated notation +Notation by Weinrich and Nelles [15] +u +˙Vf +θ1 +kch +θ2 +kpr +θ3 +kli +θ4 +kdec +θ5 +µm,ac +θ6 +KS,ac +θ7 +KI,nh3 +c1 +V −1 +l +c2 +nac +c3 +10 +− 3 +2 +pHUL,ac+pHLL,ac +pHUL,ac−pHLL,ac +c4 +4KW +c5 +kLa +c6 +kLaKH,ch4 ¯RT +c7 +kLaKH,co2 ¯RT +c8 +KS,IN +c9 +kAB,ac +c10 +kAB,co2 +c11 +kAB,IN +c12 +kLaVlV −1 +g +c13 +kpp−1 +0 +� ¯RT ¯ +M−1 +ch4 +�2 +c14 +2kpp−1 +0 +� ¯RT +�2 ¯ +M−1 +ch4 ¯ +M−1 +co2 +c15 +kpp−1 +0 +� ¯RT ¯ +M−1 +co2 +�2 +c16 +kpp−1 +0 +¯RT ¯ +M−1 +ch4 (2ph2o − p0) +c17 +kpp−1 +0 +¯RT ¯ +M−1 +co2 (2ph2o − p0) +c18 +kpp−1 +0 +(ph2o − p0) ph2o +c19 +¯RT ¯ +M−1 +ch4 +c20 +¯RT ¯ +M−1 +co2 +c21 +ρl +c22 +−kpV −1 +g +p−1 +0 +� ¯RT ¯ +M−1 +ch4 +�2 +c23 +−2kpV −1 +g +p−1 +0 +� ¯RT +�2 ¯ +M−1 +ch4 ¯ +M−1 +co2 +c24 +−kpV −1 +g +p−1 +0 +� ¯RT ¯ +M−1 +co2 +�2 +c25 +−kpV −1 +g +p−1 +0 +¯RT ¯ +M−1 +ch4 (2ph2o − p0) +c26 +−kpV −1 +g +p−1 +0 +¯RT ¯ +M−1 +co2 (2ph2o − p0) +c27 +−kLaVlV −1 +g +KH,ch4 ¯RT − kpV −1 +g +p−1 +0 +(ph2o − p0) ph2o +c28 +−kLaVlV −1 +g +KH,co2 ¯RT − kpV −1 +g +p−1 +0 +(ph2o − p0) ph2o +c29 +kAB,acKa,ac +c30 +kAB,co2Ka,co2 +c31 +kAB,INKa,IN +c32 +VlV −1 +g +c33 +−kpV −1 +g +p−1 +0 +(ph2o − p0) ph2o +24 + +Table 8: Petersen matrix of ADM1-R3, derived from [28]. +Component i → +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +j +Process ↓ +Sac +Sch4 +SIC +SIN +Sh2o +Xch +Xpr +Xli +Xbac +Xac +Process rate rj +1 Fermentation Xch +0.6555 +0.0818 +0.2245 +-0.0169 +-0.0574 +-1 +0.1125 +θ1 x6 +2 Fermentation Xpr +0.9947 +0.0696 +0.1029 +0.1746 +-0.4767 +-1 +0.1349 +θ2 x7 +3 Fermentation Xli +1.7651 +0.1913 +-0.6472 +-0.0244 +-0.4469 +-1 +0.1621 +θ3 x8 +4 Methanogenesis Sac +-26.5447 +6.7367 +18.4808 +-0.1506 +0.4778 +1 +θ5 +x1 +θ6+x1 x10 Iac +5 Decay Xbac +0.18 +0.77 +0.05 +-1 +θ4 x9 +6 Decay Xac +0.18 +0.77 +0.05 +-1 +θ4 x10 +2 +3 +. . . +11 +12 +13 +14 +15 +16 +17 +Sch4 +SIC +Xash +Sion +Sac− +Shco3− +Snh3 +Sch4,gas +Sco2,gas +7 Dissoziation Sac +-1 +c29 (x13 − x1) + c9 x13 SH+ +8 Dissoziation SIC +-1 +c30 (x14 − x3) + c10 x14 SH+ +9 Dissoziation SIN +-1 +c31 (x15 − x4) + c11 x15 SH+ +10 Phase transition Sch4 +-1 +c32 +c5 x2 − c6 x16 +11 Phase transition Sco2 +-1 +c32 +c5 (x3 − x14) − c7 x17 +25 + +Table 9: Petersen matrix of BMR3+BC, derived from [28]. +Component i → +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +j Process ↓ +Sac +Sch4 +SIC +SIN +Sh2o +Xch +Xpr +Xli +Xbac +Xac +Xash +Sch4,gas +Sco2,gas Process rate rj +1 Fermentation Xch +0.6555 +0.0818 +0.2245 +-0.0169 +-0.0574 +-1 +0.1125 +θ1 x6 +2 Fermentation Xpr +0.9947 +0.0696 +0.1029 +0.1746 +-0.4767 +-1 +0.1349 +θ2 x7 +3 Fermentation Xli +1.7651 +0.1913 +-0.6472 +-0.0244 +-0.4469 +-1 +0.1621 +θ3 x8 +4 Methanogenesis Sac +-26.5447 +6.7367 +18.4808 +-0.1506 +0.4778 +1 +θ5 +x1 +θ6+x1 x10 +x4 +x4+c8 +5 Decay Xbac +0.9722 +0.0779 +0.0873 +0.1302 +-0.3997 +-0.8678 +θ4 x9 +6 Decay Xac +0.9722 +0.0779 +0.0873 +0.1302 +-0.3997 +0.1322 +-1 +θ4 x10 +7 Phase transition Sch4 +-1 +c32 +c5 x2 − c6 x12 +8 Phase transition Sco2 +-1 +c32 +c5 x3 − c7 x13 +26 + +Derivations for neglecting individual model parts +This section details the basic ideas behind neglecting individual model parts for both ADM1-R3 +and ADM1-R4. +Neglecting model part A - stoichiometric degradation of microbial biomass to macro +nutrients +Stoichiometric pathways of the ADM1 models can easily be derived from the rows of the +petersen matrix, Tab. 8. In the original ADM1-R3, biomass (represented by the states Xbac +and Xac) is formed during degradation of macro nutrients (carbohydrates, proteins and lipids). +Biomass in turn is decomposed into macro nutrients (decay of bacteria). This feedback can +be removed by modifying the stoichiometry of biomass. Biomass is thereby not decomposed +into macro nutrients, but into the stoichiometric degradation products of the macro nutrients +directly. This delivers a modified stoichiometry, affecting the differential equations of Sac, Sch4, +SIC, SIN, Sh2o, Xch, Xpr, Xli, Xbac, and Xac. +Neglecting model part B - gas solubility of CH4 and CO2 +If CH4 and CO2 are assumed to be insoluble in the liquid phase, their stoichiometric formation +therein (see e.g. (24b) and (24c)) has to be allocated to the gas phase. Gas transfer from liquid +to gas phase is originally modeled by means of Henry’s law. The according terms can be cut +out, as well as the convection terms. Instead, the terms describing stoichioetric formation of +CH4 and CO2 are normalized with the ratio of liquid and gas volume (Vl/Vg) and appear in +the differential equations of Sch4,gas and Sco2,gas, respectively. This can be seen in e.g. (35j) +and (35k). Computation of the gas volume flow remains as in the case with model part B. +Neglecting model part C - inhibition through nitrogen limitation +The second factor in the inhibition function Iac describes inhibition through nitrogen limitation, +see (25). +Neglecting model part C is achieved by removing this factor, which reduces +nonlinearities in the model. However, this does not allow to omit any of the state variables. +27 + +Neglecting model part D - inhibition through pH and ammonia +The full inhibition function Iac is a major source of nonlinearity in the ADM1-R3. Neglecting +model part D (inhibition through pH and ammonia) allows to cut out the first and last factor +of Iac, see (25). Consequently, Snh3 can be omitted in the state vector. +Neglecting model part E - computation of pH +Measuring the pH allows to infer SH+ directly because these two variables are linked via the +negative common logarithm, (28d). Measureing the pH hence allows to interpret the variable +SH+ as a time-variant parameter (without any associated differential equation). The states +Sion, Sac− and Shco3− (x12 to x14 in (23) and (24)) only appear in the computation of the +charge balance Φ, (27) which is required to calculate SH+. However, as SH+ can be directly +determined from pH measurements, the states Sion, Sac− and Shco3− become redundant. Their +respective differential equations can be cut out of the system of equations. The resulting +model BMR3+ABCD (not shown here) only incorporates dissociation between ammonium +and ammonia. Yet, full inhibition through all three factors of Iac (pH, nitrogen limitation and +ammonia) are considered. +28 + +References +[1] Ye Chen, Jay J. 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PhD thesis, Universität Rostock, 2017. +31 + diff --git a/3NE4T4oBgHgl3EQfbAwV/content/tmp_files/load_file.txt b/3NE4T4oBgHgl3EQfbAwV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4ee6622a05a6ecb2407e4d0096178f579a375ab --- /dev/null +++ b/3NE4T4oBgHgl3EQfbAwV/content/tmp_files/load_file.txt @@ -0,0 +1,1504 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf,len=1503 +page_content='Observability and Identifiability Analyses of Models for Agricultural Anaerobic Digestion Plants Simon Hellmann1, Arne-Jens Hempel2, Stefan Streif3, Sören Weinrich1 1 DBFZ Deutsches Biomasseforschungszentrum gGmbH, Leipzig, Germany, {simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='hellmann, soeren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='weinrich}@dbfz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='de 2 Saxon University of Cooperative Education, Glauchau, Germany, arne-jens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='hempel@ba-sachsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='de 3 Chemnitz University of Technology, Chemnitz, Germany, stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='streif@etit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='tu-chemnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='de This is the extended version of a paper submitted to the 24th International Conference on Process Control on January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' It is available under a CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0 license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Abstract Dynamic operation of anaerobic digestion plants requires advanced process monitoring and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Different simplifications of the Anaerobic Digestion Model No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 1 (ADM1) have been proposed recently, which appear promising for model-based process automation and state estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' As a fundamental requirement, observability and identifiability of these models are analysed in this work, which was pursued through algebraic and geometric analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Manual algebraic assessment was successfull for small models such as the ADM1-R4 and simplified versions of the ADM1-R3, which were derived in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' However, for larger model classes the algebraic approach showed to be insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' By contrast, the geometric approach, implemented in the STRIKE_GOLDD toolbox, allowed to show observability for more complex models (including ADM1-R4 and ADM1-R3), employing two independent algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The present study lays the groundwork for state observer design, parameter estimation and advanced control resting upon ADM1-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Key words: ADM1, model simplification, Biogas technology, parameter estimation, STRIKE_GOLDD 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='05068v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='SY] 12 Jan 2023 1 Introduction Anaerobic digestion (AD) allows to convert numerous organic feedstocks into biogas, which can either be upgraded and injected into the natural gas grid or combusted to produce renewable electricity and heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The AD process naturally shows strongly nonlinear behavior and is sensitive to process inhibition [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Moreover, the process is prone to instability, especially during dynamic feeding [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' To avoid instable process behavior, monitoring and control schemes are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Many investigations focus on automated operation of domestic or industrial wastewater treatment plants [3–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Additionally, advanced control for efficient and demand-oriented biogas production of agricultural AD plants are frequently examined [2, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' A bottleneck in full-scale application of AD is the lack of online measurements, especially regarding reliable stability indicators, such as volatile fatty acids (VFA) and alkalinity [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' A remedy to overcome this shortage is to apply soft sensors (or state observers), which use readily available external measurements and a mathematical model of the process to estimate internal, non-measurable process states [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' A prominent process model was presented by Bernard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' [11], which has successfully been used in multiple monitoring and control applications [3, 4, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' However, nonlinear aspects of the AD process are described in more detail within the established Anaerobic Digestion Model No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 1 (ADM1) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' While the model proposed by Bernard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' only includes pH inhibition, the ADM1 covers process inhibition through pH, nitrogen limitation and ammonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Still, successful applications of the ADM1 in full-scale monitoring and control applications have not yet been reported, mostly because of its complexity and vast number of parameters [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Yet, in an agricultural setting, which is an important application of AD, both the orginal ADM1 and the model by Bernard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' cannot directly be used due to their underlying reference unit (chemical oxygen demand, COD) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Thus, Weinrich and Nelles have recently proposed mass-based simplifications of the ADM1 [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' These models represent a suitable alternative for application in monitoring and control of agricultural AD plants [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Individual model variations differ significantly in their number of differential equations, state variables and required parameters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Simplified models (such as the ADM1-R4) combine nutrient degradation and biogas formation based on first-order sum reactions, whereas more detailed models (such as the ADM1-R3) depict specific degradation pathways and ammonia or pH inhibition during acetoclastic methanogensis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2 Observability is a model property which indicates whether internal states can be inferred based on input-output measurements [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Likewise, identifiability implies that model parameters can be calibrated based on input-output measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Assessing observability and identifiability is therefore a fundamental requirement before implementing state and parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Numerous approaches for assessing observability and identifiability have been proposed [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' However, especially for complex models they are seldom analyzed a priori because of the computational complexity of the symbolic calculations involved [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This contribution analyses observability and identifiability of different ADM1 simplifications proposed by Weinrich and Nelles [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' For this purpose, two different approaches are pursued: the differential algebraic and differential geometric approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1 Typical measurements at full-scale AD plants are assumed to be available to ensure feasibility of future process control schemes based on these analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2 Methods In this work, we consider systems of ordinary differential equations of the form: M : � � � � � � � � � ˙x = f (x(t), u(t), θ) y = h (x(t), θ) x(t0) = x0 (1) with state variables x ∈ Rn, initial state x0, manipulated variables u ∈ Rp, measurement variables y ∈ Rq, and model parameters θ ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Generally, model parameters (such as microbial growth or decay rates) can be time-variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' However, they vary at a slow rate of change and their dynamics does not follow a defined differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Therefore, their implicit time dependence is suppressed in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1 Modelling of the Anaerobic Digestion Process Based on available model simplifications in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 1, observability and identifiability were assessed for the ADM1-R4, ADM1-R3 and ADM1-R2 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Mass concentration of ash was integrated to compute volatile solids (VS) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 1The two approaches are simply referred to as algebraic and geometric in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 3 Figure 1: Characteristics of different ADM1 simplifications [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1 Model Simplification Both ADM1-R4 and ADM1-R3 were further simplified throughout the investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' For this purpose, individual model components were isolated and omitted or incorporated systematically to assess their influence on observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2 illustrates the full ADM1-R4 and its model parts qualiltatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' These model parts are decay of microbial biomass and its stoichiometric feedback as macro nutrients (part A, in green), and gas solubility of methane and carbon dioxide (part B, in orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The ADM1-R3 allows to isolate more model parts as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Part A and B were left identical as for the ADM1-R4 (in green and orange, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Part C (in purple) describes inhibition through nitrogen limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Part D (in blue) covers inhibition through pH and ammonia, as well as dissociation of ammonium/ammonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Lastly, part E (in red) contains the computation of pH, which includes the charge balance of available anions and cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Carbohydrates Proteins Polymers Lipids First-order kine�cs Biomass decay A Liquid- as g transfer Methane Carbon dioxide Biogas Microorganisms B Figure 2: Model components of the ADM1-R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Polymers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Monomers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Acetic acid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Organic acids ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Biogas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ADM1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ch I pr I li ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='su|aa|fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='va |bulpro ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='CH4 ICO2 / H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Hydrogen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Polymers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Monomers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Organic acids ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Acetic acid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Biogas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ADM1-R1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ch |pr|li ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='su|aa|fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='va | bu |pro ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='CH4 1 CO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Polymers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Organic acids ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Acetic acid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Biogas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ADM1-R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ch | pr |li ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='va|bu|pro ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='CH4 I CO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Polymers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Acetic acid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Biogas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ADM1-R3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ch I pr |li ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='CH4 I CO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Polymers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Biogas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ADM1-R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='> ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ch |pr |li ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='CH4 1 CO2The core elements of the models ADM1-R4 and ADM1-R3 without any additional model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='parts are denoted as BMR4 and BMR3 (base model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' BM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Augmenting them with additional model parts results in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' BMR4+A or BMR4+B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The same notation applies for individual ADM1-R3 model variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The investigated models are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' A full set of the corresponding model equations is given in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2 Data Availability For observability and identifiabilty analyses, measurement signals in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 1 were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Online measurements were generally restricted to pH and gas composition, expressed as partial pressures of methane and carbon dioxide [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' For detailed assessment of observability using the algebraic approach, acetic acid was also considered as available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' For lab and pilot-scale settings, this is a reasonable assumption [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Generally however, online measurements of acetic acid were not considered, which represents the more realistic scenario in a full-scale agricultural setting [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Offline measurements of total solids (TS), volatile solids (VS) and inroganic nitrogen (IN) were assumed to be slowly time-variant in between samples [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' To obtain time-continuous signals, sample-and-hold behavior was supposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2 Observability Analyses Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' A state variable xi(t) is observable if its initial state xi(0) can be reconstructed from measurements of inputs u(τ) and outputs y(τ) over a finite time τ ∈ [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' A system M is fully observable if all of its n states are observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' If at least one state is not observable, the system is not fully observable [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' pH calcula�on Liquid- as g transfer Methane Carbon dioxide Biogas Microorganisms Ace�c acid Organic acids Carbohydrates Proteins Polymers Lipids E pH inhibi�on D N limita�on C Dissocia�on NH inhibi�on 3 D D B Monod kine�cs First-order kine�cs Biomass decay A Figure 3: Model components of the ADM1-R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 5 In this work, two pathways to assess observability were pursued: the algebraic and the geometric approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' In the former, a system of equations of output variables and their time derivatives is established and solved for individual model states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The latter assesses the observability rank condition, which relies on an observability matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1 Algebraic Approach In the algebraic approach, a system of equations of output variables y and their time derivatives y, ˙y, ¨y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' is established and solved for the state variables xi, i ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The system of equations can be summarized as Y = Y(x, u, θ), where Y = (y, ˙y, ¨y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (2) Time derivatives of the outputs are obtained by iteratively computing the Lie derivatives with respect to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' We assumed constant input signals during individual feeding events ( ˙u = 0) and constant parameters ( ˙θ = 0), resulting in: y = h(x, θ) (3) ˙y = Lfh(x, θ) = ∂h(x, θ) ∂x f(x, u, θ), (4) ¨y = L2 fh(x, θ) = ∂Lfh(x, θ) ∂x f(x, u, θ), (5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' y(k) = L(k) f h(x, θ) = ∂L(k−1) f h(x, θ) ∂x f(x, u, θ) (6) Observability is given if the system of equations (2) can be solved algebraically for all states xi, i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' n and has at least one solution [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This requires n equations, which can either be obtained by incorporating all of the q elements of each y, ˙y etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' or by building higher-order time derivatives y(k) and incorporating their elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' If the system of equations can be solved uniquely, global observability is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Multiple solutions indicate local observability [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Table 1: Measurement signals considered for individual model classes Model class Online Offlinea ADM1-R4 gas composition (CH4, CO2) TS, VS, IN ADM1-R3b gas composition (CH4, CO2), pH TS, VS, IN aTotal solids (TS), volatile solids (VS) and inorganic nitrogen (IN) bDuring algebraic oberservability analyses acetic acid was also considered as an optional online measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 6 Implementation The algebraic approach was implemented in Mathematica (version 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0, Wolfram Research, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Equations which could be solved for state variables manually were excluded from the system of equations to minimize computational demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Complexity of the system of equations was further reduced by seeking to incorporate terms with minimal orders of time derivatives first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This is because generally complexity of symbolic derivatives y(k) increases rapidly with the order of derivatives k [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Selection of terms among the q elements of y(k) followed a heuristic procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Since the algebraic approach turned out to be inconclusive for the ADM1-R3, the models ADM1-R4 and ADM1-R3 were systematically simplified (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The maximum allowable model complexity for the algebraic approach was determined by systematically adding model components, starting from the base models (BMR4 and BMR3), and terminating when the system of equations could no longer be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' In the algebraic approach, all parameters and influent concentrations were assumed to be known and time-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Moreover, for the ADM1-R3 model classes, measurements of acetic acid were assumed to be available online in order to achieve conclusive statements on observability, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2 Geometric Approach In the differential geometric approach observability is investigated by computing the rank of an observability matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Two different algorithms were considered in this context, which differ in the way the observability matrix is built: Full Input-State-Parameter Observabilty (FISPO) and Observability Rank Condition with Direct Feedthrough (ORC-DF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Both algorithms rely on Lie derivatives of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' For better understanding, computation of the observability matrix is explained by means of the FISPO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' In a second step, disparities of the ORC-DF algorithm are illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' In FISPO, the observability matrix O(x) is built by taking the Lie derivatives of the output symbolically and computing their partial derivatives with respect to the states x: O(x) = � � � � � � � � ∂h(x) ∂x ∂ ∂x (Lfh(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' ∂ ∂x � Ln−1 f h(x) � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (7) A model is locally observable in a neighborhood N(x0) of a point x0 if it holds that [18] rank (O(x0)) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (8) 7 The ORC-DF algorithm pursues a different approach in computing the observability matrix and is only applicable for input-affine systems [19], which can be described as: ˙x = f(x) + p � j=1 gj(x)uj, (9) y = h0(x) + p � j=1 hj(x)uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (10) f, gj ∈ Rn and h0, hj ∈ Rp are vectorial functions, which describe the output y uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' They are aggregated in a column vector Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The observability matrix in ORC-DF is built repeatedly using symbolic computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Initially, it is computed as partial derivatives of Ω with respect to the states x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' If the observability rank condition can be satisfied, the system is locally observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' If not, the vector Ω is extended by the Lie derivatives of the previous version of Ω with respect to f and gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This procedure continues until the rank condition is satisfied or a maximum number of iterations is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Details can be found in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Implementation Both geometric approaches were implemented in the Matlab toolbox STRIKE_GOLDD 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The toolbox allows to assess local observability (and structural local identifiability) with the FISPO and ORC-DF algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' To ensure that statements on observability of the models are independent from the numeric value of the initial state x0, no previously known initial states x0 were assumed for both procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This results in symbolic computations of O(x) and its rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' In the geometric approach, full model classes ADM1-R4 and ADM1-R3 as well as all submodels were analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Inlet concentrations were assumed to be known and time-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Computations of both algebraic and geometric approaches were conducted on a standard personal computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='3 Identifiability Analyses Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' A model parameter θi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' , m is structurally identifiable (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=') if for almost all true parameter values θ∗ the parameter estimate ˆθ can be determined from input-output behavior of the model M [26]: M �x(t), u(t), ˆθ � = M �x(t), u(t), θ∗� (11a) ⇒ ˆθ = θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (11b) 2Intel Core i5 processor (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='7 GHz), 32 GB RAM and Windows 10 operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 8 If all model parameters θi are s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=', the model is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='. A parameter θi is locally structural identifiability (l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=') if (11) holds in a neighborhood N(θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' A full model is l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' if all its parameters θi are l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='. If at least one of them is not, the model is not l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='. Identifiability was analyzed as part of the differential geometric approach and considered as augmented observability [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' For this purpose, the state vector is augmented by the model parameters, for which trivial dynamics is assumed: ˙˜x = � ˙x ˙θ � = � f(x, u, θ) 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (12) Consequently, for FISPO, computation of the extended observability matrix is augmented to: O(˜x) = � � � � � � � � ∂h(˜x) ∂˜x ∂ ∂˜x (Lfh(˜x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' ∂ ∂˜x � Ln+m−1 f h(˜x) � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (13) The system is considered observable and identifiable if the rank conditions satisfies [18] rank (O(˜x0)) = n + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (14) For ORC-DF the state differential equations (9) are augmented by a zero vector of dimension m × 1, accounting for the trivial dynamics of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The output equations remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' For the ADM1-R4, the rate constants of hydrolysis and decay were assumed as time-variant and analyzed for identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' For the ADM1-R3, the following parameters were investigated: maximum growth rate and half-saturation constant of acetoclastic methanogens as well as the inhibition constant for non-competitive ammonia inhibition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 3 Results and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1 Algebraic Approach The full ADM1-R4 (BMR4+AB) could be shown to be globally observable by using the measurements given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The underlying system of equations involves online measurements of the gas composition (CH4 and CO2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Furthermore, to guarantee observability, measurements of IN, TS and VS need to be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This follows directly from the model equations: all three states only appear in their corresponding differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Therefore, if they were not available as measurements, they could not be observable because they would not be introduced 9 Table 2: Measurements and their time derivatives required for algebraic observability of model simplifications Nominal measurementsa,b 1st derivative 2nd derivative 3rd deriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Model name nc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='t[s]d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='CH4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='CO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='IN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='TS VS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Ac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='CH4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='CO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='IN TS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='CH4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='CO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='IN TS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='CH4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='CO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='BMR4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='BMR4+A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='BMR4+B ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='BMR4+AB ' metadata={'source': 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nitrogen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' TS, VS - total and volatile solids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Ac - acetic acid bCH4 and CO2 were assumed as online measurements, as well as Ac for ADM1-R3 variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' IN, TS and VS were assumed as offline measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' cNumber of states dComputation time ederivative fOnly local observability could be shown because two solutions to the equation systems were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' into the system of equations via other measurements, regardless of the degree of time derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Model simplifications were applied to the ADM1-R4, resulting in the submodels BMR4, BMR4+A and BMR4+B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This was done to explore the effect of individual model parts on the complexity of the system of equations, indicated by the computation time required to solve the system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' However, reduction in computational complexity could not clearly be attributed to omitted model parts, as summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2: The simplest model variant (BMR4) showed the second highest computation time among the ADM1-R4 models, whereas the fastest computation time could be achieved with the less reduced variant BMR4+A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Nevertheless, all ADM1-R4 submodels could be shown to be globally observable and required only short computation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' For the full ADM1-R3 (BMR3+ABCDE), the solver of Mathematica failed to deliver conclusive statements on observability, likely due to the complexity of the resulting terms in the system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Neither a set of solutions nor an empty set could be returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Instead, the kernel systematically died in the process of solving the system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Hence, with the algebraic approach, observability of the full ADM1-R3 could not be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Therefore, model complexity of the ADM1-R3 was reduced by isolating and omitting individual model parts (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Up to this point, measurements of CH4, CO2, IN, TS and VS were assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Additionally, for all ADM1-R3 submodels online measurements of acetic acid had to be taken into account to show local observability, see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The obtained results are therefore strictly valid only 10 for lab- and pilot-scale settings [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' To find solutions of the systems of equations, first and second-order derivatives of the offline measurements IN and TS are required, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' With given nominal values of VS, it was preferred to include time derivatives of TS over VS, because both terms added the same information to the system of equations but TS delivered less complex terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' For both BMR3+AC and BMR3+BC, local observability could be shown for two combinations of equations, each only differing by one equation: the second derivative of IN or of TS, as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Both respective combinations were solved in similar computation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' BMR3+ABC is the most complex model variant which still delivered a solution of the resulting system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Two solutions of the systems of equations each could be found by either incorporating the first derivative of IN or TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The latter resulted in more than double computation times (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Based on specific model equations, this was not expected because the first derivatives of both IN and TS have the same algebraic structure and are linearly dependent except for their convection terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' By contrast, for BMR3+AC and BMR3+BC computation times of the two respective solutions did not differ much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This indicates that computational complexity of the terms resulting from the second derivatives of IN and TS does not change as much as for their first derivatives (observed in BMR3+ABC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Omitting model part B (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' BMR3+AC) resulted in a significant reduction of computation times, which is in line with the findings for ADM1-R4 model variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Omitting part A instead does not allow to draw such clear conclusions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' BMR3+BC), since computation times are in between the two observable combinations of BMR3+ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' When restricting measurements to gas composition, IN, TS and VS, none of the ADM1-R3 submodels could be shown to be observable using the algebraic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' For higher-order variants than BMR3+ABC, inhibition through pH and ammonia need to be considered, as well as computation of pH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This entails a higher number of state variables, which leads to symbolic systems of equations too complex to be solved in the presented framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Lower-order model variants of the ADM1-R3 were not considered because (i) no clear tendency of reduction in computation time could be observed by omitting model parts, and (ii) for practical applications, lower-order models were already given through the ADM1-R4 and corresponding submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2 Geometric Approach Based on the implementation in the STRIKE_GOLDD toolbox, the geometric approach proved to be successful in showing both local observability and structural local identifiability of the 11 Table 3: Model properties and computation times of FISPO and ORC-DF algorithms for ADM1-R4 and ADM1-R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Model class number of computation time in s states parameters FISPO ORC-DF ADM1-R4 11 4 3 7 ADM1-R3 17 7 11959 811 full models ADM1-R4 and ADM1-R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Moreover, this was achieved in (i) considerably shorter computation times, (ii) without assuming acetic acid as an online measurement, and (iii) by employing two independent algorithms (FISPO and ORC-DF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This is summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' As noted by [27], computational efficiency of FISPO and ORC-DF can differ vastly depending on the model structure, which is apparent for the larger ADM1-R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Furthermore, computational effort decreases significantly when neglecting model parts of the full ADM1-R3: computation times of the submodel BMR3-ABC, for instance, could be reduced to 12 and 5 s for the FISPO and ORC-DF algorithms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This is also considerably faster than those computation times achieved with the algebraic approach, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Higher-order model classes such as the ADM1-R2 are of similar structure as the ADM1-R3, but involve significantly more time-variant parameters and contain a more detailled acid spectrum (acetic to valeric acid instead of only acetic acid) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' In an agricultural setting, these acid measurements are not available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' However, even when assuming them to be available as online measurements, both algorithms of the geometric approach fail to deliver conclusive statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This is likely due to the aggravated complexity of the involved symbolic expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' A practical application of the ADM1-R2 is thus not anticipated for monitoring and control schemes due to the lack of online measurements for the individual VFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' It was therefore not further simplified into submodels and no further observability and identifiability analyses were pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 4 Conclusion The models ADM1-R4 and ADM1-R3 were analysed for observability and identifiability using differential algebraic and geometric approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' With the former, observability of the ADM1-R4 was successfully shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' However, the algebraic approach failed for the larger ADM1-R3 if only typical measurements at agricultural biogas plants were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' In this scenario however, the geometric approach succeeded to show observability and identifiability for both ADM1-R3 and ADM1-R4, and exhibited a higher computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This emphasizes that the 12 ADM1 simplifications of Weinrich and Nelles [15] are indeed promising for state and parameter estimation as well as process automation for agricultural AD plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Acknowledgment The authors are thankful for funding from German Federal Ministry of Food and Agriculture of the junior research group on simulation, monitoring and control of anaerobic digestion plants (grant 2219NR333).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' thanks Terrance Wilms for his encouragement and advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Appendix The ADM1 simplifications of Weinrich and Nelles [15] have been transformed into standard control notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This section summarizes the model equations and parameters of all successfully investigated models, starting with the simplest model structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The following notation was applied: xi - states (mass concentrations of the species involved), yi - measurements u - control variable (feed volume flow) ξi - time-variant, uncertain parameters (inlet mass concentration) θi - time-variant parameters ci - (aggregated) time-invariant parameters aij - time-invariant stoichiometric coefficients ADM1-R4 Models Aside from the ADM1-R4, the following models were analyzed: BMR4+B, BMR4+A, and BMR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' ADM1-R4 State vector: x = [Sch4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' SIC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' SIN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sh2o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xbac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xash,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sch4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='gas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sco2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='gas]T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(15) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Differential equations: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x1 = c1 (ξ1 − x1) u + a11θ1x5 + a12θ2x6 + a13θ3x7 − c2x1 + c3x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(16a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x2 = c1 (ξ2 − x2) u + a21θ1x5 + a22θ2x6 + a23θ3x7 − c2x2 + c4x11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(16b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x3 = c1 (ξ3 − x3) u − a31θ1x5 + a32θ2x6 − a33θ3x7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(16c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x4 = c1 (ξ4 − x4) u − a41θ1x5 − a42θ2x6 − a43θ3x7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(16d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x5 = c1 (ξ5 − x5) u − θ1x5 + a54θ4x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(16e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x6 = c1 (ξ6 − x6) u − θ2x6 + a64θ4x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(16f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x7 = c1 (ξ7 − x7) u − θ3x7 + a74θ4x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(16g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x8 = c1 (ξ8 − x8) u + a81θ1x5 + a82θ2x6 + a83θ3x7 − θ4x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(16h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x9 = c1 (ξ9 − x9) u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(16i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x10 = c15x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='10 + c16x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='10x11 + c17x10x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='11 + c18x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='10 + c19x10x11 + c20x10 + c5x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(16j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x11 = c17x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='11 + c16x10x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='11 + c15x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='10x11 + c19x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='11 + c18x10x11 + c21x11 + c6x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(16k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='aij are the absolute values of the entries of the petersen matrix given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' i denotes the column (component) and j the row (process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' For brevity, only those entries with an absolute value ̸= 1 or 0 were denoted specifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Measurements: y1 = ˙Vg = c6x2 10 + c7x10x11 + c8x2 11 + c9x10 + c10x11 + c11 (17a) y2 = pch4 = c12x10 (17b) y3 = pco2 = c13x11 (17c) y4 = SIN = x3 (17d) y5 = TS = 1 − 1 c14 x4 (17e) y6 = V S = 1 − 1 c14 − x4 x9 (17f) Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 4 summarizes the aggregated parameters ci and the time-variant parameters θi used for ADM1-R4 and its submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' BMR4+B The state vector remains as in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Differential equations: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x1 = c1 (ξ1 − x1) u + a11θ1x5 + a12θ2x6 + a13θ3x7 + a14θ4x8 − c2x1 + c3x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(18a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x2 = c1 (ξ2 − x2) u + a21θ1x5 + a22θ2x6 + a23θ3x7 + a24θ4x8 − c2x2 + c4x11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(18b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x3 = c1 (ξ3 − x3) u − a31θ1x5 + a32θ2x6 − a33θ3x7 + a34θ4x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(18c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x4 = c1 (ξ4 − x4) u − a41θ1x5 − a42θ2x6 − a43θ3x7 − a44θ4x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(18d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x5 = c1 (ξ5 − x5) u − θ1x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(18e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x6 = c1 (ξ6 − x6) u − θ2x6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(18f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x7 = c1 (ξ7 − x7) u − θ3x7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(18g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x8 = c1 (ξ8 − x8) u + a81θ1x5 + a82θ2x6 + a83θ3x7 − a84θ4x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(18h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x9 = c1 (ξ9 − x9) u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(18i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x10 = c15x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='10 + c16x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='10x11 + c17x10x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='11 + c18x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='10 + c19x10x11 + c20x10 + c5x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(18j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x11 = c17x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='11 + c16x10x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='11 + c15x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='10x11 + c19x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='11 + c18x10x11 + c21x11 + c6x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (18k) Note that omitting model part A entails modified stoichiometric constants aij as given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The other parameters remain the same and are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The measurements remain as in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' BMR4+A State vector: x = [SIN, Sh2o, Xch, Xpr, Xli, Xbac, Xash, Sch4,gas, Sco2,gas]T (19) Differential equations, where aij are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 5 and the remaining parameters in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x1 = c1 (ξ1 − x1) u − a31θ1x3 + a32θ2x4 − a33θ3x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(20a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x2 = c1 (ξ2 − x2) u − a41θ1x3 − a42θ2x4 − a43θ3x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(20b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x3 = c1 (ξ3 − x3) u − θ1x3 + a54θ4x6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(20c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x4 = c1 (ξ4 − x4) u − θ2x4 + a64θ4x6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(20d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x5 = c1 (ξ5 − x5) u − θ3x5 + a74θ4x6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(20e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x6 = c1 (ξ6 − x6) u + a81θ1x3 + a82θ2x4 + a83θ3x5 − θ4x6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(20f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x7 = c1 (ξ7 − x7) u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(20g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x8 = c22a11θ1x3 + c22a12θ2x4 + c22a13θ3x5 + c15x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='8 + c16x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='8x9 + c17x8x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9 + c18x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='8+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ c19x8x9 + c23x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(20h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x9 = c22a21θ1x3 + c22a22θ2x4 + c22a23θ3x5 + c17x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9 + c16x8x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9 + c15x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='8x9 + c19x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ c18x8x9 + c23x9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(20i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Measurements: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='y1 = ˙Vg = c6x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='8 + c7x8x9 + c8x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9 + c9x8 + c10x9 + c11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (21a) y2 = pch4 = c12x8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (21b) y3 = pco2 = c13x9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (21c) y4 = SIN = x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (21d) y5 = TS = 1 − 1 c14 x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (21e) 15 y6 = V S = 1 − 1 c14 − x2 x7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (21f) BMR4 The state vector remains as in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Differential equations: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x1 = c1 (ξ1 − x1) u − a31θ1x3 + a32θ2x4 − a33θ3x5 + a34θ4x6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(22a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x2 = c1 (ξ2 − x2) u − a41θ1x3 − a42θ2x4 − a43θ3x5 − a44θ4x6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(22b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x3 = c1 (ξ3 − x3) u − θ1x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(22c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x4 = c1 (ξ4 − x4) u − θ2x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(22d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x5 = c1 (ξ5 − x5) u − θ3x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(22e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x6 = c1 (ξ6 − x6) u + a81θ1x3 + a82θ2x4 + a83θ3x5 − a84θ4x6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(22f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x7 = c1 (ξ7 − x7) u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(22g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x8 = c22a11θ1x3 + c22a12θ2x4 + c22a13θ3x5 + c22a14θ4x6 + c15x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='8 + c16x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='8x9+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ c17x8x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9 + c18x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='8 + c19x8x9 + c23x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(22h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x9 = c22a21θ1x3 + c22a22θ2x4 + c22a23θ3x5 + c22a24θ4x6 + c17x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9 + c16x8x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ c15x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='8x9 + c19x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9 + c18x8x9 + c23x9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(22i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='aij are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 6 and the remaining parameters in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The measurements remain as in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 16 Table 4: Aggregated and time-variant parameters of ADM1-R4 and its submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Aggregated notation Notation by Weinrich and Nelles [15] u ˙Vf θ1 kch θ2 kpr θ3 kli θ4 kdec c1 V −1 l c2 kLa c3 kLaKH,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='¯RT ¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='M−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='co2 (2ph2o − p0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='c20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='−kLaVlV −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='KH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ch4 ¯RT − kpp−1 0 V −1 g (ph2o − p0) ph2o c21 −kLaVlV −1 g KH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='co2 ¯RT − kpp−1 0 V −1 g (ph2o − p0) ph2o c22 VlV −1 g c23 −kpp−1 0 V −1 g (ph2o − p0) ph2o 17 Table 5: Petersen matrix of ADM1-R4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' derived from [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Component i → 1 2 3 4 5 6 7 8 9 10 11 j Process ↓ Sch4 SIC SIN Sh2o Xch Xpr Xli Xbac Xash Sch4,gas Sco2,gas Process rate rj 1 Fermentation Xch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='6809 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0456 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1372 θ1 x5 2 Fermentation Xpr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='3221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='7954 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4588 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1723 θ2 x6 3 Fermentation Xli 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='6393 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='5817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0344 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4152 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2286 θ3 x7 4 Decay Xbac 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='05 1 θ4 x8 5 Phase transition Sch4 1 c22 c2x1 − c3x10 6 Phase transition SIC 1 c22 c2x2 − c4x11 Table 6: Petersen matrix of BMR4+B, derived from [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Component i → 1 2 3 4 5 6 7 8 9 10 11 j Process ↓ Sch4 SIC SIN Sh2o Xch Xpr Xli Xbac Xash Sch4,gas Sco2,gas Process rate rj 1 Fermentation Xch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='6809 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0456 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1372 θ1 x5 2 Fermentation Xpr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='3221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='7954 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4588 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1723 θ2 x6 3 Fermentation Xli 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='6393 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='5817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0344 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4152 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2286 θ3 x7 4 Decay Xbac 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='3247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='7641 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='3822 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='8312 θ4 x8 5 Phase transition Sch4 1 c22 c2x1 − c3x10 6 Phase transition SIC 1 c22 c2x2 − c4x11 18 ADM1-R3 Models Aside from ADM1-R3, the following models were analyzed: BMR3+AC, BMR3+BC, and BMR3+ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' ADM1-R3 State vector: x = �Sac, Sch4, SIC, SIN, Sh2o, Xch, Xpr, Xli, Xbac, Xac, Xash, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sac−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Shco3−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Snh3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sch4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='gas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sco2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='gas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(23) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Differential equations: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x1 = c1 (ξ1 − x1) u + a11θ1x6 + a12θ2x7 + a13θ3x8 − a14θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Iac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x2 = c1 (ξ2 − x2) u + a21θ1x6 + a22θ2x7 + a23θ3x8 − c5x2 + c6x16 + a24θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Iac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x3 = c1 (ξ3 − x3) u + a31θ1x6 + a32θ2x7 − a33θ3x8 − c5x3 + c5x14 + c7x17 + a34θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Iac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x4 = c1 (ξ4 − x4) u − a41θ1x6 + a42θ2x7 − a43θ3x8 − a44θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Iac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x5 = c1 (ξ5 − x5) u − a51θ1x6 − a52θ2x7 − a53θ3x8 + a54θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Iac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x6 = c1 (ξ6 − x6) u − θ1x6 + a65θ4x9 + a66θ4x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x7 = c1 (ξ7 − x7) u − θ2x7 + a75θ4x9 + a76θ4x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x8 = c1 (ξ8 − x8) u − θ3x8 + a85θ4x9 + a86θ4x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x9 = c1 (ξ9 − x9) u + a91θ1x6 + a92θ2x7 + a93θ3x8 − θ4x9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x10 = c1 (ξ10 − x10) u − θ4x10 + θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Iac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x11 = c1 (ξ11 − x11) u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x12 = c1 (ξ12 − x12) u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24l) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x13 = c29 (x1 − x13) − c9x13SH+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24m) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x14 = c30 (x3 − x14) − c10x14SH+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x15 = c31 (x4 − x15) − c11x15SH+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24o) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x16 = c22x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='16 + c23x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='16x17 + c24x16x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='17 + c25x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='16 + c26x16x17 + c12x2 + c27x16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24p) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x17 = c24x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='17 + c23x16x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='17 + c22x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='16x17 + c26x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='17 + c25x16x17 + c12x3 − c12x14 + c28x17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(24q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Iac and SH+ are defined as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Iac = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='c3 + Sc2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='H+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ7 + x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(25) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='SH+ = −Φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Φ2 + c4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' where (26) 19 Φ = x12 + x4 − x15 17 − x14 44 − x13 60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (27) The absolute values of the stoichiometric coefficients aij are given in the petersen matrix of ADM1-R3, Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Note that for all ADM1-R3 models, definition and indexing of parameters and stoichiometric constants is independent from all ADM1-R4 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Measurements: y1 = ˙Vg = c13x2 16 + c14x16x17 + c15x2 17 + c16x16 + c17x17 + c18 (28a) y2 = pch4 = c19x16 (28b) y3 = pco2 = c20x17 (28c) y4 = pH = − log10 SH+ (28d) y5 = SIN = x4 (28e) y6 = TS = 1 − 1 c21 x5, (28f) y7 = V S = 1 − 1 c21 − x5 x11 (28g) y8 = Sac = x1 (28h) Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 7 summarizes the aggregated parameters ci and the time-variant parameters θi used for ADM1-R3 and all of its submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' BMR3+ABC State vector: x = [Sac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sch4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' SIC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' SIN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sh2o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xbac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xash,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sch4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='gas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sco2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='gas]T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(29) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Differential equations: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x1 = c1 (ξ1 − x1) u + a11θ1x6 + a12θ2x7 + a13θ3x8 − a14θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(30a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x2 = c1 (ξ2 − x2) u + a21θ1x6 + a22θ2x7 + a23θ3x8 − c5x2 + c6x12 + a24θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(30b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x3 = c1 (ξ3 − x3) u + a31θ1x6 + a32θ2x7 − a33θ3x8 − c5x3 + c7x13 + a34θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(30c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x4 = c1 (ξ4 − x4) u − a41θ1x6 + a42θ2x7 − a43θ3x8 − a44θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(30d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x5 = c1 (ξ5 − x5) u − a51θ1x6 − a52θ2x7 − a53θ3x8 + a54θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(30e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x6 = c1 (ξ6 − x6) u − θ1x6 + a65θ4x9 + a66θ4x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(30f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x7 = c1 (ξ7 − x7) u − θ2x7 + a75θ4x9 + a76θ4x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(30g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x8 = c1 (ξ8 − x8) u − θ3x8 + a85θ4x9 + a86θ4x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(30h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x9 = c1 (ξ9 − x9) u + a91θ1x6 + a92θ2x7 + a93θ3x8 − θ4x9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(30i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x10 = c1 (ξ10 − x10) u − θ4x10 + θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(30j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x11 = c1 (ξ11 − x11) u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(30k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x12 = c22x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='12 + c23x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='12x13 + c24x12x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='13 + c25x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='12 + c26x12x13 + c12x2 + c27x12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(30l) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x13 = c24x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='13 + c23x12x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='13 + c22x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='12x13 + c26x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='13 + c25x12x13 + c12x3 + c28x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(30m) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='aij have to be taken from Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The other involved parameters are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Measurements: y1 = ˙Vg = c13x2 12 + c14x12x13 + c15x2 13 + c16x12 + c17x13 + c18 (31a) y2 = pch4 = c19x12 (31b) y3 = pco2 = c20x13 (31c) y4 = SIN = x4 (31d) y5 = TS = 1 − 1 c21 x5 (31e) y6 = V S = 1 − 1 c21 − x5 x11 (31f) y7 = Sac = x1 (31g) BMR3+BC The state vector remains as in (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Differential equations: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x1 = c1 (ξ1 − x1) u + a11θ1x6 + a12θ2x7 + a13θ3x8 − a14θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ a15θ4x9+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ a16θ4x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(32a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x2 = c1 (ξ2 − x2) u + a21θ1x6 + a22θ2x7 + a23θ3x8 − c5x2 + c6x12 + a24θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ a25θ4x9 + a26θ4x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(32b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x3 = c1 (ξ3 − x3) u + a31θ1x6 + a32θ2x7 − a33θ3x8 − c5x3 + c7x13 + a34θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ a35θ4x9 + a36θ4x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(32c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x4 = c1 (ξ4 − x4) u − a41θ1x6 + a42θ2x7 − a43θ3x8 − a44θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ a45θ4x9+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ a46θ4x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(32d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x5 = c1 (ξ5 − x5) u − a51θ1x6 − a52θ2x7 − a53θ3x8 + a54θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='− a55θ4x9+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='− a56θ4x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(32e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x6 = c1 (ξ6 − x6) u − θ1x6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(32f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x7 = c1 (ξ7 − x7) u − θ2x7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(32g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x8 = c1 (ξ8 − x8) u − θ3x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(32h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x9 = c1 (ξ9 − x9) u + a91θ1x6 + a92θ2x7 + a93θ3x8 − a95θ4x9 + a96θ4x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(32i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x10 = c1 (ξ10 − x10) u − θ4x10 + θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x4 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(32j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x11 = c1 (ξ11 − x11) u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(32k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x12 = c22x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='12 + c23x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='12x13 + c24x12x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='13 + c25x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='12 + c26x12x13 + c12x2 + c27x12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(32l) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x13 = c24x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='13 + c23x12x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='13 + c22x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='12x13 + c26x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='13 + c25x12x13 + c12x3 + c28x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(32m) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Note that omitting model part A entails modified stoichiometric constants aij as given in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The other involved parameters remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' They are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Measurements: y1 = ˙Vg = c13x2 12 + c14x12x13 + c15x2 13 + c16x12 + c17x13 + c18 (33a) y2 = pch4 = c19x12 (33b) y3 = pco2 = c20x13 (33c) y4 = SIN = x4 (33d) y5 = TS = 1 − 1 c21 x5 (33e) y6 = V S = 1 − 1 c21 − x5 x11 (33f) y7 = Sac = x1 (33g) BMR3+AC State vector: x = [Sac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' SIN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sh2o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xbac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Xash,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sch4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='gas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Sco2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='gas]T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(34) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='Differential equations: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x1 = c1 (ξ1 − x1) u + a11θ1x4 + a12θ2x5 + a13θ3x6 − a14θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x2 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(35a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x2 = c1 (ξ2 − x2) u − a41θ1x4 + a42θ2x5 − a43θ3x6 − a44θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x2 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(35b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x3 = c1 (ξ3 − x3) u − a51θ1x4 − a52θ2x5 − a53θ3x6 + a54θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x2 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(35c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x4 = c1 (ξ4 − x4) u − θ1x4 + a65θ4x7 + a66θ4x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(35d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x5 = c1 (ξ5 − x5) u − θ2x5 + a75θ4x7 + a76θ4x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(35e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x6 = c1 (ξ6 − x6) u − θ3x6 + a85θ4x7 + a86θ4x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(35f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x7 = c1 (ξ7 − x7) u + a91θ1x4 + a92θ2x5 + a93θ3x6 − θ4x7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(35g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x8 = c1 (ξ8 − x8) u + θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x2 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='− θ4x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(35h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x9 = c1 (ξ9 − x9) u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(35i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x10 = c32a21θ1x4 + c32a22θ2x5 + c32a23θ3x6 + c32a24θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x2 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='c22x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='10 + c23x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='10x11 + c24x10x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='11 + c25x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='10 + c26x10x11 + c33x10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(35j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='˙x11 = c32a31θ1x4 + c32a32θ2x5 + c32a33θ3x6 + c32a34θ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x1 x8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='θ6 + x1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='x2 + c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='c24x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='11 + c23x10x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='11 + c22x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='10x11 + c26x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='11 + c25x10x11 + c33x11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='(35k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='aij have to be taken from Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The involved parameters remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' They are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Measurements: y1 = ˙Vg = c13x2 10 + c14x10x11 + c15x2 11 + c16x10 + c17x11 + c18 (36a) y2 = pch4 = c19x10 (36b) y3 = pco2 = c20x11 (36c) y4 = SIN = x2 (36d) y5 = TS = 1 − 1 c21 x3 (36e) y6 = V S = 1 − 1 c21 − x3 x9 (36f) y7 = Sac = x1 (36g) 23 Table 7: Aggregated and time-variant parameters of ADM1-R3 and its submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Aggregated notation Notation by Weinrich and Nelles [15] u ˙Vf θ1 kch θ2 kpr θ3 kli θ4 kdec θ5 µm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ac θ6 KS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ac θ7 KI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='nh3 c1 V −1 l c2 nac c3 10 − 3 2 pHUL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ac+pHLL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ac pHUL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ac−pHLL,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='−kLaVlV −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='KH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ch4 ¯RT − kpV −1 g p−1 0 (ph2o − p0) ph2o c28 −kLaVlV −1 g KH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='co2 ¯RT − kpV −1 g p−1 0 (ph2o − p0) ph2o c29 kAB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='acKa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='ac c30 kAB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='co2Ka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='co2 c31 kAB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='INKa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='IN c32 VlV −1 g c33 −kpV −1 g p−1 0 (ph2o − p0) ph2o 24 Table 8: Petersen matrix of ADM1-R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' derived from [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Component i → 1 2 3 4 5 6 7 8 9 10 j Process ↓ Sac Sch4 SIC SIN Sh2o Xch Xpr Xli Xbac Xac Process rate rj 1 Fermentation Xch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='6555 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='2245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0574 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1125 θ1 x6 2 Fermentation Xpr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0696 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4767 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1349 θ2 x7 3 Fermentation Xli 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='7651 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='6472 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0244 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4469 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1621 θ3 x8 4 Methanogenesis Sac 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='5447 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='7367 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1506 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4778 1 θ5 x1 θ6+x1 x10 Iac 5 Decay Xbac 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='05 1 θ4 x9 6 Decay Xac 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='05 1 θ4 x10 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 11 12 13 14 15 16 17 Sch4 SIC Xash Sion Sac− Shco3− Snh3 Sch4,gas Sco2,gas 7 Dissoziation Sac 1 c29 (x13 − x1) + c9 x13 SH+ 8 Dissoziation SIC 1 c30 (x14 − x3) + c10 x14 SH+ 9 Dissoziation SIN 1 c31 (x15 − x4) + c11 x15 SH+ 10 Phase transition Sch4 1 c32 c5 x2 − c6 x16 11 Phase transition Sco2 1 c32 c5 (x3 − x14) − c7 x17 25 Table 9: Petersen matrix of BMR3+BC, derived from [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Component i → 1 2 3 4 5 6 7 8 9 10 11 12 13 j Process ↓ Sac Sch4 SIC SIN Sh2o Xch Xpr Xli Xbac Xac Xash Sch4,gas Sco2,gas Process rate rj 1 Fermentation Xch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='6555 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0818 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4767 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1349 θ2 x7 3 Fermentation Xli 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='7651 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='6472 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='4778 1 θ5 x1 θ6+x1 x10 x4 x4+c8 5 Decay Xbac 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0779 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='3997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='8678 θ4 x9 6 Decay Xac 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='9722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0779 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='0873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='3997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='1322 1 θ4 x10 7 Phase transition Sch4 1 c32 c5 x2 − c6 x12 8 Phase transition Sco2 1 c32 c5 x3 − c7 x13 26 Derivations for neglecting individual model parts This section details the basic ideas behind neglecting individual model parts for both ADM1-R3 and ADM1-R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Neglecting model part A - stoichiometric degradation of microbial biomass to macro nutrients Stoichiometric pathways of the ADM1 models can easily be derived from the rows of the petersen matrix, Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' In the original ADM1-R3, biomass (represented by the states Xbac and Xac) is formed during degradation of macro nutrients (carbohydrates, proteins and lipids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Biomass in turn is decomposed into macro nutrients (decay of bacteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This feedback can be removed by modifying the stoichiometry of biomass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Biomass is thereby not decomposed into macro nutrients, but into the stoichiometric degradation products of the macro nutrients directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This delivers a modified stoichiometry, affecting the differential equations of Sac, Sch4, SIC, SIN, Sh2o, Xch, Xpr, Xli, Xbac, and Xac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Neglecting model part B - gas solubility of CH4 and CO2 If CH4 and CO2 are assumed to be insoluble in the liquid phase, their stoichiometric formation therein (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (24b) and (24c)) has to be allocated to the gas phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Gas transfer from liquid to gas phase is originally modeled by means of Henry’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The according terms can be cut out, as well as the convection terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Instead, the terms describing stoichioetric formation of CH4 and CO2 are normalized with the ratio of liquid and gas volume (Vl/Vg) and appear in the differential equations of Sch4,gas and Sco2,gas, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' This can be seen in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' (35j) and (35k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Computation of the gas volume flow remains as in the case with model part B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Neglecting model part C - inhibition through nitrogen limitation The second factor in the inhibition function Iac describes inhibition through nitrogen limitation, see (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Neglecting model part C is achieved by removing this factor, which reduces nonlinearities in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' However, this does not allow to omit any of the state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 27 Neglecting model part D - inhibition through pH and ammonia The full inhibition function Iac is a major source of nonlinearity in the ADM1-R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Neglecting model part D (inhibition through pH and ammonia) allows to cut out the first and last factor of Iac, see (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Consequently, Snh3 can be omitted in the state vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Neglecting model part E - computation of pH Measuring the pH allows to infer SH+ directly because these two variables are linked via the negative common logarithm, (28d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Measureing the pH hence allows to interpret the variable SH+ as a time-variant parameter (without any associated differential equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The states Sion, Sac− and Shco3− (x12 to x14 in (23) and (24)) only appear in the computation of the charge balance Φ, (27) which is required to calculate SH+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' However, as SH+ can be directly determined from pH measurements, the states Sion, Sac− and Shco3− become redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Their respective differential equations can be cut out of the system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' The resulting model BMR3+ABCD (not shown here) only incorporates dissociation between ammonium and ammonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Yet, full inhibition through all three factors of Iac (pH, nitrogen limitation and ammonia) are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' 28 References [1] Ye Chen, Jay J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Cheng, and Kurt S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Creamer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Inhibition of anaerobic digestion process: A review.' 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+page_content=' Identification of parametric models from experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Communications and control engineering series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Springer, London, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' [27] Nerea Martínez and Alejandro F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Villaverde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} +page_content=' Nonlinear observability algorithms with known and unknown inputs: Analysis and implementation.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE4T4oBgHgl3EQfbAwV/content/2301.05068v1.pdf'} diff --git a/46dWaIoBww_ee65IHl0v/content/tmp_files/2301.00001v1.pdf.txt b/46dWaIoBww_ee65IHl0v/content/tmp_files/2301.00001v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..31dd38e5912c60c586baaaaad1f00593db61099f --- /dev/null +++ b/46dWaIoBww_ee65IHl0v/content/tmp_files/2301.00001v1.pdf.txt @@ -0,0 +1,629 @@ +NFTrig: Using Blockchain Technologies for Math Education +JORDAN THOMPSON, Augustana College, USA +RYAN BENAC, Augustana College, USA +KIDUS OLANA, Augustana College, USA +TALHA HASSAN, Augustana College, USA +ANDREW SWARD, Augustana College, USA +TAUHEED KHAN MOHD, Augustana College, USA +NFTrig is a web-based application created for use as an educational tool to teach trigonometry and block +chain technology. Creation of the application includes front and back end development as well as integration +with other outside sources including MetaMask and OpenSea. The primary development languages include +HTML, CSS (Bootstrap 5), and JavaScript as well as Solidity for smart contract creation. The application itself +is hosted on Moralis utilizing their Web3 API. This technical report describes how the application was created, +what the application requires, and smart contract design with security considerations in mind. The NFTrig +application has underwent significant testing and validation prior to and after deployment. Future suggestions +and recommendations for further development, maintenance, and use in other fields for education are also +described. +CCS Concepts: • Computer systems organization → Redundancy; Robotics; • Networks → Network +reliability. +Additional Key Words and Phrases: Matic, Metamask, polygon, bootstrap5, Solidity +1 +INTRODUCTION +The purpose of this report is to describe the technical details involved in the development of the +NFTrig application. This includes both the front end website design, the back end smart contract, +and NFT creation. It will mainly focus on the technical details of the project outlining software +requirements, design through programming languages, client and server side interactions, and +validation testing. This allows the reader to undertake further development, fixes, or maintenance +of the software, as this forms part of the documentation for the software. +The NFTrig project is based around the creation of a web-based game application that allows +interaction of NFTs (non-fungible token) with trigonometric function designs. NFts are digital +assets, for example a picture, that has a unique identification and can generally be freely traded +with cryptocurrency [33]. Through this application, users are able to purchase digital artwork of +many different trigonometric functions and combine them using mathematical operations. Current +supported operations include multiplication and division of the trigonometry functions, and the +output of each operation is a new NFT card that would be the result of an operation. The old cards +will then be removed from the user’s possession and burned using the smart contact. For example, +if a user combined the two cards Sin(x) and Cos(x) using multiplication, they would lose their two +old cards and receive the new card Tan(x). Further, the NFT cards are assigned one of the following +rarity levels: common, uncommon, rare, and legendary. The probability of each of these levels is +defined later in this report. +The application also allows a user to connect to MetaMask, a digital wallet capable of storing a +user’s cryptocurrency and NFTs as well as a way to connect to block chain. The NFTrig application +Authors’ addresses: Jordan Thompson, jordanthompson18@augustana.edu, Augustana College, Rock Island, USA; Ryan +Benac, ryanbenac18@augustana.edu, Augustana College, Rock Island, USA; Kidus Olana, kidusolana18@augustana.edu, +Augustana College, Rock Island, USA; Talha Hassan, talhahassan18@augustana.edu, Augustana College, Rock Island, +USA; Andrew Sward, andrewsward@augustana.edu, Augustana College, Rock Island, USA; Tauheed Khan Mohd, +tauheedkhanmohd@augustana.edu, Augustana College, Rock Island, USA. +arXiv:2301.00001v1 [cs.HC] 21 Dec 2022 + +2 +Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd +can also display the NFTs owned by the user and allow them to connect to OpenSea to sell the +NFTrig cards on a public marketplace. The application is hosted on Moralis employing their Web3 +API. Technical languages used in this project, which will be discussed in detail throughout this +paper, include front end web development languages HTML, CSS (specifically Bootstrap5), and +JavaScript as well as the back end smart contract development language Solidity. +In order to attract users, this application also allows a user to answer trivia questions and gain +experience points. These points can then be used to unlock new sets of NFT cards or upgrade existing +cards in a user’s wallet. This game-like design should appeal to a younger audience and encourage +them to answer trigonometry or math based questions. This will have an incredible educational +benefit for the user because they will be both learning and playing a game simultaneously. +2 +MOTIVATION +The purpose of this application is as an educational tool for students who are attempting to +understand the ways that trigonometric functions interact with each other. As opposed to just +graphing these functions by hand, students will be able to generate new NFTs by combining +whatever trigonometric functions they already own. In fact, using technology is shown to influence +and better educational processes by increasing interaction between those in the classroom [9]. +Technology is becoming increasingly prevalent in every sphere of daily life, so the use of technology +in a classroom setting is not only logical, but it increases the educational benefit of students [29]. +However, as the technology continues to evolve, "the gap between traditional course material +taught to students in B.S./M.S. programs at universities and the cutting edge of technology used in +industry is widening at an unprecedented rate" [30]. By creating this project, it will give students +the opportunity to gain experience with block chain, and hopefully be a starting place for narrowing +that ever growing gap. After much research, it is likely that this proposed application is the first of +its kind that utilizes NFTs to teach mathematical concepts. +Aside from user benefit of this application, there is also an intellectual merit in the block chain +and education fields. Best described by Carmen Holotescu, "As education becomes more open, +diversified, democratised, and decentralised, the block chain technology is taken in consideration +by researchers, teachers and institutions, to maintain reputation, trust in certification, and proof of +learning" [17]. Further, development of this project continues research on NFT and block chain +technologies. This application can also serve as the boilerplate basis for other NFT-based educational +tools and resources. Research for this project provides opportunities for training computer science +students on how to use NFTs in general, but more specifically in educational contexts. +NFTrig was developed by computer science students as a final senior inquiry project at Augustana +College. In conjunction and with funding by the Department of Mathematics and Computer Science, +this project employs a variety of software development skills and techniques that further the +research and understanding of the block chain and web development field. +3 +RELATED WORK +Block chain technology has enabled the formation of decentralized distributed records of digital +data which does not require any third party to moderate any transactions [34]. The decentralized +nature of block chain also renders it easy for use in a ranging variety of applications in several fields +such as healthcare [16], internet of things [7], gaming [2], banking [6], and education (explored in +greater detail in subsection 3.1). Non Fungible token (NFTs) are a relatively new phenomena within +the field of block chain based technologies, but its application in aforementioned fields are already +being studied. Specifically within the healthcare context, NFT’s are solving long term issues such as +storing patients’ private data more safely as well as maintaining better records while giving better +autonomy and privacy to both patients and healthcare providers [22]. The application of NFTs in + +NFTrig: Using Blockchain Technologies for Math Education +3 +education is still an understudied area. These next related work sections explore the broader use of +block chain based technologies for educational purposes, gamification, and overall collaborative +learning. +3.1 +Block chain Based Technologies for Educational Purposes +There has been extensive work concerning how block chain based technologies are enabling better +ownership and sharing of personal records for students and supporting collaborative learning +environments. Yumna et al. conducted a systematic literature review of the use of block chain +technologies in educational sector [35]. They also propose several uses of existing block chain +based technologies in educational sector that leverage the decentralized and traceable consensus +making mechanisms of block chain. Researchers have examined the use of block chain to allow +students to maintain educational records such as transcripts, credentials, diplomas, and learning +activities [5, 14, 31]. Similarly, research has also explored learning management systems design +based on block chain based technology. The technology can potentially verify a students records as +well as enable the design of an automatic decentralized enrollment system which does not require +moderation from school staff [31]. +Another elegant use of block chain in the field of education is the ability to support life-long +learning applications. The educational sector is becoming more diverse with a variety of different +types of classrooms and learning modalities. E-learning has also allowed students to acquire +licences and accreditation online. Therefore, it is imperative to maintain the learning journeys of +students over time to understand the different types of learning that they have been engaging in +and improving on over time. The traceable nature of block chain based technologies (defined as +one of the salient features in the aforementioned systematic review by [35]) enables all of these +applications. +The decentralized nature of block chains coupled with the consensus making algorithms also +makes it suitable for collaborative environments. Prior research has looked at how block chain +based technologies can enable better developmental experiences in the realm of business [11] but +there is very minimal work on its application within the field of education application[3]. +3.2 +Applications in Education Application and Collaborative Learning +Although preliminary in nature, limited prior work has explored the utilization of NFTs for design- +ing various different independent learning environments for students. There are some proposed +commercial systems that have analogous functioning to some of the systems described in the prior +section. For example, commercial systems are looking at leveraging NFTs to award “Pass" status +to students for different courses 1. NFTs enjoy a key advantage over conventional block chain +technologies as they are typically designed using the more secure Ethereum block chain enabling an +even more secure record and identity management. Researchers have shown that there is promise +in using NFTs as academic tokens to represent student transcripts and other records as well that +can be more easily verified [9]. However, there is still a dearth of academic literature in this field. +Student incentivization is heavily advocated in pedagogical literature [12]. NFTs make it easier +to tie incentivization to learning outcomes as they can be automatically acquired by students at +any time upon completion of learning outcomes. This gives NFTs based certifications an advantage +over the more traditional learning settings where students have to strongly adhere to semester +timelines. Elmessiry et al. has looked at designing an incentive mechanism that can be used by +teachers and students to achieve better learning outcomes in an effective and cost-efficient manner +1A teacher at Pepperdine University using NFTs to award course completion certifications to students: https://upcea.edu/tech- +trends-in-higher-ed-metaverse-nft-and-dao/ + +4 +Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd +[9]. They also concluded there was better engagement outcomes for students. On several metrics +of usability, the students reported more than 80% preference for buying, using, and collecting +NFTs. Such independent learning methods were particularly more useful during the COVID-19 +pandemic to accommodate the need of remote independent learning options. Architecturally, this +project takes inspiration from [9], and applies it to a more narrower, focused domain of learning +mathematical operations in this study. Further, these NFTs are also easier to share on social media +[20]. Therefore, it also allows students to more readily share their accomplishments. +3.3 +Gamification to Support Mathematical Learning +Since the proposed application teaches mathematical and trigonometric formulas to students, the +literature on use of gamification to support mathematical learning should be better described. +Gamification, in combination with incentivization explained in the previous section, will allow +for the success of this application. Gaming settings have traditionally been used to teach simple +mathematical operations to students. More recently, researchers have also proposed systems that +teach advanced concepts to students including College Algebra [10]. These learning environments +make it easier for students to relate the learning concepts with more daily life phenomena. While +gamification itself cannot guarantee better learning outcomes, it can improve students’ interest +and performance by encouraging them to engage with the content for a longer duration of time +[18]. The simpler, more systematic, and operational nature of mathematics as a subject also makes +it easier for incorporation in gaming environments because final answers are usually short and +numerical as opposed to long and descriptive answer that might be found in social or natural +sciences. Trigonometry especially can easily be broken down into a series of operations and steps +which simulates a similar environment found in other online games where users play to find +different “rewards" and “collectables". Despite all these benefits there are some limitations of +gamification as well. For example, it is hard to know how a student arrived a solution and give +feedback [4]. Not being able to solve trigonometric equations can also lead to frustration and +impeded learning experience. Foresight into the project’s future looks to mitigate these concerns +by fostering better communication between different game players and providing links to useful +learning resources in the application. Prior research has extensively explored the use of gamification +in different mathematical fields. This application is likely the first to extend the use of NFTs and +block chain to aid in teaching trigonometric equations. +Research shows that technology, specifically games are shown to be excellent educational tools. +In fact, "one of the most successful positive reinforcement mechanisms [in education] known is +gamification" [9]. This includes taking a topic transforming it into a game with positive reinforce- +ment. This leverages educational benefits in students and encourages them to continue playing the +game to learn. Nftrig has future plans to add a game function which will allow the user to answer +trigonometry trivia and math questions. This will aid in both their learning and the continued use +of the NFTrig application. Further, the ability to combine owned NFTs with math functions also +aids in the education of trigonometry for the student. +4 +EXPERIMENTAL SETUP +4.1 +Software Development Requirements +The NFTrig application employs a variety of software development requirements that cover the +range of the project. From front end web development to back end smart contract creation and +NFT storage, this section describes the requirements and software used to complete the project. +4.1.1 +Compiling IDE. The smart contracts created for NFTrig are hosted on Remix. Remix is an +an open source online compiler IDE that can be used to test and deploy smart contracts [1]. The + +NFTrig: Using Blockchain Technologies for Math Education +5 +platform can be accessed by any browser, and it allows the developer to write and deploy smart +contracts on an actual or test server simultaneously. The current deployment is on a test server. In +order to test and debug the smart contract, Visual Studio Code is used. Visual Studio was found +to be the best code editor because a developer can easily upload most file types, and edit them +[19]. For NFTrig, it was used to develop front end HTML and CSS files, as well as back end solidity +contract editing. The required installed plugins for Visual Studio (VS) include Solidity and Block +chain development. [21] These allowed for simple, straightforward development of code. +4.1.2 +Moralis. Moralis SDK is the primary back end platform for the project. The platform allows +connection of the front end web application to the smart contract. [8] The Moralis platform uses +a combination of server management and a JavaScript SDK to allow for maximum interaction +and simplicity. A developer can do many tasks through this including authentication of users, +getting necessary user data, and connecting with MetaMask in a non-complicated and simply coded +process. The only expectation is that a developer will need to have programming knowledge in +JavaScript as well as a familiarity with Moralis and MetaMask, experience querying a database, and +some knowledge of Web3 development to ensure maximum results and efficiency. Moralis also has +the ability to easily connect to MetaMask. +4.1.3 +MetaMask. MetaMask is the digital wallet required for participation in the NFTrig game +application. It allows the collection of purchases from the user, and it can be installed as an extension +on a browser for increased ease of use [28]. MetaMask stores all NFTs owned by the user, and in +connection with the NFTrig application, can view and upgrade or modify existing NFTs at a users +discretion. Connection to the browser extension is required for the application to access anything +owned by a user [24]. Because MetaMask is easily integrated into Moralis, and thus NFTrig, there +is little a user needs to do to create a connection aside from installing the MetaMask extension, and +clicking connect. +4.1.4 +Front End Design. Front end design was accomplished primarily through Visual Studio. The +Live Server extension was installed which allows each developer to "host" their developed website +using a native web application. Doing so allowed simplified testing and front end development. +Instead of creating CSS files from scratch, the NFTrig interface heavily employs Bootstrap5, which +simplifies the process of modifying the content layout and design of buttons and other content +[25]. Moralis and Bootstrap5 each have extensive documentation to understand and support front +end web development. These tools have been utilized to a near maximum extent. +4.1.5 +Web Hosting Platform. The initial testing of NFTrig, as previously explained, was hosted on +a local live server through Visual Studio. After initial development, the project was moved to a web +server hosted by Augustana College so that initial testing could begin. It is currently unclear how +the site will ultimately be hosted. One option for hosting the web application is directly through +Google [32]. This would allow the website to be named something easily searchable and accessible. +A second option would be to host directly through Moralis, but a limitation of this would be a +more diluted website naming convention along with a more confusion process of uploading and +modifying website content. Currently, the NFTrig application will remain on the local Augustana +College Server. +5 +SOFTWARE DESIGN +This section covers all of the decisions necessary to understand the development of NFTrig, as well +as the technical implementation of each technology used in the design process. + +6 +Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd +5.1 +Software Architecture +The architecture of this project follows the model-server design architecture [27]. Using this model, +the clients send transactions and requests to a proxy smart contract stored on the block chain +which then makes the appropriate calls to the logic smart contract which is also stored on the block +chain. This style of architecture is required for this project because the smart contracts must be +stored on the server-side chain in order to be functional. The use of proxy contracts also allows our +smart contracts to be fully upgradeable with any future updates that may need to be implemented. +5.2 +Choice of Programming Language +This section examines and explains the benefit of each chosen language employed in NFTrig. Front +end languages include HTML and CSS and the back end includes Solidity and JavaScript. Each has +been chosen because they were found to be the best option for development. +5.2.1 +Solidity. Solidity is the programming language of choice when it comes to coding smart +contracts. Solidity is "similar to JavaScript and yet has some features of object-oriented languages +such as Java and C++" [26]. This is a leading language for the development of smart contracts and +use on block chain technologies. This project utilizes the solidity library openzeppelin in order to +create a solid foundation for the smart contracts. Hardhat and Node JS are then used for the testing +and deployment of the smart contracts to the Polygon blockchain. +5.2.2 +JavaScript. In the NFTrig application, JavaScript (JS) is primarily used in the front end +application. The primary purpose of this language is generally to create dynamic and interactive +web content [15]. For the client, JS was used in the navigation bar to allow for clickable links and +resizing of the navigation bar in smaller screens. This language was also used to give buttons +functionality ranging from logging in to MetaMask to purchasing NFTrig cards. Further, JS was used +to test the logic of the front-end combination page until the smart contract was applied. Aside from +augmenting HTML and CSS application pages, JavaScript is also used in this project to connect the +back end smart contract with the from end web application. This application was also developed +using Next JS and deployed via an application known as vercel. +5.2.3 +HTML and CSS. Web development of the user interface was primarily completed using +HTML and CSS (Bootstrap5). These languages are equally popular and necessary to develop the +web pages [13]. Instead of creating all CSS requirements from scratch, Bootstrap5 was utilized to +allow for cleaner design across web pages and better alignment of web page elements. Bootstrap5 +also simplifies the need to explicitly code buttons and other interactive items. +5.3 +Security Considerations +Throughout this project, there have been several security considerations discovered that threatened +the safety and use of the application. One such discovered issue was initially, there was no code +written to block a user from looking at another users token. Further, before minting a new NFT +card, the smart contracts check to ensure that the card does not already exist, the cards used for +combining are owned by the user, and that the newly minted card follows the correct probabilities +of outcomes shows in 2. These probabilities are coded into the smart contract. +5.4 +Smart Contract Design +The smart contract for this project is broken up into two separate contracts. The first of which is +the NFTrig logic contract which contains the logic for purchasing packs of cards as well as the logic +for how cards will interact with each other. The second contract is the marketplace contract which +will allow users to trade their own NFTs with other users through the website. Within the NFTrig + +NFTrig: Using Blockchain Technologies for Math Education +7 +contract, there are functions for multiplying and dividing cards, purchasing randomized packs of +cards, and tracking the details of each individual token as transactions are made. The marketplace +contract contains information about sale history as well as the functionality to post new sales and +purchase items for sale. Both of these contracts were deployed as upgradeable contracts so they +can have updates implemented in the future. +5.5 +NFT Storage and Naming Conventions +All NFT images are stored on the server with the HTML, CSS, and JS files. The naming convention +for each image references what image it is in four numbers. The first number is the power of sin, the +second is the power of cos, the third is the rarity or color of the card (0-3 is green, blue, purple, and +red respectively), and the final number is the text variant (0-3). These files were named accordingly +to better determine the output if cards were combined using a mathematical function. For example, +a sin card might have the naming convention: 1023.jpg. 10 defines it is a sin card, 2 defines it is +rarity purple, and 3 defines it is text variant 3. The purpose of naming the files in this way is so +that the front end can easily determine which image corresponds to a particular NFT by simply +looking at the four features of each token which match the four numbers in the file name. +5.6 +Client Design +The NFTrig application interface was designed using HTML and CSS. The primary use of CSS was +often replaced by Bootstrap5. Bootstrap 5, a library for CSS, allows for easier scaling and alignment +of objects in the HTML file, and thus the computer screen [23]. Documentation on the Bootstrap5 +has utilized to a full extent. Each section examines the layout and use of each application page. +5.6.1 +NFTrig Home. The interface is designed to allow a user to access the marketplace, their +individual current collections, and their profile. The navigational bar contains links to the client-side +facing pages: NFTrigHome, MyCards, CombineCards, Marketplace, and Game. We used a total of +three colors to enable good contrast and make it easier for our users to view complex graphs and +formula without a cluttered background 2. The JavaScript elements declared are reusable across +multiple screens. They support functions and interactions such as a user hovering over a cell or +clicking a cell and providing both feedback and error handling to the user. The navigation bar is +also, the top bar changes color to indicate the tab that the user is on. +5.6.2 +Combination. The main purpose of the combination page is for users to choose cards that +they currently own, and see options for combining them using either multiplication or division. +Figure 1 displays the layout of the screen where user selected cards are shown on the left, and +potential results are shown on the right. +The page utilizes Bootstrap5 capabilities to format effectively to different screen sizes and +resolutions. It connects with a back end script to the smart contract. This provides functionality to +the buttons and easy generation of possible NFT results. Below shows the probabilities of generated +NFT outcomes based on the selected input cards. +5.6.3 +Marketplace and MyCards. Marketplace and MyCards are similar pages, as they connect to +a data source and display NFTs. The Marketplace tab shows all NFT cards available for purchase +both from other users who own NFTs and cards owned by the NFTrig project. MyCards however +specifically shows all cards owned by a user. The layout for each generates all necessary NFT +images and information about the rarity. The rarity is signified by the color and the text option of +the card. Figure 3 shows the actual layout displayed on the page. +2Background-color:#333, Color: #f2f2f2, + +8 +Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd +Fig. 1. Interface where users will combine NFTrigs +Fig. 2. Probabilities of outcomes depending on rarity of selected cards +5.6.4 +Quality attributes of client-side interface and code. In order to have an application of quality, +consistency, and accuracy, the project followed the following guidelines: +(1) The code is written in a manner that components and layouts can be rearranged to support +any structural changes in the front end. +(2) The code has consistent style and format, such as the padding used in individual NFTrig +elements and the purchase page’s color. +(3) The code contains comments and is well indented for easy maintenance and understanding. +(4) Consistent colors and feedback systems are provided so the system is easy to learn for users. +(5) Page-level styling was avoided when possible to keep design consistent. +(6) Thorough testing was completed for basic accessibility features. +5.6.5 +Testing the Client Design. Basic unit testing of different elements was initially conducted +to ensure easy navigation between front end pages. In order to ensure that testing would cover + +Combination +Choosetwo cards and a mathfunction and hit combine! +PossibleResults +SIN(x) +TAN(x) +Sin(x)*Tan(X) +SN()TAN(a) +SIN(TAN) +Sin(x)*Tan(X) +Rarity: Green +Rarity Blue +Font: 0 +Fonto +Liklihood: 75% +Liklihood: 20% +sin(r)jtao(s) +sin(z+y) +d +=sin(z)cos(y)+sin(y)cos(z) +tan(r)=sec() +dar +SIN(o)-TAN() +Sin(x)*Tan(X) +SIN(a)-TAN(a) +Sin(x)*Tan(X) +Rarity: Blue +Rarity: Pink +Borcler +Font:o +Sin(X) Green +Tan(x) Bue +Font:0 +Liklihood: 20% +Multiply +Divide +Likliho0d: 20% +(r+V +Combine +Copyright@2022-AllRightsRes +rved-Augustana CollegeNFTrigCOMBINATIONS +COMMON +UNCOMMON +RARE +LEGENDARY +Common+Common +20% +60% +15% +5% +Common+Legendary +10% +25% +35% +30% +Common+Rare +10% +50% +25% +15% +Common+Uncommon +10% +60% +20% +10% +Rare+Rare +5% +15% +30% +50% +Uncommon+Uncommon +5% +20% +60% +15% +Uncommon+Rare +5% +10% +60% +25% +Uncommon+Legendary +5% +10% +55% +30% +Rare +Legendary +0% +10% +30% +60% +Legendary+Legendary +0% +5% +20% +75%NFTrig: Using Blockchain Technologies for Math Education +9 +Fig. 3. Interface displaying NFTrig Marketplace +most application uses, three user cases were devised: a user browsing NFTrigs, a user making a +purchase, and a user combining NFTrigs. All assumptions and expected actions expected from +the system were listed and analyzed through testing. Further, testing through some edge cases +were also pursued. Currently, the application works as intended, however future plans involve +rigorous testing with JavaScript code and external APIs (if any are devised). This will ensure a fully +functional, secure, and usable application that can also be used as a boiler plate project for other +educational blockchain technologies. +5.6.6 +Future Work: Game. Future work for this project will include the ability for users to play +a trivia and trigonometric equation game. This allows a user to gain experience points that they +can then use to purchase new NFTs. This eliminates the need to always need cryptocurrency to +purchase individual or group NFT cards. Although there is not currently an interface for this page +written in HTML, functionality exists for the trivia game itself. The files are currently stored on +the server, but they are disabled and there is no navigable way to get there through the application. +6 +METHODS +Most methods for completing this project have been thoroughly explained in the sections above. +However, the final intended version of this project will be hosted in a different location than it +resides currently. The initial portion of this project had the front end website hosted on a local +Augustana College server and the back end smart contract hosted on the Polygon test net. This +allowed initial testing and validation that the smart contract operated as expected, as well as give +time and opportunity to discover security vulnerabilities. The future of this project will be hosted +on a decentralized web application online so that users can access it and begin to interact with the +smart contract. Further, a redesign of the website user interface is likely. This will require transition +from BootStrap5 to NextJS which allows cards to be generated, displayed, and interactable through +a version of JavaScript. +7 +RESULTS +This project successfully allowed the exploration and creation of applying NFT and block chain +technology to math education. Although preliminary in use and nature, this project allows for +initial project creation as a boiler plate project. The smart contract is currently deployed on the + +Marketplace +Collections +Profile +About +Metamask +Search.. +Q +NFTrig +SEC(aCSC(a) +SEC(a)CSC() +SEC(a)-CSC(a) +SEC()CSa) +SEC()CGCR +U +2 csc(2x) +to be added +sec2(x)+csc2(x) +cot(x) + tan(x) +2 csc(2x) +to be added +SEC(-CSCa) +SEC(u)CSC(a) +SEC(-CSC() +sec2(x)+csc2(x) +cot(x) + tan(x) +2 csc(2x) +to be added +sec2(x)+csc2(x) +cot(x) + tan(x)10 +Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd +Polygon testnet and can be interacted with using test Matic. Each web page has functionality to +display the user’s owned NFTs as well as the NFTs they have put for sale on the marketplace. Using +NextJS will also allow the Combination page to have functionality and smart contract use. It is also +worth noting that the created web page is not required to interact with the NFTrig smart contracts. +8 +RECOMMENDATIONS FOR FUTURE WORK +The goal for this project was a working Beta demo that shows application functionality, and correct +smart contract execution. There are many other features planned for the continued work of this +project. The first, as earlier explained, is a game option which challenges the user with trigonometry +trivia and math problems. Answering these questions successfully will increase the experience +points of a user. The user can then use these experience points to purchase individual or packs of +NFTrig cards, or they can be used to combine cards. +REFERENCES +[1] Rana M Amir Latif, Khalid Hussain, NZ Jhanjhi, Anand Nayyar, and Osama Rizwan. 2020. A remix IDE: smart +contract-based framework for the healthcare sector by using Blockchain technology. 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Springer, 191–202. + diff --git a/46dWaIoBww_ee65IHl0v/content/tmp_files/load_file.txt b/46dWaIoBww_ee65IHl0v/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d516a8f5336c2112eafc279702defe1e0d5c5522 --- /dev/null +++ b/46dWaIoBww_ee65IHl0v/content/tmp_files/load_file.txt @@ -0,0 +1,636 @@ +filepath=D:\projects\langchain-ChatGLM-master\knowledge_base\46dWaIoBww_ee65IHl0v\content\2301.00001v1.pdf,len=635 +page_content='NFTrig: Using Blockchain Technologies for Math Education JORDAN THOMPSON, Augustana College, USA RYAN BENAC, Augustana College, USA KIDUS OLANA, Augustana College, USA TALHA HASSAN, Augustana College, USA ANDREW SWARD, Augustana College, USA TAUHEED KHAN MOHD, Augustana College, USA NFTrig is a web-based application created for use as an educational tool to teach trigonometry and block chain technology.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Creation of the application includes front and back end development as well as integration with other outside sources including MetaMask and OpenSea.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The primary development languages include HTML, CSS (Bootstrap 5), and JavaScript as well as Solidity for smart contract creation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The application itself is hosted on Moralis utilizing their Web3 API.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This technical report describes how the application was created, what the application requires, and smart contract design with security considerations in mind.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The NFTrig application has underwent significant testing and validation prior to and after deployment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Future suggestions and recommendations for further development, maintenance, and use in other fields for education are also described.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' CCS Concepts: • Computer systems organization → Redundancy; Robotics; • Networks → Network reliability.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Additional Key Words and Phrases: Matic, Metamask, polygon, bootstrap5, Solidity 1 INTRODUCTION The purpose of this report is to describe the technical details involved in the development of the NFTrig application.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This includes both the front end website design, the back end smart contract, and NFT creation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' It will mainly focus on the technical details of the project outlining software requirements, design through programming languages, client and server side interactions, and validation testing.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This allows the reader to undertake further development, fixes, or maintenance of the software, as this forms part of the documentation for the software.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The NFTrig project is based around the creation of a web-based game application that allows interaction of NFTs (non-fungible token) with trigonometric function designs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' NFts are digital assets, for example a picture, that has a unique identification and can generally be freely traded with cryptocurrency .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Through this application, users are able to purchase digital artwork of many different trigonometric functions and combine them using mathematical operations.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Current supported operations include multiplication and division of the trigonometry functions, and the output of each operation is a new NFT card that would be the result of an operation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The old cards will then be removed from the user’s possession and burned using the smart contact.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' For example, if a user combined the two cards Sin(x) and Cos(x) using multiplication, they would lose their two old cards and receive the new card Tan(x).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Further, the NFT cards are assigned one of the following rarity levels: common, uncommon, rare, and legendary.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The probability of each of these levels is defined later in this report.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The application also allows a user to connect to MetaMask, a digital wallet capable of storing a user’s cryptocurrency and NFTs as well as a way to connect to block chain.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The NFTrig application Authors’ addresses: Jordan Thompson, jordanthompson18@augustana.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='edu, Augustana College, Rock Island, USA; Ryan Benac, ryanbenac18@augustana.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='edu, Augustana College, Rock Island, USA; Kidus Olana, kidusolana18@augustana.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='edu, Augustana College, Rock Island, USA; Talha Hassan, talhahassan18@augustana.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='edu, Augustana College, Rock Island, USA; Andrew Sward, andrewsward@augustana.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='edu, Augustana College, Rock Island, USA; Tauheed Khan Mohd, tauheedkhanmohd@augustana.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='edu, Augustana College, Rock Island, USA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='00001v1 [cs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='HC] 21 Dec 2022 2 Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd can also display the NFTs owned by the user and allow them to connect to OpenSea to sell the NFTrig cards on a public marketplace.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The application is hosted on Moralis employing their Web3 API.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Technical languages used in this project, which will be discussed in detail throughout this paper, include front end web development languages HTML, CSS (specifically Bootstrap5), and JavaScript as well as the back end smart contract development language Solidity.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' In order to attract users, this application also allows a user to answer trivia questions and gain experience points.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' These points can then be used to unlock new sets of NFT cards or upgrade existing cards in a user’s wallet.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This game-like design should appeal to a younger audience and encourage them to answer trigonometry or math based questions.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This will have an incredible educational benefit for the user because they will be both learning and playing a game simultaneously.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 2 MOTIVATION The purpose of this application is as an educational tool for students who are attempting to understand the ways that trigonometric functions interact with each other.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' As opposed to just graphing these functions by hand, students will be able to generate new NFTs by combining whatever trigonometric functions they already own.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' In fact, using technology is shown to influence and better educational processes by increasing interaction between those in the classroom .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Technology is becoming increasingly prevalent in every sphere of daily life, so the use of technology in a classroom setting is not only logical, but it increases the educational benefit of students .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' However, as the technology continues to evolve, "the gap between traditional course material taught to students in B.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='/M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' programs at universities and the cutting edge of technology used in industry is widening at an unprecedented rate" .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' By creating this project, it will give students the opportunity to gain experience with block chain, and hopefully be a starting place for narrowing that ever growing gap.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' After much research, it is likely that this proposed application is the first of its kind that utilizes NFTs to teach mathematical concepts.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Aside from user benefit of this application, there is also an intellectual merit in the block chain and education fields.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Best described by Carmen Holotescu, "As education becomes more open, diversified, democratised, and decentralised, the block chain technology is taken in consideration by researchers, teachers and institutions, to maintain reputation, trust in certification, and proof of learning" .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Further, development of this project continues research on NFT and block chain technologies.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This application can also serve as the boilerplate basis for other NFT-based educational tools and resources.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Research for this project provides opportunities for training computer science students on how to use NFTs in general, but more specifically in educational contexts.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' NFTrig was developed by computer science students as a final senior inquiry project at Augustana College.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' In conjunction and with funding by the Department of Mathematics and Computer Science, this project employs a variety of software development skills and techniques that further the research and understanding of the block chain and web development field.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 3 RELATED WORK Block chain technology has enabled the formation of decentralized distributed records of digital data which does not require any third party to moderate any transactions .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The decentralized nature of block chain also renders it easy for use in a ranging variety of applications in several fields such as healthcare , internet of things , gaming , banking , and education (explored in greater detail in subsection 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='1).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Non Fungible token (NFTs) are a relatively new phenomena within the field of block chain based technologies, but its application in aforementioned fields are already being studied.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Specifically within the healthcare context, NFT’s are solving long term issues such as storing patients’ private data more safely as well as maintaining better records while giving better autonomy and privacy to both patients and healthcare providers .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The application of NFTs in NFTrig: Using Blockchain Technologies for Math Education 3 education is still an understudied area.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' These next related work sections explore the broader use of block chain based technologies for educational purposes, gamification, and overall collaborative learning.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='1 Block chain Based Technologies for Educational Purposes There has been extensive work concerning how block chain based technologies are enabling better ownership and sharing of personal records for students and supporting collaborative learning environments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Yumna et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' conducted a systematic literature review of the use of block chain technologies in educational sector .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' They also propose several uses of existing block chain based technologies in educational sector that leverage the decentralized and traceable consensus making mechanisms of block chain.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Researchers have examined the use of block chain to allow students to maintain educational records such as transcripts, credentials, diplomas, and learning activities .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Similarly, research has also explored learning management systems design based on block chain based technology.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The technology can potentially verify a students records as well as enable the design of an automatic decentralized enrollment system which does not require moderation from school staff .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Another elegant use of block chain in the field of education is the ability to support life-long learning applications.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The educational sector is becoming more diverse with a variety of different types of classrooms and learning modalities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' E-learning has also allowed students to acquire licences and accreditation online.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Therefore, it is imperative to maintain the learning journeys of students over time to understand the different types of learning that they have been engaging in and improving on over time.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The traceable nature of block chain based technologies (defined as one of the salient features in the aforementioned systematic review by ) enables all of these applications.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The decentralized nature of block chains coupled with the consensus making algorithms also makes it suitable for collaborative environments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Prior research has looked at how block chain based technologies can enable better developmental experiences in the realm of business but there is very minimal work on its application within the field of education application.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='2 Applications in Education Application and Collaborative Learning Although preliminary in nature, limited prior work has explored the utilization of NFTs for design- ing various different independent learning environments for students.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' There are some proposed commercial systems that have analogous functioning to some of the systems described in the prior section.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' For example, commercial systems are looking at leveraging NFTs to award “Pass" status to students for different courses 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' NFTs enjoy a key advantage over conventional block chain technologies as they are typically designed using the more secure Ethereum block chain enabling an even more secure record and identity management.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Researchers have shown that there is promise in using NFTs as academic tokens to represent student transcripts and other records as well that can be more easily verified .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' However, there is still a dearth of academic literature in this field.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Student incentivization is heavily advocated in pedagogical literature .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' NFTs make it easier to tie incentivization to learning outcomes as they can be automatically acquired by students at any time upon completion of learning outcomes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This gives NFTs based certifications an advantage over the more traditional learning settings where students have to strongly adhere to semester timelines.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Elmessiry et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' has looked at designing an incentive mechanism that can be used by teachers and students to achieve better learning outcomes in an effective and cost-efficient manner 1A teacher at Pepperdine University using NFTs to award course completion certifications to students: https://upcea.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='edu/tech- trends-in-higher-ed-metaverse-nft-and-dao/ 4 Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' They also concluded there was better engagement outcomes for students.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' On several metrics of usability, the students reported more than 80% preference for buying, using, and collecting NFTs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Such independent learning methods were particularly more useful during the COVID-19 pandemic to accommodate the need of remote independent learning options.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Architecturally, this project takes inspiration from , and applies it to a more narrower, focused domain of learning mathematical operations in this study.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Further, these NFTs are also easier to share on social media .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Therefore, it also allows students to more readily share their accomplishments.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='3 Gamification to Support Mathematical Learning Since the proposed application teaches mathematical and trigonometric formulas to students, the literature on use of gamification to support mathematical learning should be better described.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Gamification, in combination with incentivization explained in the previous section, will allow for the success of this application.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Gaming settings have traditionally been used to teach simple mathematical operations to students.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' More recently, researchers have also proposed systems that teach advanced concepts to students including College Algebra .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' These learning environments make it easier for students to relate the learning concepts with more daily life phenomena.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' While gamification itself cannot guarantee better learning outcomes, it can improve students’ interest and performance by encouraging them to engage with the content for a longer duration of time .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The simpler, more systematic, and operational nature of mathematics as a subject also makes it easier for incorporation in gaming environments because final answers are usually short and numerical as opposed to long and descriptive answer that might be found in social or natural sciences.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Trigonometry especially can easily be broken down into a series of operations and steps which simulates a similar environment found in other online games where users play to find different “rewards" and “collectables".' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Despite all these benefits there are some limitations of gamification as well.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' For example, it is hard to know how a student arrived a solution and give feedback .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Not being able to solve trigonometric equations can also lead to frustration and impeded learning experience.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Foresight into the project’s future looks to mitigate these concerns by fostering better communication between different game players and providing links to useful learning resources in the application.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Prior research has extensively explored the use of gamification in different mathematical fields.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This application is likely the first to extend the use of NFTs and block chain to aid in teaching trigonometric equations.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Research shows that technology, specifically games are shown to be excellent educational tools.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' In fact, "one of the most successful positive reinforcement mechanisms [in education] known is gamification" .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This includes taking a topic transforming it into a game with positive reinforce- ment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This leverages educational benefits in students and encourages them to continue playing the game to learn.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Nftrig has future plans to add a game function which will allow the user to answer trigonometry trivia and math questions.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This will aid in both their learning and the continued use of the NFTrig application.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Further, the ability to combine owned NFTs with math functions also aids in the education of trigonometry for the student.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 4 EXPERIMENTAL SETUP 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='1 Software Development Requirements The NFTrig application employs a variety of software development requirements that cover the range of the project.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' From front end web development to back end smart contract creation and NFT storage, this section describes the requirements and software used to complete the project.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='1 Compiling IDE.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The smart contracts created for NFTrig are hosted on Remix.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Remix is an an open source online compiler IDE that can be used to test and deploy smart contracts .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The NFTrig: Using Blockchain Technologies for Math Education 5 platform can be accessed by any browser, and it allows the developer to write and deploy smart contracts on an actual or test server simultaneously.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The current deployment is on a test server.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' In order to test and debug the smart contract, Visual Studio Code is used.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Visual Studio was found to be the best code editor because a developer can easily upload most file types, and edit them .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' For NFTrig, it was used to develop front end HTML and CSS files, as well as back end solidity contract editing.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The required installed plugins for Visual Studio (VS) include Solidity and Block chain development.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' These allowed for simple, straightforward development of code.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='2 Moralis.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Moralis SDK is the primary back end platform for the project.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The platform allows connection of the front end web application to the smart contract.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The Moralis platform uses a combination of server management and a JavaScript SDK to allow for maximum interaction and simplicity.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' A developer can do many tasks through this including authentication of users, getting necessary user data, and connecting with MetaMask in a non-complicated and simply coded process.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The only expectation is that a developer will need to have programming knowledge in JavaScript as well as a familiarity with Moralis and MetaMask, experience querying a database, and some knowledge of Web3 development to ensure maximum results and efficiency.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Moralis also has the ability to easily connect to MetaMask.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='3 MetaMask.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' MetaMask is the digital wallet required for participation in the NFTrig game application.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' It allows the collection of purchases from the user, and it can be installed as an extension on a browser for increased ease of use .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' MetaMask stores all NFTs owned by the user, and in connection with the NFTrig application, can view and upgrade or modify existing NFTs at a users discretion.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Connection to the browser extension is required for the application to access anything owned by a user .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Because MetaMask is easily integrated into Moralis, and thus NFTrig, there is little a user needs to do to create a connection aside from installing the MetaMask extension, and clicking connect.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='4 Front End Design.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Front end design was accomplished primarily through Visual Studio.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The Live Server extension was installed which allows each developer to "host" their developed website using a native web application.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Doing so allowed simplified testing and front end development.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Instead of creating CSS files from scratch, the NFTrig interface heavily employs Bootstrap5, which simplifies the process of modifying the content layout and design of buttons and other content .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Moralis and Bootstrap5 each have extensive documentation to understand and support front end web development.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' These tools have been utilized to a near maximum extent.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='5 Web Hosting Platform.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The initial testing of NFTrig, as previously explained, was hosted on a local live server through Visual Studio.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' After initial development, the project was moved to a web server hosted by Augustana College so that initial testing could begin.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' It is currently unclear how the site will ultimately be hosted.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' One option for hosting the web application is directly through Google .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This would allow the website to be named something easily searchable and accessible.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' A second option would be to host directly through Moralis, but a limitation of this would be a more diluted website naming convention along with a more confusion process of uploading and modifying website content.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Currently, the NFTrig application will remain on the local Augustana College Server.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5 SOFTWARE DESIGN This section covers all of the decisions necessary to understand the development of NFTrig, as well as the technical implementation of each technology used in the design process.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 6 Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='1 Software Architecture The architecture of this project follows the model-server design architecture .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Using this model, the clients send transactions and requests to a proxy smart contract stored on the block chain which then makes the appropriate calls to the logic smart contract which is also stored on the block chain.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This style of architecture is required for this project because the smart contracts must be stored on the server-side chain in order to be functional.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The use of proxy contracts also allows our smart contracts to be fully upgradeable with any future updates that may need to be implemented.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='2 Choice of Programming Language This section examines and explains the benefit of each chosen language employed in NFTrig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Front end languages include HTML and CSS and the back end includes Solidity and JavaScript.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Each has been chosen because they were found to be the best option for development.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='1 Solidity.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Solidity is the programming language of choice when it comes to coding smart contracts.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Solidity is "similar to JavaScript and yet has some features of object-oriented languages such as Java and C++" .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This is a leading language for the development of smart contracts and use on block chain technologies.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This project utilizes the solidity library openzeppelin in order to create a solid foundation for the smart contracts.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Hardhat and Node JS are then used for the testing and deployment of the smart contracts to the Polygon blockchain.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='2 JavaScript.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' In the NFTrig application, JavaScript (JS) is primarily used in the front end application.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The primary purpose of this language is generally to create dynamic and interactive web content .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' For the client, JS was used in the navigation bar to allow for clickable links and resizing of the navigation bar in smaller screens.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This language was also used to give buttons functionality ranging from logging in to MetaMask to purchasing NFTrig cards.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Further, JS was used to test the logic of the front-end combination page until the smart contract was applied.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Aside from augmenting HTML and CSS application pages, JavaScript is also used in this project to connect the back end smart contract with the from end web application.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This application was also developed using Next JS and deployed via an application known as vercel.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='3 HTML and CSS.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Web development of the user interface was primarily completed using HTML and CSS (Bootstrap5).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' These languages are equally popular and necessary to develop the web pages .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Instead of creating all CSS requirements from scratch, Bootstrap5 was utilized to allow for cleaner design across web pages and better alignment of web page elements.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Bootstrap5 also simplifies the need to explicitly code buttons and other interactive items.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='3 Security Considerations Throughout this project, there have been several security considerations discovered that threatened the safety and use of the application.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' One such discovered issue was initially, there was no code written to block a user from looking at another users token.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Further, before minting a new NFT card, the smart contracts check to ensure that the card does not already exist, the cards used for combining are owned by the user, and that the newly minted card follows the correct probabilities of outcomes shows in 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' These probabilities are coded into the smart contract.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='4 Smart Contract Design The smart contract for this project is broken up into two separate contracts.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The first of which is the NFTrig logic contract which contains the logic for purchasing packs of cards as well as the logic for how cards will interact with each other.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The second contract is the marketplace contract which will allow users to trade their own NFTs with other users through the website.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Within the NFTrig NFTrig: Using Blockchain Technologies for Math Education 7 contract, there are functions for multiplying and dividing cards, purchasing randomized packs of cards, and tracking the details of each individual token as transactions are made.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The marketplace contract contains information about sale history as well as the functionality to post new sales and purchase items for sale.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Both of these contracts were deployed as upgradeable contracts so they can have updates implemented in the future.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='5 NFT Storage and Naming Conventions All NFT images are stored on the server with the HTML, CSS, and JS files.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The naming convention for each image references what image it is in four numbers.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The first number is the power of sin, the second is the power of cos, the third is the rarity or color of the card (0-3 is green, blue, purple, and red respectively), and the final number is the text variant (0-3).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' These files were named accordingly to better determine the output if cards were combined using a mathematical function.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' For example, a sin card might have the naming convention: 1023.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='jpg.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 10 defines it is a sin card, 2 defines it is rarity purple, and 3 defines it is text variant 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The purpose of naming the files in this way is so that the front end can easily determine which image corresponds to a particular NFT by simply looking at the four features of each token which match the four numbers in the file name.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='6 Client Design The NFTrig application interface was designed using HTML and CSS.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The primary use of CSS was often replaced by Bootstrap5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Bootstrap 5, a library for CSS, allows for easier scaling and alignment of objects in the HTML file, and thus the computer screen .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Documentation on the Bootstrap5 has utilized to a full extent.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Each section examines the layout and use of each application page.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='1 NFTrig Home.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The interface is designed to allow a user to access the marketplace, their individual current collections, and their profile.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The navigational bar contains links to the client-side facing pages: NFTrigHome, MyCards, CombineCards, Marketplace, and Game.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' We used a total of three colors to enable good contrast and make it easier for our users to view complex graphs and formula without a cluttered background 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The JavaScript elements declared are reusable across multiple screens.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' They support functions and interactions such as a user hovering over a cell or clicking a cell and providing both feedback and error handling to the user.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The navigation bar is also, the top bar changes color to indicate the tab that the user is on.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='2 Combination.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The main purpose of the combination page is for users to choose cards that they currently own, and see options for combining them using either multiplication or division.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Figure 1 displays the layout of the screen where user selected cards are shown on the left, and potential results are shown on the right.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The page utilizes Bootstrap5 capabilities to format effectively to different screen sizes and resolutions.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' It connects with a back end script to the smart contract.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This provides functionality to the buttons and easy generation of possible NFT results.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Below shows the probabilities of generated NFT outcomes based on the selected input cards.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='3 Marketplace and MyCards.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Marketplace and MyCards are similar pages, as they connect to a data source and display NFTs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The Marketplace tab shows all NFT cards available for purchase both from other users who own NFTs and cards owned by the NFTrig project.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' MyCards however specifically shows all cards owned by a user.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The layout for each generates all necessary NFT images and information about the rarity.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The rarity is signified by the color and the text option of the card.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Figure 3 shows the actual layout displayed on the page.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 2Background-color:#333, Color: #f2f2f2, 8 Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Interface where users will combine NFTrigs Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Probabilities of outcomes depending on rarity of selected cards 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='4 Quality attributes of client-side interface and code.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' In order to have an application of quality, consistency, and accuracy, the project followed the following guidelines: (1) The code is written in a manner that components and layouts can be rearranged to support any structural changes in the front end.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' (2) The code has consistent style and format, such as the padding used in individual NFTrig elements and the purchase page’s color.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' (3) The code contains comments and is well indented for easy maintenance and understanding.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' (4) Consistent colors and feedback systems are provided so the system is easy to learn for users.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' (5) Page-level styling was avoided when possible to keep design consistent.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' (6) Thorough testing was completed for basic accessibility features.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='5 Testing the Client Design.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Basic unit testing of different elements was initially conducted to ensure easy navigation between front end pages.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' In order to ensure that testing would cover ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Combination ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Choosetwo ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='cards ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='and ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='a ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='mathfunction ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='and ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='hit ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='combine! ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='PossibleResults ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='SIN(x) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='TAN(x) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Sin(x)*Tan(X) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='SN()TAN(a) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='SIN(TAN) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Sin(x)*Tan(X) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Rarity: ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Green ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Rarity ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Blue ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Font: ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='0 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Fonto ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Liklihood: ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='75% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Liklihood: ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='20% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='sin(r)jtao(s) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='sin(z+y) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='d ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='=sin(z)cos(y)+sin(y)cos(z) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='tan(r)=sec() ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='dar ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='SIN(o)-TAN() ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Sin(x)*Tan(X) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='SIN(a)-TAN(a) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Sin(x)*Tan(X) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Rarity: ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Blue ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Rarity: ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Pink ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Borcler ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Font:o ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Sin(X) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Green ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Tan(x) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Bue ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Font:0 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Liklihood: ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='20% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Multiply ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Divide ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Likliho0d: ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='20% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='(r+V ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Combine ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Copyright@2022-AllRightsRes ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='rved-Augustana ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='CollegeNFTrigCOMBINATIONS ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='COMMON ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='UNCOMMON ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='RARE ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='LEGENDARY ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Common+Common ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='20% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='60% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='15% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='5% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Common+Legendary ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='10% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='25% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='35% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='30% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Common+Rare ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='10% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='50% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='25% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='15% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Common+Uncommon ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='10% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='60% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='20% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='10% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Rare+Rare ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='5% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='15% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='30% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='50% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Uncommon+Uncommon ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='5% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='20% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='60% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='15% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Uncommon+Rare ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='5% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='10% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='60% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='25% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Uncommon+Legendary ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='5% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='10% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='55% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='30% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Rare ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='+Legendary ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='0% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='10% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='30% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='60% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Legendary+Legendary ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='0% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='5% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='20% ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='75%NFTrig: ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Using ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Blockchain ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Technologies ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='for ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Math ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Education ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='9 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Interface displaying NFTrig Marketplace most application uses, three user cases were devised: a user browsing NFTrigs, a user making a purchase, and a user combining NFTrigs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' All assumptions and expected actions expected from the system were listed and analyzed through testing.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Further, testing through some edge cases were also pursued.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Currently, the application works as intended, however future plans involve rigorous testing with JavaScript code and external APIs (if any are devised).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This will ensure a fully functional, secure, and usable application that can also be used as a boiler plate project for other educational blockchain technologies.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='6.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content='6 Future Work: Game.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Future work for this project will include the ability for users to play a trivia and trigonometric equation game.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This allows a user to gain experience points that they can then use to purchase new NFTs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This eliminates the need to always need cryptocurrency to purchase individual or group NFT cards.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Although there is not currently an interface for this page written in HTML, functionality exists for the trivia game itself.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The files are currently stored on the server, but they are disabled and there is no navigable way to get there through the application.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 6 METHODS Most methods for completing this project have been thoroughly explained in the sections above.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' However, the final intended version of this project will be hosted in a different location than it resides currently.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The initial portion of this project had the front end website hosted on a local Augustana College server and the back end smart contract hosted on the Polygon test net.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This allowed initial testing and validation that the smart contract operated as expected, as well as give time and opportunity to discover security vulnerabilities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The future of this project will be hosted on a decentralized web application online so that users can access it and begin to interact with the smart contract.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Further, a redesign of the website user interface is likely.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' This will require transition from BootStrap5 to NextJS which allows cards to be generated, displayed, and interactable through a version of JavaScript.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 7 RESULTS This project successfully allowed the exploration and creation of applying NFT and block chain technology to math education.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Although preliminary in use and nature, this project allows for initial project creation as a boiler plate project.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The smart contract is currently deployed on the Marketplace Collections Profile About Metamask Search..' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Q NFTrig SEC(aCSC(a) SEC(a)CSC() SEC(a)-CSC(a) SEC()CSa) SEC()CGCR U 2 csc(2x) to be added sec2(x)+csc2(x) cot(x) + tan(x) 2 csc(2x) to be added SEC(-CSCa) SEC(u)CSC(a) SEC(-CSC() sec2(x)+csc2(x) cot(x) + tan(x) 2 csc(2x) to be added sec2(x)+csc2(x) cot(x) + tan(x)10 Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd Polygon testnet and can be interacted with using test Matic.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Each web page has functionality to display the user’s owned NFTs as well as the NFTs they have put for sale on the marketplace.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Using NextJS will also allow the Combination page to have functionality and smart contract use.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' It is also worth noting that the created web page is not required to interact with the NFTrig smart contracts.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' 8 RECOMMENDATIONS FOR FUTURE WORK The goal for this project was a working Beta demo that shows application functionality, and correct smart contract execution.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' There are many other features planned for the continued work of this project.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' The first, as earlier explained, is a game option which challenges the user with trigonometry trivia and math problems.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\46dWaIoBww_ee65IHl0v\\content\\2301.00001v1.pdf'} +page_content=' Answering these questions successfully will increase the experience points of a user.' 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b/49E4T4oBgHgl3EQfbgzb/content/tmp_files/2301.05075v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e21fb806f0f940573ff4d1bcd898be61f585875 --- /dev/null +++ b/49E4T4oBgHgl3EQfbgzb/content/tmp_files/2301.05075v1.pdf.txt @@ -0,0 +1,484 @@ +Draft version January 13, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +Kinematic Evidence of an Embedded Protoplanet in HD 142666 Identified by Machine Learning +J. P. Terry +,1, 2 C. Hall +,1, 2 S. Abreau +,3, 4 and S. Gleyzer +5 +1Department of Physics and Astronomy, The University of Georgia, Athens, GA 30602, USA. +2Center for Simulational Physics, The University of Georgia, Athens, GA 30602, USA. +3Department of Medicine, Division of Cardiology, University of California, San Francisco, CA, 94143, USA. +4Cardiovascular Research Institute, San Francisco, CA, 94158, USA. +5Department of Physics and Astronomy, The University of Alabama, Tuscaloosa, AL 35487, USA +(Received January 13, 2023) +Submitted to ApJ +ABSTRACT +Observations of protoplanetary discs have shown that forming exoplanets leave characteristic im- +prints on the gas and dust of the disc. In the gas, these forming exoplanets cause deviations from +Keplerian motion, which can be detected through molecular line observations. Our previous work +has shown that machine learning can correctly determine if a planet is present in these discs. Using +our machine learning models, we identify strong, localized non-Keplerian motion within the disc HD +142666. Subsequent hydrodynamics simulations of a system with a 5 Jupiter-mass planet at 75 au +recreates the kinematic structure. By currently established standards in the field, we conclude that +HD 142666 hosts a planet. This work represents a first step towards using machine learning to identify +previously overlooked non-Keplerian features in protoplanetary discs. +Keywords: Hydrodynamics — Radiative transfer — Accretion disks — Methods: numerical — Catalogs +— Planets and satellites: formation +1. INTRODUCTION +Protoplanetary accretion discs are the sites of planet +formation. +The newest generation of telescopes, such +as the Atacama Large Millimeter/submillimeter Array +(ALMA), have unprecedented capabilities for observing +protoplanetary discs. For the first time, we can not only +resolve discs themselves, but also quantify the motion of +the dust and gas within them. Discs display a striking +variety of structures such as rings (ALMA Partnership +et al. 2015; Dipierro et al. 2018), likely caused by dust +trapping due to forming planets (Pinilla et al. 2012; +Dipierro et al. 2015), and spirals (P´erez et al. 2016), +which may be caused by forming planets (e.g. Dong et al. +2015b) or another mechanism such as gravitational in- +stability (e.g. Dong et al. 2015a; Hall et al. 2018; Meru +et al. 2017). This new information has greatly advanced +our understanding of the processes underlying the for- +mation and evolution of planetary systems. +Planets and physical processes, such as gravitational +instability, influence the motion within the disc. This +causes the material to deviate from simple Keplerian +motion. Comparing the observed motion against purely +Keplerian motion provides information on the bodies +and processes present in the disc (Hall et al. 2020; +Paneque-Carre˜no et al. 2021; Longarini et al. 2021; Pinte +et al. 2022; Bae et al. 2022; Terry et al. 2022b). Non- +Keplerian motion has been used to uncover a variety of +structures, including localized perturbations associated +with gaps and planets (Teague et al. 2018; Pinte et al. +2018, 2019, 2020) as predicted by Perez et al. (2015). +Kinematic analysis is limited by our ability to accu- +rately identify non-Keplerian motion. +The deviations +can be small and frequently occur in noisy images. It +is therefore not only difficult and slow to identify them, +but there is also the strong possibility of overlooking +their occurrence. Any signature that is overlooked is a +missed opportunity to detect either a forming planet or +some other process, such as the GI-Wiggle indicative of +gravitational instability (Hall et al. 2020) or the vertical +shear instability (Barraza-Alfaro et al. 2021). +Machine learning (ML) provides a useful tool for this +task. ML has quickly become ubiquitous in both society +and the sciences, everything from self-driving cars (Bo- +arXiv:2301.05075v1 [astro-ph.EP] 12 Jan 2023 + +ID2 +Terry et al. +jarski et al. 2016) to medicine (Parmar et al. 2015). Re- +cent efforts in astronomy have made it clear that ma- +chine learning is a powerful method even with simulated +training data (Jo & Kim 2019; M¨oller & de Boissi`ere +2020; Alexander et al. 2020). Machine learning, and in +particular computer vision, excels at the analysis of im- +ages (Voulodimos et al. 2018). In some cases, it has even +been shown to outperform humans (Zhou et al. 2021). It +is therefore naturally suited for application to the noisy +datasets in observational astronomy. +Using ML models developed in a previous work (Terry +et al. 2022a), we identify a strong and localized deviation +from Keplerian motion in HD 142666. Using the current +widely accepted field standard method (Teague et al. +2018; Pinte et al. 2018, 2019), we perform smoothed par- +ticle hydrodynamic (SPH) simulations to recreate the +kinematic structure of the disc. The agreement is sig- +nificant when a 5 MJ planet is included at 75 au. We +conclude that HD 142666 hosts a planet. +The paper is arranged as follows: Section 2 describes +the models and simulations used. Section 3 shows the +results of applying the models and simulating the sys- +tem. Section 4 gives our conclusions. +2. METHODS +2.1. Machine Learning +We use the ML models described in Terry et al. +(2022a) and describe them here for completeness. We +use two different architectures: EfficientNetV2 (Tan & +Le 2021) and RegNet (Xu et al. 2022). All models were +made using PyTorch (Paszke et al. 2019), albeit with +significant modifications to the default models and hy- +perparameters. We denote these models as EN47, EN61, +EN75, RN47, RN61, and RN75. Several performance +metrics are given in Table 1. +The models were trained using synthetic observations +from the MCFOST (Pinte et al. 2006, 2009) radiative +transfer code. +MCFOST inputs were drawn from 1000 +PHANTOM (Price et al. 2018) SPH simulations of sys- +tems with and without planets (Terry et al. 2022a). +Each MCFOST calculation outputs a position-position- +velocity cube from 13CO transition lines (J = 2 → 1 +and J = 3 → 2). The cubes were convolved spatially +and spectrally and noise was added in order to replicate +current observational capabilities. +The model inputs (i.e. radiative transfer outputs) are +images of dimension C ×H ×W, where C is the number +of input channels, H is the height of the image, and W +is the width of the image (here, H = W = 600 pixels). +A typical grayscale or RGB image will have C=1 or +3, respectively. +We instead input an entire position- +position-velocity cube. +Observations vary significantly in the number of chan- +nels that cover the disc, but the typical range is between +≈40-100. To address this, we train three different imple- +mentations of each model, which gives us a total of six +models. The difference between each implementation is +the number of input velocity channels (C=47, 61, or 75). +Each model outputs a two-component vector such that +the sum of the components is 1, i.e. it has undergone +softmax activation (Goodfellow et al. 2016). This can be +interpreted as the probability that the given input be- +longs to a certain class, i.e. planet- vs no-planet class. +The models also output images of their internal activa- +tion structure, which we consider to be the more impor- +tant output in this context. While the models were not +trained to pinpoint the locations of planets —a job more +suited for semantic image segmentation (Minaee et al. +2022) —the activation structure can inform us which re- +gions the model finds important when making its clas- +sification decision. Terry et al. (2022a) found that the +activations were able to highlight velocity channels with +non-Keplerian motion in systems that host planet(s). +To this end, we apply our previously trained models to +ALMA data of the HD 142666 system. We inspect the +softmax values and activation structures to gain insight +into whether a planet might be in the system and, if so, +where its signature is the strongest. +2.2. Observational Data +The HD 142666 data was taken from the DSHARP +catalogue (Andrews et al. 2018; Huang et al. 2018). +Data includes 12CO line emission (J = 2 → 1) and 1.25 +mm continuum images. The system was imaged with a +beam with FWHM of 77 × 61 mas (≈11 × 9 au) with an +RMS noise of 1.3 mJy beam−1; channels have a 0.35 km +s−1 resolution (Andrews et al. 2018). Figure 1 shows +selected velocity channels overlaid on the continuum. +The image was cropped to focus on the disc, and a +subset of velocity channels was used. The channels were +reshaped to 600 × 600 pixels and normalized such that +all pixel values were between 0 and 1. +2.3. Hydrodynamical Simulations +We run an SPH simulation using PHANTOM and create +channel maps using MCFOST in the same way that the +original training data was made. The kink is approxi- +mately 75 au from the center of the disc, so we place a +planet at this distance. +System parameters are taken from Rubinstein et al. +(2018); Andrews et al. (2018); Huang et al. (2018). The +stellar mass, temperature, and radius are 2.0 M⊙, 7500 +K, and 2.2 R⊙, respectively. +The disc has a mass of +0.0533 M⊙, an inner radius of 1.3 au, and an outer radius + +Identification of Planet in HD 142666 +3 +of 150 au. The system is inclined at 62 degrees with a +position angle of 162 and an azimuth of 72 degrees. It +is located 148 pc from Earth. +The SPH outputs are used to create line emission +maps to mimic ALMA capabilities. These calculations +are done using the MCFOST radiative transfer code (Pinte +et al. 2006, 2009). +Each calculation uses 108 photon +packets and includes carbon/silicate dust (Draine & Lee +1984) with a dust-to-gas ratio of 1:100. +The result- +ing outputs were convolved spatially and spectrally to +match the observed line emission resolution. +3. RESULTS AND DISCUSSION +Figure 2 shows that HD 142666 has a strong, localized +kink that is detected by the ML models. The kink is par- +ticularly visible in the upper middle (∆v = −1.75 km/s) +channel. The lower row shows activation structures that +roughly correspond to the above channels. +The average softmax value is over 0.84, which means +that the models predict the probability that the input +for HD 142666 contains a planet to be over 84%. This +prompts further scrutiny of the activations, which we use +to determine the most probable channel that contains +the kink. +The strength and localization of the newly identified +kink are reminiscent of the kinks in HD 163296 and HD +97048. As with HD 163296, the kink in the gas is out- +side of the radial extent of the continuum disk. Both of +these disks were found to host planets after SPH simula- +tions containing a planet recreated the kinematic struc- +ture observed in CO observations (Teague et al. 2018; +Pinte et al. 2018, 2019). We apply this same method to +HD 142666 to demonstrate that the kink identified by +our models is consistent with kinks identified by conven- +tional means in HD 163296 and HD 97048. +We found that a simulation of a protoplanetary disk +with a 5 MJ planet reproduced the observation. Fig- +ure 3 shows the results. A localized kink in the vicinity +of the planet is clear in the upper left panel Figure 3 +(∆v = −2.3 km/s). This kink is visible to a lesser ex- +tent in the ∆v = −2.0 km/s in the upper right panel +of Figure 3, which is also the case in Figure 1. There +is strong agreement between this feature and the non- +Keplerian channel identified by our models: both dis- +play a kink of approximately the same shape and size +at approximately the same radial location. +This can +be seen in the lower left and right panels of Figure 3. +Note that the simulation and observation do not display +the strongest kink in the same velocity channel. This +is simply a relic of the finite temporal resolution of the +simulation, which makes it extremely unlikely that the +simulation will be saved when the planet is exactly co- +incident with the observation. The temporal resolution +of the simulation was increased to mitigate this effect, +but it persists to some extent. +We conclude that HD 142666 hosts a planet. +We note that our conclusion is confirmed using the +same methods described by Teague et al. (2018); Pinte +et al. (2018, 2019). However, what is new about our ap- +proach is that the non-Keplerian motion was first identi- +fied by ML models, highlighting a protoplanet candidate +that had previously been missed upon visual analysis. +Verification of the evidence is still done using the same +methodology as previous works (Pinte et al. 2018, 2019). +3.1. Future Work and Limitations +This work shows that machine learning can effectively +identify non-Keplerian motion even if it is missed by +humans. +However, our work can be improved upon. +The primary limitation is the fact that localising non- +Keplerian motion was not the explicit goal of these mod- +els when they were trained. Their purpose was classi- +fication without any attempt of segmentation or object +detection. Models specifically designed to pinpoint de- +viations would likely be more effective. +Rather than +inspecting activation structures —of which there can be +hundreds —the model would directly output a predic- +tion of the location. This would be a more precise and +straightforward method to detect the non-Keplerian sig- +nature, but it would not remove the need to perform +follow-up simulations. We intend to explore this possi- +bility in future works. +4. CONCLUSION +We have applied ML models created by Terry et al. +(2022a) to the DSHARP data of HD 142666. All models +strongly predict the presence of at least one planet. The +activation structures highlight a strong, unreported, and +localized kink. An SPH simulation with a 5 MJ planet +at 75 au is able to recreate the newly identified kine- +matic structure. By the previously established bench- +marks and methods for kinematic planet detection, we +conclude that HD 142666 hosts a planet. +This work demonstrates the utility of applying ma- +chine learning to the analysis of protoplanetary discs. +By highlighting non-Keplerian features in the disc, ML +models are able to guide planet-detection efforts. The +signatures of the planet were previously overlooked by +human analysts, and the traditional analysis was only +performed because of the information given by the mod- +els. We anticipate that this method can identify new +non-Keplerian features in both existing and future pro- +toplanetary observations. +5. ACKNOWLEDGEMENTS + +4 +Terry et al. +Δv = − 1.4km/s +Δv = − 1.75km/s +Δv = − 2.1km/s +Figure 1. +Line emission overlaid on continuum. +Left: ∆v = −1.4 km/s channel. +Middle: ∆v = −1.75 channel. +Right: +∆v = −2.1 channel. The continuum beam is in magenta, and the line emission beam is in cyan. +Δv = − 1.4km/s +Δv = − 1.75km/s +Δv = − 2.1km/s +EN61 Activation +EN47 Activation +RN61 Activation +12CO +12CO +12CO +Figure 2. HD 142666 structure (12CO: J = 2 → 1) and activations. Upper left: ∆v = −1.4 km/s channel with kink circled in +white. Upper middle: ∆v = −1.75 channel with kink circled in white. Upper right: ∆v = −2.1 channel. Bottom row: selected +mean-subtracted activations that roughly correspond to the channels in the upper row. Activations are from three different +models (EN61, EN47, and RN61, respectively). Line emission beams are the cyan ellipses in the lower right of the upper row +panels. + +Identification of Planet in HD 142666 +5 +Value +EN47 +EN61 +EN75 +RN47 +RN61 +RN75 +Accuracy at 50% cutoff (%) +97 ± 0.5 +97 ± 0.5 +93 ± 0.7 +78 ± 1.1 +98 ± 0.4 +95 ± 0.6 +Accuracy at 95% cutoff (%) +96 ± 0.5 +94 ± 0.5 +88 ± 0.9 +65 ± 1.3 +96 ± 0.6 +92 ± 0.7 +AUC +0.99 ± 0.002 +0.99 ± 0.003 +0.98 ± 0.003 +0.86 ± 0.010 +> 0.99 ± 0.001 +0.98 ± 0.032 +Table 1. Model performance metrics from Terry et al. (2022a). +HD 142666 Sims +Δv = − 2.3km/s +Δv = − 2.0km/s +Simulated Channel +Simulated Channel +Observed Channel +Figure 3. HD 142666 simulation results. Upper left: ∆v = −2.3 km/s channel from the simulation (convolved beam in lower +right). Upper right: ∆v = −2.0 km/s channel from the simulation (convolved beam in lower right). Lower left: observed +continuum overlaid with contours of simulated ∆v = −2.3 km/s channel. +Lower right: observed continuum overlaid with +simulated (cyan) and observed (white) channels. Continuum beam is in magenta, and the line emission (simulated and observed) +beam is in cyan. The system includes a 5 MJ planet at 75 au. The simulated channels have the continuum and background +subtracted for clarity. The planet’s location is indicated with an x. + +6 +Terry et al. +This paper makes use of the following ALMA data: +ADS/JAO.ALMA #2016.1.00484.L. ALMA is a part- +nership of ESO (representing its member states), +NSF (USA) and NINS (Japan), together with NRC +(Canada), MOST and ASIAA (Taiwan), and KASI (Re- +public of Korea), in cooperation with the Republic of +Chile. +The Joint ALMA Observatory is operated by +ESO, AUI/NRAO and NAOJ. J.T. was a participant +in the 2022 Machine Learning for Science (ML4SCI) +Google Summer of Code program. 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Kinematic Evidence of an Embedded Protoplanet in HD 142666 Identified by Machine Learning J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Terry ,1, 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Hall ,1, 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Abreau ,3, 4 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Gleyzer 5 1Department of Physics and Astronomy, The University of Georgia, Athens, GA 30602, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2Center for Simulational Physics, The University of Georgia, Athens, GA 30602, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 3Department of Medicine, Division of Cardiology, University of California, San Francisco, CA, 94143, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 4Cardiovascular Research Institute, San Francisco, CA, 94158, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 5Department of Physics and Astronomy, The University of Alabama, Tuscaloosa, AL 35487, USA (Received January 13, 2023) Submitted to ApJ ABSTRACT Observations of protoplanetary discs have shown that forming exoplanets leave characteristic im- prints on the gas and dust of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' In the gas, these forming exoplanets cause deviations from Keplerian motion, which can be detected through molecular line observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Our previous work has shown that machine learning can correctly determine if a planet is present in these discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Using our machine learning models, we identify strong, localized non-Keplerian motion within the disc HD 142666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Subsequent hydrodynamics simulations of a system with a 5 Jupiter-mass planet at 75 au recreates the kinematic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' By currently established standards in the field, we conclude that HD 142666 hosts a planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' This work represents a first step towards using machine learning to identify previously overlooked non-Keplerian features in protoplanetary discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Keywords: Hydrodynamics — Radiative transfer — Accretion disks — Methods: numerical — Catalogs — Planets and satellites: formation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' INTRODUCTION Protoplanetary accretion discs are the sites of planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The newest generation of telescopes, such as the Atacama Large Millimeter/submillimeter Array (ALMA), have unprecedented capabilities for observing protoplanetary discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' For the first time, we can not only resolve discs themselves, but also quantify the motion of the dust and gas within them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Discs display a striking variety of structures such as rings (ALMA Partnership et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Dipierro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018), likely caused by dust trapping due to forming planets (Pinilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Dipierro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2015), and spirals (P´erez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2016), which may be caused by forming planets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2015b) or another mechanism such as gravitational in- stability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2015a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Meru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' This new information has greatly advanced our understanding of the processes underlying the for- mation and evolution of planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Planets and physical processes, such as gravitational instability, influence the motion within the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' This causes the material to deviate from simple Keplerian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Comparing the observed motion against purely Keplerian motion provides information on the bodies and processes present in the disc (Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Paneque-Carre˜no et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Longarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Terry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Non- Keplerian motion has been used to uncover a variety of structures, including localized perturbations associated with gaps and planets (Teague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018, 2019, 2020) as predicted by Perez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Kinematic analysis is limited by our ability to accu- rately identify non-Keplerian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The deviations can be small and frequently occur in noisy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' It is therefore not only difficult and slow to identify them, but there is also the strong possibility of overlooking their occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Any signature that is overlooked is a missed opportunity to detect either a forming planet or some other process, such as the GI-Wiggle indicative of gravitational instability (Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2020) or the vertical shear instability (Barraza-Alfaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Machine learning (ML) provides a useful tool for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' ML has quickly become ubiquitous in both society and the sciences, everything from self-driving cars (Bo- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='05075v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='EP] 12 Jan 2023 ID2 Terry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' jarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2016) to medicine (Parmar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Re- cent efforts in astronomy have made it clear that ma- chine learning is a powerful method even with simulated training data (Jo & Kim 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' M¨oller & de Boissi`ere 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Machine learning, and in particular computer vision, excels at the analysis of im- ages (Voulodimos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' In some cases, it has even been shown to outperform humans (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' It is therefore naturally suited for application to the noisy datasets in observational astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Using ML models developed in a previous work (Terry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2022a), we identify a strong and localized deviation from Keplerian motion in HD 142666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Using the current widely accepted field standard method (Teague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018, 2019), we perform smoothed par- ticle hydrodynamic (SPH) simulations to recreate the kinematic structure of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The agreement is sig- nificant when a 5 MJ planet is included at 75 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' We conclude that HD 142666 hosts a planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The paper is arranged as follows: Section 2 describes the models and simulations used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Section 3 shows the results of applying the models and simulating the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Section 4 gives our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' METHODS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Machine Learning We use the ML models described in Terry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' (2022a) and describe them here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' We use two different architectures: EfficientNetV2 (Tan & Le 2021) and RegNet (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' All models were made using PyTorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2019), albeit with significant modifications to the default models and hy- perparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' We denote these models as EN47, EN61, EN75, RN47, RN61, and RN75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Several performance metrics are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The models were trained using synthetic observations from the MCFOST (Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2006, 2009) radiative transfer code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' MCFOST inputs were drawn from 1000 PHANTOM (Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018) SPH simulations of sys- tems with and without planets (Terry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Each MCFOST calculation outputs a position-position- velocity cube from 13CO transition lines (J = 2 → 1 and J = 3 → 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The cubes were convolved spatially and spectrally and noise was added in order to replicate current observational capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The model inputs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' radiative transfer outputs) are images of dimension C ×H ×W, where C is the number of input channels, H is the height of the image, and W is the width of the image (here, H = W = 600 pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' A typical grayscale or RGB image will have C=1 or 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' We instead input an entire position- position-velocity cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Observations vary significantly in the number of chan- nels that cover the disc, but the typical range is between ≈40-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' To address this, we train three different imple- mentations of each model, which gives us a total of six models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The difference between each implementation is the number of input velocity channels (C=47, 61, or 75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Each model outputs a two-component vector such that the sum of the components is 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' it has undergone softmax activation (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' This can be interpreted as the probability that the given input be- longs to a certain class, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' planet- vs no-planet class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The models also output images of their internal activa- tion structure, which we consider to be the more impor- tant output in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' While the models were not trained to pinpoint the locations of planets —a job more suited for semantic image segmentation (Minaee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2022) —the activation structure can inform us which re- gions the model finds important when making its clas- sification decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Terry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' (2022a) found that the activations were able to highlight velocity channels with non-Keplerian motion in systems that host planet(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' To this end, we apply our previously trained models to ALMA data of the HD 142666 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' We inspect the softmax values and activation structures to gain insight into whether a planet might be in the system and, if so, where its signature is the strongest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Observational Data The HD 142666 data was taken from the DSHARP catalogue (Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Data includes 12CO line emission (J = 2 → 1) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='25 mm continuum images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The system was imaged with a beam with FWHM of 77 × 61 mas (≈11 × 9 au) with an RMS noise of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='3 mJy beam−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' channels have a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='35 km s−1 resolution (Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Figure 1 shows selected velocity channels overlaid on the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The image was cropped to focus on the disc, and a subset of velocity channels was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The channels were reshaped to 600 × 600 pixels and normalized such that all pixel values were between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Hydrodynamical Simulations We run an SPH simulation using PHANTOM and create channel maps using MCFOST in the same way that the original training data was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The kink is approxi- mately 75 au from the center of the disc, so we place a planet at this distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' System parameters are taken from Rubinstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The stellar mass, temperature, and radius are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='0 M⊙, 7500 K, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='2 R⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The disc has a mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='0533 M⊙, an inner radius of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='3 au, and an outer radius Identification of Planet in HD 142666 3 of 150 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The system is inclined at 62 degrees with a position angle of 162 and an azimuth of 72 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' It is located 148 pc from Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The SPH outputs are used to create line emission maps to mimic ALMA capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' These calculations are done using the MCFOST radiative transfer code (Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2006, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Each calculation uses 108 photon packets and includes carbon/silicate dust (Draine & Lee 1984) with a dust-to-gas ratio of 1:100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The result- ing outputs were convolved spatially and spectrally to match the observed line emission resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' RESULTS AND DISCUSSION Figure 2 shows that HD 142666 has a strong, localized kink that is detected by the ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The kink is par- ticularly visible in the upper middle (∆v = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='75 km/s) channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The lower row shows activation structures that roughly correspond to the above channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The average softmax value is over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='84, which means that the models predict the probability that the input for HD 142666 contains a planet to be over 84%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' This prompts further scrutiny of the activations, which we use to determine the most probable channel that contains the kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The strength and localization of the newly identified kink are reminiscent of the kinks in HD 163296 and HD 97048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' As with HD 163296, the kink in the gas is out- side of the radial extent of the continuum disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Both of these disks were found to host planets after SPH simula- tions containing a planet recreated the kinematic struc- ture observed in CO observations (Teague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' We apply this same method to HD 142666 to demonstrate that the kink identified by our models is consistent with kinks identified by conven- tional means in HD 163296 and HD 97048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' We found that a simulation of a protoplanetary disk with a 5 MJ planet reproduced the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Fig- ure 3 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' A localized kink in the vicinity of the planet is clear in the upper left panel Figure 3 (∆v = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='3 km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' This kink is visible to a lesser ex- tent in the ∆v = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='0 km/s in the upper right panel of Figure 3, which is also the case in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' There is strong agreement between this feature and the non- Keplerian channel identified by our models: both dis- play a kink of approximately the same shape and size at approximately the same radial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' This can be seen in the lower left and right panels of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Note that the simulation and observation do not display the strongest kink in the same velocity channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' This is simply a relic of the finite temporal resolution of the simulation, which makes it extremely unlikely that the simulation will be saved when the planet is exactly co- incident with the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The temporal resolution of the simulation was increased to mitigate this effect, but it persists to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' We conclude that HD 142666 hosts a planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' We note that our conclusion is confirmed using the same methods described by Teague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' (2018, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' However, what is new about our ap- proach is that the non-Keplerian motion was first identi- fied by ML models, highlighting a protoplanet candidate that had previously been missed upon visual analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Verification of the evidence is still done using the same methodology as previous works (Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 2018, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Future Work and Limitations This work shows that machine learning can effectively identify non-Keplerian motion even if it is missed by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' However, our work can be improved upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The primary limitation is the fact that localising non- Keplerian motion was not the explicit goal of these mod- els when they were trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Their purpose was classi- fication without any attempt of segmentation or object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Models specifically designed to pinpoint de- viations would likely be more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Rather than inspecting activation structures —of which there can be hundreds —the model would directly output a predic- tion of the location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' This would be a more precise and straightforward method to detect the non-Keplerian sig- nature, but it would not remove the need to perform follow-up simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' We intend to explore this possi- bility in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' CONCLUSION We have applied ML models created by Terry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' (2022a) to the DSHARP data of HD 142666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' All models strongly predict the presence of at least one planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The activation structures highlight a strong, unreported, and localized kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' An SPH simulation with a 5 MJ planet at 75 au is able to recreate the newly identified kine- matic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' By the previously established bench- marks and methods for kinematic planet detection, we conclude that HD 142666 hosts a planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' This work demonstrates the utility of applying ma- chine learning to the analysis of protoplanetary discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' By highlighting non-Keplerian features in the disc, ML models are able to guide planet-detection efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The signatures of the planet were previously overlooked by human analysts, and the traditional analysis was only performed because of the information given by the mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' We anticipate that this method can identify new non-Keplerian features in both existing and future pro- toplanetary observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' ACKNOWLEDGEMENTS 4 Terry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Δv = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='4km/s Δv = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='75km/s Δv = − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='1km/s Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Line emission overlaid on continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Left: ∆v = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='4 km/s channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Middle: ∆v = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='75 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Right: ∆v = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='1 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The continuum beam is in magenta, and the line emission beam is in cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Δv = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='4km/s Δv = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='75km/s Δv = − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='1km/s EN61 Activation EN47 Activation RN61 Activation 12CO 12CO 12CO Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' HD 142666 structure (12CO: J = 2 → 1) and activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Upper left: ∆v = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='4 km/s channel with kink circled in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Upper middle: ∆v = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='75 channel with kink circled in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Upper right: ∆v = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='1 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Bottom row: selected mean-subtracted activations that roughly correspond to the channels in the upper row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Activations are from three different models (EN61, EN47, and RN61, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Line emission beams are the cyan ellipses in the lower right of the upper row panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Identification of Planet in HD 142666 5 Value EN47 EN61 EN75 RN47 RN61 RN75 Accuracy at 50% cutoff (%) 97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='5 97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='5 93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='7 78 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='1 98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='4 95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='6 Accuracy at 95% cutoff (%) 96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='5 94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='5 88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='9 65 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='3 96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='6 92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='7 AUC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='010 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='032 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Model performance metrics from Terry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' HD 142666 Sims Δv = − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='3km/s Δv = − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='0km/s Simulated Channel Simulated Channel Observed Channel Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' HD 142666 simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Upper left: ∆v = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='3 km/s channel from the simulation (convolved beam in lower right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Upper right: ∆v = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='0 km/s channel from the simulation (convolved beam in lower right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Lower left: observed continuum overlaid with contours of simulated ∆v = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='3 km/s channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Lower right: observed continuum overlaid with simulated (cyan) and observed (white) channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' Continuum beam is in magenta, and the line emission (simulated and observed) beam is in cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The system includes a 5 MJ planet at 75 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The simulated channels have the continuum and background subtracted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The planet’s location is indicated with an x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' 6 Terry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' This paper makes use of the following ALMA data: ADS/JAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='ALMA #2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='00484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' ALMA is a part- nership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Re- public of Korea), in cooperation with the Republic of Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQfbgzb/content/2301.05075v1.pdf'} +page_content=' The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.' metadata={'source': 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new file mode 100644 index 0000000000000000000000000000000000000000..9c81e06b32e9ef4dcc2f1c0e5f2b70fc88a346b8 --- /dev/null +++ b/6tAyT4oBgHgl3EQf2vk-/content/tmp_files/2301.00755v1.pdf.txt @@ -0,0 +1,901 @@ +Highly Sensitive and Self Powered Ultraviolet Photo Detector based on ZnO Nanorods +Coated with TiO2 +Shashi Pandey1, Alok Shukla2*, Anurag Tripathi1 +1Department of Electrical Engineering IET Lucknow, Uttar Pradesh 226021, India +2Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India + +Abstract +Nanorods (NRs) of crystalline ZnO coated with thin layers of TiO2 (ZnO@TiO2) were fabricated +with the help of spin coating technique followed by hydrothermal method. Scanning electron +microscopy (SEM) and X-ray diffraction analysis confirms the morphology and structural stability +of as-prepared NRs. The optical band gaps of the NRs were estimated, and a clear blue-shift +towards the UV region has been detected. When UV light falls on as-prepared device (i.e., in the +“ON” state), a significant increase in photocurrent (IUV) at zero voltage supply was observed from +6 µA to 17 µA, while in the “OFF” state, the dark current (Idark), increases from 0.08 µA to 0.6 µA +with ZnO@TiO2 NRs as compared to bare ZnO NRs respectively. Responsivity and detectivity of +TiO2 coated ZnO NRs based device found maximum in UV region unlike bare ZnO NRs. +Enhanced photocurrent achieved by the growth of TiO2 layers on ZnO NRs is 250 µA as compared +to bare ZnO NRs for which it is 35 µA at 10V voltage supply under the ultraviolet irradiation +(illumination intensity of 1 mW/cm2). Furthermore, theoretical calculations have been performed +using the first-principles density-functional theory to understand the effects of heterostructure NRs +on the electronic and optical properties of TiO2 coated ZnO. + +Keywords: Zinc Oxide, Titanium Dioxide, Nanorods, hydrothermal method, Spin coating, Photo +detector. +Email: 2512@ietlucknow.ac.in, shukla@phy.iitb.ac.in, anurag.tripathi@ietlucknow.ac.in +*corresponding author + + +Introduction +Ultraviolet photo detection has many applications in the field of optical imaging, +optoelectronic circuits, military surveillance, air quality monitoring, and even in space +communication [1]–[4]. Conventional photo detectors based on materials such as silicon, +germanium, gallium arsenide, and silicon carbide, etc. become expensive because they require +high temperature conditions for device fabrication, as well as visible light filters. Therefore, the +invention of fast, sensitive, direct UV detectors, that are easy to synthesize, is important. For high +UV sensitivity, we need materials that are transparent in the lower-energy region of the spectrum, +and have strong optical response in the UV region. Devices based on several semiconducting +oxides are very useful for the purpose because of their environment friendly nature, non-toxic +character, large band-gap leading to high UV sensitivity, low cost, and excellent thermal stability. +The sensitivity to the UV light in these materials can be further tuned by constructing +nanostructures of different shapes and sizes, and also by managing oxygen adsorption and native +defects [5]–[7]. + +In this work we have explored the optical response of nano-rods (NRs) of a transition metal +oxide, namely, crystalline ZnO, coated with thin layers of TiO2 (ZnO@TiO2). Pure ZnO is not a +good candidate for such devices because it has inherent deep levels and surface defects [8] which +enhance undesirable photocurrent leading to low sensitivity and switching stability during device +operation. Recently, a few groups studied the synthesis and optical response of TiO2 and ZnO- +based nanocomposites, with the purpose of demonstrating improved device performance [9], [10], +as compared to bare ZnO nanostructures. TiO2 has a wide range of applications [8], [11]–[13] such +as in photo detectors, gas sensors, and thermal catalysis. In several studies, enhancement in UV +photo detection properties of ZnO, by changing its morphology, was pursued [14]–[19]. On the +other hand, many works have suggested that doping of different materials in ZnO can improve its +optoelectronic and photoconductive properties [3], [20]–[22]. Because of its excellent thermal and +photochemical stability, TiO2 has been investigated as a candidate for enhancing the optoelectronic +properties of nanocomposite devices [23]. In order to increase the sensitivity of photodetectors, +researchers have focused more on the coupling of semiconductors, particularly a heterojunction +based on two types of semiconductors with differing energy band structures [24]. As a result of +the coupling of various energy level structures, TiO2 and ZnO are strong candidates to produce a +heterostructure with improved characteristics compared to those of either material alone. Wang et + +al [24] reported that ZnO@TiO2 nanostructures also play an important role in dye-sensitized solar +cells applications. ZnO is a well-known n-type semiconductor due to the presence of native defects +[25]–[27], as a result of which electrons (i.e., its majority charge carriers) already exist in its +conduction band at the room temperature[28]. As a result of this, while measuring the +photocurrent; one sees a drop in the photocurrent in the UV “OFF” state of the bare ZnO samples. +However, in the UV “ON” state, the photocurrent increases because of the photon-induced +promotion of the valence band electrons to the conduction band in bare ZnO[28], [29]. As far as +ZnO@TiO2 NRs are concerned, because of the high reactivity of anatase TiO2, a significant O2 +desorption takes place, thus freeing up electrons, leading to a large rise in the photocurrent in the +UV “ON” state, as compared to the bare ZnO NRs [16], [18], [19], [24]. Hence, this work will +improve the understanding of heterostructures based on semiconducting oxides, which leads to +enhance the optoelectronic properties of devices. +Experimental Procedure +A. Synthesis and Characterization of ZnO@TiO2 Nano rods +Firstly, cleaning of ITO substrate was done using acetone and methanol or IPA. ITO +substrate cleaned with firstly under acetone and then put in methanol quickly before it dries after +taking out from acetone. Then the mixture of zinc acetate dehydrates and ethanolamines (with +ratio 1:2) were dissolved in 25 mL 2-methoxy ethanol. Next, the prepared solution was stirred at +50 0C for 1 hour, and then spin-coated with 2500 rpm for 1 min onto indium-tin oxide (ITO) +glass substrates. The resultant sample was further heated at 350 0C for 6 hours to obtain ZnO +NRs on the ITO substrate. Next, the prepared NRs of ZnO were placed inside an autoclave, which +contained diethylenetriamine, isopropyl alcohol, and a titanium (IV) isopropoxide. Autoclave, +containing the mixture, was given a heat treatment (200 0C for 12 hours), after which the as- +prepared sample was smoothly rinsed with ethanol and acetone at the room temperature. Finally, +the rinsed sample was annealed at 450 0C for 6 hours to confirm the deposition (i.e. coating) of +crystalline phases of TiO2 on ZnO NRs. For structural and morphological characterizations, Zeiss +Supra-55 field emission scanning electron microscope (FESEM) was used. In order to examine +the structural phase purity of the prepared samples, the powder x-ray diffraction (XRD) +experiments were carried out on Bruker D8 diffractometer equipped with Cu target having +LYNXEYE detector. The high temperature x-ray diffraction measurements were performed to + +confirm the structural phases of the prepared samples. The optical band gap of prepared sample +has been measured using diffuse reflectivity measurements. These measurements have been +performed in the 200 nm to 800 nm wavelength range using Perkin Elmer LAMBDA 950 UV- +Vis-NIR Spectrophotometer. For device fabrication Ag (thickness 50 nm) has been deposited as +an electrode using thermal evaporation. Shadow masking was used to keep the width to 1 mm +and the distance between the Ag electrodes (channel length) to 100 µm. Electrical +characterizations were performed by Keithley 2612A source meter. +B. Computational details +The first-principles calculations to support the experimental results were performed within +the framework of plane-wave density-functional theory (PW-DFT) [30], [31] as implemented in +Vienna ab-initio simulation package (VASP)[32], [33]. To match our calculated bandgap with +the experimental results, the generalized gradient approximation coupled with the Hubbard U +(GGA+U) level of theory was used [34], [35]. We performed calculations both on the bare ZnO +NRs, and the ZnO@TiO2 NRs, and the super cells used for the two systems are shown in Fig. 1. +To model the bare NRs, we used a one-dimensional super cell of dimension 12x1x1 (see Fig. +1(a)) of bulk ZnO, while for simulating ZnO@TiO2 NRs, we added one-unit cell of the anatase +phase of TiO2 containing twelve atoms (4 Ti and 8 O) on one of the ends of the ZnO super cell. +Furthermore, to model the finite size of the NRs, 10 Å of vacuum was used in two directions, as +shown in Fig. 1. One can ask as to why did we model ZnO@TiO2 NRs by attaching a unit cell +of TiO2 only on one tip of the bare NRs, and not on the second tip and other places? The reason +behind this choice is that the SEM images presented in the next section clearly show that the +ZnO@TiO2 NRs have TiO2 attached only on one tip of the bare NRs, and nowhere else. As far +as the optimized geometries of the two types of NRs presented in Fig. 1 are concerned: (a) bare +NRs were found to be 39.28 Å long, with a diameter of 5.20 Å (Fig. 1(a)), and (b) the total length + +of ZnO@TiO2 NRs is approximately 43.01 Å, with the diameter on the ZnO side of around 5.20 +Å, and on the TiO2 side 3.73 Å (Fig. 1(b)). + +FIGURE 1: Structural Images of (a) bare ZnO having 12x1x1 supercell and (b) composite of ZnO +having supercell 12x1x1 along with TiO2 with single unit cell. Blue and red color indicates Zn, +Ti and O atoms respectively. +Results and Discussion +Figure 2(a) shows the SEM image of the as-prepared ZnO@TiO2 on ITO glass substrate. +Firstly, the ZnO NRs have been deposited on ITO glass substrate using spin coating method (see +Figure 2(b)). Figure 2(b) shows the vertical orientation of ZnO NRs (top view). Deposition of TiO2 +on ZnO NRs was done using the hydrothermal process, and as-prepared TiO2 nanocomposites on +ZnO[37] are also shown in Figure 2(a) and 2(b). The average length and diameter of the NRs were +around 1.80 μm and 150 nm with particle size 3-23.2 nm. From the top view of SEM image, TiO2 +has been deposited only on the exposed tips of ZnO NRs (see figure 2(b)), The nanostructures +(NRs) are typically about 1 µm in length and 150–200 nm in diameter. Hence, the SEM analysis +confirms the fabrication of ZnO@TiO2 NRs. + + +FIGURE 2: FESEM images of (a) deposited TiO2 on ZnO NRs, (b) shows enlarged view (yellow +color) of deposited TiO2 nanoparticles on ZnO NRs. + +FIGURE 3: X-Ray diffraction patterns of as-prepared ZnO and ZnO@TiO2 NRs. + +(a) +(b) +TiO,nanoparticles +ZnONanorods +ITO +500nmZnO@TiO,NRs +450°C +ZnO +(00) +(002) +ZnO@TiO,NRs +TiO2 +(b) +ZnONRS +ITOSubstrate +(101) +(6)- +2 +(102) +(110) +(a.u.) +(201) +Intensity +ZnONR's +350°C +ZnO +(00t) +(002) +ZnONRS +(a) +ITOSubstrate +(101) +(102) +-(103) +(110) +(201) +20 +30 +40 +50 +60 +70 +80 +20 (degree) +To probe the structural stability of our prepared ZnO@TiO2 nanocomposites, we have performed +X-ray diffraction analysis (see Figure-3). X-ray diffraction patterns of the as-prepared ZnO NRs +have been shown in figure-3(a) and indexing of all the diffraction peaks confirms the hexagonal +wurtzite structure of ZnO[38], [39]. The x-ray diffraction pattern of ZnO@TiO2 is shown in Figure +3(b) and additional peaks corresponding to (101) and (004) planes of the anatase phase of TiO2 are +clearly observed. These coupled with the SEM images further confirm the formation of the +ZnO@TiO2 composite NRs, with the TiO2 deposited on the tips. We have probed the optical +absorption for bare ZnO NRs and TiO2 coated ZnO NRs using diffuse reflectance spectroscopy. +The spectra obtained from DRS is converted into equivalent absorption spectra through Kubelka– +Munk equation [8,13], +𝐹(𝑅∞) = +(1−𝑅∞)2 +2𝑅∞ , (i) +Where 𝐹(𝑅∞) is the Kubelka–Munk function,𝑅∞ = 𝑅𝑠𝑎𝑚𝑝𝑙𝑒/𝑅𝑠 𝑡𝑎𝑛 𝑑𝑎𝑟𝑑. The Kubelka–Munk +function can be related (proportional) to the absorption coefficient (α) as- + F(R∞) ∝ α ∝ +(ℎ𝜈−𝐸𝑔)1/𝑛 +ℎ𝜈 + , (ii) +In order to calculate the Eg, the obtained absorption coefficient is converted in to Tauc equation +[13] and plotted in figure 4. +(𝛼ℎ𝜈)𝑛 = 𝐴(ℎ𝜈 − 𝐸𝑔) , (iii) +Here in equation (iii) ‘n’ has the value of 2 for direct bandgap transitions (used for ZnO bare NRs), +while n is equal to 1/2 for an indirect transition (used for ZnO@TiO2 NRs) [13]. To see the change +in the optical band gap of deposited ZnO[8], [16], [17], and ZnO@TiO2 NRs, optical absorption +measurements have been carried out in the range 300 nm to 900 nm. It is well known that bulk +ZnO has a band gap of 3.3 eV, while our measured value of the optical gap of ZnO NRs is 3.46 +(see Fig. 4). Clearly, band gap increases in the NR phase, as compared to the bulk [40], [41]. Our +measured optical gap of as prepared ZnO@TiO2 NRs further increases to 3.56 eV (see Fig. 4), +which is closer to the UV range. This increase can clearly be attributed to the presence of TiO2 in +the sample. + + +FIGURE 4: Measured optical absorption spectra of ZnO and ZnO@TiO2 NRs. +To verify the experimental band gap of nanocomposites of ZnO and TiO2, systematic first- +principles DFT calculations of total density of states (TDOS)[42] and optical absorption spectra +have been performed. Figure 5 show TDOS of bare ZnO NRs with an electronic band gap of 3.46 +eV calculated using the GGA+U method [43] with U=4.3 eV (at Zn sites), while the inset shows +the optical absorption spectrum obtained from the same calculations. We have performed similar +calculations for ZnO@TiO2 NRs, and observed a clear blue shift in band gap as compared to bare +ZnO NRs. In Fig. 6 we show TDOS of ZnO@TiO2 NRs with an electronic band gap of 3.58 eV +obtained the using the GGA+U calculations with U=4.3 eV at the Zn sites and U=5.6 eV at the Ti +sites, along with the computed optical absorption spectrum shown in the inset. Very good +quantitative agreement between the theoretically computed band gaps with the experimentally +measured ones indicates that our choice of U parameters in the GGA+U calculations is correct (see +Figs. 4, 5 and 6). + + +24- +ZnONRs +24 +ZnO@TiO,NRs +20- +-20 +16 +=3.46ev +16 +12- +12 +8- +-8 +o +4- +0 +3.0 +3.2 +3.4 +3.6 +3.8 +PhotonEnergy(eV) +FIGURE 5: TDOS of ZnO NRs exhibits an electronic band gap of 3.46 eV obtained from the +GGA+U calculations (U=4.3 eV). The inset shows the calculated optical absorption spectrum of +bare ZnO NRs. + +FIGURE 6: TDOS of ZnO@TiO2 NRs shows electonic band gap of 3.58 eV using GGA+U +calculations (U=4.3 eV for Zn and U=5.6 eV for Ti). The inset shows simulated optical +absorption spectrum of ZnO@TiO2 NRs. + +20 +ZnONRs +AbsorptionCoefficient(cm-1 +OpticalAbsorption +(states/eV) +15 +States +10 +3.03.1 +3.23.33.43.53.63.7 +Energy (eV) +Density of +TDOS +5 +VB +BandGap(E)=3.46eV +0 +CB +10 +-2 +-1 +1 +2 +3 +4 +5 +Energy(eV)6 +ZnO@TiONRs +Density of States (states/eV) +OpticalAbsorption +5 +4 +3 +2 +3.2 +3.3 +3.43.53.63.73.8 +Energy (ev) +TDOS +0 +VB +CB +Band Gap (E.)=3.58 eV +3.0-1.5 +0.01.5 +3.0 +4.5 +6.0 +7.5 +Energy (ev)The measurement setups for photocurrent spectra for ZnO NRs and ZnO@TiO2 are shown +in Figs. 7(a) and 7(b), respectively. Before performing the measurements, first we heated the +prepared device up to a temperature of 3000C for 8 hours to avoid moisture, and then kept it in +dark on vacuum desiccator for several hours to reach the equilibrium condition. I-V characteristics +of ZnO NRs and ZnO@TiO2 NRs measured using a UV lamp (with wavelength 365 nm) have +been plotted in Figs.-7(c) and 7 (d), and it is clearly observed that with the increase in the voltage, +the device current also increases for both cases. When UV is “ON”, the device current increases +sharply, while for UV “OFF”, the increase in the current with the applied voltage is quite +negligible. On comparing the I-V characteristics of bare ZnO NRs and ZnO@TiO2 NRs (Figs.- +7(c) and 7(d)), we conclude that the TiO2 coated on ZnO NRs enhances the value of the current +with applied voltages. This result indicates that bare ZnO NRs as well as ZnO@TiO2 NRs are +highly UV sensitive; however, the photoelectric response of ZnO@TiO2 NRs is much more +intense. Insets (i) and (ii) of Figs. 7(c) and 7(d) respectively show that there are still small amounts +of charge carriers inside the device at V=0, irrespective of whether the UV light is “ON” or “OFF”. +To determine the performance of device, responsivity, R = (IUV/IP) [19], is an important +parameter where, IP is the incident power and IUV is the maximum current under UV irradiation. +From Fig. 8 (a) and 8 (b) it is clearly seen that responsivity of our device is maximum in UV region +while it is decreasing in visible region. Detectivity (D = 1/NEPB) [19] has been depicted in Fig. +8(a) and 8(b), and peak value reached at 1.8 × 1014 (at ~ 450 nm) and 6.4 × 1014 Hz1/2/W (at 380 +nm) for bare ZnO NRs and TiO2 coated ZnO NRs respectively. It is clearly seen that in case of +TiO2 coated ZnO NRs shows enhancement in responsivity and detectivity both as compared to +bare ZnO NRs. + + + + +Figure 7: (a)-(b) Schematic of measurement setup for as prepared composite NRs using +semiconducting oxides on ITO glass substrate (mechanism of producing photocurrent with +applied field). Current-voltage characteristic of (c) ZnO NRs and (d) ZnO@TiO2 NRs in +presence of UV light, while insets of figures (c) & (d) show enlarged views of I-V characteristics. + +UV-Vis +UV-Vis +AMONRS +ZhO NRS40 +FUVON +(i) +(c) +300 ++-UVON +(i) +(d) +30 +AtV=o +200 +At V=o +20 +lwow*2μA +lwow*17 pA +urrent (μA) +10 +- +UVON +A +100 +UVON +UVOFF +10 +geV +UVOFF +0- +(i) +UWOFF +(ii) +-10- +At.V-O +-100- +AtVO +-20 +-200- +-30- +300 +-10 +-5 +5 +10 +-10 +5 +0 +5 +10 +Voltage (V) +Voltage (V) +Figure 8: Responsivity and Detectivity of (a) bare ZnO and (b) TiO2 coated ZnO NRs at different +wavelengths. +In order to probe the stability of our devices, UV photocurrents of bare ZnO NRs and ZnO@TiO2 +NRs, at a constant bias voltage of 5 V, were measured under ambient conditions. For the purpose, +the photons of wave lengths in the range 250nm - 750nm were alternatively switched ‘‘ON’’ and +‘‘OFF’’ for 10 minute each, and the results of our measurements are shown in Figs. 9 (a) and 9(b). +This growth and decay of current can be used to sense UV as well as visible wavelengths. +Interestingly, it is found that in case of visible light, photo current response is very low while in +case of UV light photocurrent response increases ten times for prepared device. Cycles are similar +and repeatable in nature, thus, confirming the reliability of both types of devices. Increase in the +photocurrent under UV light is due to the promotion of a large number of electrons from the +valence band to the conduction band because the photon energy exceeds the band gaps of our +prepared devices. However, photons of larger wave lengths, i.e., those in the visible range, do not +have sufficient energy to transfer the electrons from the valence to the conduction band, leading +to considerable drop in the number of charge carriers, and, thus, the photocurrent. The photocurrent +in the visible region is mainly because of the charge carriers generated due to native defects and +impurities in the devices. Native defects such vacancies, self-interstitials, and anti-sites are +inescapable during the synthesis of crystal lattices and have a significant impact on the +performance of semiconducting oxide-based devices [42,44-47]. Significant changes in device +performance, such as electronic bandgap, photocurrent, response time, etc., can be seen with a +nominal change in defect concentration. This photocurrent detected in the visible area may be +caused by the intrinsic oxygen deficiency found mostly in semiconducting oxides-based devices. + +16(a)ZnONRs +2.0 +-Responsivity +(b)ZnO@TiO2NRs +W) +--Responsivity +Detectivity +100 +e-Detectivity +(W) +14 +1.6 +Responsivity (A/W) +1.2 +75 +-A +8- +0.8 +50 +3 +6 +0.4 +4- +25 +2- +0.0 +200 +300400 +500 +700 +800 +0- +600 +200 +300 +400 +500 +600 +700 +800 +Wavelength (nm) +Wavelength(nm) + +FIGURE 9: Photocurrent response of (a) ZnO NRs and (b) ZnO@TiO2 NRs. In both case +photocurrent response is maximum with UV light, while photocurrent decreases with increase in +wavelength (i.e. for Visible light). +From Figs. 7, 8 and 9 it is obvious that ZnO@TiO2 NRs has a significantly larger photocurrent +response as compared to bare ZnO NRs, with a many-fold enhancement of current in the UV +region. In order to systematically investigate the photo-response of the two types of devices as a +function of the wavelength of the incident light, we performed measurements of the photocurrents +of as prepared bare ZnO and ZnO@TiO2 NRs by varying the wavelength and the results are +presented in Fig. 10. From the figure it is obvious that: (a) in the entire region of wavelength probed +in our experiment, the photocurrents measured in ZnO@TiO2 NRs based devices is larger than +that of ZnO NR based devices, and (b) the difference in two responses becomes large with the +decreasing wavelengths, and for the shortest wavelength the photocurrent in the ZnO@TiO2 NRs +devices (53 μA) is almost five times of that in the bare ZnO devices (11.24 μA). Hence it is clearly +seen that with coating of TiO2 on ZnO NRs not only increases the photocurrent but also increases +photosensitivity and switching stability of prepared device as compare to bare ZnO NRs. + +11.24 +OFF +OFF +52.82 +11.22 +11.20 +52.63 +5.64 +ON +30.87 +OFE +445nm +445 +5.62 +(μA) +5.60 +30.66 +兰 +3.76 +wu +13.20 +OFF +545 +[LA) +3.75 +OFF +3.74 +Current +13.09 +/is-Light +1.89- +wu +645nm +645 +OFF +1.88 +7.92 +1.87 +7.81 +1.54 +wu +ZnO@TiO2NR's +ZnONR's +wu +745 +1.52 +4.51 +745 +1.50 +OFF +4.40 +OFF +0 +500 +1000 +1500 +2000 +0 +500 +1000 +1500 +Time (s) +2000 +Time (s)Photocurrent in the visible region is mainly because of the charge carriers generated due to native +defects and impurities in the devices. From the figure it is obvious that: (a) in the entire region of +wavelength probed in our experiment, the photocurrents measured in ZnO@TiO2 NRs based +devices is larger than that of ZnO NR based devices, and (b) the difference in two responses +becomes large with the decreasing wavelengths, and for the shortest wavelength the photocurrent +in the ZnO@TiO2 NRs devices (53 μA) is almost five times of that in the bare ZnO devices (11.24 +μA). Hence it is clearly seen that with coating of TiO2 on ZnO NRs not only increases the +photocurrent but also increases photosensitivity and switching stability of prepared device as +compare to bare ZnO NRs. + +FIGURE 10: Current vs wavelength plot shows photocurrent decreases with increase in the +wavelength of the incident light on bare ZnO NRs and ZnO@TiO2 NRs with applied bias of 1 V. +ZnO@TiO2 shows significantly larger photocurrent response as compared to bare ZnO NRs. + +60-Applied Bias=1V +ZnO@TiO,NRs +ZnONRs +50 +40 +30 +20 +10- +0 +300 +400 +500 +600 +700 +800 +Wavelenght (nm)Moreover, a comparison of the performance metrics of ZnO-based photodetectors [48-50] has been +summarized in Table 1. Therefore, we believe that ZnO NRs coated with TiO2 appear as one of +the highly-sensitive self-powered ultraviolet photo detectors. +Table 1: Comparison of performance against other ZnO@TiO2-based photodetectors. +Photo- detectors + +R +(mA/W) +D (Hz1/2/W) Response + Time +ON/OFF +Ratio + Ref. +TiO2/ZnO + +514 +3.2 x 109 33.7/12 s + 305 +[48] +ZnO/TiO2 + +540 +1.1 x 1010 9/20 s + 388 +[48] +Y‑ZnO/TiO2 +NWs-Au + +27 +6.2 x 1010 30 s +1786 +[49] +ZnO–TiO2/Si + +̴ 4500 +- + - +1122 +[50] +ZnO NRs +ZnO@TiO2 + +15 +90 +1.8 x 1014 < 30 s +6.4 x 1014 < 10s + 375 + 416 +This +Work +Conclusion +In summary, photosensitive devices based on bare ZnO NRs, as well as those coated with TiO2 +(ZnO@TiO2), were successfully fabricated. SEM analysis and X-ray diffraction confirm the +morphology and structural stability of as-prepared NRs. The optical band gap was measured and +a comparative blue shift was observed in ZnO@TiO2 NRs. First-principles DFT calculations at +the GGA+U level of theory were performed to understand the geometry, electronic structure, and +optical properties of both bare as well TiO2 coated NRs. Our DFT calculations, coupled with +suitable parameters, correctly predict the slight blue-shift in the band gap for the ZnO@TiO2 NRs, +as compared to the bare ones. In addition to the band-gap shift, strong enhancements in the +photoconductivity were observed ZnO@TiO2, as compared to bare ZnO NRs. With the UV light +falling on the sample (“ON” state), there is a significant increase in photocurrent, while in the case +of no incident UV light (“OFF” state), the dark current (Idark), decreases in nanocomposites. +Responsivity and detectivity of TiO2 coated ZnO NRs based device found maximum in UV region + +than bare ZnO NRs. ZnO@TiO2 NRs show significant growth and decay in photocurrent for +different wavelengths, leading to their increased photo-detection sensitivity and switching stability +in the UV region, as compared to the bare ZnO NRs. Therefore, we believe that ZnO NRs coated +with TiO2 will prove to be very useful in fabrication of highly-sensitive self-powered ultraviolet +photo detectors. +Acknowledgments +One of the authors (SP) acknowledges the Homi Bhabha Research Cum Teaching Fellowship +(A.K.T.U.), Lucknow, India for providing financial support Through Teaching Assistantship. +Authors sincerely thank Dr. Tejendra Dixit (Assistant Prof. IIIT kancheepuram) for his help during +synthesis. One of the authors (SP) likes to thank Dr. Shivendra Pandey (Assistant Prof. NIT +Silchar) for optical characterizations and his valuable suggestions in manuscript. +Compliance with ethical standards: +Conflict of interest: The authors declare that they do not have any conflict of interest. +Data availability statement – +The raw/processed data required to reproduce these findings cannot be shared at this time as the +data also forms part of an ongoing study. +References +[1] E. Monroy, F. O., and F. Calle, “Wide-bandgap semiconductor ultraviolet photodetectors,” +Semicond. Sci. Technol., vol. 18, no. 4, pp. R33–R51, Mar. 2003, doi: 10.1088/0268- +1242/18/4/201. +[2] K. Liu, M. Sakurai, M. Liao, and M. Aono, “Giant Improvement of the Performance of ZnO +Nanowire Photodetectors by Au Nanoparticles,” J. Phys. Chem. C, vol. 114, no. 46, pp. 19835– +19839, Nov. 2010, doi: 10.1021/jp108320j. +[3] T. Srivastava, G. Bajpai, G. Rathore, S. W. Liu, S. Biring, and S. 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Abdi, Enhancement of +responsivity and sensitivity of p-Silicon/n-Zinc oxide-based photodetector using titanium +dioxide nanoparticles, IEEE Transactions on Nanotechnology vol. 19, p. 744–748, sept. 2020, +doi: 10.1109/TNANO.2020.3022662. + + diff --git a/6tAyT4oBgHgl3EQf2vk-/content/tmp_files/load_file.txt b/6tAyT4oBgHgl3EQf2vk-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0709380a4813cd1ce9adebea012b685dacf84146 --- /dev/null +++ b/6tAyT4oBgHgl3EQf2vk-/content/tmp_files/load_file.txt @@ -0,0 +1,914 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf,len=913 +page_content='Highly Sensitive and Self Powered Ultraviolet Photo Detector based on ZnO Nanorods Coated with TiO2 Shashi Pandey1, Alok Shukla2*, Anurag Tripathi1 1Department of Electrical Engineering IET Lucknow, Uttar Pradesh 226021, India 2Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India Abstract Nanorods (NRs) of crystalline ZnO coated with thin layers of TiO2 (ZnO@TiO2) were fabricated with the help of spin coating technique followed by hydrothermal method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Scanning electron microscopy (SEM) and X-ray diffraction analysis confirms the morphology and structural stability of as-prepared NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The optical band gaps of the NRs were estimated, and a clear blue-shift towards the UV region has been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' When UV light falls on as-prepared device (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=', in the “ON” state), a significant increase in photocurrent (IUV) at zero voltage supply was observed from 6 µA to 17 µA, while in the “OFF” state, the dark current (Idark), increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='08 µA to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='6 µA with ZnO@TiO2 NRs as compared to bare ZnO NRs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Responsivity and detectivity of TiO2 coated ZnO NRs based device found maximum in UV region unlike bare ZnO NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Enhanced photocurrent achieved by the growth of TiO2 layers on ZnO NRs is 250 µA as compared to bare ZnO NRs for which it is 35 µA at 10V voltage supply under the ultraviolet irradiation (illumination intensity of 1 mW/cm2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Furthermore, theoretical calculations have been performed using the first-principles density-functional theory to understand the effects of heterostructure NRs on the electronic and optical properties of TiO2 coated ZnO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Keywords: Zinc Oxide, Titanium Dioxide, Nanorods, hydrothermal method, Spin coating, Photo detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Email: 2512@ietlucknow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='in, shukla@phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='in, anurag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='tripathi@ietlucknow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='in corresponding author Introduction Ultraviolet photo detection has many applications in the field of optical imaging, optoelectronic circuits, military surveillance, air quality monitoring, and even in space communication [1]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Conventional photo detectors based on materials such as silicon, germanium, gallium arsenide, and silicon carbide, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' become expensive because they require high temperature conditions for device fabrication, as well as visible light filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Therefore, the invention of fast, sensitive, direct UV detectors, that are easy to synthesize, is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' For high UV sensitivity, we need materials that are transparent in the lower-energy region of the spectrum, and have strong optical response in the UV region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Devices based on several semiconducting oxides are very useful for the purpose because of their environment friendly nature, non-toxic character, large band-gap leading to high UV sensitivity, low cost, and excellent thermal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The sensitivity to the UV light in these materials can be further tuned by constructing nanostructures of different shapes and sizes, and also by managing oxygen adsorption and native defects [5]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' In this work we have explored the optical response of nano-rods (NRs) of a transition metal oxide, namely, crystalline ZnO, coated with thin layers of TiO2 (ZnO@TiO2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Pure ZnO is not a good candidate for such devices because it has inherent deep levels and surface defects [8] which enhance undesirable photocurrent leading to low sensitivity and switching stability during device operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Recently, a few groups studied the synthesis and optical response of TiO2 and ZnO- based nanocomposites, with the purpose of demonstrating improved device performance [9], [10], as compared to bare ZnO nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' TiO2 has a wide range of applications [8], [11]–[13] such as in photo detectors, gas sensors, and thermal catalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' In several studies, enhancement in UV photo detection properties of ZnO, by changing its morphology, was pursued [14]–[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' On the other hand, many works have suggested that doping of different materials in ZnO can improve its optoelectronic and photoconductive properties [3], [20]–[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Because of its excellent thermal and photochemical stability, TiO2 has been investigated as a candidate for enhancing the optoelectronic properties of nanocomposite devices [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' In order to increase the sensitivity of photodetectors, researchers have focused more on the coupling of semiconductors, particularly a heterojunction based on two types of semiconductors with differing energy band structures [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' As a result of the coupling of various energy level structures, TiO2 and ZnO are strong candidates to produce a heterostructure with improved characteristics compared to those of either material alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Wang et al [24] reported that ZnO@TiO2 nanostructures also play an important role in dye-sensitized solar cells applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' ZnO is a well-known n-type semiconductor due to the presence of native defects [25]–[27], as a result of which electrons (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=', its majority charge carriers) already exist in its conduction band at the room temperature[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' As a result of this, while measuring the photocurrent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' one sees a drop in the photocurrent in the UV “OFF” state of the bare ZnO samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' However, in the UV “ON” state, the photocurrent increases because of the photon-induced promotion of the valence band electrons to the conduction band in bare ZnO[28], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' As far as ZnO@TiO2 NRs are concerned, because of the high reactivity of anatase TiO2, a significant O2 desorption takes place, thus freeing up electrons, leading to a large rise in the photocurrent in the UV “ON” state, as compared to the bare ZnO NRs [16], [18], [19], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Hence, this work will improve the understanding of heterostructures based on semiconducting oxides, which leads to enhance the optoelectronic properties of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Experimental Procedure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Synthesis and Characterization of ZnO@TiO2 Nano rods Firstly, cleaning of ITO substrate was done using acetone and methanol or IPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' ITO substrate cleaned with firstly under acetone and then put in methanol quickly before it dries after taking out from acetone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Then the mixture of zinc acetate dehydrates and ethanolamines (with ratio 1:2) were dissolved in 25 mL 2-methoxy ethanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Next, the prepared solution was stirred at 50 0C for 1 hour, and then spin-coated with 2500 rpm for 1 min onto indium-tin oxide (ITO) glass substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The resultant sample was further heated at 350 0C for 6 hours to obtain ZnO NRs on the ITO substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Next, the prepared NRs of ZnO were placed inside an autoclave, which contained diethylenetriamine, isopropyl alcohol, and a titanium (IV) isopropoxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Autoclave, containing the mixture, was given a heat treatment (200 0C for 12 hours), after which the as- prepared sample was smoothly rinsed with ethanol and acetone at the room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Finally, the rinsed sample was annealed at 450 0C for 6 hours to confirm the deposition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' coating) of crystalline phases of TiO2 on ZnO NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' For structural and morphological characterizations, Zeiss Supra-55 field emission scanning electron microscope (FESEM) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' In order to examine the structural phase purity of the prepared samples, the powder x-ray diffraction (XRD) experiments were carried out on Bruker D8 diffractometer equipped with Cu target having LYNXEYE detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The high temperature x-ray diffraction measurements were performed to confirm the structural phases of the prepared samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The optical band gap of prepared sample has been measured using diffuse reflectivity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' These measurements have been performed in the 200 nm to 800 nm wavelength range using Perkin Elmer LAMBDA 950 UV- Vis-NIR Spectrophotometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' For device fabrication Ag (thickness 50 nm) has been deposited as an electrode using thermal evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Shadow masking was used to keep the width to 1 mm and the distance between the Ag electrodes (channel length) to 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Electrical characterizations were performed by Keithley 2612A source meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Computational details The first-principles calculations to support the experimental results were performed within the framework of plane-wave density-functional theory (PW-DFT) [30], [31] as implemented in Vienna ab-initio simulation package (VASP)[32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' To match our calculated bandgap with the experimental results, the generalized gradient approximation coupled with the Hubbard U (GGA+U) level of theory was used [34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' We performed calculations both on the bare ZnO NRs, and the ZnO@TiO2 NRs, and the super cells used for the two systems are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' To model the bare NRs, we used a one-dimensional super cell of dimension 12x1x1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 1(a)) of bulk ZnO, while for simulating ZnO@TiO2 NRs, we added one-unit cell of the anatase phase of TiO2 containing twelve atoms (4 Ti and 8 O) on one of the ends of the ZnO super cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Furthermore, to model the finite size of the NRs, 10 Å of vacuum was used in two directions, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' One can ask as to why did we model ZnO@TiO2 NRs by attaching a unit cell of TiO2 only on one tip of the bare NRs, and not on the second tip and other places?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The reason behind this choice is that the SEM images presented in the next section clearly show that the ZnO@TiO2 NRs have TiO2 attached only on one tip of the bare NRs, and nowhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' As far as the optimized geometries of the two types of NRs presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 1 are concerned: (a) bare NRs were found to be 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='28 Å long, with a diameter of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='20 Å (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 1(a)), and (b) the total length of ZnO@TiO2 NRs is approximately 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='01 Å, with the diameter on the ZnO side of around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='20 Å, and on the TiO2 side 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='73 Å (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' FIGURE 1: Structural Images of (a) bare ZnO having 12x1x1 supercell and (b) composite of ZnO having supercell 12x1x1 along with TiO2 with single unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Blue and red color indicates Zn, Ti and O atoms respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Results and Discussion Figure 2(a) shows the SEM image of the as-prepared ZnO@TiO2 on ITO glass substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Firstly, the ZnO NRs have been deposited on ITO glass substrate using spin coating method (see Figure 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Figure 2(b) shows the vertical orientation of ZnO NRs (top view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Deposition of TiO2 on ZnO NRs was done using the hydrothermal process, and as-prepared TiO2 nanocomposites on ZnO[37] are also shown in Figure 2(a) and 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The average length and diameter of the NRs were around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='80 μm and 150 nm with particle size 3-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='2 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' From the top view of SEM image, TiO2 has been deposited only on the exposed tips of ZnO NRs (see figure 2(b)), The nanostructures (NRs) are typically about 1 µm in length and 150–200 nm in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Hence, the SEM analysis confirms the fabrication of ZnO@TiO2 NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' FIGURE 2: FESEM images of (a) deposited TiO2 on ZnO NRs, (b) shows enlarged view (yellow color) of deposited TiO2 nanoparticles on ZnO NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' FIGURE 3: X-Ray diffraction patterns of as-prepared ZnO and ZnO@TiO2 NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' (a) (b) TiO,nanoparticles ZnONanorods ITO 500nmZnO@TiO,NRs 450°C ZnO (00) (002) ZnO@TiO,NRs TiO2 (b) ZnONRS ITOSubstrate (101) (6)- 2 (102) (110) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=") (201) Intensity ZnONR's 350°C ZnO (00t) (002) ZnONRS (a) ITOSubstrate (101) (102) (103) (110) (201) 20 30 40 50 60 70 80 20 (degree) To probe the structural stability of our prepared ZnO@TiO2 nanocomposites, we have performed X-ray diffraction analysis (see Figure-3)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' X-ray diffraction patterns of the as-prepared ZnO NRs have been shown in figure-3(a) and indexing of all the diffraction peaks confirms the hexagonal wurtzite structure of ZnO[38], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The x-ray diffraction pattern of ZnO@TiO2 is shown in Figure 3(b) and additional peaks corresponding to (101) and (004) planes of the anatase phase of TiO2 are clearly observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' These coupled with the SEM images further confirm the formation of the ZnO@TiO2 composite NRs, with the TiO2 deposited on the tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' We have probed the optical absorption for bare ZnO NRs and TiO2 coated ZnO NRs using diffuse reflectance spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The spectra obtained from DRS is converted into equivalent absorption spectra through Kubelka– Munk equation [8,13], 𝐹(𝑅∞) = (1−𝑅∞)2 2𝑅∞ , (i) Where 𝐹(𝑅∞) is the Kubelka–Munk function,𝑅∞ = 𝑅𝑠𝑎𝑚𝑝𝑙𝑒/𝑅𝑠 𝑡𝑎𝑛 𝑑𝑎𝑟𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The Kubelka–Munk function can be related (proportional) to the absorption coefficient (α) as- F(R∞) ∝ α ∝ (ℎ𝜈−𝐸𝑔)1/𝑛 ℎ𝜈 , (ii) In order to calculate the Eg, the obtained absorption coefficient is converted in to Tauc equation [13] and plotted in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' (𝛼ℎ𝜈)𝑛 = 𝐴(ℎ𝜈 − 𝐸𝑔) , (iii) Here in equation (iii) ‘n’ has the value of 2 for direct bandgap transitions (used for ZnO bare NRs), while n is equal to 1/2 for an indirect transition (used for ZnO@TiO2 NRs) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' To see the change in the optical band gap of deposited ZnO[8], [16], [17], and ZnO@TiO2 NRs, optical absorption measurements have been carried out in the range 300 nm to 900 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' It is well known that bulk ZnO has a band gap of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='3 eV, while our measured value of the optical gap of ZnO NRs is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='46 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Clearly, band gap increases in the NR phase, as compared to the bulk [40], [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Our measured optical gap of as prepared ZnO@TiO2 NRs further increases to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='56 eV (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 4), which is closer to the UV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' This increase can clearly be attributed to the presence of TiO2 in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' FIGURE 4: Measured optical absorption spectra of ZnO and ZnO@TiO2 NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' To verify the experimental band gap of nanocomposites of ZnO and TiO2, systematic first- principles DFT calculations of total density of states (TDOS)[42] and optical absorption spectra have been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Figure 5 show TDOS of bare ZnO NRs with an electronic band gap of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='46 eV calculated using the GGA+U method [43] with U=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='3 eV (at Zn sites), while the inset shows the optical absorption spectrum obtained from the same calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' We have performed similar calculations for ZnO@TiO2 NRs, and observed a clear blue shift in band gap as compared to bare ZnO NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 6 we show TDOS of ZnO@TiO2 NRs with an electronic band gap of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='58 eV obtained the using the GGA+U calculations with U=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='3 eV at the Zn sites and U=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='6 eV at the Ti sites, along with the computed optical absorption spectrum shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Very good quantitative agreement between the theoretically computed band gaps with the experimentally measured ones indicates that our choice of U parameters in the GGA+U calculations is correct (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 4, 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 24- ZnONRs 24 ZnO@TiO,NRs 20- 20 16 =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='46ev 16 12- 12 8- 8 o 4- 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='8 PhotonEnergy(eV) FIGURE 5: TDOS of ZnO NRs exhibits an electronic band gap of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='46 eV obtained from the GGA+U calculations (U=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='3 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The inset shows the calculated optical absorption spectrum of bare ZnO NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' FIGURE 6: TDOS of ZnO@TiO2 NRs shows electonic band gap of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='58 eV using GGA+U calculations (U=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='3 eV for Zn and U=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='6 eV for Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The inset shows simulated optical absorption spectrum of ZnO@TiO2 NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 20 ZnONRs AbsorptionCoefficient(cm-1 OpticalAbsorption (states/eV) 15 States 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='7 Energy (eV) Density of TDOS 5 VB BandGap(E)=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='46eV 0 CB 10 2 1 1 2 3 4 5 Energy(eV)6 ZnO@TiONRs Density of States (states/eV) OpticalAbsorption 5 4 3 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='8 Energy (ev) TDOS 0 VB CB Band Gap (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=')=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='58 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='5 Energy (ev)The measurement setups for photocurrent spectra for ZnO NRs and ZnO@TiO2 are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 7(a) and 7(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Before performing the measurements, first we heated the prepared device up to a temperature of 3000C for 8 hours to avoid moisture, and then kept it in dark on vacuum desiccator for several hours to reach the equilibrium condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' I-V characteristics of ZnO NRs and ZnO@TiO2 NRs measured using a UV lamp (with wavelength 365 nm) have been plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='-7(c) and 7 (d), and it is clearly observed that with the increase in the voltage, the device current also increases for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' When UV is “ON”, the device current increases sharply, while for UV “OFF”, the increase in the current with the applied voltage is quite negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' On comparing the I-V characteristics of bare ZnO NRs and ZnO@TiO2 NRs (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='- 7(c) and 7(d)), we conclude that the TiO2 coated on ZnO NRs enhances the value of the current with applied voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' This result indicates that bare ZnO NRs as well as ZnO@TiO2 NRs are highly UV sensitive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' however, the photoelectric response of ZnO@TiO2 NRs is much more intense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Insets (i) and (ii) of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 7(c) and 7(d) respectively show that there are still small amounts of charge carriers inside the device at V=0, irrespective of whether the UV light is “ON” or “OFF”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' To determine the performance of device, responsivity, R = (IUV/IP) [19], is an important parameter where, IP is the incident power and IUV is the maximum current under UV irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 8 (a) and 8 (b) it is clearly seen that responsivity of our device is maximum in UV region while it is decreasing in visible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Detectivity (D = 1/NEPB) [19] has been depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 8(a) and 8(b), and peak value reached at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='8 × 1014 (at ~ 450 nm) and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='4 × 1014 Hz1/2/W (at 380 nm) for bare ZnO NRs and TiO2 coated ZnO NRs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' It is clearly seen that in case of TiO2 coated ZnO NRs shows enhancement in responsivity and detectivity both as compared to bare ZnO NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Figure 7: (a)-(b) Schematic of measurement setup for as prepared composite NRs using semiconducting oxides on ITO glass substrate (mechanism of producing photocurrent with applied field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Current-voltage characteristic of (c) ZnO NRs and (d) ZnO@TiO2 NRs in presence of UV light, while insets of figures (c) & (d) show enlarged views of I-V characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' UV-Vis UV-Vis AMONRS ZhO NRS40 FUVON (i) (c) 300 +-UVON (i) (d) 30 AtV=o 200 At V=o 20 lwow*2μA lwow*17 pA urrent (μA) 10 UVON A 100 UVON UVOFF 10 geV UVOFF 0- (i) UWOFF (ii) 10- At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='V-O 100- AtVO 20 200- 30- 300 10 5 5 10 10 5 0 5 10 Voltage (V) Voltage (V) Figure 8: Responsivity and Detectivity of (a) bare ZnO and (b) TiO2 coated ZnO NRs at different wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' In order to probe the stability of our devices, UV photocurrents of bare ZnO NRs and ZnO@TiO2 NRs, at a constant bias voltage of 5 V, were measured under ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' For the purpose, the photons of wave lengths in the range 250nm - 750nm were alternatively switched ‘‘ON’’ and ‘‘OFF’’ for 10 minute each, and the results of our measurements are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 9 (a) and 9(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' This growth and decay of current can be used to sense UV as well as visible wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Interestingly, it is found that in case of visible light, photo current response is very low while in case of UV light photocurrent response increases ten times for prepared device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Cycles are similar and repeatable in nature, thus, confirming the reliability of both types of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Increase in the photocurrent under UV light is due to the promotion of a large number of electrons from the valence band to the conduction band because the photon energy exceeds the band gaps of our prepared devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' However, photons of larger wave lengths, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=', those in the visible range, do not have sufficient energy to transfer the electrons from the valence to the conduction band, leading to considerable drop in the number of charge carriers, and, thus, the photocurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The photocurrent in the visible region is mainly because of the charge carriers generated due to native defects and impurities in the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Native defects such vacancies, self-interstitials, and anti-sites are inescapable during the synthesis of crystal lattices and have a significant impact on the performance of semiconducting oxide-based devices [42,44-47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Significant changes in device performance, such as electronic bandgap, photocurrent, response time, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=', can be seen with a nominal change in defect concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' This photocurrent detected in the visible area may be caused by the intrinsic oxygen deficiency found mostly in semiconducting oxides-based devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 16(a)ZnONRs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='0 Responsivity (b)ZnO@TiO2NRs W) --Responsivity Detectivity 100 e-Detectivity (W) 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='6 Responsivity (A/W) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='2 75 A 8- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='8 50 3 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='4 4- 25 2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='0 200 300400 500 700 800 0- 600 200 300 400 500 600 700 800 Wavelength (nm) Wavelength(nm) FIGURE 9: Photocurrent response of (a) ZnO NRs and (b) ZnO@TiO2 NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' In both case photocurrent response is maximum with UV light, while photocurrent decreases with increase in wavelength (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' for Visible light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' From Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 7, 8 and 9 it is obvious that ZnO@TiO2 NRs has a significantly larger photocurrent response as compared to bare ZnO NRs, with a many-fold enhancement of current in the UV region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' In order to systematically investigate the photo-response of the two types of devices as a function of the wavelength of the incident light, we performed measurements of the photocurrents of as prepared bare ZnO and ZnO@TiO2 NRs by varying the wavelength and the results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' From the figure it is obvious that: (a) in the entire region of wavelength probed in our experiment, the photocurrents measured in ZnO@TiO2 NRs based devices is larger than that of ZnO NR based devices, and (b) the difference in two responses becomes large with the decreasing wavelengths, and for the shortest wavelength the photocurrent in the ZnO@TiO2 NRs devices (53 μA) is almost five times of that in the bare ZnO devices (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='24 μA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Hence it is clearly seen that with coating of TiO2 on ZnO NRs not only increases the photocurrent but also increases photosensitivity and switching stability of prepared device as compare to bare ZnO NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='24 OFF OFF 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='82 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='22 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='20 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='63 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='64 ON 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='87 OFE 445nm 445 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='62 (μA) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='60 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='66 兰 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='76 wu 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='20 OFF 545 [LA) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='75 OFF 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='74 Current 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='09 /is-Light 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='89- wu 645nm 645 OFF 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='88 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='87 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content="54 wu ZnO@TiO2NR's ZnONR's wu 745 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='51 745 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='50 OFF 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='40 OFF 0 500 1000 1500 2000 0 500 1000 1500 Time (s) 2000 Time (s)Photocurrent in the visible region is mainly because of the charge carriers generated due to native defects and impurities in the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' From the figure it is obvious that: (a) in the entire region of wavelength probed in our experiment, the photocurrents measured in ZnO@TiO2 NRs based devices is larger than that of ZnO NR based devices, and (b) the difference in two responses becomes large with the decreasing wavelengths, and for the shortest wavelength the photocurrent in the ZnO@TiO2 NRs devices (53 μA) is almost five times of that in the bare ZnO devices (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='24 μA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Hence it is clearly seen that with coating of TiO2 on ZnO NRs not only increases the photocurrent but also increases photosensitivity and switching stability of prepared device as compare to bare ZnO NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' FIGURE 10: Current vs wavelength plot shows photocurrent decreases with increase in the wavelength of the incident light on bare ZnO NRs and ZnO@TiO2 NRs with applied bias of 1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' ZnO@TiO2 shows significantly larger photocurrent response as compared to bare ZnO NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' 60-Applied Bias=1V ZnO@TiO,NRs ZnONRs 50 40 30 20 10- 0 300 400 500 600 700 800 Wavelenght (nm)Moreover, a comparison of the performance metrics of ZnO-based photodetectors [48-50] has been summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Therefore, we believe that ZnO NRs coated with TiO2 appear as one of the highly-sensitive self-powered ultraviolet photo detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Table 1: Comparison of performance against other ZnO@TiO2-based photodetectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Photo- detectors R (mA/W) D (Hz1/2/W) Response Time ON/OFF Ratio Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' TiO2/ZnO 514 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='2 x 109 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='7/12 s 305 [48] ZnO/TiO2 540 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='1 x 1010 9/20 s 388 [48] Y‑ZnO/TiO2 NWs-Au 27 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='2 x 1010 30 s 1786 [49] ZnO–TiO2/Si ̴ 4500 1122 [50] ZnO NRs ZnO@TiO2 15 90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='8 x 1014 < 30 s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='4 x 1014 < 10s 375 416 This Work Conclusion In summary, photosensitive devices based on bare ZnO NRs, as well as those coated with TiO2 (ZnO@TiO2), were successfully fabricated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' SEM analysis and X-ray diffraction confirm the morphology and structural stability of as-prepared NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' The optical band gap was measured and a comparative blue shift was observed in ZnO@TiO2 NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' First-principles DFT calculations at the GGA+U level of theory were performed to understand the geometry, electronic structure, and optical properties of both bare as well TiO2 coated NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Our DFT calculations, coupled with suitable parameters, correctly predict the slight blue-shift in the band gap for the ZnO@TiO2 NRs, as compared to the bare ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' In addition to the band-gap shift, strong enhancements in the photoconductivity were observed ZnO@TiO2, as compared to bare ZnO NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' With the UV light falling on the sample (“ON” state), there is a significant increase in photocurrent, while in the case of no incident UV light (“OFF” state), the dark current (Idark), decreases in nanocomposites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Responsivity and detectivity of TiO2 coated ZnO NRs based device found maximum in UV region than bare ZnO NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' ZnO@TiO2 NRs show significant growth and decay in photocurrent for different wavelengths, leading to their increased photo-detection sensitivity and switching stability in the UV region, as compared to the bare ZnO NRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Therefore, we believe that ZnO NRs coated with TiO2 will prove to be very useful in fabrication of highly-sensitive self-powered ultraviolet photo detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Acknowledgments One of the authors (SP) acknowledges the Homi Bhabha Research Cum Teaching Fellowship (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' ), Lucknow, India for providing financial support Through Teaching Assistantship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Authors sincerely thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Tejendra Dixit (Assistant Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' IIIT kancheepuram) for his help during synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' One of the authors (SP) likes to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Shivendra Pandey (Assistant Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' NIT Silchar) for optical characterizations and his valuable suggestions in manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Compliance with ethical standards: Conflict of interest: The authors declare that they do not have any conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Data availability statement – The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQf2vk-/content/2301.00755v1.pdf'} +page_content=' Monroy, F.' metadata={'source': 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+Cornell University +e-mail: gs18@cornell.edu +Abstract: We investigate how large deviations events cluster in the frame- +work of an infinite moving average process with light-tailed noise and long +memory. The long memory makes clusters larger, and the asymptotic be- +haviour of the size of the cluster turns out to be described by the first +hitting time of a randomly shifted fractional Brownian motion with drift. +AMS 2000 subject classifications: Primary 60F10. +Keywords and phrases: large deviations, clustering, infinite moving av- +erage, long memory. +1. Introduction +We consider an infinite moving average process of the form +Xn = +∞ +� +i=0 +aiZn−i , n ≥ 0 , +(1.1) +where the noise variables (Zn : n ∈ Z) are assumed to bef i.i.d. non-degenerate +random variables. The noise distribution FZ is assumed have finite exponential +moments: +� +R +etz FZ(dz) < ∞ for all t ∈ R . +(1.2) +Furthermore, assuming that the noise is centred: +� +R +z FZ(dz) = 0 , +(1.3) +∗∗The corresponding author. Research partially supported by NSF grant DMS-2015242 +at Cornell University. Part of this work was performed when GS was visiting Department +of Mathematics of National University of Singapore, whose hospitality is gratefully acknowl- +edged. +1 + +Chakrabarty and Samorodnitsky/Clustering of large deviations +2 +the series defining the process in (1.1) converges if and only if the coefficients +a0, a1, a2 . . . satisfy +∞ +� +j=0 +a2 +j < ∞ . +(1.4) +In this case (Xn) is a zero mean stationary ergodic process. For ε > 0 we consider +the sequence of large deviation events +Ej(n, ε) = + + + +1 +n +n+j−1 +� +i=j +Xi ≥ ε + + + , j ≥ 0. +(1.5) +By stationarity, each event Ej(n, ε) is equally rare, and we are interested in the +cluster of these events that occur given that the event E0(n, ε) occurs. +In Chakrabarty and Samorodnitsky (2022) the short memory case was con- +sidered. In this context, “short memory” corresponds to the case +∞ +� +n=0 +|an| < ∞ and +∞ +� +n=0 +an ̸= 0. +(1.6) +In this short memory case the conditional on E0(n, ε) law of the sequence +� +1(Ej(n, ε), j = 1, 2, . . .) converges weakly, as n → ∞, to the law of a sequence +with a.s. finitely many non-zero entries. the total number Dε of the non-zero en- +tries turns out to scale as ε−2, and ε2Dε has an interesting weak limit as ε → 0. +We refer the reader to Chakrabarty and Samorodnitsky (2022) for details, and +a minor technical condition required for the above statements. +In the present paper we are interested in the long memory case. For the mov- +ing average processes (1.1) “long memory” refers to the case when the coeffi- +cients (aj) satisfy the square summability assumption (1.4) but not the absolute +summability assumption in (1.6). A typical assumption in this is +(an) is regularly varying with exponent − α, 1/2 < α < 1; +(1.7) +see Samorodnitsky (2016). It turns out that, in this case (under certain technical +assumptions, an example of which is below), the conditional on E0(n, ε) law of +the sequence +� +1(Ej(n, ε), j = 1, 2, . . .) converges weakly, as n → ∞, to the +degenerate probability measure δ(1,1,...). That is, once the event E0(n, ε) occurs, +the events (Ej(n, ε)) become very likely. In order to understand their structure +we concentrate on the random variables +In(ε) = inf {j ≥ 1 : Ej(n, ε) does not occur} , n ≥ 1 +(1.8) +and establish a weak limit theorem for this sequence under a proper scaling. +Interestingly, the limit turns out to be the law of the first hitting time of a +randomly shifted fractional Brownian motion with drift. +The main result containing the above limit theorem and the technical as- +sumptions it requires are in Section 2. The proof of the main theorem requires a +long sequence of preliminary results, all of which are presented in that section. +Finally, some useful facts needed for the proofs are collected in Section 3. + +Chakrabarty and Samorodnitsky/Clustering of large deviations +3 +2. The assumptions and the main result +Our result on clustering of large deviation events in the long memory case will +require a number of assumptions that we state next. First of all, we will replace +the assumption of regular variation (1.7) by the asymptotic power function +assumption +an ∼ n−α, 1/2 < α < 1, and is eventually monotone. +(2.1) +There is no doubt that the results of the paper hold under the more general +regular variation assumption as well. The extra generality will, however, require +making an already highly technical argument even more so. The potentially +resulting lack of clarity makes the added generality less valuable. The same is +true about the eventual monotonicity assumption. +We will need additional assumptions on the distribution of the noise variables. +We will assume that some θ0 > 0, +sup +|θ|≤θ0 +� ∞ +−∞ +t2 +���� +� ∞ +−∞ +e(it+θ)z FZ(dz) +���� dt < ∞ . +(2.2) +Next, let +σ2 +Z = +� +R +z2 FZ(dz) +(2.3) +be the variance of the noise. Denote +κ = the smallest integer > 4α − 1 +2 − 2α. +(2.4) +In other words, κ = +� +(1+2α)/(2−2α) +� +. We assume that a generic noise variable +Z satisfies +EZi = EGi for 1 ≤ i ≤ κ, +(2.5) +where G ∼ N(0, σ2 +Z). +Remark 2.1. It is standard to verify that (2.2) implies that the noise distri- +bution has a twice continuously differentiable density fZ. One the other hand, +a sufficient condition for (2.2) is that the noise distribution has a four times +continuously differentiable density fZ such that +� ∞ +−∞ +eθ0|x| +���� +di +dxi fZ(x) +���� dx < ∞ for i = 1, 2, 3, 4. +The moment equality assumption (2.5) restricts how far the the noise distri- +bution can be from a normal distribution. Note that in the range 1/2 < α < 5/8 +we have κ = 2, in which case the assumption is vacuous. Since κ ≥ 2 for all +α ∈ (1/2, 1), (1.3) is implied by (2.5). + +Chakrabarty and Samorodnitsky/Clustering of large deviations +4 +To state our main result, we need to introduce several key quantities. Let +β = 4 − 4α +3 − 2α ∈ (0, 1) +(2.6) +and +H = 3/2 − α ∈ (1/2, 1). +(2.7) +We denote by (BH(t) : t ≥ 0) the standard fractional Brownian motion with +Hurst index H, i.e. a zero mean Gaussian process with continuous paths and +covariance function +E (BH(s)BH(t)) = 1 +2 +� +s2H + t2H − |s − t|2H� +, s, t ≥ 0 . +(2.8) +If T0 is a standard exponential random variable independent of the fractional +Brownian motion, then +τε = inf +� +t ≥ 0 : BH(t) ≤ (2Cα)−1/2εt2H − (Cα/2)1/2σ2 +Zε−1T0 +� +, ε > 0, (2.9) +is an a.s. finite and strictly positive random variable. Here σ2 +Z is the variance of +the noise in (2.3) and +Cα = B(1 − α, 2α − 1) +(1 − α)(3 − 2α) , +(2.10) +with B(·, ·) the standard Beta function. +We are now in a position to state the main result of this paper. +Theorem 2.1. Assume the finite exponential moment condition (1.2), that the +coefficients satisfy the power-type condition (2.1), the regularity condition (2.2) +and the moment equality condition (2.5). Then for every ε > 0 the first non- +occurrence times (1.8) satisfy +P +� +n−βIn(ε) ∈ · +��E0(n, ε) +� +⇒ P (τε ∈ ·) , n → ∞ . +(2.11) +Remark 2.2. It is worthwhile to observe that the limit law obtained in Theorem +2.1 depends on the noise distribution only through its variance σ2 +Z. This can be +understood by noticing that in the long memory case considered in this paper +we have Var(X1 + · · · + Xn) ≫ n; see Lemma 2.1 below. Therefore, the events +Ej(n, ε) should be viewed as moderate deviation events, not large deviation +events. It has been observed in many situations that moderate deviation events +are influenced by the Gaussian weak limit of the quantities of interest. At the +intuitive level, this explains why it is the variance of the process that appears +in the limit. +For comparison, in the short memory case (1.6), we have Var(X1+· · ·+Xn) ∼ +cn for some c > 0, the events Ej(n, ε) should be viewed as large deviation events, +and their behaviour depends on much more than just the variance of the noise. +See Chakrabarty and Samorodnitsky (2022) for details. + +Chakrabarty and Samorodnitsky/Clustering of large deviations +5 +We start on the road to proving Theorem 2.1 by establishing certain basic +estimates that will be used throughout the paper. Denote +Aj = +j +� +i=0 +ai, j ∈ Z , +(2.12) +with the convention that a sum (or an integral) is zero if the lower limit exceeds +the upper limit (so that Aj = 0 for j ≤ −1, for example). Let +Sn = +n−1 +� +i=0 +Xi, n ≥ 1 , +(2.13) +and denote +σ2 +n = Var(Sn), n ≥ 1 . +(2.14) +In the sequel we use the following notation. We will denote by +ϕZ(t) = log +�� +R +etz FZ(dz) +� +, t ∈ R +(2.15) +the log-Laplace transform of a noise variable. We will frequently use the obvious +facts +ϕ is convex and ϕZ(x) ∼ x2σ2 +Z/2, x → 0, +(2.16) +and +ϕ′ +Z is continuous, nondecreasing and ϕ′ +Z(x) = xσ2 +Z + O(x2), x → 0. +(2.17) +We will write Gθ for the probability measure obtained by exponentially tilting +the law FZ by θ ∈ R. That is, +Gθ(dz) = +� +EeθZ�−1eθzFZ(dz). +(2.18) +It is clear that, as θ → 0, +� +R +z Gθ(dz) ∼ θσ2 +Z, +���� +� +R +z Gθ(dz) − θσ2 +Z +���� = O(θ2) and = O(|θ|3) if κ ≥ 3, +(2.19) +� +R +|z|k Gθ(dz) → +� +R +|z|k F(dz), k = 1, 2, . . .. +Lemma 2.1. Asymptotically we have +Aj ∼ (1 − α)−1j1−α, j → ∞ +(2.20) +and +σ2 +n ∼ Cασ2 +Zn3−2α, n → ∞ . +(2.21) + +Chakrabarty and Samorodnitsky/Clustering of large deviations +6 +Furthermore, for any t > 0, as n → ∞, +[nβt] +� +i=0 +(Ai − Ai−n)2 ∼ K1t3−2αn4−4α , +(2.22) +and +n +� +i=n−[nβt]+1 +(Ai − Ai−n)2 ∼ +n+[nβt] +� +i=n+1 +(Ai − Ai−n)2 ∼ (1 − α)−2n2−2α+βt , (2.23) +with +K1 = (1 − α)−2(3 − 2α)−1 . +(2.24) +Finally, for any t > 0, as n → ∞, +σ2 +Z +σ2n +∞ +� +i=0 +(Ai − Ai−n) +� +Ai+[nβt] − Ai+[nβt]−n +� += 1−n1−2αt3−2α(1+o(1)) . (2.25) +Proof. The claim (2.20) is, of course, an immediate consequence of the assump- +tion (2.1). For (2.21), first note that +Rn = Cov(X0, Xn) ∼ σ2 +Z +∞ +� +j=1 +j−α(j + n)−α +∼ n1−2ασ2 +Z +� ∞ +0 +x−α(1 + x)−α dx += Cασ2 +Z(1 − α)(3 − 2α)n1−2α +as n → ∞. Therefore, +σ2 +n = +n−1 +� +i=−(n−1) +(n − |i|)R|i| ∼ 2Cασ2 +Z(1 − α)(3 − 2α) +n−1 +� +i=0 +(n − i)i1−2α +∼ 2Cασ2 +Z(1 − α)(3 − 2α)n3−2α +� 1 +0 +(1 − x)x1−2α dx = Cασ2 +Zn3−2α , +which is (2.21). Next, for a fixed t > 0 and large n, by (2.20) and the fact that +β < 1, +[nβt] +� +i=0 +(Ai − Ai−n)2 = +[nβt] +� +i=0 +A2 +i ∼ (1 − α)−2 +[nβt] +� +i=1 +i2−2α ∼ K1 +� +nβt +�3−2α , +proving (2.22). Similarly, +n +� +i=n−[nβt]+1 +(Ai − Ai−n)2 ∼ +n +� +i=n−[nβt]+1 +A2 +n ∼ (1 − α)−2nβ+2−2αt , + +Chakrabarty and Samorodnitsky/Clustering of large deviations +7 +showing the first equivalence in (2.23) and the second equivalence can be shown +in the same way. +For (2.25), we start by writing +Sn = +∞ +� +j=0 +(Aj − Aj−n)Zn−1−j , n ≥ 1 , +(2.26) +so that +σ2 +n = σ2 +Z +∞ +� +j=0 +(Aj − Aj−n)2 , n ≥ 1 . +(2.27) +Therefore, for large n, +σ2 +n +σ2 +Z +− +∞ +� +i=0 +(Ai − Ai−n)(Ai+[nβt] − Ai+[nβt]−n) += 1 +2 + + +[nβt]−1 +� +i=0 +(Ai − Ai−n)2 + +∞ +� +i=0 +� +Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n +�2 + + += 1 +2 +�n−1 +� +i=0 +� +Ai − Ai−[nβt] +�2 +(2.28) ++ +∞ +� +i=n−[nβt] +� +Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n +�2 +� +. +By (2.20), +n−1 +� +i=0 +� +Ai − Ai−[nβt] +�2 ∼ (1 − α)−2 +n−1 +� +i=1 +� +i1−α − (i − [nβt])1−α ++ +�2 +∼ n4−4αt3−2α(1 − α)−2 +� ∞ +0 +� +y1−α − (y − 1)1−α ++ +�2 dy +as n → ∞. By (3.1) with H = 3/2 − α, +� ∞ +0 +� +y1−α − (y − 1)1−α ++ +�2 dy +(2.29) += [(3 − 2α) (1 − α)]−1 sin(πα) +π +Γ(2α − 1)Γ(2 − α)2 += 1 − α +3 − 2αB (2α − 1, 1 − α) = (1 − α)2Cα, +so +n−1 +� +i=0 +� +Ai − Ai−[nβt] +�2 ∼ Cαt3−2αn4−4α, n → ∞ . +(2.30) + +Chakrabarty and Samorodnitsky/Clustering of large deviations +8 +Since +∞ +� +i=n +� +Ai − Ai−[nβt] +�2 = O +� +n2β +∞ +� +i=n +i−2α +� += O +� +n2β+1−2α� += o +� +n4−4α� +, (2.31) +we conclude also that +∞ +� +i=0 +� +Ai − Ai−[nβt] +�2 ∼ Cαt3−2αn4−4α, n → ∞ . +(2.32) +It follows from (2.31) and (2.32) that +∞ +� +i=n−[nβt] +� +Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n +�2 += +∞ +� +j=0 +� +−Aj + Aj−[nβt] + +� +Aj+n − Aj+n−[nβt] +��2 ∼ Cαt3−2αn4−4α . +In combination with (2.28) and (2.30) we obtain +σ2 +n +σ2 +Z +− +∞ +� +i=0 +(Ai − Ai−n)(Ai+[nβt]Ai+[nβt]−n) ∼ Cαt3−2αn4−4α . +Dividing both sides by σ−2 +Z σ2 +n and appealing to (2.21), (2.25) follows. +We now consider certain large deviations of the partial sum Sn under a change +of measure. With an eye towards a subsequent application, we allow the partial +sum, given in the form (2.26), to be “corrupted”. For n ≥ 1 and t ≥ 0 we define +ξ1 +n(t) = +[nβt] +� +i=1 +(Ai − Ai−n) Zn−i−1 , +(2.33) +ξ2 +n(t) = +n−1 +� +i=n−[nβt] +(Ai − Ai−n) Zn−i−1 , +(2.34) +ξ3 +n(t) = +n+[nβt] +� +i=n+1 +(Ai − Ai−n) Zn−i−1 . +(2.35) +Lemma 2.2. Fix t1, t2, t3 > 0 and denote +¯Sn = Sn − +3 +� +i=1 +ξi +n(ti), n ≥ 1 . +(2.36) +Let (γn), (θn) and (ηn) be real sequences satisfying +γn = o +� +n3/2−α� +, θn = o +� +n−(1−α)� +, 1 ≪ ηn ≪ n1/2 . + +Chakrabarty and Samorodnitsky/Clustering of large deviations +9 +If ˜Sn is a random variable with the law +P +� +˜Sn ∈ dx +� += +� +E(eθn ¯Sn) +�−1 +eθnxP +� ¯Sn ∈ dx +� +, n ≥ 1 , +(2.37) +then for all x ∈ R and h > 0, +P +� +ηnσ−1 +n +� +˜Sn − E( ˜Sn) + γn +� +∈ [x, x + h] +� +∼ η−1 +n (2π)−1/2h, n → ∞. +(2.38) +Furthermore, +sup +n≥1, x∈R +ηnP +� +ηnσ−1 +n ˜Sn ∈ [x, x + 1] +� +< ∞ . +(2.39) +Proof. Let ( ˜Zni, n ≥ 1, i ≥ 0) be a collection of independent random variables +such that the law of ˜Zni is G(Ai−Ai−n)θn in the notation of (2.18). Then for +large n, +˜Sn +d= A0 ˜Zn0 + (An − A0) ˜Znn + +n−[nβt2]−1 +� +i=[nβt1]+1 +Ai ˜Zni + +∞ +� +i=n+[nβt3]+1 +(Ai − Ai−n) ˜Zni . +(2.40) +The proof applies to (2.40) the bound (3.2) in the appendix, with n = ∞. +For any z ∈ R +����P +� +˜Sn − E( ˜Sn) ≤ z +� +Var( ˜Sn) +� +− Φ(z) +���� +(2.41) +≤ Cu +� +Var( ˜Sn) +�−3/2 ∞ +� +i=0 +|Ai − Ai−n|3E +� +| ˜Zni − E ˜Zni|3� +, n ≥ 1 . +It is immediate from (2.1) that +sup +i≥0 +|Ai − Ai−n| = O(n1−α) , +(2.42) +so that +lim +n→∞ θn sup +i≥0 +|Ai − Ai−n| = 0 . +It follows from (2.19) that +E ˜Zni → 0, +Var( ˜Zni) → σ2 +Z, +E +� +| ˜Zni − E ˜Zni|3� +→ +� ∞ +−∞ +|z3| FZ(dz) +(2.43) +uniformly in i as n → ∞. Since it is an elementary conclusion from Lemma 2.1 +that for any κ > 1/α, +∞ +� +i=0 +|Ai − Ai−n|κ = O +� +nκ+1−κα� +, +(2.44) + +Chakrabarty and Samorodnitsky/Clustering of large deviations +10 +it follows from (2.41) that +sup +z∈R +����P +� +˜Sn − E( ˜Sn) ≤ z +� +Var( ˜Sn) +� +− Φ(z) +���� += O +� +n4−3α � +Var( ˜Sn) +�−3/2� +. +Using (2.43) again we see that +Var( ˜Sn) ∼ σ2 +n − +3 +� +i=1 +Var +� +ξi +n(ti) +� +∼ Cασ2 +Zn3−2α, +(2.45) +with the second equivalence following from various claims in Lemma 2.1. Thus, +sup +z∈R +����P +� +˜Sn − E( ˜Sn) ≤ z +� +Var( ˜Sn) +� +− Φ(z) +���� = O(n−1/2) = o +� +η−1 +n +� +. +(2.46) +Therefore, for x ∈ R and h > 0, as n → ∞, +P +� +ηnσ−1 +n +� +˜Sn − E( ˜Sn) + γn +� +∈ [x, x + h] +� += o +� +η−1 +n +� ++ +� +R +1 +� +Var( ˜Sn)−1/2(xη−1 +n σn − γn) ≤ z +≤ Var( ˜Sn)−1/2((x + h)η−1 +n σn − γn) +� +φ(z) dz, +where φ is the standard normal density. The assumptions on γn and ηn along +with (2.45) imply that the integration interval shrinks towards the origin. Thus, +the integral above is asymptotically equivalent to η−1 +n φ(0)h, and (2.38) follows. +Boundedness of φ in the above integral establishes (2.39). +We now look more closely at the processes defined in (2.33), (2.34) and (2.35). +The next lemma describes the limiting distribution of their increments under +the same change of measure as in the previous lemma. +Lemma 2.3. Suppose that θn ∈ R satisfies θn = o +� +n−(1−α)� +. Fix 0 ≤ s < t +and consider random variables with the laws +P(Uni ∈ dx) = cnieθnxP +� +ξi +n(t) − ξi +n(s) ∈ dx +� +, i = 1, 2, 3, n ≥ 1 , +with appropriate cni. Then, as n → ∞, +n−(2−2α) (Un1 − E(Un1)) ⇒ N +� +0, K1σ2 +Z +� +t3−2α − s3−2α�� +, +(2.47) +where K1 is given in (2.24), and for i = 2, 3, +n−(1−α+β/2) (Uni − E(Uni)) ⇒ N +� +0, (1 − α)−2σ2 +Z(t − s) +� +. +(2.48) + +Chakrabarty and Samorodnitsky/Clustering of large deviations +11 +Proof. For large n, +Un1 +d= +[nβt] +� +i=[nβs]+1 +Ai ˜Zni +with ( ˜Zni) as in the previous lemma. That is, Un1 − E(Un1) is the sum of +independent zero mean random variables. By (2.43) and (2.22), +Var(Un1) ∼ σ2 +Z +[nβt] +� +i=[nβs]+1 +A2 +i ∼ K1σ2 +Zn4−4α � +t3−2α − s3−2α� +, +and a similar calculation using the third moment bound in (2.43) verifies the +Lindeberg conditions of the central limit theorem. Hence (2.47) follows, and the +calculations for (2.48) are similar. +Consider the overshoot defined by +T ∗ +n = Sn − nε, n ≥ 1 . +(2.49) +Conditionally on the event E0 = E0(n, ε) in (1.5) the overshoot is nonnegative. +The next lemma is a joint weak limit theorem for the joint law of the overshoot +and the processes defined in (2.33), (2.34) and (2.35). The joint law is computed +conditionally on E0. +Lemma 2.4. Let +ζn = nε/σ2 +n, n ≥ 1 , +(2.50) +Conditionally on E0, as n → ∞, +� +ζnT ∗ +n, + +n2α−2 + +ξ1 +n(t) − +[nβt] +� +i=1 +Ai +� ∞ +−∞ +z GζnAi(dz) + + , t ≥ 0 + + , +� +nα−β/2−1 + +ξ2 +n(t) − +n−1 +� +i=n−[nβt] +Ai +� ∞ +−∞ +z GζnAi(dz) + + , t ≥ 0 +� +, +� +nα−β/2−1 + +ξ3 +n(t) − +n+[nβt] +� +i=n+1 +(Ai − Ai−n) +� ∞ +−∞ +z Gζn(Ai−Ai−n)(dz) + + , t ≥ 0 +�� +⇒ +� +T0, +� +K1/2 +1 +σZB1(t3−2α), t ≥ 0 +� +, +� +(1 − α)−1σZB2(t), t ≥ 0 +� +, +� +(1 − α)−1σZB3(t), t ≥ 0 +�� +, +in finite dimensional distributions, where T0 is a standard exponential random +variable independent of independent standard Brownian motions B1, B2, and +B3, K1 is the constant in (2.24) and Gθ is the exponentially tilted law in (2.18). + +Chakrabarty and Samorodnitsky/Clustering of large deviations +12 +Proof. Denote +ψn(s) = σ2 +n +n2 log E +� +exp +� +s n +σ2n +Sn +�� += σ2 +n +n2 +∞ +� +j=0 +ϕZ +� +σ−2 +n n(Aj − Aj−n)s +� +, (2.51) +where the second equality follows from (2.26). By (2.16), (2.21) and (2.42) we +see that +lim +n→∞ ψn(s) = s2/2 +(2.52) +uniformly for s in a compact set. Furthermore, the sum in (2.51) can be differ- +entiated term by term, and it follows by (2.17), (2.21) and (2.42) that +lim +n→∞ ψ′ +n(s) = s, +(2.53) +also uniformly on compact sets. Since ψ′ +n is increasing and continuous, for large +n there exists a unique τn > 0 such that +ψ′ +n(τn) = ε . +(2.54) +It is immediate that τn → ε as n → ∞. Denoting +θn = σ−2 +n nτn, n ≥ 1 , +(2.55) +we have +� +E +� +eθnSn��−1 E +� +SneθnSn� += nε. +(2.56) +Fix k ≥ 1 and for each i = 1, 2, 3 fix points 0 = ti0 < ti1 < . . . < tik. Denote +¯Sn = Sn − +3 +� +i=1 +ξi +n(tik), n ≥ 1 . +Let Unij, n ≥ 1, i = 1, 2, 3, j = 1, . . . , k, ˜Sn, n ≥ 1 be independent random +variables, with +P (Unij ∈ dx) += +� +E +� +eθn(ξi +n(tij)−ξi +n(ti j−1))��−1 +eθnxP +� +ξi +n(tij) − ξi +n(ti j−1) ∈ dx +� +, +and +P +� +˜Sn ∈ dx +� += +� +E +� +eθn ¯Sn��−1 +eθnxP +� ¯Sn ∈ dx +� +for n ≥ 1, i = 1, 2, 3 and j = 1, . . . , k. Let +µnij = E (Unij) , µn = E( ˜Sn). +(2.57) + +Chakrabarty and Samorodnitsky/Clustering of large deviations +13 +It follows from (2.56) that +µn + +3 +� +i=1 +k +� +j=1 +µnij = nε, n ≥ 1 . +(2.58) +Let t > 0 and (αij) ⊂ R. We have +P +�� +T ∗ +n > tσ2 +n/nε +� +∩ +� k� +j=1 +� +n2α−2 � +ξ1 +n(t1j) − ξ1 +n(t1 j−1) − µn1j +� +> α1j +�� +∩ +� +� +2≤i≤3, 1≤j≤k +� +nα−β/2−1 � +ξi +n(tij) − ξi +n(ti j−1) − µnij +� +> αij +��� += +� +R3k+1 1 +� +x > nε + tσ2 +n/nε − +3 +� +i=1 +k +� +j=1 +sij +� +1 +� +s1j > n2−2αα1j + µn1j , 1 ≤ j ≤ k +� +1 +� +sij > n1−α+β/2αij + µnij , i = 2, 3 , j = 1, . . . , k +� +P( ¯Sn ∈ dx) +3 +� +i=1 +k +� +j=1 +P +� +ξi +n(tij) − ξi +n(ti j−1) ∈ dsij +� += +� +R3k+1 1 +� +x > nε + tσ2 +n/nε − +3 +� +i=1 +k +� +j=1 +sij +� +1 +� +s1j > n2−2αα1j + µn1j , 1 ≤ j ≤ k +� +1 +� +sij > n1−α+β/2αij + µnij , i = 2, 3 , 1 ≤ j ≤ k +� +exp +� +−θnx − θn +3 +� +i=1 +k +� +j=1 +sij +� +P +� +˜Sn ∈ dx +� +E +� +eθnSn� +3 +� +i=1 +k +� +j=1 +P(Unij ∈ dsij) += cn +� +R3k 1 +� +min +i,j (uij − αij) > 0 +� +k +� +j=1 +P +� +n2α−2� +Un1j − µn1j +� +∈ du1j +� +3 +� +i=2 +k +� +j=1 +P +� +nα−β/2−1� +Unij − µnij +� +∈ duij +� +� +R +e−z1 +� +z > tθnσ2 +n/nε +� +P +� +θn +� ˜Sn − µn + γn(u11, . . . , u3k) +� +∈ dz +� +, + +Chakrabarty and Samorodnitsky/Clustering of large deviations +14 +with +cn = e−θnnεE +� +eθnSn� +(2.59) +and +γn(u11, . . . , u3k) = n2−2α +k +� +j=1 +u1j + n1−α+β/2 +3 +� +i=2 +k +� +j=1 +uij . +Let θn be as above and ηn = σnθn. For n ≥ 1, we introduce the notation +fn(u11, . . . , u3k) +=ηn +� ∞ +0 +e−z1 +� +z > tθnσ2 +n/nε +� +P +� +θn +� ˜Sn − µn + γn(u11, . . . , u3k) +� +∈ dz +� +. +Fix (uij) and let u(n) +ij +→ uij as n → ∞ for all i, j. Let us denote γn = +γn +� +u(n) +11 , . . . , u(n) +3k +� +. With θn and ηn already defined, we use Lemma 2.2 with this +γn. It is elementary to check that the hypothesis of the lemma are satisfied. +Since tθnσ2 +n/nε → t, it follows from (2.38) that for all fixed T > t, +� +R +e−z1 +� +tθnσ2 +n/nε < z ≤ T +� +P +� +θn +� ˜Sn − µn + γn +� +∈ dz +� +∼ η−1 +n (2π)−1/2 +� T +t +e−z dz, +and if follows from (2.39) that +lim +T →∞ lim sup +n→∞ ηn +� +R +e−z1 +� +z > T +� +P +� +θn +� ˜Sn − µn + γn +� +∈ dz +� += 0 , +showing that +lim +n→∞ fn +� +u(n) +11 , . . . , u(n) +3k +� += (2π)−1/2e−t . +Another application of (2.39) implies that +sup +{uij}⊂R +fn(u11, . . . , u3k) < ∞ . +It follows immediately from Lemma 2.3 and bounded convergence theorem that +E +� +f +� +n2α−2(Un11 − µn11), . . . , nα−β/2−1(Un3k − µn3k) +� +(2.60) +1 +� +n2α−2(Un1j − µn1j) > α1j, nα−β/2−1(Unij − µnij) > αij, i = 2, 3, +j = 1, . . . , k, +�� +→(2π)−1/2P (T0 > t , Gij > αij for all i, j) , +with T0 standard exponential and (Gij : i = 1, 2, 3, j = 1, . . . , k) independent +zero mean Gaussian random variables, independent of T0, with +Var(G1j) = K1σ2 +Z +� +t3−2α +1j +− t3−2α +1 j−1 +� +, 1 ≤ j ≤ k , + +Chakrabarty and Samorodnitsky/Clustering of large deviations +15 +and for i = 2, 3, +Var(Gij) = (1 − α)−2σ2 +Z(tij − ti ,j−1), 1 ≤ j ≤ k . +A simple way to verify the convergence above is to appeal to the Skorohod +representation and replace the weak convergence in Lemma 2.3 by the a.s. con- +vergence. +Notice that using (2.60) with t = 0 and αij = −∞ for all i, j tells us that +P(E0) ∼ (2π)−1/2cn/ηn = (2π)−1/2e−θnnεE +� +eθnSn� +/(σnθn). +(2.61) +Dividing (2.60) by (2.61) gives us the statement of the lemma apart from a +possibly different centring. In order to complete the proof, it suffices to show +that as n → ∞, for j = 1, . . . , k, +µn1j = +[nβt1j] +� +i=[nβt1j−1]+1 +Ai +� ∞ +−∞ +z GζnAi(dz) + o +� +n2−2α� +, +(2.62) +µn2j = +n−[nβtnj−1] +� +i=n−[nβtnj] +Ai +� ∞ +−∞ +z GζnAi(dz) + o +� +n1+β/2−α� +, +(2.63) +µn3j = +n+[nβtnj] +� +i=n+[nβtnj−1] +(Ai − Ai−n) +� ∞ +−∞ +z Gζn(Ai−Ai−n)(dz) + o +� +n1+β/2−α� +. +(2.64) +For simplicity of notation we prove these statements for j = 1. For θn as in +(2.55), let ( ˜Zni, n ≥ 1, i ≥ 0) be a collection of independent random variables +such that the law of ˜Zni is G(Ai−Ai−n)θn. Since both θnAi and ζnAi converge +to zero uniformly in i ≤ nβt11, we can use (2.19) to write +µn11 = +[nβt11] +� +i=1 +AiE +� +˜Zni +� += +[nβt11] +� +i=1 +Ai +� ∞ +−∞ +z GθnAi(dz) += +[nβt11] +� +i=1 +Ai +� ∞ +−∞ +z GζnAi(dz) + o + +ζn +[nβt11] +� +i=1 +A2 +i + + . +It follows from (2.21) and (2.22) that +ζn +[nβt11] +� +i=1 +A2 +i = o +� +n2−2α� +, +and we obtain (2.62) (for j = 1). +For (2.63) with j = 1 we notice that by (2.17), +E +� +˜Zni +� += θn(Ai − Ai−n)σ2 +Z + O +� +θ2 +n(Ai − Ai−n)2� +, +(2.65) + +Chakrabarty and Samorodnitsky/Clustering of large deviations +16 +uniformly in i ≥ 0, as n → ∞. Thus, +µn21 = σ2 +Zσ−2 +n nτn +n−1 +� +i=n−[nβt21] +A2 +i + O + +θ2 +n +n−1 +� +i=n−[nβt21] +A3 +i + + . +It follows from Lemma 2.1 that +θ2 +n +n−1 +� +i=n−[nβt21] +A3 +i = O +� +nα+β−1� += o +� +n1−α+β/2� +. +Therefore, +µn21 = σ2 +Zσ−2 +n nτn +n−1 +� +i=n−[nβt21] +A2 +i + o +� +n1−α+β/2� +(2.66) +and, similarly, +n−1 +� +i=n−[nβt21] +Ai +� ∞ +−∞ +z GζnAi(dz) = σ2 +Zζn +n−1 +� +i=n−[nβt21] +A2 +i + o +� +n1−α+β/2� +. +Another appeal to Lemma 2.1 shows that for (2.63) we only need to argue that +τn = ε + o +� +n1−α−β/2� +, n → ∞ . +(2.67) +However, by (2.19), +ψ′ +n(s) = s + O + +nσ−4 +n +∞ +� +j=0 +(Aj − Aj−n)3 + + , +uniformly for s in compact sets. Using this and (2.44), we obtain +ε = ψ′ +n(τn) += τn + O + +nσ−4 +n +∞ +� +j=0 +(Aj − Aj−n)3 + + += τn + O(nα−1) = τn + o +� +n1−α−β/2� +. +This establishes (2.67) and, hence, (2.63) with j = 1. The proof of (2.64) is +similar. +None of the statements proved so far required the additional assumptions +stated at the beginning of this section. These assumptions start to play a role +now. + +Chakrabarty and Samorodnitsky/Clustering of large deviations +17 +The next several lemmas require additional notation designed to focus on the +contribution of individual noise variables on Sn. For n ≥ 1 and i, j ≥ 0, i ̸= j, +we set +S′ +n(i) = Sn − (Ai − Ai−n)Zn−i−1 , +S′ +n(i, j) = Sn − (Ai − Ai−n)Zn−i−1 − (Aj − Aj−n)Zn−j−1 , +and, with ζn given by (2.50), we let ˆSn, ˆSni, ˆSn(i, j) be random variables with +distributions +P( ˆSn ∈ ds) ∝ eζnsP(Sn ∈ ds) , +P( ˆSn(i) ∈ ds) ∝ eζnsP(S′ +n(i) ∈ ds) , +P( ˆSn(i, j) ∈ ds) ∝ eζnsP(S′ +n(i, j) ∈ ds) . +Denote the characteristic functions of σ−1 +n ( ˆSn − nε), σ−1 +n ( ˆSn(i) − nε) and +σ−1 +n ( ˆSn(i, j) − nε) by φn, φni and φnij, respectively. For µ ∈ R and σ ≥ 0 we +denote by φG(µ; σ2; ·) the characteristic function of N(µ, σ2). +Lemma 2.5. Let κ be given by (2.4) and assume that (2.5) holds. Then the +following statements hold uniformly in t ∈ R: +|φn(t) − φG(0; 1; t)| = O +� +n1/2−κ(1−α)(1 + |t|)κ+1� +, +(2.68) +sup +i≥0 +��φni(t) − φG +� +σ−1 +n nε(λni − 1); λni; t +��� = O +� +n1/2−κ(1−α)(1 + |t|)κ+1� +, +(2.69) +sup +i,j≥0 +i̸=j +��φnij(t) − φG +� +σ−1 +n nε(λnij − 1); λnij; t +��� +(2.70) += O +� +n1/2−κ(1−α)(1 + |t|)κ+1� +, +where for n ≥ 1 and i, j ≥ 0, i ̸= j, we set +λni = 1 − σ2 +Z +σ2n +(Ai − Ai−n)2, +λnij = 1 − σ2 +Z +σ2n +� +(Ai − Ai−n)2 + (Aj − Aj−n)2� +. +Proof. It is an elementary conclusion from (2.5) that, for each 1 ≤ i ≤ κ, +�� +R +eδz Fz(dz) +�−1 � +R +zieδz Fz(dz) = σi +ZE +� +(G + δσZ)i� ++ O +� +|δ|κ−i+1� +(2.71) +as δ → 0, where G is a standard Gaussian random variable. +Let ( ˆZni : n ≥ 1, i ≥ 0) be a family of independent random variables with +each ˆZni ∼ G(Ai−Ai−n)ζn, so that for n ≥ 1 and i, j ≥ 0, i ̸= j we have +ˆSn +d= +∞ +� +k=0 +(Ak − Ak−n) ˆZnk , + +Chakrabarty and Samorodnitsky/Clustering of large deviations +18 +ˆSn(i) +d= +� +k∈{0,1,2,...}\{i} +(Ak − Ak−n) ˆZnk , +ˆSn(i, j) +d= +� +k∈{0,1,2,...}\{i,j} +(Ak − Ak−n) ˆZnk . +Let now (Gni : n ≥ 1, i ≥ 0) be a collection of independent random variables, +also independent of ( ˆZni : n ≥ 1, i ≥ 0), where +Gni ∼ N +� +(Ai − Ai−n)ζnσ2 +Z , σ2 +Z +� +, for all n ≥ 1, i ≥ 0 . +It follows from Lemma 2.1 and (2.42) that (2.71) can be reformulated as +E +� +ˆZi +nj +� +− E +� +Gi +nj +� += O +� +|Aj − Aj−n|κ−i+1n−2(1−α)(κ−i+1)� +(2.72) +uniformly in j ≥ 0 and 1 ≤ i ≤ κ. For a fixed t ∈ R we use telescoping to write +������ +E exp + + +i + +tσ−1 +n +∞ +� +j=0 +(Aj − Aj−n)Gnj + + + + + − E exp +� +i +� +tσ−1 +n ˆSn +�� +������ +(2.73) +≤ +∞ +� +j=0 +������ +E exp + + +i + +tσ−1 +n + + +j−1 +� +k=0 +(Aj − Aj−n) ˆZnj + +∞ +� +k=j +(Aj − Aj−n)Gnj + + + + + + + +−E exp + + +i + +tσ−1 +n + + +j +� +k=0 +(Aj − Aj−n) ˆZnj + +∞ +� +k=j+1 +(Aj − Aj−n)Gnj + + + + + + + +������ +. +Fix j ≥ 0 and denote +U = tσ−1 +n + + +j−1 +� +k=0 +(Aj − Aj−n) ˆZnj + +∞ +� +k=j+1 +(Aj − Aj−n)Gnj + + , +V = tσ−1 +n (Aj − Aj−n)Gnj , +so that by expanding in the Taylor series around U, +E exp + + +i + +tσ−1 +n + + +j−1 +� +k=0 +(Aj − Aj−n) ˆZnj + +∞ +� +k=j +(Aj − Aj−n)Gnj + + + + + + + += Eei(U+V ) = +κ +� +m=0 +im +m!E (V m) EeiU + R1 , +with |R1| ≤ E(|V |κ+1)/(κ + 1)!. Similarly, +E exp + + +i + +tσ−1 +n + + +j +� +k=0 +(Aj − Aj−n) ˆZnj + +∞ +� +k=j+1 +(Aj − Aj−n)Gnj + + + + + + + += +κ +� +m=0 +im +m!E (W m) EeiU + R2 , + +Chakrabarty and Samorodnitsky/Clustering of large deviations +19 +with |R2| ≤ E(|W|κ+1)/(κ + 1)!, where +W = (Aj − Aj−n) ˆZnj . +We conclude that +������ +E exp + + +i + +tσ−1 +n + + +j−1 +� +k=0 +(Aj − Aj−n) ˆZnj + +∞ +� +k=j +(Aj − Aj−n)Gnj + + + + + + + +−E exp + + +i + +tσ−1 +n + + +j +� +k=0 +(Aj − Aj−n) ˆZnj + +∞ +� +k=j+1 +(Aj − Aj−n)Gnj + + + + + + + +������ +≤ +κ +� +i=1 +|t|i +i! +���(Aj − Aj−n)iσ−i +n E +� +ˆZi +nj − Gi +nj +���� ++ +|t|κ+1 +(κ + 1)! |Aj − Aj−n|κ+1 σ−(κ+1) +n +E +� +|Gnj|κ+1 + | ˆZnj|κ+1� +. +(2.74) +Note that by (2.44) and Lemma 2.1, +σ−(κ+1) +n +∞ +� +j=0 +|Aj − Aj−n|κ+1 E +� +|Gnj|κ+1 + | ˜Znj|κ+1� += O +� +n−(κ−1)/2� += o +� +n1/2−κ(1−α)� +. +For 1 ≤ i ≤ κ we use, in addition. (2.72) to write +σ−i +n +∞ +� +j=0 +���(Aj − Aj−n)iE +� +˜Zi +nj − Gi +nj +���� += O +� +n−κ(1−α)+α−i(α−1/2)� += O +� +n1/2−κ(1−α)� +. +Putting these bounds into (2.74) we obtain +E +� +eιtσ−1 +n +˜Sn� += φG +� +σ−1 +n nε; 1; t +� ++ O +� +n1/2−κ(1−α) � +1 + |t|κ+1�� +uniformly for t ∈ R, which is equivalent to (2.68). The argument for (2.69) and +(2.70) is the same. +By the assumption (2.2), for large n, the random variables σ−1 +n ( ˆSn − nε), +σ−1 +n ( ˆSn(i) − nε) and σ−1 +n ( ˆSn(i, j) − nε) have densities which we denote by fn, +fni and fnij, correspondingly. +Lemma 2.6. Suppose that (2.5) and(2.2) hold. Then for large n, the densities +fni and fnij are twice differentiable. Furthermore, as n → ∞, +fni(0) = (2π)−1/2 + o +� +n1−2α� +, +(2.75) +f ′ +ni(0) = o +� +n1/2−α� +(2.76) + +Chakrabarty and Samorodnitsky/Clustering of large deviations +20 +uniformly in i, and for some n0 ∈ N, +sup {|f ′′ +ni(x)| : n ≥ n0, i ≥ 0, x ∈ R} < ∞ . +(2.77) +All three statements also hold if fni is replaced by fnij, i < j. Finally, as n → ∞, +sup +x∈R +���fn(x) − (2π)−1/2e−x2/2��� = o +� +n1−2α� +. +(2.78) +Proof. We start with the proof of (2.78) which would follow from the inversion +formula for densities once it is shown that +� ∞ +−∞ +|φn(t) − φG(0; 1; t)| dt = o +� +n1−2α� +. +By Lemma 2.5 and (2.4), +� log n +− log n +|φn(t) − φG(0; 1; t)| dt = O +� +n1/2−κ(1−α)(log n)κ+2� += o +� +n1−2α� +. +Furthermore, +� +[− log n,log n]c φG(0; 1; t) dt = O +� +e−(log n)2/2� += o +� +n1−2α� +, +Thus, (2.78) will follow once we show that +� +[− log n,log n]c |φn(t)| dt = o +� +n1−2α� +. +(2.79) +With ( ˆZni : n ≥ 1, i ≥ 0) as above, we set +Uni = σ−1 +n (Ai − Ai−n) +� +ˆZni − E( ˆZni) +� +, n ≥ 1, i ≥ 0 , +so that +|φn(t)| = +∞ +� +i=0 +��E +� +eιtUni��� , n ≥ 1, t ∈ R . +(2.80) +Set +H(x, t) = +�� ∞ +−∞ +exzfZ(z) dz +�−1 � ∞ +−∞ +e(x+ιt)zfZ(z) dz, (x, t) ∈ R2 , +which is a characteristic function for any fixed x. A consequence of that is +∂|H(x, t)|/∂t|t=0 ≤ 0 for any x ∈ R. Furthermore, +∂2 +∂t2 |H(0, t)| +��� +t=0 = −σ2 +Z < 0 + +Chakrabarty and Samorodnitsky/Clustering of large deviations +21 +and by continuity of the second partial derivative we conclude that there is +δ0 > 0 such that +∂2 +∂t2 |H(x, t)| +��� < 0 whenever 0 ≤ |t|, |x| ≤ δ0. +That means we also have +∂ +∂t|H(x, t)| +��� ≤ 0 whenever 0 ≤ |t|, |x| ≤ δ0. +(2.81) +We may and will choose δ0 ∈ (0, θ0], with θ0 as in (2.2). By (2.2) we can appeal +to (3.3) to conclude that +lim +t→∞ sup +|x|≤δ0 +|H(x, t)| = 0 . +Thus, there is M > 0 large enough so that +sup +t>M,|x|≤δ0 +|H(x, t)| < 1. +Since by continuity of H and compactness we have +sup +δ0≤t≤M,|x|≤δ0 +|H(x, t)| < 1, +it follows that +η = +sup +t≥δ0,|x|≤δ0 +|H(x, t)| < 1 . +The continuity argument also shows that there is δ1 ∈ (0, δ0] such that +min +|x|≤δ0 |H(x, δ1)| ≥ η . +Therefore, for |x| ≤ δ0 and 0 ≤ t ≤ δ1, (2.81) implies that +|H(x, t)| ≥ |H(x, δ1)| ≥ η ≥ sup +s≥δ0 +|H(x, s)| . +Since by (2.81) we also have +|H(x, t)| = +sup +s∈[t,δ0] +|H(x, s)| , +we conclude that +|H(x, t)| = sup +s≥t +|H(x, s)|, |x| ≤ δ0, 0 ≤ t ≤ δ1 . +(2.82) +By (2.80) +|φn(t)| ≤ +��E(eιtUnn) +�� +n−1 +� +i=[n/2] +��E(eιtUni) +�� += +��E(eιtUnn) +�� +n−1 +� +i=[n/2] +��H +� +ζnAi, σ−1 +n Ait +��� . +(2.83) + +Chakrabarty and Samorodnitsky/Clustering of large deviations +22 +It follows from Lemma 2.1 that there exists s0 > 0 such that for all n large +enough, +Ai ≥ s0σnn−1/2, [n/2] ≤ i ≤ n − 1 . +Thus, for n large enough and t ≥ log n, (2.82) implies that +n−1 +� +i=[n/2] +��H +� +ζnAi, σ−1 +n Ait +��� ≤ +n−1 +� +i=[n/2] +���H +� +ζnAi, s0n−1/2 log n +���� . +Since any partial derivative of H is bounded on a compact set, we can use the +bound (3.4) to conclude that there exists s1 > 0 such that +sup +|x|≤δ0 +|H(x, t)| ≤ (1 − s1t2)1/2, 0 ≤ t ≤ 1 . +Thus, there is s2 > 0 such that for all large n and all t ≥ log n we have +n−1 +� +i=[n/2] +��H +� +ζnAi, σ−1 +n Ait +��� ≤ +� +1 − s2 +0s1n−1(log n)2�n/4 = O +� +e−s2(log n)2� +. +Using this bound in (2.83), and appealing to (2.2) we obtain +� ∞ +log n +|φn(t)| dt = O +� +e−s2(log n)2� � ∞ +log n +��E +� +eitUnn��� dt += O +� +n1/2e−s2(log n)2� += o +� +n1−2α� +. +Since we can switch from t to −t, (2.79) follows, which establishes (2.78). +A similar calculation with the aid of (2.69) shows that +fni(0) = (2πλni)−1/2 exp +� +−σ−2 +n n2ε2(λni − 1)2/2λni +� ++ o +� +n1−2α� +, +uniformly in i ≥ 0. Since λni − 1 = O(1/n) uniformly in i ≥ 0, it follows that +λ−1/2 +ni +exp +� +−σ−2 +n n2ε2(λni − 1)2/2λni +� += 1 + O +� +n−1 + σ−2 +n +� += 1 + o +� +n1−2α� +, +uniformly for i ≥ 0, which proves (2.75). For (2.77) we write +f ′′ +nk(x) = −(2π)−1/2 +� ∞ +−∞ +e−itxt2φnk(t) dt +and repeat the arguments used above in the proof of (2.78), applying (2.69) and +the full force of the assumption (2.2). +Finally, for (2.76) we use the identity +f ′ +nk(0) = −i(2π)−1/2 +� ∞ +−∞ +tφnk(t) dt. + +Chakrabarty and Samorodnitsky/Clustering of large deviations +23 +Since +���� +� ∞ +−∞ +t φG +� +σ−1 +n nε(λnk − 1); λnk; t +� +dt +���� = O +� +σ−1 +n +� += o +� +n1/2−α� +, +uniformly in k ≥ 0, (2.76) follows. +The arguments with fnij replacing fni are similar. This completes the proof. +The next lemma tackles certain expectations conditionally on E0; its state- +ment should be compared to (2.61). +Lemma 2.7. Suppose that (2.5) and(2.2) hold. Then +E (Zn−i−11(E0)) = Kn +�� ∞ +−∞ +z Gζn(Ai−Ai−n)(dz) + o +� +ζ−1 +n σ−2 +n |Ai − Ai−n| +�� +(2.84) +and +E (Zn−i−1Zn−j−11(E0)) +(2.85) += Kn +�� ∞ +−∞ +z1 Gζn(Ai−Ai−n)(dz1) +� ∞ +−∞ +z2 Gζn(Ai−Ai−n)(dz2) ++ o +� +σ−2 +n |(Ai − Ai−n)(Aj − Aj−n)| +� +� +, n → ∞, +uniformly for i, j ≥ 0 with i ̸= j, where +Kn = (2π)−1/2ζ−1 +n σ−1 +n e−nεζnE +� +eζnSn� +, n ≥ 1 . +(2.86) +Proof. We only prove (2.85); the proof of (2.84) is similar and easier. Write +E (Zn−i−1Zn−j−11(E0)) += +� ∞ +−∞ +z1 FZ(dz1) +� ∞ +−∞ +z2 FZ(dz2) +P (S′ +n(i, j) ≥ nε − (Ai − Ai−n)z1 − (Aj − Aj−n)z2) += σ−1 +n E +� +eζnS′ +n(i,j)� � ∞ +−∞ +z1 FZ(dz1) +� ∞ +−∞ +z2 FZ(dz2) +� ∞ +nε−(Ai−Ai−n)z1−(Aj−Aj−n)z2 +fnij +� +s − nε)/σn +� +e−ζns ds. +We adopt the convention +� b +a ≡ − +� a +b , and denote +cnij = ζ−1 +n σ−1 +n e−nεζnE +� +eζnS′ +n(i,j)� += Kn(2π)1/2 +�� ∞ +−∞ +eζn(Ai−Ai−n)zFZ(dz) +� ∞ +−∞ +eζn(Aj−Aj−n)zFZ(dz) +�−1 +. + +Chakrabarty and Samorodnitsky/Clustering of large deviations +24 +Changing the variable and using the fact that EZ = 0, we obtain +E (Zn−i−1Zn−j−11(E0)) += cnij +� ∞ +−∞ +z1 FZ(dz1) +� ∞ +−∞ +z2 FZ(dz2) +� ζn(Ai−Ai−n)z1+ζn(Aj−Aj−n)z2 +0 +exfnij +� +−x/(σnζn) +� +dx += cnij +� ∞ +−∞ +z1 FZ(dz1) +� ∞ +−∞ +z2 FZ(dz2) +(2.87) +�� ζn(Ai−Ai−n)z1+ζn(Aj−Aj−n)z2 +0 +exfnij +� +−x/(σnζn) +� +dx +− +� ζn(Ai−Ai−n)z1 +0 +exfnij +� +−x/(σnζn) +� +dx +− +� ζn(Aj−Aj−n)z2 +0 +exfnij +� +−x/(σnζn) +� +dx +� +. +For fixed z1, z2 ∈ R, the expression inside the square brackets can be rewritten +as +� +eζn(Ai−Ai−n)z1 − 1 +� � ζn(Aj−Aj−n)z2 +0 +ex +fnij +� +−(x + ζn(Ai − Ai−n)z1)/(σnζn) +� +dx ++ +� ζn(Aj−Aj−n)z2 +0 +ex +� +fnij +� +−(x + ζn(Ai − Ai−n)z1)/(σnζn) +� +− fnij +� +−x/(σnζn) +� +� +dx. +By Taylor’s theorem, +fnij +� +− x + ζn(Ai − Ai−n)z1 +σnζn +� += fnij(0) − x + ζn(Ai − Ai−n)z1 +σnζn +f ′ +nij(0) ++O +�(x + ζn(Ai − Ai−n)z1)2 +σ2nζ2n +∥f ′′ +nij∥∞ +� +. + +Chakrabarty and Samorodnitsky/Clustering of large deviations +25 +Using this and (2.42), straightforward algebra gives us +� ζn(Aj−Aj−n)z2 +0 +exfnij +� +−(x + ζn(Ai − Ai−n)z1)/(σnζn) +� +dx += fnij(0) +� +eζn(Aj−Aj−n)z2 − 1 +� ++ O +� +eζn|Aj−Aj−n||z2|� +|f ′ +nij(0)|σ−1 +n ζnn1−α|Aj − Aj−n||z2| +� +|z1| + |z2| +� ++ ∥f ′′ +nij∥∞σ−2 +n ζnn2−2α|Aj − Aj−n||z2| +� +|z1| + |z2| +�2�� +. +The obvious inequality |ex − 1| ≤ |x|e|x| for x ∈ R along with Lemma 2.6 now +show that +� +eζn(Ai−Ai−n)z1 − 1 +� � ζn(Aj−Aj−n)z2 +0 +ex +fnij +� +−(x + ζn(Ai − Ai−n)z1)/(σnζn) +� +dx += fnij(0) +� +eζn(Ai−Ai−n)z1 − 1 +� � +eζn(Aj−Aj−n)z2 − 1 +� ++ o +� +σ−2 +n |(Ai − Ai−n)(Aj − Aj−n)z1z2| (|z1| + |z2|)2 +eζn(|Ai−Ai−n||z1|+|Aj−Aj−n||z2|)� +, +uniformly for i, j ≥ 0 with i ̸= j and z1, z2 ∈ R. +Treating in a similar manner the second term, we conclude that the expression +inside the square brackets in the right hand side of (2.87) equals +fnij(0) +� +eζn(Ai−Ai−n)z1 − 1 +�� +eζn(Aj−Aj−n)z2 − 1 +� ++ o +� +σ−2 +n |(Ai − Ai−n)(Aj − Aj−n)| (1 + |z1|3)(1 + |z2|3) +eζn|(Ai−Ai−n)z1|+ζn|(Aj−Aj−n)z2|� +, +uniformly for i, j ≥ 0 with i ̸= j and z1, z2 ∈ R, and substitution into (2.87) +gives us +E (Zn−i−1Zn−j−11(E0)) += cnij +� +fnij(0) +� ∞ +−∞ +z1eζn(Ai−Ai−n)z1FZ(dz1) +� ∞ +−∞ +z2eζn(Aj−Aj−n)z2FZ(dz2) ++ o +� +σ−2 +n |(Ai − Ai−n)(Aj − Aj−n)| +�� + +Chakrabarty and Samorodnitsky/Clustering of large deviations +26 += Kn(2π)1/2fnij(0) +� ∞ +−∞ +z1 Gζn(Ai−Ai−n)(dz1) +� ∞ +−∞ +z2 Gζn(Aj−Aj−n)(dz2) +(2.88) ++ cnijo +� +σ−2 +n |(Ai − Ai−n)(Aj − Aj−n)| +� +, +as n → ∞, uniformly for i, j ≥ 0 with i ̸= j. Recalling that EZ = 0, we see that +� ∞ +−∞ +z1 Gζn(Ai−Ai−n)(dz1) = O (ζn(Ai − Ai−n)) , +and likewise for the second integral in (2.88). Since Kn = O(cnij), the claim +(2.85) follows from Lemma 2.6. +The next lemma is an important step in the proof of the main result; the +previous lemmas 2.5, 2.6 and 2.7 are needed for this lemma. We denote +Yni = Zn−i−1 − +� +1 + ζ−2 +n σ−2 +n +� � ∞ +−∞ +z Gζn(Ai−Ai−n)(dz), i ∈ Z, n ≥ 1 . +(2.89) +Lemma 2.8. Suppose that (2.5) and(2.2) hold. Then +sup +n≥1,i≥0 +E +� +Y 2 +ni +��E0 +� +< ∞ , +(2.90) +and +E +� +YniYnj +��E0 +� += −σ−2 +n σ4 +Z (Ai − Ai−n) (Aj − Aj−n) (1 + o(1)) +(2.91) +as n → ∞, uniformly in i, j ≥ 0 with i ̸= j. +Proof. We prove (2.91); the proof of (2.90) is similar (and much easier). We +write +P(E0) = Kn(2π)1/2 +� ∞ +0 +e−xfn +� +x/(ζnσn) +� +dx, +with Kn as in (2.86). By (2.78) and simple integration, +P(E0) =Kn(2π)1/2 +� +o +� +ζ−2 +n σ−2 +n +� ++ (2π)−1/2 +� ∞ +0 +exp +� +−x − x2/(2ζ2 +nσ2 +n) +� +dx +� +(2.92) +=Kn +� +1 − ζ−2 +n σ−2 +n (1 + o(1)) +� +, n → ∞ . +In combination with (2.85) this means that +E (Zn−i−1Zn−j−11(E0)) P(E0) += K2 +n +� +� +1 − ζ−2 +n σ−2 +n +� � ∞ +−∞ +z1 Gζn(Ai−Ai−n)(dz1) +� ∞ +−∞ +z2 Gζn(Aj−Aj−n)(dz2) ++ o +� +σ−2 +n |(Ai − Ai−n)(Aj − Aj−n)| +� +� +, n → ∞, + +Chakrabarty and Samorodnitsky/Clustering of large deviations +27 +uniformly in i, j ≥ 0 with i ̸= j. Since by (2.84), +E (Zn−i−11(E0)) E (Zn−j−11(E0)) += K2 +n +� ∞ +−∞ +z1 Gζn(Ai−Ai−n)(dz1) +� ∞ +−∞ +z2 Gζn(Aj−Aj−n)(dz2) ++ o +� +K2 +nσ−2 +n |Ai − Ai−n||Aj − Aj−n| +� +, +we conclude that +E (Zn−i−1Zn−j−11(E0)) P(E0) − E (Zn−i−11(E0)) E (Zn−j−11(E0)) += −K2 +nζ−2 +n σ−2 +n +� ∞ +−∞ +z1 Gζn(Ai−Ai−n)(dz1) +� ∞ +−∞ +z2 Gζn(Aj−Aj−n)(dz2) ++ o +� +K2 +nσ−2 +n |Ai − Ai−n||Aj − Aj−n| +� += −K2 +nσ−2 +n σ4 +Z(Ai − Ai−n)(Aj − Aj−n) (1 + o(1)) +as n → ∞, uniformly in i, j ≥ 0 with i ̸= j. Dividing both sides by P(E0)2 and +using (2.92), we obtain +E +�� +Zn−i−1 − E(Zn−i−1|E0) +�� +Zn−j−1 − E(Zn−j−1|E0) +����E0 +� +(2.93) += −σ−2 +n σ4 +Z(Ai − Ai−n)(Aj − Aj−n) (1 + o(1)) , +as n → ∞, again uniformly for i, j ≥ 0 with i ̸= j. Since by (2.92) with (2.84) +E (Zn−i−1|E0) = +� +1 + ζ−2 +n σ−2 +n +� � ∞ +−∞ +z Gζn(Ai−Ai−n)(dz) ++ o +� +ζ−1 +n σ−2 +n |Ai − Ai−n| +� +, +with a similar statement for Zn−j−1, (2.93) implies (2.91). +We proceed with establishing conditional distributional limits of certain trun- +cated sums. +Lemma 2.9. Suppose that (2.5) and(2.2) hold. For 0 < δ < L denote +Sn(j, δ, L) = +[nβL]−1 +� +i=[nβδ] +(Ai+j − Ai)Yni + +n−1 +� +i=n−j +(Ai+j − Ai+j−n − Ai)Yni +(2.94) ++ +n+[nβL] +� +i=n +(Ai+j − Ai+j−n − Ai + Ai−n)Yni, n ≥ 1, j ≥ 0. +With the overshoot T ∗ +n as in (2.49), we have, conditionally on E0, +� +ζnT ∗ +n, +� +n2α−2Sn([nβt], δ, L), t ≥ 0 +�� +⇒ +� +T0, +� +(1 − α)−1σZ +�� L +δ +� +(s + t)1−α − s1−α� +dB1(s) +(2.95) ++ +� t +0 +(t − s)1−αdB2(s) + +� L +0 +� +s1−α − (s + t)1−α� +dB3(s) +� +, t ≥ 0 +�� + +Chakrabarty and Samorodnitsky/Clustering of large deviations +28 +in finite dimensional distributions as n → ∞, where T0 is a standard exponen- +tial random variable independent of independent standard Brownian motions +B1, B2, B3, +Proof. For n ≥ 1 and t ≥ 0 we write +ξ1◦ +n (t) = +[nβt] +� +i=1 +AiYni, +ξ2◦ +n (t) = +n−1 +� +i=n−[nβt] +AiYni, +ξ3◦ +n (t) = +n+[nβt] +� +i=n+1 +(Ai − Ai−n) Yni. +It follows from Lemma 2.4 that, conditionally on E0, +� +ζnT ∗ +n, +� +n2α−2ξ1◦ +n (t) : t ≥ 0 +� +, +� +nα−β/2−1ξ2◦ +n (t) : t ≥ 0 +� +, +(2.96) +� +nα−β/2−1ξ3◦ +n (t) : t ≥ 0 +�� +⇒ +� +T0, +� +K1/2 +1 +σZB1(t3−2α) : t ≥ 0 +� +, +� +(1 − α)−1σZB2(t) : t ≥ 0 +� +, +� +(1 − α)−1σZB3(t) : t ≥ 0 +�� +because the difference between the two processes vanishes in the limit. For ex- +ample, +n2α−2ζ−2 +n σ−2 +n +[nβt] +� +i=1 +Ai +� ∞ +−∞ +z GζnAi(dz) = O +� +n1−2α� += o(1) , +and similarly with the other two components. Furthermore, for large n, +Sn([nβt], δ, L) += +[nβL]−1 +� +i=[nβδ] +(Ai+[nβt] − Ai)Yni + +n−1 +� +i=n−[nβt] +(Ai+[nβt] − Ai+[nβt]−n − Ai)Yni ++ +n+[nβL] +� +i=n +(Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n)Yni =: V 1 +n (t) + V 2 +n (t) + V 3 +n (t). +Starting with V 3 +n , we write +V 3 +n (t) = n−(1−α)(1−β) +[nβL] +� +i=1 +fn +� +n−βi, t +� +(An+i − Ai) Yn,n+i , +(2.97) + +Chakrabarty and Samorodnitsky/Clustering of large deviations +29 +where for 0 ≤ s ≤ L, +fn(s, t) = n(1−α)(1−β) An+[nβs]+[nβt] − A[nβs]+[nβt] − An+[nβs] + A[nβs] +An+[nβs] − A[nβs] +. +It is elementary that for fixed s, t, as n → ∞, +An+[nβs]+[nβt] − An+[nβs] ≪ A[nβs]+[nβt] − A[nβs] +∼ (1 − α)−1nβ(1−α) � +(s + t)1−α − s1−α� +, +while An+[nβs] − A[nβs] ∼ (1 − α)−1n1−α. Therefore, +lim +n→∞ fn(s, t) = s1−α − (s + t)1−α =: f(s, t), +(2.98) +and the limit is easily seen to be uniform in 0 ≤ s ≤ L and t in a compact +interval. We will show that, conditionally on E0, +� +n2α−2V 3 +n (t), t ≥ 0 +� +(2.99) +⇒ +� +σZ(1 − α)−1 +� L +0 +� +s1−α − (s + t)1−α� +dB3(s), t ≥ 0 +� +in finite-dimensional distributions, as n → ∞. To this end, set +cnj(k, t) = +inf +(j−1)L/k≤s≤jL/k fn(s, t), k ≥ 1, 1 ≤ j ≤ k , +and +eni(k, t) = fn +� +n−βi, t +� +− cn,⌈L−1n−βki⌉(k, t) ≥ 0, k ≥ 1, 1 ≤ i ≤ [nβL] . +By (2.98) and monotonicity, +lim +n→∞ cnj(k, t) = f +� +(j − 1)k−1L, t +� +, 1 ≤ j ≤ k . +(2.100) +A standard continuity argument shows that +lim +k→∞ lim sup +n→∞ sup +t∈A +max +1≤i≤[nβL] eni(k, t) = 0 +(2.101) +for any compact set A. We have +[nβL] +� +i=1 +cn,⌈L−1n−βki⌉(k, t)(An+i − Ai)Yn,n+i += +k′ +� +j=1 +cnj(k, t) +� +i∈ +� +k−1Lnβ(j−1),k−1Lnβj +� +∩Z +(An+i − Ai)Yn,n+i += +k′ +� +j=1 +cnj(k, t) +� +ξ3◦ +n +� +k−1Lj +� +− ξ3◦ +n +� +k−1L(j − 1) +�� +=: Wnk(t) , + +Chakrabarty and Samorodnitsky/Clustering of large deviations +30 +where k′ = ⌈L−1n−βk[nβL]⌉. This, together with (2.96) and (2.100), implies +that for fixed k, as n → ∞, +� +nα−β/2−1Wnk(t), t ≥ 0 +� +(2.102) +⇒ +� +(1 − α)−1σZ +k +� +j=1 +f +� +(j − 1)k−1L, t +� � +B3(k−1jL) − B3(k−1(j − 1)L) +� +, +t ≥ 0 +� +in finite-dimensional distributions. We have +[nβL] +� +i=1 +fn +� +n−βi, t +� +(An+i − Ai) Yn,n+i − Wnk(t) += +[nβL] +� +i=1 +eni(k, t) (An+i − Ai) Yn,n+i . +It follows from (2.91) that, for large n, +sup +i,j≥0:i̸=j +(Ai − Ai−n) (Aj − Aj−n) E (YniYnj|E0) ≤ 0 . +This, along with (2.90) and the non-negativity of each eni, implies that for large +n, +E + + + + + +[nβL] +� +i=1 +eni(k, t) (An+i − Ai) Yn,n+i + + +2�����E0 + + + +≤ +[nβL] +� +i=1 +[eni(k, t) (An+i − Ai)]2 E(Y 2 +n,n+i|E0) += O + + +max +1≤j≤[nβL] enj(k, t)2 +[nβL] +� +i=1 +(An+i − Ai)2 + + += O +� +n2−2α+β +max +1≤j≤[nβL] enj(k, t)2 +� +. +Invoking (2.101) we conclude that for any compact set A, +lim +k→∞ lim sup +n→∞ n2α−β−2 sup +t∈A +E +�� +Wnk(t) +(2.103) +− +[nβL] +� +i=1 +fn +� +n−βi, t +� +(An+i − Ai) Yn,n+i +�2�����E0 +� += 0 . + +Chakrabarty and Samorodnitsky/Clustering of large deviations +31 +As k → ∞, the process in the right hand side of (2.102) converges in finite- +dimensional distributions to the process in the right-hand side of (2.99). Since +(2α− 2)− (1 − α)(1 − β) = α− β/2 − 1, the claim (2.99) follows from (2.97) and +(2.103) by the “convergence together” argument; see Theorem 3.2 in Billingsley +(1999). +A nearly identical argument shows that, conditionally on E0, +� +n2α−2V 2 +n (t), t ≥ 0 +� +⇒ +� +−σZ(1 − α)−1 +� t +0 +(t − s)1−αdB2(s), t ≥ 0 +� +(2.104) +d= +� +σZ(1 − α)−1 +� t +0 +(t − s)1−αdB2(s), t ≥ 0 +� +in finite-dimensional distributions. +The situation with the term V 1 +n is, once again, similar, with a small twist. +Since +lim +n→∞ +A[nβs]+[nβt] − A[nβs] +A[nβs] += (s + t)1−α − s1−α +s1−α +uniformly for δ ≤ s ≤ L and t, our argument now shows that, conditionally on +E0, +� +n−(2−2α)V 1 +n , t ≥ 0 +� +⇒ +� +σZK1/2 +1 +� L +δ +(s + t)1−α − s1−α +s1−α +M(ds), t ≥ 0 +� +in finite-dimensional distributions, where M is a centred Gaussian random mea- +sure with the variance measure with the density (3 − 2α)s2−2α, s > 0. Since +the centred Gaussian random measures (1 − α)−1B3(ds) and K1/2 +1 +M(ds)/s1−α +have the same variance measure, this means that, conditionally on E0, +� +n2α−2V 2 +n (t), t ≥ 0 +� +(2.105) +⇒ +� +σZ(1 − α)−1 +� L +δ +� +(s + t)1−α − s1−α� +dB3(s), t ≥ 0 +� +in finite-dimensional distributions. +Since (2.99), (2.104) and (2.105) are all consequences of (2.96), the conver- +gence statements they contain hold jointly, and jointly with ζnT ∗ +n ⇒ T0. The +claim (2.95) follows. +The next lemma treats the sequence of shifts appearing due to conditioning +on E0. +Lemma 2.10. Define +µn(t) += n2α−2 +∞ +� +i=0 +� +Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n +� � ∞ +−∞ +z Gζn(Ai−Ai−n)(dz), +for t ≥ 0 and n ≥ 1. Then µn → µ∞ as n → ∞, in D([0, ∞)) equipped with the +Skorohod J1 topology, where µ∞(t) = −εt3−2α, t ≥ 0. + +Chakrabarty and Samorodnitsky/Clustering of large deviations +32 +Proof. Writing +µn(t) =n2α−2ζn +∞ +� +i=0 +� +Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n +� � +Ai − Ai−n +� ++n2α−2 +∞ +� +i=0 +� +Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n +� +�� ∞ +−∞ +z Gζn(Ai−Ai−n)(dz) − ζn +� +Ai − Ai−n +�� +=: µ(1) +n (t) + µ(2) +n (t), t ≥ 0, +the claim of the lemma will follow once we prove that +µ(1) +n +→ µ∞ in D([0, ∞)) +(2.106) +and +µ(2) +n (t) → 0 uniformly on compact intervals. +(2.107) +We start by proving (2.107). Fix L > 0 so that 0 ≤ t ≤ L. Suppose first that +1/2 < α < 5/6. By (2.19) +��µ(2) +n (t) +�� +=O +� +n2α−2ζ2 +n +∞ +� +i=0 +��Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n +�� � +Ai − Ai−n +�2 +� +=O +� +n2α−2ζ2 +nnβ +∞ +� +i=1 +i−α� +Ai − Ai−n +�2 +� += O +� +n2α−2ζ2 +nnβn3−3α� +→ 0 +uniformly in 0 ≤ t ≤ L, showing (2.107). On the other hand, if α ≥ 5/6, then +κ ≥ 3 in (2.5), so by (2.19) +��µ(2) +n (t) +�� +=O +� +n2α−2ζ3 +n +∞ +� +i=0 +��Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n +�� � +Ai − Ai−n +�3 +� +=O +� +n2α−2ζ3 +nnβ +∞ +� +i=1 +i−α� +Ai − Ai−n +�3 +� += O +� +n2α−2ζ3 +nnβn4−4α� +→ 0 +uniformly in 0 ≤ t ≤ L, again showing (2.107). +We now prove (2.106). The pointwise convergence is clear: for fixed t, +µ(1) +n (t) = σ2 +Zσ−2 +n n2α−1ε +∞ +� +i=0 +� +Ai+[nβt] − Ai+[nβt]−n +� +(Ai − Ai−n) − n2α−1ε +→ −εt3−2α + +Chakrabarty and Samorodnitsky/Clustering of large deviations +33 +as n → ∞, where we have used (2.25). Next, as in (2.28) we can write for t ≥ 0, +µ(1) +n (t) =n2α−2ζn +2 +�n−1 +� +i=0 +� +Ai − Ai−[nβt] +�2 ++ +∞ +� +i=n−[nβt] +� +Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n +�2 +� +=: µ(11) +n +(t) + µ(12) +n +(t). +The claim (2.106) will follow once we show that both µ(11) +n +and µ(12) +n +converge +in D([0, ∞)) to continuous limits (both constant factors of µ∞). The fact that +µ(11) +n +converges pointwise to a constant factor of of the pointwise limit of µ(1) +n +is +an intermediate step in the proof of (2.25). Since µ(11) +n +is a monotone function, +its convergence in D([0, ∞)) follows. +We already know that µ(12) +n +converges pointwise to a continuous limit. Let i0 +be such that ai is monotone for i ≥ i0. Write for t ≥ 0 +µ(12) +n +(t) =n2α−2ζn +2 +� +∞ +� +i=n+i0 +� +Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n +�2 +− +n+i0−1 +� +i=n−[nβt] +� +Ai+[nβt] − Ai+[nβt]−n − Ai +�2 +� +=: µ(121) +n +(t) − µ(122) +n +(t), +so it is enough to show that both µ(121) +n +and µ(122) +n +converge in D([0, ∞)) to +continuous limits. Splitting further, we write for t ≥ 0, +µ(122) +n +(t) = n2α−2ζn +2 +� n+i0−1 +� +i=n−[nβt] +A2 +i+[nβt]−n ++ +n+i0−1 +� +i=n−[nβt] +� +Ai − Ai+[nβt] +�� +Ai − Ai+[nβt] − 2Ai+[nβt]−n +� +� +=: µ(1221) +n +(t) + µ(1222) +n +(t). +Clearly, +µ(1221) +n +(t) = n2α−2ζn +2 +[nβt]+i0−1 +� +i=0 +A2 +i +converges pointwise to a constant factor of µ∞. Since µ(1221) +n +is monotone, we +conclude that µ(1221) +n +converges in D([0, ∞)) to a continuous limit. In order to + +Chakrabarty and Samorodnitsky/Clustering of large deviations +34 +prove that so does µ(122) +n +, we will show that µ(1222) +n +(t) → 0 uniformly on compact +intervals. Considering once again 0 ≤ t ≤ L, we have +��µ(1222) +n +(t) +�� +≤ n2α−2ζn +2 +n+i0−1 +� +i=n−[nβt] +� +Ai+[nβt] − Ai +��� +Ai+[nβt] − Ai +� ++ 2Ai+[nβt]−n +� += O + +n2α−2ζn +n+i0−1 +� +i=n−[nβt] +nβn−α� +nβn−α + nβ(1−α)� + + += O +� +nα−2ζnn3β−βα� +→ 0 +uniformly over 0 ≤ t ≤ L, as required. +Finally, we already know that µ(121) +n +converges pointwise to a continuous limit. +Furthermore, by the choice of i0, µ(121) +n +is a monotone function. Therefore, it +converges in D([0, ∞)), and the proof is complete. +The following is the final lemma before we prove Theorem 2.1. +Lemma 2.11. Suppose that (2.5) and(2.2) hold. Let +Sn(j) = +j+n−1 +� +i=j +Xi, j ≥ 0, n ≥ 1. +(2.108) +As n → ∞, conditionally on E0, +� +n−(2−2α) � +Sn([nβt]) − nε +� +, t ≥ 0 +� +⇒ +� +(2Cα)1/2BH(t) + ε−1Cασ2 +ZT0 − εt3−2α, t ≥ 0 +� +in finite-dimensional distributions, where (BH(t) : t ≥ 0) is the standard frac- +tional Brownian motion (2.8) with the Hurst exponent H given in (2.7), Cα is +the constant defined in (2.10), and T0 is a standard exponential random variable +independent of the fractional Brownian motion. +Proof. It follows from (2.91) and the eventual monotonicity of the sequence +(An) that there is i0 ≥ 0 such that for all large n, +sup +i0≤i 0 this and (2.90) imply that +E + + + + +n−[nβt]−1 +� +i=[nβL] +� +Ai+[nβt] − Ai +� +Yni + + +2�����E0 + + +=O + + +∞ +� +i=[nβL] +� +Ai+[nβt] − Ai +�2 + + +=O + + +∞ +� +j=[nβL] +�� +j + [nβt] +�1−α − j1−α�2 + + +≤O +� +n4−4α +� ∞ +L +� +(x + t)1−α − x1−α�2 dx +� +. +Therefore, for fixed t, +lim +L→∞ lim sup +n→∞ E + + + +n2α−2 +n−[nβt]−1 +� +i=[nβL] +� +Ai+[nβt] − Ai +� +Yni + + +2�����E0 + + = 0 . +(2.110) +Since the sequence (an) is eventually monotone, we can increase, if necessary, +i0 to guarantee that Aj+k − Aj ≤ Ai+k − Ai for all i0 ≤ i ≤ j and k ≥ 0. By +(2.109), for fixed L, t > 0, large n and i, j ≥ n + [nβL], +� +Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n +� +� +Aj+[nβt] − Aj+[nβt]−n − Aj + Aj−n +� +E +� +YniYnj +��E0 +� +≤ 0 , +and the same argument as above implies that +lim +L→∞ lim sup +n→∞ E +�� +n2α−2 +(2.111) +∞ +� +i=n+[nβL]+1 +� +Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n +� +Yni +�2 +�����E0 +� += 0 . +Similarly, for a fixed t > 0, +lim +δ→0 lim sup +n→∞ E + + + +n2α−2 +[nβδ]−1 +� +i=i0 +� +Ai+[nβt] − Ai +� +Yni + + +2�����E0 + + = 0 , +(2.112) +and it is elementary that for a fixed t > 0, +lim +n→∞ E + + +� +n2α−2 +i0−1 +� +i=0 +� +Ai+[nβt] − Ai +� +Yni +�2�����E0 + + = 0 . +(2.113) + +Chakrabarty and Samorodnitsky/Clustering of large deviations +36 +It follows from (2.110), (2.111), (2.112), (2.113) and Lemma 2.9 that, condi- +tionally on E0, +� +ζnT ∗ +n, +� +n−(2−2α) +∞ +� +i=0 +� +Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n +� +Yni, t ≥ 0 +�� +(2.114) +⇒ +� +T0, +� +(1 − α)−1σZ +�� ∞ +0 +� +(s + t)1−α − s1−α� +dB1(s) ++ +� t +0 +(t − s)1−α dB2(s) + +� ∞ +0 +� +(s + t)1−α − s1−α� +dB3(s) +� +, t ≥ 0 +�� +, +in finite-dimensional distributions, as n → ∞. Furthermore, one can easily check +the Lindeberg conditions of the central limit theorem to see that + +n−(2−2α) +−1 +� +i=−[nβt] +Ai+[nβt]Zn−1−i, t ≥ 0 + + +(2.115) +⇒ +� +(1 − α)−1σZ +� t +0 +(t − s)1−α dB0(s), t ≥ 0 +� +in finite-dimensional distributions, as n → ∞, where B0 is a standard Brownian +motion. Note that the random variables in the left hand side of (2.115) are +independent of the the random variables in the left hand side of (2.114) and, in +particular, independent of E0. +Using (2.26) we conclude by (2.114) and (2.115) that, in the notation of +Lemma 2.10, conditionally on E0, +� +ζnT ∗ +n, +� +n−(2−2α) � +Sn([nβt]) − Sn +� +− +� +1 + ζ−2 +n σ−2 +n +� +µn(t), t ≥ 0 +�� +⇒ +� +T0, +� +(1 − α)−1σZ +�� t +0 +(t − s)1−α dB0(s) ++ +� ∞ +0 +� +(s + t)1−α − s1−α� +dB1(s) ++ +� t +0 +(t − s)1−α dB2(s) + +� ∞ +0 +� +(s + t)1−α − s1−α� +dB3(s) +� +, t ≥ 0 +�� +d= +� +T0, +� +21/2(1 − α)−1σZ +� ∞ +−∞ +� +(t − s)1−α ++ +− (−s)1−α ++ +� +dW(s), t ≥ 0 +�� +in finite-dimensional distributions as n → ∞, where at the intermediate step +the four standard Brownian motions, B0, B1, B2 and B3 are independent (and +independent of T0), and in the final expression (W(s), s ∈ R) is a two-sided +standard Brownian motion, independent of T0. By (2.29), this can be restated +as saying that, conditionally on E0, +� +ζnT ∗ +n, +� +n−(2−2α) � +Sn([nβt]) − Sn +� +− µn(t), t ≥ 0 +�� +⇒ +� +T0, +� +(2Cα)1/2BH(t), t ≥ 0 +�� +, + +Chakrabarty and Samorodnitsky/Clustering of large deviations +37 +and by Lemma 2.10 also +� +ζnT ∗ +n, +� +n−(2−2α) � +Sn([nβt]) − Sn +� +, t ≥ 0 +�� +⇒ +� +T0, +� +(2Cα)1/2BH(t) − εt3−2α, t ≥ 0 +�� +in finite-dimensional distributions, as n → ∞. Since +n−(2−2α)� +Sn([nβt]) − nε +� += n−(2−2α)� +Sn([nβt]) − Sn +� ++ +� +n2α−2ζ−1 +n +� +ζnTn∗, +the claim of the lemma follows from the definition (2.50) of ζn and (2.22). +Now we are in a position to prove Theorem 2.1. +Proof of Theorem 2.1. We will prove that +� +P +�� +n−(2−2α) � +Sn([nβt]) − nε +� +, 0 ≤ t < ∞ +� +∈ · +���E0 +� +, n ≥ 1 +� +(2.116) +is a tight family of probability measures on D([0, ∞)) equipped with the Skoro- +hod J1 topology. Assuming for a moment that this is true, it would follow from +Lemma 2.11 that, conditionally on E0, +� +n−(2−2α) � +Sn([nβt]) − nε +� +: t ≥ 0 +� +⇒ +� +(2Cα)1/2BH(t) + ε−1Cασ2 +ZT0 − εt3−2α : t ≥ 0 +� +weakly in D([0, ∞)), as n → ∞. Since the functional x �→ inf{t ≥ 0 : x(t) ≤ 0} +on D([0, ∞)) is, clearly, a.s. continuous with respect to the law induced on that +space by the limiting process, the continuous mapping theorem would imply +that, conditionally on E0, +n−βIn(ε) = inf +� +t ≥ 0 : n−(2−2α) � +Sn([nβt]) − nε +� +≤ 0 +� +⇒ inf +� +t ≥ 0 : (2Cα)1/2BH(t) + ε−1Cασ2 +ZT0 − εt3−2α ≤ 0 +� += τε +as n → ∞. Therefore, establishing tightness of the family (2.116) suffices to +complete the proof of Theorem 2.1, and by Lemma 2.10 it is enough to prove +that the family +� +P +�� +n−(2−2α) � +Sn([nβt]) − nε +� +− µn(t), 0 ≤ t < ∞ +� +∈ · +���E0 +� +, n ≥ 1 +� +(2.117) +is a tight family of probability measures on D([0, ∞)). +We have to prove tightness of the restriction of the family (2.117) to the +interval [0, L] for any L > 0, so fix L. We start by showing that +E +�� +Sn +� +[nβt] +� +− n2α−2µn(t) − Sn +� +[nβs] +� ++ n2α−2µn(s) +�2����E0 +� += O +�� +[nβt] − [nβs] +�3−2α� +, +(2.118) + +Chakrabarty and Samorodnitsky/Clustering of large deviations +38 +uniformly for 0 ≤ s ≤ t ≤ L. We write +Sn +� +[nβt] +� +− n2α−2µn(t) − Sn +� +[nβs] +� ++ n2α−2µn(s) += +−1 +� +i=−[nβt] +� +Ai+[nβt] − Ai+[nβs] +� +Zn−i−1 ++ +∞ +� +i=0 +� +Ai+[nβt] − Ai+[nβt]−n − Ai+[nβs] + Ai+[nβs]−n +� +Yni. +Since Zn, Zn+1, . . . are independent of E0, by Lemma 2.8, +E +�� +Sn +� +[nβt] +� +− n2α−2µn(t) − Sn +� +[nβs] +� ++ n2α−2µn(s) +�2����E0 +� +=O +�[nβt]−1 +� +j=0 +� +Aj − Aj+[nβs]−[nβt] +�2 ++ +∞ +� +i=0 +� +Ai+[nβt] − Ai+[nβt]−n − Ai+[nβs] + Ai+[nβs]−n +�2 +� +=O +�� +[nβt] − [nβs] +�3−2α� +uniformly for 0 ≤ s ≤ t ≤ L by (2.71) with κ = 2, and (2.118) follows. +Let now 0 ≤ r ≤ s ≤ t ≤ L. If t − r ≤ n−β, then +E +� +��Sn([nβs]) − µn(s) − Sn([nβr]) + µn(r) +�� +��Sn([nβt]) − µn(t) − Sn([nβs]) + µn(s) +�� +���E0 +� +vanishes. On the other hand, if t − r > n−β, then by (2.118) and the Cauchy- +Schwarz inequality, the conditional expectation can be bounded by +O +�� +[nβt] − [nβr] +�3−2α� += O +� +n4−4α(t − r)3−2α� +uniformly for 0 ≤ r ≤ s ≤ t ≤ L. Since 3 − 2α > 1, the required tightness of the +family in (2.117) follows, which completes the proof of Theorem 2.1. +3. Some useful facts +We collect in this section for easy reference a number of known or easily derivable +results. +The following integral evaluation follows from (2), (6) and (51) in Pickard +(2011). If H ∈ (0, 1), H ̸= 1/2, then +� ∞ +0 +� +xH−1/2 − (x − 1)H−1/2 ++ +�2 +dx = cos(πH)Γ(2 − 2H) +πH(1 − 2H) +Γ(H + 1/2)2 . +(3.1) + +Chakrabarty and Samorodnitsky/Clustering of large deviations +39 +Next, we will need the following version of the Berry-Essen theorem valid for +independent not necessarily identically distributed summands; see Batirov et al. +(1977). +Let X1, . . . , Xn be independent zero mean random variables with finite third +moments. Denote +A = +n +� +i=1 +E|X3 +i |, B = +� +� +� +� +n +� +i=1 +E(X2 +i ). +Assuming B > 0 we have +�����P +� n +� +i=1 +Xi ≤ Bz +� +− Φ(z) +����� ≤ CuAB−3, z ∈ R , +(3.2) +with Cu a universal constant, and Φ the standard normal CDF. The fact that +the constant is universal means that (3.2) remains valid for n = ∞ as long the +series in the left hand side converges and A, B are finite. +The following generalization of the Riemann-Lebesgue lemma can be proven +in the same way as the original statement. If f : R → R is a measurable function +such that for some δ > 0, +� ∞ +−∞ +eθx|f(x)|dx < ∞ for all θ ∈ [−δ, δ] , +then +lim +t→∞ sup +|θ|≤δ +���� +� ∞ +−∞ +e(θ+it)xf(x) dx +���� = 0 . +(3.3) +We will use a simple bound on the characteristic function φ of a random +variable X with a finite third moment. Let X′ be an independent copy of X +and Y = X − X′. Using the bound cos t ≤ 1 − t2/2 + |t|3/6 for t ∈ R, we have +EeitY ≤ 1 − t2E(Y 2)/2 + |t|3E|Y |3/6 +≤ 1 − t2Var(X) + 4|t|3E|X|3/3 . +This implies that +|φ(t)| ≤ +� +1 − t2Var(X) + 4|t|3E|X|3/3 +�1/2 , t ∈ R . +(3.4) +Acknowledgment +AC is thankful to Rudra Sarkar for helpful discussions. + +Chakrabarty and Samorodnitsky/Clustering of large deviations +40 +References +K. Batirov, D. Manevich and S. Nagaev (1977): The Esseen inequality +for sums of a random number of differently distributed random variables. +Mathematical Notes of the Academy of Sciences of the USSR 22:569–571. +P. Billingsley (1999): Convergence of Probability Measures. Wiley, New York, +2nd edition. +A. Chakrabarty and G. Samorodnitsky (2022): Clustering of large de- +viations in moving average processes: the short memory regime. +arXiv +2208.04582. +G. Pickard (2011): Representation formulae for the Fractional Brownian mo- +tion. In S´eminaire de Probabilit´es XLIII , C. Donati-Martin, A. Lejay and +A. Rounalt, editors, number 2006 in Lecture Notes in Mathemtics. Springer, +Berlin, pp. 3–70. +G. Samorodnitsky (2016): Stochastic Processes and Long Range Dependence. +Springer, Cham, Switzerland. + diff --git a/79AzT4oBgHgl3EQf-f7W/content/tmp_files/load_file.txt b/79AzT4oBgHgl3EQf-f7W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ace6e76dd3bbf5d90e6830503aae316702328b01 --- /dev/null +++ b/79AzT4oBgHgl3EQf-f7W/content/tmp_files/load_file.txt @@ -0,0 +1,931 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf,len=930 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='01936v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='PR] 5 Jan 2023 Clustering of large deviations in moving average processes: the long memory regime Arijit Chakrabarty Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Kolkata e-mail: arijit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='isi@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='com and Gennady Samorodnitsky ∗ School of Operations Research and Information Engineering Cornell University e-mail: gs18@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='edu Abstract: We investigate how large deviations events cluster in the frame- work of an infinite moving average process with light-tailed noise and long memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The long memory makes clusters larger, and the asymptotic be- haviour of the size of the cluster turns out to be described by the first hitting time of a randomly shifted fractional Brownian motion with drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' AMS 2000 subject classifications: Primary 60F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Keywords and phrases: large deviations, clustering, infinite moving av- erage, long memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Introduction We consider an infinite moving average process of the form Xn = ∞ � i=0 aiZn−i , n ≥ 0 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1) where the noise variables (Zn : n ∈ Z) are assumed to bef i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' non-degenerate random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The noise distribution FZ is assumed have finite exponential moments: � R etz FZ(dz) < ∞ for all t ∈ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) Furthermore, assuming that the noise is centred: � R z FZ(dz) = 0 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3) ∗∗The corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Research partially supported by NSF grant DMS-2015242 at Cornell University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Part of this work was performed when GS was visiting Department of Mathematics of National University of Singapore, whose hospitality is gratefully acknowl- edged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1 Chakrabarty and Samorodnitsky/Clustering of large deviations 2 the series defining the process in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1) converges if and only if the coefficients a0, a1, a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' satisfy ∞ � j=0 a2 j < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4) In this case (Xn) is a zero mean stationary ergodic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For ε > 0 we consider the sequence of large deviation events Ej(n, ε) = \uf8f1 \uf8f2 \uf8f3 1 n n+j−1 � i=j Xi ≥ ε \uf8fc \uf8fd \uf8fe , j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) By stationarity, each event Ej(n, ε) is equally rare, and we are interested in the cluster of these events that occur given that the event E0(n, ε) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In Chakrabarty and Samorodnitsky (2022) the short memory case was con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In this context, “short memory” corresponds to the case ∞ � n=0 |an| < ∞ and ∞ � n=0 an ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6) In this short memory case the conditional on E0(n, ε) law of the sequence � 1(Ej(n, ε), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=') converges weakly, as n → ∞, to the law of a sequence with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' finitely many non-zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' the total number Dε of the non-zero en- tries turns out to scale as ε−2, and ε2Dε has an interesting weak limit as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We refer the reader to Chakrabarty and Samorodnitsky (2022) for details, and a minor technical condition required for the above statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In the present paper we are interested in the long memory case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For the mov- ing average processes (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1) “long memory” refers to the case when the coeffi- cients (aj) satisfy the square summability assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4) but not the absolute summability assumption in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' A typical assumption in this is (an) is regularly varying with exponent − α, 1/2 < α < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='7) see Samorodnitsky (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It turns out that, in this case (under certain technical assumptions, an example of which is below), the conditional on E0(n, ε) law of the sequence � 1(Ej(n, ε), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=') converges weakly, as n → ∞, to the degenerate probability measure δ(1,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' That is, once the event E0(n, ε) occurs, the events (Ej(n, ε)) become very likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In order to understand their structure we concentrate on the random variables In(ε) = inf {j ≥ 1 : Ej(n, ε) does not occur} , n ≥ 1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='8) and establish a weak limit theorem for this sequence under a proper scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Interestingly, the limit turns out to be the law of the first hitting time of a randomly shifted fractional Brownian motion with drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The main result containing the above limit theorem and the technical as- sumptions it requires are in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The proof of the main theorem requires a long sequence of preliminary results, all of which are presented in that section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Finally, some useful facts needed for the proofs are collected in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The assumptions and the main result Our result on clustering of large deviation events in the long memory case will require a number of assumptions that we state next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' First of all, we will replace the assumption of regular variation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='7) by the asymptotic power function assumption an ∼ n−α, 1/2 < α < 1, and is eventually monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1) There is no doubt that the results of the paper hold under the more general regular variation assumption as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The extra generality will, however, require making an already highly technical argument even more so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The potentially resulting lack of clarity makes the added generality less valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The same is true about the eventual monotonicity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We will need additional assumptions on the distribution of the noise variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We will assume that some θ0 > 0, sup |θ|≤θ0 � ∞ −∞ t2 ���� � ∞ −∞ e(it+θ)z FZ(dz) ���� dt < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) Next, let σ2 Z = � R z2 FZ(dz) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3) be the variance of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Denote κ = the smallest integer > 4α − 1 2 − 2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4) In other words, κ = � (1+2α)/(2−2α) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We assume that a generic noise variable Z satisfies EZi = EGi for 1 ≤ i ≤ κ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) where G ∼ N(0, σ2 Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It is standard to verify that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) implies that the noise distri- bution has a twice continuously differentiable density fZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' One the other hand, a sufficient condition for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) is that the noise distribution has a four times continuously differentiable density fZ such that � ∞ −∞ eθ0|x| ���� di dxi fZ(x) ���� dx < ∞ for i = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The moment equality assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) restricts how far the the noise distri- bution can be from a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Note that in the range 1/2 < α < 5/8 we have κ = 2, in which case the assumption is vacuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since κ ≥ 2 for all α ∈ (1/2, 1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3) is implied by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 4 To state our main result, we need to introduce several key quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let β = 4 − 4α 3 − 2α ∈ (0, 1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6) and H = 3/2 − α ∈ (1/2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='7) We denote by (BH(t) : t ≥ 0) the standard fractional Brownian motion with Hurst index H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' a zero mean Gaussian process with continuous paths and covariance function E (BH(s)BH(t)) = 1 2 � s2H + t2H − |s − t|2H� , s, t ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='8) If T0 is a standard exponential random variable independent of the fractional Brownian motion, then τε = inf � t ≥ 0 : BH(t) ≤ (2Cα)−1/2εt2H − (Cα/2)1/2σ2 Zε−1T0 � , ε > 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='9) is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' finite and strictly positive random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Here σ2 Z is the variance of the noise in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3) and Cα = B(1 − α, 2α − 1) (1 − α)(3 − 2α) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='10) with B(·, ·) the standard Beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We are now in a position to state the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Assume the finite exponential moment condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2), that the coefficients satisfy the power-type condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1), the regularity condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) and the moment equality condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then for every ε > 0 the first non- occurrence times (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='8) satisfy P � n−βIn(ε) ∈ · ��E0(n, ε) � ⇒ P (τε ∈ ·) , n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='11) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It is worthwhile to observe that the limit law obtained in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 depends on the noise distribution only through its variance σ2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' This can be understood by noticing that in the long memory case considered in this paper we have Var(X1 + · · · + Xn) ≫ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Therefore, the events Ej(n, ε) should be viewed as moderate deviation events, not large deviation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It has been observed in many situations that moderate deviation events are influenced by the Gaussian weak limit of the quantities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' At the intuitive level, this explains why it is the variance of the process that appears in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For comparison, in the short memory case (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6), we have Var(X1+· · ·+Xn) ∼ cn for some c > 0, the events Ej(n, ε) should be viewed as large deviation events, and their behaviour depends on much more than just the variance of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' See Chakrabarty and Samorodnitsky (2022) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 5 We start on the road to proving Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 by establishing certain basic estimates that will be used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Denote Aj = j � i=0 ai, j ∈ Z , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='12) with the convention that a sum (or an integral) is zero if the lower limit exceeds the upper limit (so that Aj = 0 for j ≤ −1, for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let Sn = n−1 � i=0 Xi, n ≥ 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='13) and denote σ2 n = Var(Sn), n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='14) In the sequel we use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We will denote by ϕZ(t) = log �� R etz FZ(dz) � , t ∈ R (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='15) the log-Laplace transform of a noise variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We will frequently use the obvious facts ϕ is convex and ϕZ(x) ∼ x2σ2 Z/2, x → 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='16) and ϕ′ Z is continuous, nondecreasing and ϕ′ Z(x) = xσ2 Z + O(x2), x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='17) We will write Gθ for the probability measure obtained by exponentially tilting the law FZ by θ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' That is, Gθ(dz) = � EeθZ�−1eθzFZ(dz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='18) It is clear that, as θ → 0, � R z Gθ(dz) ∼ θσ2 Z, ���� � R z Gθ(dz) − θσ2 Z ���� = O(θ2) and = O(|θ|3) if κ ≥ 3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='19) � R |z|k Gθ(dz) → � R |z|k F(dz), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='. Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Asymptotically we have Aj ∼ (1 − α)−1j1−α, j → ∞ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='20) and σ2 n ∼ Cασ2 Zn3−2α, n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21) Chakrabarty and Samorodnitsky/Clustering of large deviations 6 Furthermore, for any t > 0, as n → ∞, [nβt] � i=0 (Ai − Ai−n)2 ∼ K1t3−2αn4−4α , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='22) and n � i=n−[nβt]+1 (Ai − Ai−n)2 ∼ n+[nβt] � i=n+1 (Ai − Ai−n)2 ∼ (1 − α)−2n2−2α+βt , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='23) with K1 = (1 − α)−2(3 − 2α)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='24) Finally, for any t > 0, as n → ∞, σ2 Z σ2n ∞ � i=0 (Ai − Ai−n) � Ai+[nβt] − Ai+[nβt]−n � = 1−n1−2αt3−2α(1+o(1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='25) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='20) is, of course, an immediate consequence of the assump- tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21), first note that Rn = Cov(X0, Xn) ∼ σ2 Z ∞ � j=1 j−α(j + n)−α ∼ n1−2ασ2 Z � ∞ 0 x−α(1 + x)−α dx = Cασ2 Z(1 − α)(3 − 2α)n1−2α as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Therefore, σ2 n = n−1 � i=−(n−1) (n − |i|)R|i| ∼ 2Cασ2 Z(1 − α)(3 − 2α) n−1 � i=0 (n − i)i1−2α ∼ 2Cασ2 Z(1 − α)(3 − 2α)n3−2α � 1 0 (1 − x)x1−2α dx = Cασ2 Zn3−2α , which is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Next, for a fixed t > 0 and large n, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='20) and the fact that β < 1, [nβt] � i=0 (Ai − Ai−n)2 = [nβt] � i=0 A2 i ∼ (1 − α)−2 [nβt] � i=1 i2−2α ∼ K1 � nβt �3−2α , proving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Similarly, n � i=n−[nβt]+1 (Ai − Ai−n)2 ∼ n � i=n−[nβt]+1 A2 n ∼ (1 − α)−2nβ+2−2αt , Chakrabarty and Samorodnitsky/Clustering of large deviations 7 showing the first equivalence in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='23) and the second equivalence can be shown in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='25), we start by writing Sn = ∞ � j=0 (Aj − Aj−n)Zn−1−j , n ≥ 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='26) so that σ2 n = σ2 Z ∞ � j=0 (Aj − Aj−n)2 , n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='27) Therefore, for large n, σ2 n σ2 Z − ∞ � i=0 (Ai − Ai−n)(Ai+[nβt] − Ai+[nβt]−n) = 1 2 \uf8ee \uf8f0 [nβt]−1 � i=0 (Ai − Ai−n)2 + ∞ � i=0 � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �2 \uf8f9 \uf8fb = 1 2 �n−1 � i=0 � Ai − Ai−[nβt] �2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='28) + ∞ � i=n−[nβt] � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='20), n−1 � i=0 � Ai − Ai−[nβt] �2 ∼ (1 − α)−2 n−1 � i=1 � i1−α − (i − [nβt])1−α + �2 ∼ n4−4αt3−2α(1 − α)−2 � ∞ 0 � y1−α − (y − 1)1−α + �2 dy as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1) with H = 3/2 − α, � ∞ 0 � y1−α − (y − 1)1−α + �2 dy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='29) = [(3 − 2α) (1 − α)]−1 sin(πα) π Γ(2α − 1)Γ(2 − α)2 = 1 − α 3 − 2αB (2α − 1, 1 − α) = (1 − α)2Cα, so n−1 � i=0 � Ai − Ai−[nβt] �2 ∼ Cαt3−2αn4−4α, n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='30) Chakrabarty and Samorodnitsky/Clustering of large deviations 8 Since ∞ � i=n � Ai − Ai−[nβt] �2 = O � n2β ∞ � i=n i−2α � = O � n2β+1−2α� = o � n4−4α� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='31) we conclude also that ∞ � i=0 � Ai − Ai−[nβt] �2 ∼ Cαt3−2αn4−4α, n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='32) It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='31) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='32) that ∞ � i=n−[nβt] � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �2 = ∞ � j=0 � −Aj + Aj−[nβt] + � Aj+n − Aj+n−[nβt] ��2 ∼ Cαt3−2αn4−4α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In combination with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='28) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='30) we obtain σ2 n σ2 Z − ∞ � i=0 (Ai − Ai−n)(Ai+[nβt]Ai+[nβt]−n) ∼ Cαt3−2αn4−4α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Dividing both sides by σ−2 Z σ2 n and appealing to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='25) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We now consider certain large deviations of the partial sum Sn under a change of measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' With an eye towards a subsequent application, we allow the partial sum, given in the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='26), to be “corrupted”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For n ≥ 1 and t ≥ 0 we define ξ1 n(t) = [nβt] � i=1 (Ai − Ai−n) Zn−i−1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='33) ξ2 n(t) = n−1 � i=n−[nβt] (Ai − Ai−n) Zn−i−1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='34) ξ3 n(t) = n+[nβt] � i=n+1 (Ai − Ai−n) Zn−i−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='35) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Fix t1, t2, t3 > 0 and denote ¯Sn = Sn − 3 � i=1 ξi n(ti), n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='36) Let (γn), (θn) and (ηn) be real sequences satisfying γn = o � n3/2−α� , θn = o � n−(1−α)� , 1 ≪ ηn ≪ n1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 9 If ˜Sn is a random variable with the law P � ˜Sn ∈ dx � = � E(eθn ¯Sn) �−1 eθnxP � ¯Sn ∈ dx � , n ≥ 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='37) then for all x ∈ R and h > 0, P � ηnσ−1 n � ˜Sn − E( ˜Sn) + γn � ∈ [x, x + h] � ∼ η−1 n (2π)−1/2h, n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='38) Furthermore, sup n≥1, x∈R ηnP � ηnσ−1 n ˜Sn ∈ [x, x + 1] � < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='39) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let ( ˜Zni, n ≥ 1, i ≥ 0) be a collection of independent random variables such that the law of ˜Zni is G(Ai−Ai−n)θn in the notation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then for large n, ˜Sn d= A0 ˜Zn0 + (An − A0) ˜Znn + n−[nβt2]−1 � i=[nβt1]+1 Ai ˜Zni + ∞ � i=n+[nβt3]+1 (Ai − Ai−n) ˜Zni .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='40) The proof applies to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='40) the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) in the appendix, with n = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For any z ∈ R ����P � ˜Sn − E( ˜Sn) ≤ z � Var( ˜Sn) � − Φ(z) ���� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='41) ≤ Cu � Var( ˜Sn) �−3/2 ∞ � i=0 |Ai − Ai−n|3E � | ˜Zni − E ˜Zni|3� , n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It is immediate from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1) that sup i≥0 |Ai − Ai−n| = O(n1−α) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='42) so that lim n→∞ θn sup i≥0 |Ai − Ai−n| = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='19) that E ˜Zni → 0, Var( ˜Zni) → σ2 Z, E � | ˜Zni − E ˜Zni|3� → � ∞ −∞ |z3| FZ(dz) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='43) uniformly in i as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since it is an elementary conclusion from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 that for any κ > 1/α, ∞ � i=0 |Ai − Ai−n|κ = O � nκ+1−κα� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='44) Chakrabarty and Samorodnitsky/Clustering of large deviations 10 it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='41) that sup z∈R ����P � ˜Sn − E( ˜Sn) ≤ z � Var( ˜Sn) � − Φ(z) ���� = O � n4−3α � Var( ˜Sn) �−3/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='43) again we see that Var( ˜Sn) ∼ σ2 n − 3 � i=1 Var � ξi n(ti) � ∼ Cασ2 Zn3−2α, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='45) with the second equivalence following from various claims in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Thus, sup z∈R ����P � ˜Sn − E( ˜Sn) ≤ z � Var( ˜Sn) � − Φ(z) ���� = O(n−1/2) = o � η−1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='46) Therefore, for x ∈ R and h > 0, as n → ∞, P � ηnσ−1 n � ˜Sn − E( ˜Sn) + γn � ∈ [x, x + h] � = o � η−1 n � + � R 1 � Var( ˜Sn)−1/2(xη−1 n σn − γn) ≤ z ≤ Var( ˜Sn)−1/2((x + h)η−1 n σn − γn) � φ(z) dz, where φ is the standard normal density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The assumptions on γn and ηn along with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='45) imply that the integration interval shrinks towards the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Thus, the integral above is asymptotically equivalent to η−1 n φ(0)h, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='38) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Boundedness of φ in the above integral establishes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We now look more closely at the processes defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='33), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='34) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The next lemma describes the limiting distribution of their increments under the same change of measure as in the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose that θn ∈ R satisfies θn = o � n−(1−α)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Fix 0 ≤ s < t and consider random variables with the laws P(Uni ∈ dx) = cnieθnxP � ξi n(t) − ξi n(s) ∈ dx � , i = 1, 2, 3, n ≥ 1 , with appropriate cni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then, as n → ∞, n−(2−2α) (Un1 − E(Un1)) ⇒ N � 0, K1σ2 Z � t3−2α − s3−2α�� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='47) where K1 is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='24), and for i = 2, 3, n−(1−α+β/2) (Uni − E(Uni)) ⇒ N � 0, (1 − α)−2σ2 Z(t − s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='48) Chakrabarty and Samorodnitsky/Clustering of large deviations 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For large n, Un1 d= [nβt] � i=[nβs]+1 Ai ˜Zni with ( ˜Zni) as in the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' That is, Un1 − E(Un1) is the sum of independent zero mean random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='43) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='22), Var(Un1) ∼ σ2 Z [nβt] � i=[nβs]+1 A2 i ∼ K1σ2 Zn4−4α � t3−2α − s3−2α� , and a similar calculation using the third moment bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='43) verifies the Lindeberg conditions of the central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='47) follows, and the calculations for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='48) are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Consider the overshoot defined by T ∗ n = Sn − nε, n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='49) Conditionally on the event E0 = E0(n, ε) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) the overshoot is nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The next lemma is a joint weak limit theorem for the joint law of the overshoot and the processes defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='33), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='34) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The joint law is computed conditionally on E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let ζn = nε/σ2 n, n ≥ 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='50) Conditionally on E0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' as n → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' � ζnT ∗ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' \uf8eb \uf8edn2α−2 \uf8eb \uf8edξ1 n(t) − [nβt] � i=1 Ai � ∞ −∞ z GζnAi(dz) \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ≥ 0 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' � nα−β/2−1 \uf8eb \uf8edξ2 n(t) − n−1 � i=n−[nβt] Ai � ∞ −∞ z GζnAi(dz) \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ≥ 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' � nα−β/2−1 \uf8eb \uf8edξ3 n(t) − n+[nβt] � i=n+1 (Ai − Ai−n) � ∞ −∞ z Gζn(Ai−Ai−n)(dz) \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ≥ 0 �� ⇒ � T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' � K1/2 1 σZB1(t3−2α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ≥ 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' � (1 − α)−1σZB2(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ≥ 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' � (1 − α)−1σZB3(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ≥ 0 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' in finite dimensional distributions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' where T0 is a standard exponential random variable independent of independent standard Brownian motions B1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' B2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' and B3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' K1 is the constant in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='24) and Gθ is the exponentially tilted law in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Denote ψn(s) = σ2 n n2 log E � exp � s n σ2n Sn �� = σ2 n n2 ∞ � j=0 ϕZ � σ−2 n n(Aj − Aj−n)s � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='51) where the second equality follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='16), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='42) we see that lim n→∞ ψn(s) = s2/2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='52) uniformly for s in a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Furthermore, the sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='51) can be differ- entiated term by term, and it follows by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='17), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='42) that lim n→∞ ψ′ n(s) = s, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='53) also uniformly on compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since ψ′ n is increasing and continuous, for large n there exists a unique τn > 0 such that ψ′ n(τn) = ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='54) It is immediate that τn → ε as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Denoting θn = σ−2 n nτn, n ≥ 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='55) we have � E � eθnSn��−1 E � SneθnSn� = nε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='56) Fix k ≥ 1 and for each i = 1, 2, 3 fix points 0 = ti0 < ti1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' < tik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Denote ¯Sn = Sn − 3 � i=1 ξi n(tik), n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let Unij, n ≥ 1, i = 1, 2, 3, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , k, ˜Sn, n ≥ 1 be independent random variables, with P (Unij ∈ dx) = � E � eθn(ξi n(tij)−ξi n(ti j−1))��−1 eθnxP � ξi n(tij) − ξi n(ti j−1) ∈ dx � , and P � ˜Sn ∈ dx � = � E � eθn ¯Sn��−1 eθnxP � ¯Sn ∈ dx � for n ≥ 1, i = 1, 2, 3 and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let µnij = E (Unij) , µn = E( ˜Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='57) Chakrabarty and Samorodnitsky/Clustering of large deviations 13 It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='56) that µn + 3 � i=1 k � j=1 µnij = nε, n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='58) Let t > 0 and (αij) ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We have P �� T ∗ n > tσ2 n/nε � ∩ � k� j=1 � n2α−2 � ξ1 n(t1j) − ξ1 n(t1 j−1) − µn1j � > α1j �� ∩ � � 2≤i≤3, 1≤j≤k � nα−β/2−1 � ξi n(tij) − ξi n(ti j−1) − µnij � > αij ��� = � R3k+1 1 � x > nε + tσ2 n/nε − 3 � i=1 k � j=1 sij � 1 � s1j > n2−2αα1j + µn1j , 1 ≤ j ≤ k � 1 � sij > n1−α+β/2αij + µnij , i = 2, 3 , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' k � P( ¯Sn ∈ dx) 3 � i=1 k � j=1 P � ξi n(tij) − ξi n(ti j−1) ∈ dsij � = � R3k+1 1 � x > nε + tσ2 n/nε − 3 � i=1 k � j=1 sij � 1 � s1j > n2−2αα1j + µn1j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1 ≤ j ≤ k � 1 � sij > n1−α+β/2αij + µnij ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' i = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1 ≤ j ≤ k � exp � −θnx − θn 3 � i=1 k � j=1 sij � P � ˜Sn ∈ dx � E � eθnSn� 3 � i=1 k � j=1 P(Unij ∈ dsij) = cn � R3k 1 � min i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='j (uij − αij) > 0 � k � j=1 P � n2α−2� Un1j − µn1j � ∈ du1j � 3 � i=2 k � j=1 P � nα−β/2−1� Unij − µnij � ∈ duij � � R e−z1 � z > tθnσ2 n/nε � P � θn � ˜Sn − µn + γn(u11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u3k) � ∈ dz � , Chakrabarty and Samorodnitsky/Clustering of large deviations 14 with cn = e−θnnεE � eθnSn� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='59) and γn(u11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u3k) = n2−2α k � j=1 u1j + n1−α+β/2 3 � i=2 k � j=1 uij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let θn be as above and ηn = σnθn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For n ≥ 1, we introduce the notation fn(u11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u3k) =ηn � ∞ 0 e−z1 � z > tθnσ2 n/nε � P � θn � ˜Sn − µn + γn(u11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u3k) � ∈ dz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Fix (uij) and let u(n) ij → uij as n → ∞ for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let us denote γn = γn � u(n) 11 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u(n) 3k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' With θn and ηn already defined, we use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2 with this γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It is elementary to check that the hypothesis of the lemma are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since tθnσ2 n/nε → t, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='38) that for all fixed T > t, � R e−z1 � tθnσ2 n/nε < z ≤ T � P � θn � ˜Sn − µn + γn � ∈ dz � ∼ η−1 n (2π)−1/2 � T t e−z dz, and if follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='39) that lim T →∞ lim sup n→∞ ηn � R e−z1 � z > T � P � θn � ˜Sn − µn + γn � ∈ dz � = 0 , showing that lim n→∞ fn � u(n) 11 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u(n) 3k � = (2π)−1/2e−t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Another application of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='39) implies that sup {uij}⊂R fn(u11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u3k) < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows immediately from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3 and bounded convergence theorem that E � f � n2α−2(Un11 − µn11), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , nα−β/2−1(Un3k − µn3k) � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='60) 1 � n2α−2(Un1j − µn1j) > α1j, nα−β/2−1(Unij − µnij) > αij, i = 2, 3, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , k, �� →(2π)−1/2P (T0 > t , Gij > αij for all i, j) , with T0 standard exponential and (Gij : i = 1, 2, 3, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , k) independent zero mean Gaussian random variables, independent of T0, with Var(G1j) = K1σ2 Z � t3−2α 1j − t3−2α 1 j−1 � , 1 ≤ j ≤ k , Chakrabarty and Samorodnitsky/Clustering of large deviations 15 and for i = 2, 3, Var(Gij) = (1 − α)−2σ2 Z(tij − ti ,j−1), 1 ≤ j ≤ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' A simple way to verify the convergence above is to appeal to the Skorohod representation and replace the weak convergence in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3 by the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' con- vergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Notice that using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='60) with t = 0 and αij = −∞ for all i, j tells us that P(E0) ∼ (2π)−1/2cn/ηn = (2π)−1/2e−θnnεE � eθnSn� /(σnθn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='61) Dividing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='60) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='61) gives us the statement of the lemma apart from a possibly different centring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In order to complete the proof, it suffices to show that as n → ∞, for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , k, µn1j = [nβt1j] � i=[nβt1j−1]+1 Ai � ∞ −∞ z GζnAi(dz) + o � n2−2α� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='62) µn2j = n−[nβtnj−1] � i=n−[nβtnj] Ai � ∞ −∞ z GζnAi(dz) + o � n1+β/2−α� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='63) µn3j = n+[nβtnj] � i=n+[nβtnj−1] (Ai − Ai−n) � ∞ −∞ z Gζn(Ai−Ai−n)(dz) + o � n1+β/2−α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='64) For simplicity of notation we prove these statements for j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For θn as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='55), let ( ˜Zni, n ≥ 1, i ≥ 0) be a collection of independent random variables such that the law of ˜Zni is G(Ai−Ai−n)θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since both θnAi and ζnAi converge to zero uniformly in i ≤ nβt11, we can use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='19) to write µn11 = [nβt11] � i=1 AiE � ˜Zni � = [nβt11] � i=1 Ai � ∞ −∞ z GθnAi(dz) = [nβt11] � i=1 Ai � ∞ −∞ z GζnAi(dz) + o \uf8eb \uf8edζn [nβt11] � i=1 A2 i \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='22) that ζn [nβt11] � i=1 A2 i = o � n2−2α� , and we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='62) (for j = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='63) with j = 1 we notice that by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='17), E � ˜Zni � = θn(Ai − Ai−n)σ2 Z + O � θ2 n(Ai − Ai−n)2� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='65) Chakrabarty and Samorodnitsky/Clustering of large deviations 16 uniformly in i ≥ 0, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Thus, µn21 = σ2 Zσ−2 n nτn n−1 � i=n−[nβt21] A2 i + O \uf8eb \uf8edθ2 n n−1 � i=n−[nβt21] A3 i \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 that θ2 n n−1 � i=n−[nβt21] A3 i = O � nα+β−1� = o � n1−α+β/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Therefore, µn21 = σ2 Zσ−2 n nτn n−1 � i=n−[nβt21] A2 i + o � n1−α+β/2� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='66) and, similarly, n−1 � i=n−[nβt21] Ai � ∞ −∞ z GζnAi(dz) = σ2 Zζn n−1 � i=n−[nβt21] A2 i + o � n1−α+β/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Another appeal to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 shows that for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='63) we only need to argue that τn = ε + o � n1−α−β/2� , n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='67) However, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='19), ψ′ n(s) = s + O \uf8eb \uf8ednσ−4 n ∞ � j=0 (Aj − Aj−n)3 \uf8f6 \uf8f8 , uniformly for s in compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Using this and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='44), we obtain ε = ψ′ n(τn) = τn + O \uf8eb \uf8ednσ−4 n ∞ � j=0 (Aj − Aj−n)3 \uf8f6 \uf8f8 = τn + O(nα−1) = τn + o � n1−α−β/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' This establishes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='67) and, hence, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='63) with j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='64) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' None of the statements proved so far required the additional assumptions stated at the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' These assumptions start to play a role now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 17 The next several lemmas require additional notation designed to focus on the contribution of individual noise variables on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For n ≥ 1 and i, j ≥ 0, i ̸= j, we set S′ n(i) = Sn − (Ai − Ai−n)Zn−i−1 , S′ n(i, j) = Sn − (Ai − Ai−n)Zn−i−1 − (Aj − Aj−n)Zn−j−1 , and, with ζn given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='50), we let ˆSn, ˆSni, ˆSn(i, j) be random variables with distributions P( ˆSn ∈ ds) ∝ eζnsP(Sn ∈ ds) , P( ˆSn(i) ∈ ds) ∝ eζnsP(S′ n(i) ∈ ds) , P( ˆSn(i, j) ∈ ds) ∝ eζnsP(S′ n(i, j) ∈ ds) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Denote the characteristic functions of σ−1 n ( ˆSn − nε), σ−1 n ( ˆSn(i) − nε) and σ−1 n ( ˆSn(i, j) − nε) by φn, φni and φnij, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For µ ∈ R and σ ≥ 0 we denote by φG(µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' σ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' ·) the characteristic function of N(µ, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let κ be given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4) and assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then the following statements hold uniformly in t ∈ R: |φn(t) − φG(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t)| = O � n1/2−κ(1−α)(1 + |t|)κ+1� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='68) sup i≥0 ��φni(t) − φG � σ−1 n nε(λni − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' λni;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ��� = O � n1/2−κ(1−α)(1 + |t|)κ+1� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='69) sup i,j≥0 i̸=j ��φnij(t) − φG � σ−1 n nε(λnij − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' λnij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ��� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='70) = O � n1/2−κ(1−α)(1 + |t|)κ+1� , where for n ≥ 1 and i, j ≥ 0, i ̸= j, we set λni = 1 − σ2 Z σ2n (Ai − Ai−n)2, λnij = 1 − σ2 Z σ2n � (Ai − Ai−n)2 + (Aj − Aj−n)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It is an elementary conclusion from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) that, for each 1 ≤ i ≤ κ, �� R eδz Fz(dz) �−1 � R zieδz Fz(dz) = σi ZE � (G + δσZ)i� + O � |δ|κ−i+1� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='71) as δ → 0, where G is a standard Gaussian random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let ( ˆZni : n ≥ 1, i ≥ 0) be a family of independent random variables with each ˆZni ∼ G(Ai−Ai−n)ζn, so that for n ≥ 1 and i, j ≥ 0, i ̸= j we have ˆSn d= ∞ � k=0 (Ak − Ak−n) ˆZnk , Chakrabarty and Samorodnitsky/Clustering of large deviations 18 ˆSn(i) d= � k∈{0,1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='}\\{i} (Ak − Ak−n) ˆZnk , ˆSn(i, j) d= � k∈{0,1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='}\\{i,j} (Ak − Ak−n) ˆZnk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let now (Gni : n ≥ 1, i ≥ 0) be a collection of independent random variables, also independent of ( ˆZni : n ≥ 1, i ≥ 0), where Gni ∼ N � (Ai − Ai−n)ζnσ2 Z , σ2 Z � , for all n ≥ 1, i ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='42) that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='71) can be reformulated as E � ˆZi nj � − E � Gi nj � = O � |Aj − Aj−n|κ−i+1n−2(1−α)(κ−i+1)� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='72) uniformly in j ≥ 0 and 1 ≤ i ≤ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For a fixed t ∈ R we use telescoping to write ������ E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n ∞ � j=0 (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe − E exp � i � tσ−1 n ˆSn �� ������ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='73) ≤ ∞ � j=0 ������ E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n \uf8eb \uf8ed j−1 � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe −E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n \uf8eb \uf8ed j � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j+1 (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Fix j ≥ 0 and denote U = tσ−1 n \uf8eb \uf8ed j−1 � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j+1 (Aj − Aj−n)Gnj \uf8f6 \uf8f8 , V = tσ−1 n (Aj − Aj−n)Gnj , so that by expanding in the Taylor series around U, E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n \uf8eb \uf8ed j−1 � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe = Eei(U+V ) = κ � m=0 im m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='E (V m) EeiU + R1 , with |R1| ≤ E(|V |κ+1)/(κ + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='. Similarly, E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n \uf8eb \uf8ed j � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j+1 (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe = κ � m=0 im m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='E (W m) EeiU + R2 , Chakrabarty and Samorodnitsky/Clustering of large deviations 19 with |R2| ≤ E(|W|κ+1)/(κ + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=', where W = (Aj − Aj−n) ˆZnj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We conclude that ������ E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n \uf8eb \uf8ed j−1 � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe −E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n \uf8eb \uf8ed j � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j+1 (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe ������ ≤ κ � i=1 |t|i i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' ���(Aj − Aj−n)iσ−i n E � ˆZi nj − Gi nj ���� + |t|κ+1 (κ + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' |Aj − Aj−n|κ+1 σ−(κ+1) n E � |Gnj|κ+1 + | ˆZnj|κ+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='74) Note that by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='44) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1, σ−(κ+1) n ∞ � j=0 |Aj − Aj−n|κ+1 E � |Gnj|κ+1 + | ˜Znj|κ+1� = O � n−(κ−1)/2� = o � n1/2−κ(1−α)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For 1 ≤ i ≤ κ we use, in addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='72) to write σ−i n ∞ � j=0 ���(Aj − Aj−n)iE � ˜Zi nj − Gi nj ���� = O � n−κ(1−α)+α−i(α−1/2)� = O � n1/2−κ(1−α)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Putting these bounds into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='74) we obtain E � eιtσ−1 n ˜Sn� = φG � σ−1 n nε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t � + O � n1/2−κ(1−α) � 1 + |t|κ+1�� uniformly for t ∈ R, which is equivalent to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The argument for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='69) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='70) is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By the assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2), for large n, the random variables σ−1 n ( ˆSn − nε), σ−1 n ( ˆSn(i) − nε) and σ−1 n ( ˆSn(i, j) − nε) have densities which we denote by fn, fni and fnij, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) and(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then for large n, the densities fni and fnij are twice differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Furthermore, as n → ∞, fni(0) = (2π)−1/2 + o � n1−2α� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='75) f ′ ni(0) = o � n1/2−α� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='76) Chakrabarty and Samorodnitsky/Clustering of large deviations 20 uniformly in i, and for some n0 ∈ N, sup {|f ′′ ni(x)| : n ≥ n0, i ≥ 0, x ∈ R} < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='77) All three statements also hold if fni is replaced by fnij, i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Finally, as n → ∞, sup x∈R ���fn(x) − (2π)−1/2e−x2/2��� = o � n1−2α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='78) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We start with the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='78) which would follow from the inversion formula for densities once it is shown that � ∞ −∞ |φn(t) − φG(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t)| dt = o � n1−2α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4), � log n − log n |φn(t) − φG(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t)| dt = O � n1/2−κ(1−α)(log n)κ+2� = o � n1−2α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Furthermore, � [− log n,log n]c φG(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t) dt = O � e−(log n)2/2� = o � n1−2α� , Thus, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='78) will follow once we show that � [− log n,log n]c |φn(t)| dt = o � n1−2α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='79) With ( ˆZni : n ≥ 1, i ≥ 0) as above, we set Uni = σ−1 n (Ai − Ai−n) � ˆZni − E( ˆZni) � , n ≥ 1, i ≥ 0 , so that |φn(t)| = ∞ � i=0 ��E � eιtUni��� , n ≥ 1, t ∈ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='80) Set H(x, t) = �� ∞ −∞ exzfZ(z) dz �−1 � ∞ −∞ e(x+ιt)zfZ(z) dz, (x, t) ∈ R2 , which is a characteristic function for any fixed x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' A consequence of that is ∂|H(x, t)|/∂t|t=0 ≤ 0 for any x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Furthermore, ∂2 ∂t2 |H(0, t)| ��� t=0 = −σ2 Z < 0 Chakrabarty and Samorodnitsky/Clustering of large deviations 21 and by continuity of the second partial derivative we conclude that there is δ0 > 0 such that ∂2 ∂t2 |H(x, t)| ��� < 0 whenever 0 ≤ |t|, |x| ≤ δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' That means we also have ∂ ∂t|H(x, t)| ��� ≤ 0 whenever 0 ≤ |t|, |x| ≤ δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='81) We may and will choose δ0 ∈ (0, θ0], with θ0 as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) we can appeal to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3) to conclude that lim t→∞ sup |x|≤δ0 |H(x, t)| = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Thus, there is M > 0 large enough so that sup t>M,|x|≤δ0 |H(x, t)| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since by continuity of H and compactness we have sup δ0≤t≤M,|x|≤δ0 |H(x, t)| < 1, it follows that η = sup t≥δ0,|x|≤δ0 |H(x, t)| < 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The continuity argument also shows that there is δ1 ∈ (0, δ0] such that min |x|≤δ0 |H(x, δ1)| ≥ η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Therefore, for |x| ≤ δ0 and 0 ≤ t ≤ δ1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='81) implies that |H(x, t)| ≥ |H(x, δ1)| ≥ η ≥ sup s≥δ0 |H(x, s)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='81) we also have |H(x, t)| = sup s∈[t,δ0] |H(x, s)| , we conclude that |H(x, t)| = sup s≥t |H(x, s)|, |x| ≤ δ0, 0 ≤ t ≤ δ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='82) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='80) |φn(t)| ≤ ��E(eιtUnn) �� n−1 � i=[n/2] ��E(eιtUni) �� = ��E(eιtUnn) �� n−1 � i=[n/2] ��H � ζnAi, σ−1 n Ait ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='83) Chakrabarty and Samorodnitsky/Clustering of large deviations 22 It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 that there exists s0 > 0 such that for all n large enough, Ai ≥ s0σnn−1/2, [n/2] ≤ i ≤ n − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Thus, for n large enough and t ≥ log n, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='82) implies that n−1 � i=[n/2] ��H � ζnAi, σ−1 n Ait ��� ≤ n−1 � i=[n/2] ���H � ζnAi, s0n−1/2 log n ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since any partial derivative of H is bounded on a compact set, we can use the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4) to conclude that there exists s1 > 0 such that sup |x|≤δ0 |H(x, t)| ≤ (1 − s1t2)1/2, 0 ≤ t ≤ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Thus, there is s2 > 0 such that for all large n and all t ≥ log n we have n−1 � i=[n/2] ��H � ζnAi, σ−1 n Ait ��� ≤ � 1 − s2 0s1n−1(log n)2�n/4 = O � e−s2(log n)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Using this bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='83), and appealing to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) we obtain � ∞ log n |φn(t)| dt = O � e−s2(log n)2� � ∞ log n ��E � eitUnn��� dt = O � n1/2e−s2(log n)2� = o � n1−2α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since we can switch from t to −t, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='79) follows, which establishes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' A similar calculation with the aid of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='69) shows that fni(0) = (2πλni)−1/2 exp � −σ−2 n n2ε2(λni − 1)2/2λni � + o � n1−2α� , uniformly in i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since λni − 1 = O(1/n) uniformly in i ≥ 0, it follows that λ−1/2 ni exp � −σ−2 n n2ε2(λni − 1)2/2λni � = 1 + O � n−1 + σ−2 n � = 1 + o � n1−2α� , uniformly for i ≥ 0, which proves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='77) we write f ′′ nk(x) = −(2π)−1/2 � ∞ −∞ e−itxt2φnk(t) dt and repeat the arguments used above in the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='78), applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='69) and the full force of the assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Finally, for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='76) we use the identity f ′ nk(0) = −i(2π)−1/2 � ∞ −∞ tφnk(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 23 Since ���� � ∞ −∞ t φG � σ−1 n nε(λnk − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' λnk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t � dt ���� = O � σ−1 n � = o � n1/2−α� , uniformly in k ≥ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='76) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The arguments with fnij replacing fni are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The next lemma tackles certain expectations conditionally on E0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' its state- ment should be compared to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) and(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then E (Zn−i−11(E0)) = Kn �� ∞ −∞ z Gζn(Ai−Ai−n)(dz) + o � ζ−1 n σ−2 n |Ai − Ai−n| �� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='84) and E (Zn−i−1Zn−j−11(E0)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='85) = Kn �� ∞ −∞ z1 Gζn(Ai−Ai−n)(dz1) � ∞ −∞ z2 Gζn(Ai−Ai−n)(dz2) + o � σ−2 n |(Ai − Ai−n)(Aj − Aj−n)| � � , n → ∞, uniformly for i, j ≥ 0 with i ̸= j, where Kn = (2π)−1/2ζ−1 n σ−1 n e−nεζnE � eζnSn� , n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='86) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We only prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='85);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='84) is similar and easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Write E (Zn−i−1Zn−j−11(E0)) = � ∞ −∞ z1 FZ(dz1) � ∞ −∞ z2 FZ(dz2) P (S′ n(i, j) ≥ nε − (Ai − Ai−n)z1 − (Aj − Aj−n)z2) = σ−1 n E � eζnS′ n(i,j)� � ∞ −∞ z1 FZ(dz1) � ∞ −∞ z2 FZ(dz2) � ∞ nε−(Ai−Ai−n)z1−(Aj−Aj−n)z2 fnij � s − nε)/σn � e−ζns ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We adopt the convention � b a ≡ − � a b , and denote cnij = ζ−1 n σ−1 n e−nεζnE � eζnS′ n(i,j)� = Kn(2π)1/2 �� ∞ −∞ eζn(Ai−Ai−n)zFZ(dz) � ∞ −∞ eζn(Aj−Aj−n)zFZ(dz) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 24 Changing the variable and using the fact that EZ = 0, we obtain E (Zn−i−1Zn−j−11(E0)) = cnij � ∞ −∞ z1 FZ(dz1) � ∞ −∞ z2 FZ(dz2) � ζn(Ai−Ai−n)z1+ζn(Aj−Aj−n)z2 0 exfnij � −x/(σnζn) � dx = cnij � ∞ −∞ z1 FZ(dz1) � ∞ −∞ z2 FZ(dz2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='87) �� ζn(Ai−Ai−n)z1+ζn(Aj−Aj−n)z2 0 exfnij � −x/(σnζn) � dx − � ζn(Ai−Ai−n)z1 0 exfnij � −x/(σnζn) � dx − � ζn(Aj−Aj−n)z2 0 exfnij � −x/(σnζn) � dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For fixed z1, z2 ∈ R, the expression inside the square brackets can be rewritten as � eζn(Ai−Ai−n)z1 − 1 � � ζn(Aj−Aj−n)z2 0 ex fnij � −(x + ζn(Ai − Ai−n)z1)/(σnζn) � dx + � ζn(Aj−Aj−n)z2 0 ex � fnij � −(x + ζn(Ai − Ai−n)z1)/(σnζn) � − fnij � −x/(σnζn) � � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By Taylor’s theorem, fnij � − x + ζn(Ai − Ai−n)z1 σnζn � = fnij(0) − x + ζn(Ai − Ai−n)z1 σnζn f ′ nij(0) +O �(x + ζn(Ai − Ai−n)z1)2 σ2nζ2n ∥f ′′ nij∥∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 25 Using this and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='42), straightforward algebra gives us � ζn(Aj−Aj−n)z2 0 exfnij � −(x + ζn(Ai − Ai−n)z1)/(σnζn) � dx = fnij(0) � eζn(Aj−Aj−n)z2 − 1 � + O � eζn|Aj−Aj−n||z2|� |f ′ nij(0)|σ−1 n ζnn1−α|Aj − Aj−n||z2| � |z1| + |z2| � + ∥f ′′ nij∥∞σ−2 n ζnn2−2α|Aj − Aj−n||z2| � |z1| + |z2| �2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The obvious inequality |ex − 1| ≤ |x|e|x| for x ∈ R along with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6 now show that � eζn(Ai−Ai−n)z1 − 1 � � ζn(Aj−Aj−n)z2 0 ex fnij � −(x + ζn(Ai − Ai−n)z1)/(σnζn) � dx = fnij(0) � eζn(Ai−Ai−n)z1 − 1 � � eζn(Aj−Aj−n)z2 − 1 � + o � σ−2 n |(Ai − Ai−n)(Aj − Aj−n)z1z2| (|z1| + |z2|)2 eζn(|Ai−Ai−n||z1|+|Aj−Aj−n||z2|)� , uniformly for i, j ≥ 0 with i ̸= j and z1, z2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Treating in a similar manner the second term, we conclude that the expression inside the square brackets in the right hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='87) equals fnij(0) � eζn(Ai−Ai−n)z1 − 1 �� eζn(Aj−Aj−n)z2 − 1 � + o � σ−2 n |(Ai − Ai−n)(Aj − Aj−n)| (1 + |z1|3)(1 + |z2|3) eζn|(Ai−Ai−n)z1|+ζn|(Aj−Aj−n)z2|� , uniformly for i, j ≥ 0 with i ̸= j and z1, z2 ∈ R, and substitution into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='87) gives us E (Zn−i−1Zn−j−11(E0)) = cnij � fnij(0) � ∞ −∞ z1eζn(Ai−Ai−n)z1FZ(dz1) � ∞ −∞ z2eζn(Aj−Aj−n)z2FZ(dz2) + o � σ−2 n |(Ai − Ai−n)(Aj − Aj−n)| �� Chakrabarty and Samorodnitsky/Clustering of large deviations 26 = Kn(2π)1/2fnij(0) � ∞ −∞ z1 Gζn(Ai−Ai−n)(dz1) � ∞ −∞ z2 Gζn(Aj−Aj−n)(dz2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='88) + cnijo � σ−2 n |(Ai − Ai−n)(Aj − Aj−n)| � , as n → ∞, uniformly for i, j ≥ 0 with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Recalling that EZ = 0, we see that � ∞ −∞ z1 Gζn(Ai−Ai−n)(dz1) = O (ζn(Ai − Ai−n)) , and likewise for the second integral in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since Kn = O(cnij), the claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='85) follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The next lemma is an important step in the proof of the main result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' the previous lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='7 are needed for this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We denote Yni = Zn−i−1 − � 1 + ζ−2 n σ−2 n � � ∞ −∞ z Gζn(Ai−Ai−n)(dz), i ∈ Z, n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='89) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) and(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then sup n≥1,i≥0 E � Y 2 ni ��E0 � < ∞ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='90) and E � YniYnj ��E0 � = −σ−2 n σ4 Z (Ai − Ai−n) (Aj − Aj−n) (1 + o(1)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='91) as n → ∞, uniformly in i, j ≥ 0 with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='91);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='90) is similar (and much easier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We write P(E0) = Kn(2π)1/2 � ∞ 0 e−xfn � x/(ζnσn) � dx, with Kn as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='86).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='78) and simple integration, P(E0) =Kn(2π)1/2 � o � ζ−2 n σ−2 n � + (2π)−1/2 � ∞ 0 exp � −x − x2/(2ζ2 nσ2 n) � dx � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='92) =Kn � 1 − ζ−2 n σ−2 n (1 + o(1)) � , n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In combination with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='85) this means that E (Zn−i−1Zn−j−11(E0)) P(E0) = K2 n � � 1 − ζ−2 n σ−2 n � � ∞ −∞ z1 Gζn(Ai−Ai−n)(dz1) � ∞ −∞ z2 Gζn(Aj−Aj−n)(dz2) + o � σ−2 n |(Ai − Ai−n)(Aj − Aj−n)| � � , n → ∞, Chakrabarty and Samorodnitsky/Clustering of large deviations 27 uniformly in i, j ≥ 0 with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='84), E (Zn−i−11(E0)) E (Zn−j−11(E0)) = K2 n � ∞ −∞ z1 Gζn(Ai−Ai−n)(dz1) � ∞ −∞ z2 Gζn(Aj−Aj−n)(dz2) + o � K2 nσ−2 n |Ai − Ai−n||Aj − Aj−n| � , we conclude that E (Zn−i−1Zn−j−11(E0)) P(E0) − E (Zn−i−11(E0)) E (Zn−j−11(E0)) = −K2 nζ−2 n σ−2 n � ∞ −∞ z1 Gζn(Ai−Ai−n)(dz1) � ∞ −∞ z2 Gζn(Aj−Aj−n)(dz2) + o � K2 nσ−2 n |Ai − Ai−n||Aj − Aj−n| � = −K2 nσ−2 n σ4 Z(Ai − Ai−n)(Aj − Aj−n) (1 + o(1)) as n → ∞, uniformly in i, j ≥ 0 with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Dividing both sides by P(E0)2 and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='92), we obtain E �� Zn−i−1 − E(Zn−i−1|E0) �� Zn−j−1 − E(Zn−j−1|E0) ����E0 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='93) = −σ−2 n σ4 Z(Ai − Ai−n)(Aj − Aj−n) (1 + o(1)) , as n → ∞, again uniformly for i, j ≥ 0 with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='92) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='84) E (Zn−i−1|E0) = � 1 + ζ−2 n σ−2 n � � ∞ −∞ z Gζn(Ai−Ai−n)(dz) + o � ζ−1 n σ−2 n |Ai − Ai−n| � , with a similar statement for Zn−j−1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='93) implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We proceed with establishing conditional distributional limits of certain trun- cated sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) and(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For 0 < δ < L denote Sn(j, δ, L) = [nβL]−1 � i=[nβδ] (Ai+j − Ai)Yni + n−1 � i=n−j (Ai+j − Ai+j−n − Ai)Yni (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='94) + n+[nβL] � i=n (Ai+j − Ai+j−n − Ai + Ai−n)Yni, n ≥ 1, j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' With the overshoot T ∗ n as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='49), we have, conditionally on E0, � ζnT ∗ n, � n2α−2Sn([nβt], δ, L), t ≥ 0 �� ⇒ � T0, � (1 − α)−1σZ �� L δ � (s + t)1−α − s1−α� dB1(s) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='95) + � t 0 (t − s)1−αdB2(s) + � L 0 � s1−α − (s + t)1−α� dB3(s) � , t ≥ 0 �� Chakrabarty and Samorodnitsky/Clustering of large deviations 28 in finite dimensional distributions as n → ∞, where T0 is a standard exponen- tial random variable independent of independent standard Brownian motions B1, B2, B3, Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For n ≥ 1 and t ≥ 0 we write ξ1◦ n (t) = [nβt] � i=1 AiYni, ξ2◦ n (t) = n−1 � i=n−[nβt] AiYni, ξ3◦ n (t) = n+[nβt] � i=n+1 (Ai − Ai−n) Yni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4 that, conditionally on E0, � ζnT ∗ n, � n2α−2ξ1◦ n (t) : t ≥ 0 � , � nα−β/2−1ξ2◦ n (t) : t ≥ 0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='96) � nα−β/2−1ξ3◦ n (t) : t ≥ 0 �� ⇒ � T0, � K1/2 1 σZB1(t3−2α) : t ≥ 0 � , � (1 − α)−1σZB2(t) : t ≥ 0 � , � (1 − α)−1σZB3(t) : t ≥ 0 �� because the difference between the two processes vanishes in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For ex- ample, n2α−2ζ−2 n σ−2 n [nβt] � i=1 Ai � ∞ −∞ z GζnAi(dz) = O � n1−2α� = o(1) , and similarly with the other two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Furthermore, for large n, Sn([nβt], δ, L) = [nβL]−1 � i=[nβδ] (Ai+[nβt] − Ai)Yni + n−1 � i=n−[nβt] (Ai+[nβt] − Ai+[nβt]−n − Ai)Yni + n+[nβL] � i=n (Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n)Yni =: V 1 n (t) + V 2 n (t) + V 3 n (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Starting with V 3 n , we write V 3 n (t) = n−(1−α)(1−β) [nβL] � i=1 fn � n−βi, t � (An+i − Ai) Yn,n+i , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='97) Chakrabarty and Samorodnitsky/Clustering of large deviations 29 where for 0 ≤ s ≤ L, fn(s, t) = n(1−α)(1−β) An+[nβs]+[nβt] − A[nβs]+[nβt] − An+[nβs] + A[nβs] An+[nβs] − A[nβs] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It is elementary that for fixed s, t, as n → ∞, An+[nβs]+[nβt] − An+[nβs] ≪ A[nβs]+[nβt] − A[nβs] ∼ (1 − α)−1nβ(1−α) � (s + t)1−α − s1−α� , while An+[nβs] − A[nβs] ∼ (1 − α)−1n1−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Therefore, lim n→∞ fn(s, t) = s1−α − (s + t)1−α =: f(s, t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='98) and the limit is easily seen to be uniform in 0 ≤ s ≤ L and t in a compact interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We will show that, conditionally on E0, � n2α−2V 3 n (t), t ≥ 0 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='99) ⇒ � σZ(1 − α)−1 � L 0 � s1−α − (s + t)1−α� dB3(s), t ≥ 0 � in finite-dimensional distributions, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' To this end, set cnj(k, t) = inf (j−1)L/k≤s≤jL/k fn(s, t), k ≥ 1, 1 ≤ j ≤ k , and eni(k, t) = fn � n−βi, t � − cn,⌈L−1n−βki⌉(k, t) ≥ 0, k ≥ 1, 1 ≤ i ≤ [nβL] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='98) and monotonicity, lim n→∞ cnj(k, t) = f � (j − 1)k−1L, t � , 1 ≤ j ≤ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='100) A standard continuity argument shows that lim k→∞ lim sup n→∞ sup t∈A max 1≤i≤[nβL] eni(k, t) = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='101) for any compact set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We have [nβL] � i=1 cn,⌈L−1n−βki⌉(k, t)(An+i − Ai)Yn,n+i = k′ � j=1 cnj(k, t) � i∈ � k−1Lnβ(j−1),k−1Lnβj � ∩Z (An+i − Ai)Yn,n+i = k′ � j=1 cnj(k, t) � ξ3◦ n � k−1Lj � − ξ3◦ n � k−1L(j − 1) �� =: Wnk(t) , Chakrabarty and Samorodnitsky/Clustering of large deviations 30 where k′ = ⌈L−1n−βk[nβL]⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' This, together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='96) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='100), implies that for fixed k, as n → ∞, � nα−β/2−1Wnk(t), t ≥ 0 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='102) ⇒ � (1 − α)−1σZ k � j=1 f � (j − 1)k−1L, t � � B3(k−1jL) − B3(k−1(j − 1)L) � , t ≥ 0 � in finite-dimensional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We have [nβL] � i=1 fn � n−βi, t � (An+i − Ai) Yn,n+i − Wnk(t) = [nβL] � i=1 eni(k, t) (An+i − Ai) Yn,n+i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='91) that, for large n, sup i,j≥0:i̸=j (Ai − Ai−n) (Aj − Aj−n) E (YniYnj|E0) ≤ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' This, along with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='90) and the non-negativity of each eni, implies that for large n, E \uf8eb \uf8ec \uf8ed \uf8ee \uf8f0 [nβL] � i=1 eni(k, t) (An+i − Ai) Yn,n+i \uf8f9 \uf8fb 2�����E0 \uf8f6 \uf8f7 \uf8f8 ≤ [nβL] � i=1 [eni(k, t) (An+i − Ai)]2 E(Y 2 n,n+i|E0) = O \uf8eb \uf8ed max 1≤j≤[nβL] enj(k, t)2 [nβL] � i=1 (An+i − Ai)2 \uf8f6 \uf8f8 = O � n2−2α+β max 1≤j≤[nβL] enj(k, t)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Invoking (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='101) we conclude that for any compact set A, lim k→∞ lim sup n→∞ n2α−β−2 sup t∈A E �� Wnk(t) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='103) − [nβL] � i=1 fn � n−βi, t � (An+i − Ai) Yn,n+i �2�����E0 � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 31 As k → ∞, the process in the right hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='102) converges in finite- dimensional distributions to the process in the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since (2α− 2)− (1 − α)(1 − β) = α− β/2 − 1, the claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='99) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='97) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='103) by the “convergence together” argument;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2 in Billingsley (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' A nearly identical argument shows that, conditionally on E0, � n2α−2V 2 n (t), t ≥ 0 � ⇒ � −σZ(1 − α)−1 � t 0 (t − s)1−αdB2(s), t ≥ 0 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='104) d= � σZ(1 − α)−1 � t 0 (t − s)1−αdB2(s), t ≥ 0 � in finite-dimensional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The situation with the term V 1 n is, once again, similar, with a small twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since lim n→∞ A[nβs]+[nβt] − A[nβs] A[nβs] = (s + t)1−α − s1−α s1−α uniformly for δ ≤ s ≤ L and t, our argument now shows that, conditionally on E0, � n−(2−2α)V 1 n , t ≥ 0 � ⇒ � σZK1/2 1 � L δ (s + t)1−α − s1−α s1−α M(ds), t ≥ 0 � in finite-dimensional distributions, where M is a centred Gaussian random mea- sure with the variance measure with the density (3 − 2α)s2−2α, s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since the centred Gaussian random measures (1 − α)−1B3(ds) and K1/2 1 M(ds)/s1−α have the same variance measure, this means that, conditionally on E0, � n2α−2V 2 n (t), t ≥ 0 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='105) ⇒ � σZ(1 − α)−1 � L δ � (s + t)1−α − s1−α� dB3(s), t ≥ 0 � in finite-dimensional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='99), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='104) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='105) are all consequences of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='96), the conver- gence statements they contain hold jointly, and jointly with ζnT ∗ n ⇒ T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='95) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The next lemma treats the sequence of shifts appearing due to conditioning on E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Define µn(t) = n2α−2 ∞ � i=0 � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n � � ∞ −∞ z Gζn(Ai−Ai−n)(dz), for t ≥ 0 and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then µn → µ∞ as n → ∞, in D([0, ∞)) equipped with the Skorohod J1 topology, where µ∞(t) = −εt3−2α, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 32 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Writing µn(t) =n2α−2ζn ∞ � i=0 � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n � � Ai − Ai−n � +n2α−2 ∞ � i=0 � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n � �� ∞ −∞ z Gζn(Ai−Ai−n)(dz) − ζn � Ai − Ai−n �� =: µ(1) n (t) + µ(2) n (t), t ≥ 0, the claim of the lemma will follow once we prove that µ(1) n → µ∞ in D([0, ∞)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='106) and µ(2) n (t) → 0 uniformly on compact intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='107) We start by proving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Fix L > 0 so that 0 ≤ t ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose first that 1/2 < α < 5/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='19) ��µ(2) n (t) �� =O � n2α−2ζ2 n ∞ � i=0 ��Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �� � Ai − Ai−n �2 � =O � n2α−2ζ2 nnβ ∞ � i=1 i−α� Ai − Ai−n �2 � = O � n2α−2ζ2 nnβn3−3α� → 0 uniformly in 0 ≤ t ≤ L, showing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' On the other hand, if α ≥ 5/6, then κ ≥ 3 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5), so by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='19) ��µ(2) n (t) �� =O � n2α−2ζ3 n ∞ � i=0 ��Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �� � Ai − Ai−n �3 � =O � n2α−2ζ3 nnβ ∞ � i=1 i−α� Ai − Ai−n �3 � = O � n2α−2ζ3 nnβn4−4α� → 0 uniformly in 0 ≤ t ≤ L, again showing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We now prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The pointwise convergence is clear: for fixed t, µ(1) n (t) = σ2 Zσ−2 n n2α−1ε ∞ � i=0 � Ai+[nβt] − Ai+[nβt]−n � (Ai − Ai−n) − n2α−1ε → −εt3−2α Chakrabarty and Samorodnitsky/Clustering of large deviations 33 as n → ∞, where we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Next, as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='28) we can write for t ≥ 0, µ(1) n (t) =n2α−2ζn 2 �n−1 � i=0 � Ai − Ai−[nβt] �2 + ∞ � i=n−[nβt] � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �2 � =: µ(11) n (t) + µ(12) n (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='106) will follow once we show that both µ(11) n and µ(12) n converge in D([0, ∞)) to continuous limits (both constant factors of µ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The fact that µ(11) n converges pointwise to a constant factor of of the pointwise limit of µ(1) n is an intermediate step in the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since µ(11) n is a monotone function, its convergence in D([0, ∞)) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We already know that µ(12) n converges pointwise to a continuous limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let i0 be such that ai is monotone for i ≥ i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Write for t ≥ 0 µ(12) n (t) =n2α−2ζn 2 � ∞ � i=n+i0 � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �2 − n+i0−1 � i=n−[nβt] � Ai+[nβt] − Ai+[nβt]−n − Ai �2 � =: µ(121) n (t) − µ(122) n (t), so it is enough to show that both µ(121) n and µ(122) n converge in D([0, ∞)) to continuous limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Splitting further, we write for t ≥ 0, µ(122) n (t) = n2α−2ζn 2 � n+i0−1 � i=n−[nβt] A2 i+[nβt]−n + n+i0−1 � i=n−[nβt] � Ai − Ai+[nβt] �� Ai − Ai+[nβt] − 2Ai+[nβt]−n � � =: µ(1221) n (t) + µ(1222) n (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Clearly, µ(1221) n (t) = n2α−2ζn 2 [nβt]+i0−1 � i=0 A2 i converges pointwise to a constant factor of µ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since µ(1221) n is monotone, we conclude that µ(1221) n converges in D([0, ∞)) to a continuous limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In order to Chakrabarty and Samorodnitsky/Clustering of large deviations 34 prove that so does µ(122) n , we will show that µ(1222) n (t) → 0 uniformly on compact intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Considering once again 0 ≤ t ≤ L, we have ��µ(1222) n (t) �� ≤ n2α−2ζn 2 n+i0−1 � i=n−[nβt] � Ai+[nβt] − Ai ��� Ai+[nβt] − Ai � + 2Ai+[nβt]−n � = O \uf8eb \uf8edn2α−2ζn n+i0−1 � i=n−[nβt] nβn−α� nβn−α + nβ(1−α)� \uf8f6 \uf8f8 = O � nα−2ζnn3β−βα� → 0 uniformly over 0 ≤ t ≤ L, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Finally, we already know that µ(121) n converges pointwise to a continuous limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Furthermore, by the choice of i0, µ(121) n is a monotone function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Therefore, it converges in D([0, ∞)), and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The following is the final lemma before we prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) and(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let Sn(j) = j+n−1 � i=j Xi, j ≥ 0, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='108) As n → ∞, conditionally on E0, � n−(2−2α) � Sn([nβt]) − nε � , t ≥ 0 � ⇒ � (2Cα)1/2BH(t) + ε−1Cασ2 ZT0 − εt3−2α, t ≥ 0 � in finite-dimensional distributions, where (BH(t) : t ≥ 0) is the standard frac- tional Brownian motion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='8) with the Hurst exponent H given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='7), Cα is the constant defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='10), and T0 is a standard exponential random variable independent of the fractional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='91) and the eventual monotonicity of the sequence (An) that there is i0 ≥ 0 such that for all large n, sup i0≤i