dataesr/scientific-paragraphs-categorization huggingface dataset:
Time complexity The time complexity of our method can be divided into two steps: the preprocessing time of the construction of the trie of occurrences and the drawing time of an occurrence. Preprocessing time. It is the most expensive phase of TPSpace. A first pass on the database is necessary to retrieve the items from the database T and compute their frequencies in O(||T ||) where ||T || is the sum of the lengths transactions from the database T. The previously retrieved items are ordered according to the total order relation chosen > I in O(|I| × log(|I|)). Then, before adding a transaction to the trie, we order its items according to the total order relation > I in O(T max × log(T max )) where T max is the maximum length of transactions in the database T. Finally, if z is the total number of nodes in the trie following the total order relation > I , µ and M the minimum and maximum length constraints respectively, then the weighting of the nodes is done in O(z × 2 × (M × µ)).
Thus, the total complexity for building the trie of occurrence of the transactional database T built on the set of literals I is in O(||T || + |I| × log(|I|) + |T | × T max × log(T max ) + z × (M × µ)). However, remember that this preprocessing of a database is done only once. Reprocessing time for utility change. When the utility changes without updating the length constraints µ and M , the complexity of the reprocessing time is in O(Mµ), which is particularly fast. This is because only the array of the root of the trie is traveled to compute the new weight of each length ℓ ∈ [µ. M ]. It's clear that in this paper we deal only with utility change for the reprocessing because we suppose that M is large and no longer needed to be increased to avoid the long tail problem. But, when the user needs to increase the maximum length constraint from M to M ′ , the complexity of the reprocessing is bounded by O((M ′ -M ) × z), where z is the total number of nodes in the trie.
Experiments This experimental section aims to assess the efficiency of our approach to large transactional datasets. It's important to note that our main goal is not to compare pattern sampling algorithms with exhaustive pattern mining methods based on compact structure  [32, 33]. Previous works  [3, 12, 10]  already justified the advantages of output pattern sampling over exhaustive pattern mining. The difficulties encountered in exhaustive pattern mining, such as setting the threshold or controlling the output size, are not resolved by the compact representation. So, we first evaluate the memory size occupied by the trie of occurrences according to the maximum length constraint and the total order relation on the items > I ∈ {> f req I , > lexico I }. Second, we study the building time of a trie of occurrences according to the maximum length constraint on different datasets and then evaluate the speed of drawing a pattern by TPSampling.
Finally, we show the scalability of TPSampling on large databases. The experiments were conducted with 4 UCI databases for Connect, Pumsb, Susy, and USCensus. We also used 2 synthetic datasets built with the IBMGenerator 1  T10I4D2000K and T10I6D3000K. These synthetic datasets allow us to evaluate our approach with large transactional databases. Table  3  details on the one hand the characteristics of the benchmarks according to the number of items, of transactions, and the maximum and the average length of the transactions. On the other hand, it presents the number of nodes in the trie corresponding to each database and according to the total order of the items. For sampling with length constraint, the value of the minimum length constraint is fixed at µ = 1 throughout the experiments. The prototype of our method is implemented in Python version 3 and all the experiments are performed on a 2.
Storage cost of the trie of occurrences The cost 2  of storing a trie of occurrences in a database depends on both the total order relation > I and the maximum length constraint. As we can see in Table  4 , the number of nodes depends on the order in which the items are inserted in the trie. Overall, we noted that the number of nodes is much smaller with the > f req I relation than with the > lexico I relation according to the gain obtained in the last column. For example, with the Connect database, the number of nodes in the trie built with the order relation > f req I is 64. 6% less than the number of nodes in the trie built with the total order relation. This characteristic is very important in our proposal because each node must contain two arrays whose size depends on the maximum length constraint.
Thus, the smaller the number of nodes, the lower the storage cost. With the gain obtained in Table  4 , we can already note that the total order relation plays a very important role in optimizing the cost of memory storage, especially with large datasets like Susy, T10I4D2000K, and T10I6D3000K. In the two last datasets, it is not possible to run TPSampling with le lexicographical order due to an exception of "Out of memory". Figure  7  shows the evolution of the memory size required by the tries of each database according to the maximum length constraint M ∈ [2. 10] and the chosen order relation. We take as a baseline the storage cost of the tabular representations of the databases weighted according to the maximum length constraints of the two-step method  [4] (with length constraints). We first observe that the memory size of tries of occurrences increases according to the maximum length constraint.
Indeed, the two-dimensional matrixes that make it possible to count the patterns by length increase in size according to the maximum length constraint. This increase is more remarkable in the case where we considered the relation of total order > lexico I because the number of nodes in the trie constructed with the relation of total order > lexico I is higher than that of the trie built with the relation > f req I. Therefore, the size of the trie is more expensive with the > lexico I relation than with the > f req I relation. Thus, in the case where the database is very large, for example with Susy, T10I4D2000K, and T10I6D3000K, it is preferable to insert the items of the instances of the database in the trie following the relation total order > f req I. The evolution of the size of the trie according to the maximum length constraint in memory is almost stationary with the UCI datasets Connect, Pumsb, USCensus, and Susy.
When the maximum length constraint is greater than 6, T10I4D2000K becomes more expensive than Susy while the latter has more nodes in its trie of occurrences. This is because T10I4D2000K has more nodes near the root of its trie than that of Susy. These experimental results show that our approach is sensitive to the total number of items on which the database is built. For instance, TPSampling +> f req I returns an "Out of memory" exception with T10I6D3000K when the maximum length constraint is greater than 7. We also noted that if the database is large Connect (67,557 instances having an average length of 43. 0) and Pumsb (49,096 instances having an average length of 74. 0), the memory size of the trie of occurrences constructed according to the order relation > f req I is 200 times smaller than that of the representation of the database used by the two-step approach  [4].
Speed of the approach This section studies also the speed of our trie-based sampling method by distinguishing between preprocessing time, reprocessing time, and that of drawing a pattern. We will start evaluating the preprocessing time of each dataset of Table  3  and then compare it with that of the two-step method. We finish by computing the average time to draw a pattern with TPSampling in the 6 datasets according to the maximum length constraint and the chosen interestingness measure. In our experiments, we consider the maximum length constraints M ∈ {2, 6}. These values seem sufficient to avoid drawing patterns from the long tail. Table  5  presents the preprocessing times for the construction of the tries of occurrences and those for weighting the databases with the two-step algorithm according to the maximum length constraint.
Each experiment is repeated 10 times to have the average preprocessing times and the standard deviations. As we can see, the two-step method is faster than the trie-based method for  weighting a database. This can be explained by the fact that the preprocessing of the two-step method is done in one pass on the database with linear complexity relative to the size of the database. Contrariwise, TPSpace needs 2 passes in the case where we want to insert the items following a total order relation > I. To this, is added the time to insert the items of each transaction in the trie of occurrence according to the total order relation chosen by the user. We recall that with Susy, T10I4D2000K, and T10I6D3000K databases, an "Out of memory" exception was thrown by the implementation of the two-step algorithm because of their very large sizes after preprocessing which do not hold in memory.
With the largest database, T10I6D3000K, the construction of the trie of occurrences according to the total order relation > f req I lasts on average 13 and 18 minutes with the maximum length constraints M = 2 and M = 6 respectively. Interestingly, we do this preprocessing once only, then we can use the resulted trie of occurrences with any length-based utility. Evaluation of the reprocessing time for utility change. Our approach allows users to draw patterns according to any length-based utility measures that do not change the minimum µ and the maximum M length constraints. The reprocessing time, when utility changes, depends only on the minimum and maximum length constraint. It is linear to the difference between the minimum and the maximum length constraints whatever the size of the database because of the reusability of the trie of occurrences.
Utility measures like frequency, area, and exponential decay have not a notorious impact on the speed of the reprocessing phase. Contrariwise, Two-Step  [4] +length constraint should do a new preprocessing when utility changes. Table  6  shows that in the reprocessing phase when the utility changes while our approach needs a few time, less than 10 × 10 -6 seconds with M = 10. Decay is more consuming than Freq and Area due to the cost of the power function. Interestingly, the reprocessing time is the same for all datasets with the same length constraints. These results show how important is the trie data structure for user-centered pattern mining. Evaluation of the drawing time per pattern. Now, we will evaluate the speed of drawing a pattern with our approach on the 6 datasets of Table  3.
Precisely, we are going to compute the average drawing time of a pattern according to the maximum length constraint belonging to the interval [1. 10] and an exponential decay with a fixed value α = 0. 1. When M > 7, TPSampling returns an "Out of memory" exception with T10I6D3000K dataset. On Connect, Pumsb, and USCensus datasets, we can note that the speeds of the two methods are almost equal. Indeed, for the drawing of a pattern of length ℓ within a transaction t, the two-step algorithm exactly visits ℓ items of t. Conversely, TPSampling can visit more than ℓ nodes before returning the corresponding pattern. We note on the one hand that the drawing times of TPSampling+Area and TPSampling+Freq evolve almost in the same way. On the other hand, the drawing time of a pattern increases slightly according to the maximum length constraint with Connect, Pumsb, and USCensus.
Conclusion This paper proposed a new method of output pattern sampling based on a compact data structure called "trie". Considering a trie of occurrence built by our first algorithm TPSpace, it shows how to directly sample patterns according to any length-based utility measure with the second algorithm TPSampling. It is a generic algorithm that draws patterns from a trie of occurrences following a distribution proportional to a length-based utility measure chosen by the user. Experiments show that TPSpace is very parsimonious in storage cost and TPSampling is as good as the two-step method with length constraints to draw thousands of patterns per second. Unfortunately, the effectiveness of TPSpace decreases when the distinct items used in the database are very numerous. But unlike the two-step approaches, the trie of occurrences is the same for any lengthbased utility measure provided that the same values are kept for the maximum length constraint.
For instance, once the trie of occurrences is built with a given minimum and maximum length constraints, the can then draw patterns with frequency, area, and any exponential decay α ∈]0, 1] with the same built trie. They also show that our proposed method scales very well with large transactional databases. In future work, we would like to draw sequential patterns from a trie of occurrences. Indeed, the pattern numbering system we used to draw a sample proportionally to a length-based utility measure is a promising avenue for avoiding the rejection method used for sequential pattern mining  [12]. By resolving the problem of rejection in sequential pattern sampling, the numbering system we present in this paper becomes a very promising perspective to draw a representative set of patterns from the sequential data stream.
Considering the dataset T , we have freq(A, T ) = 3, freq(B, T ) = 3, freq(C, T ) = 3 and freq(D, T ) = 1. In this case, we can already say that A > f req I D, B > f req I D and C > f req I D because freq(A, T ) = freq(B, T ) = freq(C, T ) > freq(D, T ). Now, if we consider the lexicographic order between the items, we have A > lexico I B > lexico I C. So, continuing with the order relation > f req I , we have A > f req I B > f req I C > f req I D.
-a label denoted by η. label belonging to I ∪ {ǫ}, ǫ being the label reserved for the root. -a list of children denoted by η. child. Subsequently, we denote |η. child| the number of children of node η and η. child[i] the i-th child of η for i ∈ [1. k] with k = |η. child|. -an array of positive weights denoted by η. Φ +. If P is the identifier of the nodeη in τ , then η. Φ + [ℓ] = Φ + ℓ (P, T ) for ℓ ∈ [µ. M ], µ and M being the minimum and maximum length constraints considered during the construction of the trie. -an array of negative weights denoted by η. Φ -. If P is the identifier of the nodeη in τ , then η. Φ -[ℓ] = Φ - ℓ (P, T ) for ℓ ∈ [µ. M ], µ and M being the minimum and maximum length constraints considered during the construction of the trie. Let us consider the transactional database of Figure2, the minimum µ = 1 and maximum M = 3 length constraints, we build the enriched trie according to the total order relation > I in Figure3.
Having uniformly drawn a rank x = 5 of an occurrence among those of length 2, we are going to show at Figure5, with Algorithm 2, how to find the corresponding occurrence in the trie of occurrences of Figure3. The search for the fifth occurrence among the set of occurrences of length 2 requires 3 iterations. During the first iteration, we select the first child c of the root of the trie τ of identifierP = A because 0 < x ≤ (c. Φ -[2] + c. Φ + [2] = 3 + 5) (line7), and also we have x > c. Φ -[2] at line 9. Therefore, item A is part of the occurrence to be returned as output and the current content of the occurrence is ϕ = A. The value of ℓ becomes 1 because we are now going to look at the patterns of length ℓ -1 = 1 considering the sub-trie τ A then return the 5 -3 = Algorithm TPSampling Input : A trie τ of occurrences of a database T , a length-based utility u ∈ U and the minimum and maximum length constraints µ and M 2: Output : A pattern ϕ drawn proportionally to its interest ϕ ∼ f u (|ϕ|) × freq(ϕ, T ) 3: ϕ ← ∅ Draw a length ℓ proportionally to τ.
Example 4. 4 Considering the dataset T , we have freq(A, T ) = 3, freq(B, T ) = 3, freq(C, T ) = 3 and freq(D, T ) = 1. In this case, we can already say that A > f req I D, B > f req I D and C > f req I D because freq(A, T ) = freq(B, T ) = freq(C, T ) > freq(D, T ). Now, if we consider the lexicographic order between the items, we have A > lexico I B > lexico I C. So, continuing with the order relation > f req I , we have A > f req I B > f req I C > f req I D.
Example 8. 8 -a label denoted by η. label belonging to I ∪ {ǫ}, ǫ being the label reserved for the root. -a list of children denoted by η. child. Subsequently, we denote |η. child| the number of children of node η and η. child[i] the i-th child of η for i ∈ [1. k] with k = |η. child|. -an array of positive weights denoted by η. Φ +. If P is the identifier of the nodeη in τ , then η. Φ + [ℓ] = Φ + ℓ (P, T ) for ℓ ∈ [µ. M ], µ and M being the minimum and maximum length constraints considered during the construction of the trie. -an array of negative weights denoted by η. Φ -. If P is the identifier of the nodeη in τ , then η. Φ -[ℓ] = Φ - ℓ (P, T ) for ℓ ∈ [µ. M ], µ and M being the minimum and maximum length constraints considered during the construction of the trie. Let us consider the transactional database of Figure2, the minimum µ = 1 and maximum M = 3 length constraints, we build the enriched trie according to the total order relation > I in Figure3.
Example 12. 2 - 2 1 : 4 : 5 : 1222145 Having uniformly drawn a rank x = 5 of an occurrence among those of length 2, we are going to show at Figure5, with Algorithm 2, how to find the corresponding occurrence in the trie of occurrences of Figure3. The search for the fifth occurrence among the set of occurrences of length 2 requires 3 iterations. During the first iteration, we select the first child c of the root of the trie τ of identifierP = A because 0 < x ≤ (c. Φ -[2] + c. Φ + [2] = 3 + 5) (line7), and also we have x > c. Φ -[2] at line 9. Therefore, item A is part of the occurrence to be returned as output and the current content of the occurrence is ϕ = A. The value of ℓ becomes 1 because we are now going to look at the patterns of length ℓ -1 = 1 considering the sub-trie τ A then return the 5 -3 = Algorithm TPSampling Input : A trie τ of occurrences of a database T , a length-based utility u ∈ U and the minimum and maximum length constraints µ and M 2: Output : A pattern ϕ drawn proportionally to its interest ϕ ∼ f u (|ϕ|) × freq(ϕ, T ) 3: ϕ ← ∅ Draw a length ℓ proportionally to τ.
A rapid 2D acquisition (< 4 s) ensured negligible variations in the state of charge (SoC) during the signal acquisition. This approach in turn allowed much higher operational currents and, as a result, a higher sensitivity  [18]  as compared to 3D io-MRI  [16, 17]. Surface-Scan MRI can be one of the most promising battery diagnostic methods to date. Its wide-scale implementation would benefit from an experimental database serving as a "biometric archive" for batteries. Although such a database can be acquired within realistic time frames, establishing MF patterns theoretically based on known cell designs and common failure modes should be considered. In this preliminary study, we attempt to validate the Surface-Scan MRI concepts by modeling electromagnetic fields of operating cells (e. g. flat jelly roll and prismatic designs) using COMSOL Multi-physics.
Simulations COMSOL (v. 5. 6) simulations were performed using AC/DC module (Electromagnetic Fields > Vector Formulations > Magnetic and Electric Fields). In the stationary case, current density and associated magnetic field distributions are described with the following Eqs. (5-9): ∇ × H = J, ( 5 ) B = ∇ × A, ( 6 ) J = σE, ( 7 ) ∇⋅J = 0, (8) E = -∇V, (9) Here H -magnetic field intensity, J -current density, E -electric field intensity, B -magnetic flux density, A -magnetic vector potential, Velectric scalar potential. The COMSOL algorithm solves Ampère's law (Eq. (  4 )) for the magnetic vector potential (A) together with a current conservation equation (Eq. (  7 )) for the electric potential (V). The computation time varied between 15 min and 160 h depending on the number of layers in a cell model.
Sampling and hydrography During the GEOTRACES GIpr07 cruise, which took place in early austral winter (28 June-13 July 2017) on board the R/V SA Agulhas II, seven stations were sampled along 30 • E, from 58. 5 to 41. 0 • S (WOCE I06S, Fig. 1a ). At each station between 15 and 21 samples were collected from 25 m down to 1500 m, for shallow stations, and down to 4250 m, for deep stations, to be analysed for multiple parameters. Positions of the fronts during the cruise were determined using the July monthly mean absolute dynamic topography data from the CLS/AVISO product  (Rio et al. , 2011) , with boundary definitions from  Swart et al. (2010). From north to south the identified fronts are the subtropical front (STF), the subantarctic front (SAF), the polar front (PF), the Southern Antarctic Circumpolar Current front (SACCf) and the southern boundary (SBdy) (Fig.
, 2017) , to ensure homogenous sampling. Although the large fast-sinking fraction of particles may be under-sampled by using bottles  (Bishop and Edmond, 1976; Planquette and Sherrell, 2012) , comparing data that were generated using the same internationally validated sampling systems and protocols  (Cutter et al. , 2017)  as we do in this study minimises potential bias. After filtration, filters were placed in trace-metal-clean petri slides (Pall) and kept frozen at -20 • C until further processing on land. Sample processing was conducted under a class 100 HEPA-filtered laminar flow and extraction hood in a clean laboratory. The pBa and pAl samples were processed and analysed 6 months after sample collection at LEMAR (France). Unused blank filters and filters containing the samples were acid reflux digested at 130 • C in acid-cleaned Savillex vials using a mixture of HF and HNO 3 (both Ultrapur grade, Merck) solutions  (Planquette and Sherrell, 2012).
Integrated remotely sensed PP The integrated remotely sensed PP (mg C m -2 d -1 ) within the surface mixed layer was calculated using the CbPM algorithm  (Behrenfeld et al. , 2005) , which requires chlorophyll concentration (mg m -3 ), particulate backscatter (λ 443 nm, m -1 ), photosynthetically active radiation (PAR; µmol photons m -2 d -1 ) and the MLD (m). Ocean Colour Climate Change Initiative (OC-CCI) data (https: //esa-oceancolour-cci. org/, last access: 14 August 2020), which blend existing data streams into a coherent record, meeting the quality requirements for climate assessment  (Sathyendranath et al. , 2019) , were used for chlorophyll and particulate backscatter. PAR was taken from GlobColour (http://www. globcolour. info/, last access: 14 August 2020), and the MLD was taken from the climatology of  de Boyer Montegut et al.
(2004). The integrated remotely sensed PP data were regridded to 0. 25 • spatially, using bilinear interpolation using the Python programming package xESMF  (Zhuang, 2018) , and averaged monthly. The area-averaged PP was averaged over a 6 × 1 • rectangular sample area, starting 6 • upstream extending up to the sampled station longitudinally, and 1 • latitudinally centred around each sampled station (see "Discussion" for details). In order to assess the validity of the remotely sensed PP data and demonstrate no meridional bias across the SO, the percentage of valid pixels was calculated for data north (90 ± 20 %; mean ± SD, n = 370) and south (82 ± 29 %; mean ± SD, n = 488) of the PF, revealing no bias.
Timescale of the mesopelagic Ba xs signal The Ba xs signal that we observed in winter is in agreement with the suggestion by  Dehairs et al. (1997)  that there can be significant carry-over between bloom seasons. Other studies have also pointed out that the timescale of this proxy is longer than a snapshot view  (Cardinal et al. , 2005)  and have highlighted a seasonal increase in mesopelagic Ba xs  (Jacquet et al. , 2011). This strongly suggests that the Ba xs signal is not directly linked to synoptic measurements of PP at the time of sampling. In order to investigate this hypothesis, for the first time, we compiled a SO mesopelagic Ba xs stock dataset with all available literature data including data from this study (Fig. 4a , Table  S3 ). The mesopelagic Ba xs stock was integrated over the Ba xs peak depth range (as identified in each study).