Supervised feature extraction algorithm based on improved polynomial entropy Based on information entropy theory, a novel feature extraction algorithm based on improved polynomial entropy (IPE) is set up. Firstly, the concepts and their properties of information entropy and cross entropy (CE) are analysed and studied. On this foundation, we prove that symmetrical cross entropy (SCE) proposed here based on CE satisfies three axioms of the distance, i.e. nonnegativity, symmetry and triangle inequation. So SCE is a kind of distance measure, which can be used to measure the degree of variation between two random variables. Secondly, for convenience, we propose a new concept of improved polynomial entropy (IPE) based on polynomial entropy (PE), and explain that IPE is equivalent to SCE. Thirdly, we make IPE separability criterion of the classes for feature extraction, and design a novel feature extraction algorithm based on IPE. Fianlly, the experimental results demonstrate that the algorithm proposed here is valid and reliable, and it provides a new research approach for feature extraction, data mining and pattern recognition. It is well known that the discovery and selection of an effective feature set are most important and difficult problems in pattern recognition (PR). The choice of features to represent the patterns affects several aspects of the PR algorithm, such as accuracy, required learning time, and the necessary numbers of samples [1, 2] . The difficulties in selecting effective features are due to several facts. Firstly, it is necessary to select a feature combination that separates the class volumes so that the classes can be distinguished, but this property of a feature set is not generally obvious without carrying out a PR or data visualization analysis of the training set using the selected features. Secondly, there are often many features, and combinatorial large numbers of feature combinations to select from them. Note that the number of feature subset combinations with m features from a According to the definitions and properties of Shannon entropy and cross entropy, we have studied and discussed the feature extraction problem in this paper. We have proved that the SCE is a kind of distance measure, and proposed a new concept of IPE based on PE, which is equivalent to SCE and can be used as the criterion of class separability. Based on IPE, we have designed a new supervised feature extraction algorithm, and applied it in practice. The experimental results show that the algorithm presented here is valid, and the compression effect is significant.

B I I O O O O O O O O B I I O O O O O O O B I I I O O O O O O O O O O O O O B I I O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O


Adaptive scanning - a proposal how to scan theoretical predictions over a multi-dimensional parameter space efficiently A method is presented to exploit adaptive integration algorithms using importance sampling, like VEGAS, for the task of scanning theoretical predictions depending on a multi-dimensional parameter space. Usually, a parameter scan is performed with emphasis on certain features of a theoretical prediction. Adaptive integration algorithms are wellsuited to perform this task very efficiently. Predictions which depend on parameter spaces with many dimensions call for such an adaptive scanning algorithm. Most observations in elementary particle physics so far are well described by the Standard Model of particle physics. However, it is widely believed that the Standard Model is only an effective low-energy limit embedded in a larger theory which is not known at present. Extensions of the Standard Model of particle physics usually are equipped with a lot of additional parameters compared to the Standard Model which describe new physics at energy scales not probed so far by experiment. The predictions of promising extensions of the Standard Model for physical observables should be calculated and studied in detail, either to find hints for new physics or to exclude alternative models or, at least, to constrain their parameters.. A way to study such predictions is performing parameter scans. But what exactly is a parameter scan? A parameter scan may be defined as the process of calculating numerical values for a theoretical prediction for a sufficient amount of points in parameter space, such that, sufficient smoothness of the prediction's dependence on the parameters assumed, a clear understanding of the range of values for the prediction arises. Usually, in this process all other known restrictions on the parameters are taken into account. Thus, only meaningful, i.e. not otherwise excluded, points in parameter space are sampled. In high energy physics the most common goal of a parameter scan over a prediction of an alternative model is to restrict the parameter space of the model by comparison of the prediction with experimental results. Specifically, one wants to find the regions in parameter space where a theoretical prediction for an observable either matches the measured value within error bars, or, if nothing has been observed, respects exclusion limits on possible observables drawn from experiments. This is a typical situation. In practice, of course, things are sometimes more involved.. Common techniques for parameter scans are sampling a rigid grid of points in parameter space or choosing points at random with a uniform distribution. For the latter sometimes special non-uniform probability functions are used for plotting reasons, e.g. to have the sample points uniformly distributed on a logarithmic scale. But, in the same way as a rigid grid of samples or the naive Monte Carlo approach are not the best options for evaluating numerically multi-dimensional integrals efficiently these parameter scan techniques are not optimal for multi-dimensional parameter spaces. The amount of calculation time needed for a parameter scan may easily get out of hand for parameter spaces with many dimensions, for instance the minimal supersymmetric standard model (MSSM) with general parametersespecially, if time-consuming calculations are necessary already for each sampled parameter point, as for instance in many predictions for hadron colliders.. The basic observation made in this note is that finding regions of interest for a function in a multi-dimensional parameter space and finding a good approximation to the integral of a multi-dimensional function can be attacked by the same method in a very efficient way. That means, essentially, existing algorithms for adaptive integration by importance sampling (implemented e.g. in VEGAS [1] ) can also be used to perform an what will here be called "adaptive parameter scan".. In section 2 a formulation of our proposal is given, furnished in section 3 with a concrete example from the study of an MSSM cross section prediction, and summarized in section 4. We present a new method to perform parameter scans for theoretical predictions: "adaptive scanning". The method is designed to handle parameter spaces of large dimensionality and is very efficient if there is an emphasis on certain features of the prediction. Furthermore, in cases where parts of the parameter space are excluded or forbidden by further restrictions our method of scanning can focus adaptively on the allowed region. The scan of a prediction f (p i ) over (a portion of) a parameter space {p i } with emphasis on certain features is realized by defining an importance function G(f, p i ) and sampling adaptively more points in regions where G is large. This is done by numerically estimating the integral of G over (a portion of) parameter space with an adaptive Monte Carlo integration algorithm which is based on importance sampling and by recording the sampled points. The method can be straightforwardly implemented as a computer program using VEGAS with a suitable random number generator and a function to be called by VEGAS which calculates the desired function G and stores the sampled parameter points. The method is modular and flexible: One is free to choose the function G and it is simple to use different integration algorithms. Any sophistication present in implementations of adaptive integration algorithms, for instance parallelization, may be exploited for our method of parameter scanning. An exemplary FORTRAN code using VEGAS is available from the author upon request. By an adaptive parameter scan we mean the task to scan a function f , depending on a set of N parameters (p 1 , . . . , p N ), over a hypercube 1 . . . , N } in order to find regions of parameter space where some "importance function" G(f, p 1 , . . . , p N ) becomes large. To that end an approximation to the integral. is evaluated by an adaptive integration algorithm using importance sampling (e.g. VEGAS) and each point in parameter space is stored during the process. 2 The idea is to choose G in such a way that the (for some specific reasons) preferred values of f (and possibly p i ) lead to large values of G and that therefore the parameter regions where G(f, p i ) is large (i.e. where f and possibly the p i take on preferred values according to the specific reasons which lead to the choice of G) are more thoroughly sampled by an adaptive integration algorithm based on importance sampling. Examples are:. • G(f ) = f to emphasize regions with large values of |f |,. • or G(f ) = 1/f to emphasize regions with small values of |f |,. • or G(f ) = Θ(−f )f to emphasize regions with large |f | but with f negative.. The end result is a set of points in parameter space which may allow to give meaningful answers to questions like. • "What are the regions of large cross section ?",. • "Where are the regions in parameter space where the observable is small enough to be consistent with present data ?",. • "In what regions of parameter space are the radiative corrections to some observable negative ?".. If there are complicated parameter restrictions to be taken into account the method proves to be very powerful. By setting the importance function zero in disallowed regions and non-zero otherwise, the algorithm can adapt to the allowed region of parameter space. Thus, even if no other feature of the theoretical prediction is emphasized (i.e. G(f ) = 1 in the allowed region), the algorithm focuses iteratively on the allowed region, which is not the case in the naive Monte Carlo approach.. For the integration of a function by an adaptive Monte Carlo algorithm the approximate value and error of the integral estimate is most important, while the sampled points are essentially of no interest. For adaptive scanning it's just the opposite. Most interesting are the sampled points, while the approximate value of the estimate and its error are quantities to control the thoroughness of the scan.

O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Nonparametric Bayesian clustering to detect bipolar methylated genomic loci Background: With recent development in sequencing technology, a large number of genome-wide DNA methylation studies have generated massive amounts of bisulfite sequencing data. The analysis of DNA methylation patterns helps researchers understand epigenetic regulatory mechanisms. Highly variable methylation patterns reflect stochastic fluctuations in DNA methylation, whereas well-structured methylation patterns imply deterministic methylation events. Among these methylation patterns, bipolar patterns are important as they may originate from allele-specific methylation (ASM) or cell-specific methylation (CSM). Results: Utilizing nonparametric Bayesian clustering followed by hypothesis testing, we have developed a novel statistical approach to identify bipolar methylated genomic regions in bisulfite sequencing data. Simulation studies demonstrate that the proposed method achieves good performance in terms of specificity and sensitivity. We used the method to analyze data from mouse brain and human blood methylomes. The bipolar methylated segments detected are found highly consistent with the differentially methylated regions identified by using purified cell subsets. Conclusions: Bipolar DNA methylation often indicates epigenetic heterogeneity caused by ASM or CSM. With allele-specific events filtered out or appropriately taken into account, our proposed approach sheds light on the identification of cell-specific genes/pathways under strong epigenetic control in a heterogeneous cell population. We developed a statistical approach to detect bipolar DNA methylation in bisulfite sequencing data, through nonparametric Bayesian clustering and hypothesis testing. Simulation studies demonstrated that our method achieves high specificity under all settings. The sensitivity increases as cell-type mixture proportion becomes more balanced and as depth of coverage increases. With allele-specific events filtered out, this approach can be used to identify cell-specific genes/pathways under epigenetic control within a heterogeneous cell population. We applied the method to analyze data from mouse brain and human blood methylomes. Results on real data analysis demonstrated that the predicted CSMs are highly consistent with the DMRs identified by using purified neuron-glia cells and neutrophil-B cells.. Methylation differences observed among sequence reads mapped to the same locus may result from: 1) allele specific DNA methylation; 2) asymmetric DNA methylation; and 3) cell-subset specific DNA methylation. ASM may be classified into imprinted (parent-oforigin dependent ASM) and sequence dependent ASM. Recent genome analyses revealed 55 discrete genomic loci imprinted in the mouse genome [17] and 51 loci imprinted in the human genome [40] . Meanwhile, the majority of sequence dependent ASM CpG sites are scattered throughout the genome [17] . The asymmetric DNA methylation can be determined with the genome-wide hairpin bisulfite sequencing technique recently developed by our lab [41] . We found that 88.5% and 91.9% CpG sites sequenced for mouse renewing and differentiating ES cells are symmetrically methylated, respectively. In addition, the asymmetric methylated CpG sites are frequently associated with random DNA methylation patterns resulting from stochastic DNA methylation events [12] . Thus, after the filtering of imprinted loci, bipolar methylated regions identified in this study are likely to be cell-subset specifically methylated.. Our approach provides new insights to the identification and interpretation of partially methylated genomic loci. It is advantageous over currently available methods for CSM identification in two aspects. 1) The detection of bipolar DNA methylation is based on bisulfite sequencing data from a heterogeneous cell population, while traditional CSM identification relies on detecting DMRs, i.e., testing for differential methylation levels between highly purified cell subsets. The proposed method thus provides a feasible solution to the analysis of DNA methylation in higher organisms where cells are difficult to be dissociated. Apropos, a growing body of evidence from single cell analysis reveals significant cellular heterogeneity even within a cloned population. Therefore, CSM regions identified by our method generally include not only DMRs but also partially methylated genomic regions caused by the same type of cells at different stages, and the latter are of particular interest as they cannot be detected by traditional methods. 2) Besides identifying CSM regions, the proposed method simultaneously estimates the proportions of heterogeneous epigenomes in bisulfite sequencing data. In recently published literature, some efforts have been made to estimate the proportions of different cell types in unfractionated samples by employing the identified DMRs as markers of cell identity. For example, Guintivano et al. (2013) [42] incorporated cell epigenotype specific (CETS) markers in a linear slope model to quantify neuronal and glial proportions. Houseman et al. (2012) [43] and Montaño et al. (2013) [44] employed DMRs and adopted a regression calibration model to estimate the proportions of different cell types in unfractionated blood sample and in brain tissue, respectively. Compared to the prediction by identified DMRs, our method provides a straightforward solution to this problem.. Our method for detecting bipolar DNA methylation is also related to statistical methods recently developed for ASM prediction. In particular, Fang et al. (2012) [26] proposed a model selection method to identify ASM by comparing Bayesian information criterion (BIC) under non-allele specific and allele-specific models. Peng and Ecker (2012) [25] adopted supervised learning to classify candidate regions into ASM or non-ASM by using features (estimated methylation levels and allele frequencies) extracted from a mixed model. Compared with these two methods, our bipolar detection framework has several advantages: 1) Our model makes no restriction on the ratio between number of hyper-methylated and hypo-methylated reads, therefore, it is more appropriate for predicting CSM. 2) By treating bipolar detection as a hypothesis testing problem, our method is more suitable for sensitivity and specificity evaluation than the BIC-based model selection method [26] and the supervised learning method [25] . 3) Because neither EM algorithm nor Markov chain Monte Carlo (MCMC) is required, the DPM search algorithm is easy to implement and requires the least amount of computation load.. The method developed in this study will greatly extend our capacity to dissect the epigenetic heterogeneity in a cell population, in particular for the ones with limited or no prior knowledge of cell-type composition. Detecting bipolar methylation patterns is an important step for unlocking the biological meaning of epigenetic heterogeneity; however, the methodology development is still in initial stages. Due to the limitation on depth of coverage and read length, with current bisulfite sequencing data, bipolar DNA methylation can only be detected in small segments (e.g., 4-CpG). In general, more complicated methylation patterns arising from a mixture of various cell types or cells at various stages may be decomposed into a combination of bipolar DNA methylation patterns presented at multiple loci. Our future work will be directed to investigate the functional relevance of partially methylated genomic loci and incorporate their combinations as epigenetic signatures for complex phenotypic and functional variability. In bisulfite sequencing, the methylation state of each CpG site on every mapped read is determined by the presence of cytosines or thymidines. For a particular sequence read, the combination of methylation states on neighboring CpG sites defines a certain methylation pattern, which, for a better understanding, can be thought of as methylcytosine "haplotype" (see Figure 1 for an example). Our purpose is to identify genomic regions with bipolar methylation patterns. To fulfill the purpose, two critical questions need to be answered: 1) How to build a feasible model for bipolar methylation? 2) How to detect bipolar methylation with sufficient statistical power? In what follows, we first introduce a hierarchical model modified from Peng and Ecker (2012) [25] and from Fang et al. (2012) [26] , then describe an effective statistical method for the detection of bipolar methylated genomic loci.

B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


GLOUDS: Representing tree-like graphs The Graph Level Order Unary Degree Sequence (GLOUDS) is a new succinct data structure for directed graphs that are "tree-like," in the sense that the number of "additional" edges (w.r.t. a spanning tree) is not too high. The algorithmic idea is to represent a BFS-spanning tree of the graph (consisting of n nodes) with a well known succinct data structure for trees, named LOUDS, and enhance it with additional information that accounts for the non-tree edges. In practical tests, our data structure performs well for graphs containing up to m = 5n edges, while still having competitive running times for listing adjacent nodes. Succinct data structures have been one of the key contributions to the algorithms community in the past two decades. Their goal is to represent objects from a universe of size u in information-theoretical optimal space lg u bits of space. 1 Apart from the bare representation of the object, fast operations should also be supported, ideally in time no worse than with a "conventional" data structure for the object. For this, one usually allows extra space o(lg u) bits [25] .. A prime example of succinct data structures are ordered rooted trees, where with n nodes we have lg u ≈ 2n. In 1989, Jacobson made a first step towards achieving this goal, by giving a data structure using 10n + o(n) bits, while supporting the most common navigational operations in O (lg n) time [25] . This was further improved to the asymptotically optimal 2n + o(n) bits and optimal O (1) navigation time by Munro and Raman [35] . Note that a conventional, pointer-based data structure for trees requires (n lg n) bits, which is off by a factor of lg n from the information-theoretical minimum.. Since the work of Munro and Raman, the research on succinct data structures has blossomed. We now have succinct data structures for bit-vectors [37] , permutations [33] , binary relations [2] , dictionaries [36] , suffix trees [39] , to name just a few.. The practical value of those data structures has sometimes been disputed. However, as far as we know, in all cases where genuine attempts were made at practical implementations, the results have mostly been successful [18,26,22, etc. , to cite some recent papers presented in the algorithm engineering community]. Further examples of well-performing practical succinct tree implementations will be mentioned throughout this paper. In this paper we presented a framework and implementation for a new succinct data structure for "tree-like" graphs based on the LOUDS representation for trees, which we called GLOUDS. We took a three-pronged approach to evaluate our representation. Firstly, we created random data structures inspired by phylogenetic networks with 10% of non-tree edges and compared the runtime of the operations children and parents on every node matched to a pointer based representation. The evaluation confirmed that our succinct data structure is practically feasible with a space reduction of around 95%. Secondly, a test on general graphs was performed by the use of a succinct library, also compressing the pointer-based data-structure. The practical evaluations again confirmed that the GLOUDS achieves a significant space reduction. Lastly, we showed how well our data-structure performs against a framework for web-graphs. The evaluation on "real-world" graphs shows that the GLOUDS performs well for graphs where m < 10n. In total, a trade-off between space and time can be observed, which is common in the world of succinct data structures.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I


FPT-ALGORITHMS FOR MINIMUM-BENDS TOURS This paper discusses the κ-BENDS TRAVELING SALESMAN PROBLEM. In this NP-complete problem, the inputs are n points in the plane and a positive integer κ, and we are asked whether we can travel in straight lines through these n points with at most κ bends. There are a number of applications where minimizing the number of bends in the tour is desirable because bends are considered very costly. We prove that this problem is fixed-parameter tractable (FPT). The proof is based on the kernelization approach. We also consider the RECTILINEAR κ-BENDS TRAVELING SALESMAN PROBLEM, which requires that the line-segments be axis-parallel.1 Note that a rectilinear tour with κ bends is a cover with κ-line segments, and therefore a cover by lines. We introduce two types of constraints derived from the distinction between line-segments and lines. We derive FPT-algorithms with different techniques and improved time complexity for these cases. We have presented several FPT-algorithms for a hard geometric problem where we have to answer if there is a tour through a set S of n points with at most k bends. We summarize the results of this problem in Table 1 . It is apparent that the complexity of the algorithms is slightly better as more constraints are placed on the solution. In the rectilinear tour, for example, a solution for the general case may not be a solution for either of the constrained cases, and a solution of the constrained cases may require more bends although it is a solution for the general case. Therefore, the fact that one variant is FPT does not imply the other variant is FPT. The kernelization approach required additional analysis of the witness structure and could be considered to follow the ideas of M. Fellows on extremal structure. While we have discussed the decision-version of the problem, it is not hard to adapt to an FPT-algorithm for the optimization version. For example, we can use an optimization-version FPT-algorithm to find a k -cover, and the number k of covering lines is bounded by O(k) where k is the minimum number of bends. That is, we can approximate the minimum in FPT-time with respect to the sought minimum. Then the decision problem can be used to find the exact value of the minimum in FPT-time as well. We note that to further improve the time complexity of the algorithms we may need further algorithmic engineering. However, even vertex cover was first found FPT by a total impractical argument using the Minors Theorem and steady progress in the parameterized complexity field has now delivered practical implementations. Our aim for this paper is to establish that the NP-complete problems we presented are in the class FPT.

O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Ensuring semantic spatial constraints in virtual environments using UML/OCL Spatial objects and relationships between them, compose a spatial model that is the backbone of virtual environments (VEs). However, due to the natural complexity of both spatial objects and spatial information, the modeling of such spatial relationships is still a difficult task. This paper presents a novel approach for representing semantic spatial relations in VEs using the Unified Modeling Language (UML) and the Object Constraint Language (OCL). Our approach first uses the UML class model as a conceptual model for VEs. We then propose a spatial extension of OCL named VRX-OCL as a high-level and flexible language to cover multidimensional, manifold, and reference frame-dependent spatial constraints. We mainly focus on two important classes of spatial relations, namely, topological and projective relations that allow nonmetric representation of space. The applicability of our approach is demonstrated in the Virtual Physics Laboratory, a VE for learning physics. Based on the constraints satisfaction, the system is able to visualize abstract spatial information and thus provides educational assistance to the learners. Spatial relationships between objects provide crucial knowledge about virtual environments (VEs). Particularly, in VEs dedicated to enhancing spatial behaviors such as exploring a space or searching an item in the space, an explicit representation of spatial knowledge is a need [Durlach et al. 2000] . Spatial expressions found in such applications could be "let's move to the statue in front of the picture", "the cube is on the bench". However, spatial objects populating virtual world are often described in a low-level way within 3D scenes built upon standard formats (e.g., VRML, X3D). As a consequence, spatial relations are implicit and hard to define. * e-mail:trinh@enib.fr † e-mail:querrec@enib.fr ‡ e-mail:deloor@enib.fr § e-mail:chevaillier@enib.fr. To overcome these issues, it becomes clear that a high-level representation of spatial objects and notably an explicit description of spatial relationships between them is required. Recent work has focused on defining a semantic model for VEs [Latoschik and Blach 2008] . The main idea is to represent spatial objects not only as animatable 3D shapes but also with their functions, roles, and the relationships between them [Gutierrez et al. 2005] . However, there still lacks a conceptual base and a common language to precisely specify spatial relations and to support the ability to query the spatial knowledge. Difficulties increase when spatial relationships form a constraints network that must be satisfied when users interact with VEs. The problem is not only to express constraints but also to decide whether these objectives are achieved or not. For example, how to compute that a user is near the statue, and consequently, in front of the picture? How to ensure that the user has put the cube on the bench? Especially, spatial constraints can be used as a condition for spatial interactions (e.g., "Before activating the laser, verify that the cube is between the lens and the mirror"). To do so, an indepth formalization of both semantics and computational methods of spatial constraints is needed.. This paper presents a general approach for ensuring semantic spatial constraints in VEs. The novelty relies on the use of the Unified Modeling Language (UML) for the conceptual modeling of VEs and the Object Constraint Language (OCL) for representing spatial constraints. Our motive is that UML is a relevant basis for metamodeling, recognized by both industrial domain and the scientific community. However, this language has not been fully exploited for modeling semantics of VEs. Being an integral part of UML, OCL is a textual language to express and query constraints that diagrams can not cover by themselves [OMG 2006] . Because of the necessity to manage not only logical constraints in the conceptual model but also spatial constraints specialized to 3D dynamic environments, we propose VRX-OCL, a Virtual Reality eXtension of OCL. Our extension aims at ensuring different semantic constraints in VEs: (1) internal properties of spatial object (e.g., type, value of attributes), (2) spatial relationships among a set of spatial objects, and (3) semantics of spatial object interactions (e.g., pre-and post-condition of spatial tasks). In addition, when associated with adequate spatial reasoning techniques, such a constraint language could be used within an inference system to deduce automatically new knowledge from given one. For example, given "the statue is in front of the picture" and "the table is in front of the statue", one can conclude that "the table is in front of the picture".. By coupling 3D scenes representation with a semantic model, using UML/OCL, our approach gives a novel perspective to define spatial relations between objects that should be satisfied during users's manipulations and to make domain specific knowledge visible to users. Especially, this knowledge can be directly interpreted as instructions displayed textually or graphically, e.g., by visualizing the acceptance areas for the localization of manipulated objects in the 3D environment. VEs for training, learning, or spatial assistance systems are typical use cases that can benefit from our approach.. In the next section, we briefly discuss open problems in the modeling of spatial constraints in VEs and related researches. We focus on two important classes of spatial relationships, namely, topological and projective constraints that allow nonmetric representation of space. We raise the need for a spatial language to model these constraints through examples borrowed from the Virtual Physics Laboratory, a VE for learning physics. Section 3 gives an overview of our approach. Sections 4 and 5 respectively detail how nonmetric spatial constraints are modeled in VRX-OCL. We present both formalization and computational model of topological and projective relations. Section 6 draws our conclusions and outlines future work. In this paper, we presented a general approach to ensure semantic spatial constraints in VEs. Our first finding was concerned with the use of UML to conceptualize 3D space, thus provided an abstract, implementation independent, and semantic representation of VEs. We then defined VRX-OCL, a spatial extension of OCL to formally specify spatial constraints that are inherently manifold and dependent on frames of reference. We presented an in-depth definition of the semantics of nonmetric spatial constraints, i.e., topological and projective constraints. In order to tackle the lack of precision when manipulating 3D objects within VEs, we introduced the concept of thick boundary objects and then defined 17 topological relations between them. Our method for formalizing and computing projective relations allowed one to define semantic constraints between three objects (i.e., "before", "between", "after", "leftside", "rightside") and four objects (i.e., "above", "below", "coplanar"), both in orthographic and immersive views. Through the VPLab, we argued that the combination of these two families of spatial constraint offered the ability to define complex constraints without any metric information.. Metric spatial constraints, namely, distance and directional relations will be our future focus. We are particularly interested in qualitative approaches that abstract from metrical details of the physical world and thus provide a spatial representation closer to mental model of humans [Cohn and Hazarika 2001] . Further, we plan to define and incorporate into our model suitable reasoning techniques. The main goal is to enable virtual agents to find out new relationships from existing ones. This research forms part of our project that aims at defining UML/OCL as the basis for a generic language dedicated to the semantic modeling of VEs. As semantic model intensively requires the reification and introspection of the conceptual model, we have been investigating our efforts on the design of a UML-based meta-model that might be used as a semantic reflection layer for VEs.

O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


LeView:

automatic

and

interactive

generation

of

diagrams

for

biomacromolecule/ligand

interactions

diagrams

are

widely

used

in

the

scientific

literature

to

represent

interactions

between

ligands

and

biomacromolecules.

Such

schematic

diagrams

are

very

helpful

to

better

understand

the

chemical

interactions

and

biological

processes

in

which

ligands

are

involved.

Here,

a

new

tool

for

automatic

and

interactive

generation

of

diagrams

for

biomacromolecule/ligand

interactions

is

presented.

LeView

(Ligand-Environment

Viewer)

produces

customised

and

high-quality

figures,

with

a

good

compromise

between

a

faithful

representation

of

the

data

(structures

and

interactions)

and

aesthetic

criteria.

LeView

can

be

freely

downloaded

at

LeView

produces

customisable

and

high-quality

figures

often

necessary

for

scientific

publications,

in

several

popular

formats.

It

is

able

to

work

with

a

wide

range

of

complex

ligands

and

environments

not

possible

with

existing

tools.

This

tool

can

be

freely

downloaded

and

easily

used

without

installation

and

offers

the

user

an

interactive

graphical

interface

with

numerous

options

to

customise

the

figure

such

as:

varying

the

cut-off

distances,

changing

the

diagram

element

colours,

moving

and

deleting

elements.

The

diagrams

are

a

good

compromise

between

a

faithful

representation

of

the

data

(structures

and

interactions)

and

aesthetic

criteria.

Ligand,

hydrogen

bonds

and

nearby

residues

are

included

and

the

list

of

possible

water-mediated

interactions

is

also

available.

Non-covalent

interactions,

such

as

hydrogen

bonds,

between

ligands

and

biomacromolecules

forming

a

complex

provide

important

clues

in

the

understanding

of

biological

processes

and

the

design

of

compounds

with

desirable

binding

properties.

Schematic

diagrams

are

widely

used

to

visualise

the

binding

interactions

of

ligands,

which

can

provide

important

clues

about

their

role

and

function.

A

lot

of

software

packages

are

available

for

visualising

structure

in

but

only

a

handful

exists

for

generating

protein/ligand

interaction

diagrams.

LIG-PLOT

is

the

most

commonly

used

and

its

successor

LigPlot+

includes

a

new

interface,

superposition

of

related

diagrams

and

links

to

PyMOL

and

RasMol.

PoseView

automatically

generates

structure

diagrams

of

molecular

complexes.

A

module

of

the

commercial

software

MOE

-Molecular

Operating

Environment

generates

schematic

diagrams

for

protein-ligand

complexes,

and

other

commercial

software

vendors

offer

similar

tools.

In

this

paper,

LeView

(Ligand-Environment

Viewer)

is

introduced.

This

new

tool

automatically

detects

ligands

in

a

PDB

file,

generates

diagrams

for

biomacromolecule/ligand

interactions

and

allows

the

user

to

customise

diagrams

through

an

interactive

and

intuitive

graphical

interface.

LeView

is

written

in

Java

and

distributed

as

an

executable

file

(source

code

is

also

available

under

GNU

Public

Licence,

Additional

file

in

two

versions:

a

version

with

an

interactive

graphical

interface

and

a

command-line

version

for

use

in

pipelines.

LeView

does

not

require

any

special

installation

steps,

can

be

run

on

all

operating

systems.

LeView

was

written

in

Java

in

order

to

be

environmentindependent,

it

is

open-source

and

can

be

freely

downloaded

(GNU

Public

Licence).

Figure

shows

the

main

steps

of

the

LeView

implementation.

In

a

first

step

LeView

automatically

identifies

ligands

and

metal

ions

directly

from

the

PDB

file

with

an

algorithm

based

on

the

method

used

in

PDBSum

Explicit

residues,

i.e.

residues

explicitly

connected

to

the

ligand

in

the

PDB

file,

are

also

identified.

For

a

ligand,

the

layout

algorithm

is

then

used

to

obtain

the

diagram

with

which

the

user

can

interact

via

a

graphical

interface

to

customise

it.

The

layout

algorithm

is

divided

into

three

steps:

first,

the

coordinates

of

the

ligand

atoms

are

computed,

then

the

interactions

are

calculated

from

the

PDB

file

and

placed

around

the

ligand

and

finally

residue

names

are

added

to

the

diagram.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O


Multisource Bayesian sequential binary hypothesis testing problem We consider the problem of testing two simple hypotheses about unknown local characteristics of several independent Brownian motions and compound Poisson processes. All of the processes may be observed simultaneously as long as desired before a final choice between hypotheses is made. The objective is to find a decision rule that identifies the correct hypothesis and strikes the optimal balance between the expected costs of sampling and choosing the wrong hypothesis. Previous work on Bayesian sequential hypothesis testing in continuous time provides a solution when the characteristics of these processes are tested separately. However, the decision of an observer can improve greatly if multiple information sources are available both in the form of continuously changing signals (Brownian motions) and marked count data (compound Poisson processes). In this paper, we combine and extend those previous efforts by considering the problem in its multisource setting. We identify a Bayes optimal rule by solving an optimal stopping problem for the likelihood-ratio process. Here, the likelihood-ratio process is a jump-diffusion, and the solution of the optimal stopping problem admits a two-sided stopping region. Therefore, instead of using the variational arguments (and smooth-fit principles) directly, we solve the problem by patching the solutions of a sequence of optimal stopping problems for the pure diffusion part of the likelihood-ratio process. We also provide a numerical algorithm and illustrate it on several examples. On some probability space (Ω, F , P), let (X (i) t ) t≥0 , 1 ≤ i ≤ d be d independent Brownian motions with constant drifts μ (i) , 1 ≤ i ≤ d, and (T (j ) n , Z (j ) n ) n≥1 , 1 ≤ j ≤ m be m independent compound Poisson processes independent of the Brownian motions. For every 1 ≤ j ≤ m, (T (j ) n ) n≥1 are the arrival times, and (Z (j ) n ) n≥1 are the marks on some measurable space (E, E), with arrival rates λ (j ) and mark distributions ν (j ) (·) on (E, E). Suppose that μ (i) , 1 ≤ i ≤ d and (λ (j ) , ν (j ) ) 1≤j ≤m are unknown, but exactly one of the following two simple hypotheses, H 0 : 1 , ν (j ) 1 ) for every 1 ≤ j ≤ m, where probability measures ν (j ) 0 and ν (j ) 1 on (E, E) are equivalent. Let Θ be the index of correct hypothesis, which is a {0, 1}-valued random variable with prior distribution P{Θ = 1} = 1 − P{Θ = 0} = π for some known π ∈ (0, 1).. The problem is to find a stopping time τ and a terminal decision rule d which depend only on the observations of Brownian motions (X (i) n ) n≥0 , 1 ≤ i ≤ d and compound Poisson processes (T (j ) n , Z (j ) n ) n≥1 , 1 ≤ j ≤ m, and which minimizes the Bayes risk where a and b are known positive constants and correspond to the costs of making wrong terminal decisions. If such a decision rule (τ, d) exists, then it strikes optimal balance between the expected total sampling cost and the expected cost of selecting the wrong hypothesis. Sequential hypothesis testing problems have been studied extensively in the literature due to their practical applications in different fields. These include target detection in radar and sonar systems, threat identification in homeland security, fault identification and isolation in industrial processes, testing the riskiness of financial assets; see, e.g., Marcus and Swerling (1962) , Fu (1968) , Veeravalli and Baum (1996) , Ernisse et al. (1997) , Dragalin et al. (2000) , Lai (2001) , and the references therein.. The non-Bayes formulation of the sequential hypothesis testing problem has been studied by many authors, both in discrete-and continuous-time, and can be found in the recent reviews and contributions made by Lai (2000 Lai ( , 2001 , Dragalin et al. (1999 Dragalin et al. ( , 2000 , Lorden (1977) . In the Bayesian framework, sequential hypothesis testing problems were studied in discrete-time for the identification of the distribution of i.i.d. observations by Wald and Wolfowitz (1950) , Blackwell and Girshick (1979) , Zacks (1971) , Shiryaev (1978) . In continuous-time, on the other hand, Bayesian formulations are studied and solved for the identification of the drift of a Brownian motion in Shiryaev (1978) , Peskir and Gapeev (2004) , Shiryaev and Zhitlukhin (2011) , for the identification of the drift term of more general diffusion processes in Shiryaev and Gapeev (2011) , for the identification of the arrival rate of a simple Poisson process in Shiryaev (2000, 2006) , and for the identification of the arrival rate and mark distribution of a compound Poisson process in Gapeev (2002) , Dayanik and Sezer (2006) , Dayanik et al. (2008b) , Ludkovski and Sezer [2012, Sect. 5.2] . The problem has not been addressed earlier for joint identification of local characteristics of concurrently observed independent Brownian motions and compound Poisson processes, and its solution is the main contribution of this paper.. Multisource detection problems with Brownian motions and compound Poisson processes appear naturally in many fields. In finance, for example, stock prices are modeled with exponential Brownian motions, and the firm defaults or credit derivatives are modeled with compound Poisson processes. Stock prices, firm defaults, and credit derivatives move together in a random economic environment, which may either be in recession or in a booming state. For a fund's asset manager interested in the best short-term financial plans, it is important to identify as quickly and confidently as possible if the economy is in recession or in a booming regime. For that purpose, the asset manager can trace the stock prices of certain companies operating in different industries or even in different countries. It is then reasonable to expect that those stock price processes change independently conditionally on the common market indicators of a recession or a booming economy. In addition to those stocks, the asset manager may also trace the number of defaults in some other key industries. To the extend that the conditionally independent industry default processes and the stock return rate processes of companies operating in different and diverse industries can be modeled with compound Poisson processes and Brownian motions with arrival rates, mark distributions, and drift rates modulated by the given common unobserved market indicators of economic state, our paper provides a simple formulation and solution for the asset manager's problem. Similar problem also arises when we test the reliability of mechanical systems; we can monitor both the occurrence/depth of cracks as marked count data and the vibrations in the system as continuously changing signals for a better diagnosis.. When the observations come from both several Brownian motions and compound Poisson processes simultaneously, finding the best multisource sequential identification rule becomes a very difficult dynamic programming problem because of an unfavorable secondorder integro-differential operator. Instead of following the standard variational arguments, we develop an alternative solution method that takes specifically into account the special structure of the sample-paths of a suitable sufficient statistic for the problem. This method enables us to solve the problem completely for the most general case-without any need for specific simplifying assumptions about the relationships between drift rates, arrival rates, and mark distributions.. We show that an optimal decision rule (τ, d) always exists. The optimal stopping time τ is when the likelihood-ratio process. exits for the first time a bounded interval (φ 1 (1 − π)/π, φ 2 (1 − π)/π) for some suitable constants 0 < φ 1 < b/a < φ 2 < ∞, and optimal terminal decision rule d is to choose the null hypothesis if πL τ /(1 − π) ≤ b/a and the alternative hypothesis otherwise. We describe a provably convergent numerical method to calculate both the minimum Bayes risk and the decision boundaries φ 1 and φ 2 of the optimal stopping rule τ . The minimum Bayes risk is shown to be the uniform limit of a decreasing sequence of successive approximations, which are obtained by applying a contraction mapping iteratively to a suitable initial function. The maximum absolute difference between successive approximations is bounded by an explicit bound, which decays at a known exponential rate with the number of iterations. Thus, one can always determine the necessary number of iterations ex-ante for any desired level of accuracy in the approximations of the minimum Bayes risk and optimal decision boundaries. We address the problem by reducing it to the optimal stopping of the likelihood-ratio process, which we solve later. The likelihood-ratio process is a jump-diffusion with an infinitesimal generator which is a second-order integro-differential operator, and the conventional method of variational inequalities is very unlikely to succeed. Instead, we solve the problem by means of a jump operator, which is obtained by applying the dynamic programming principle at the jump times. The role of this operator is to patch, at successive jump times, the solutions of suitably modified optimal stopping problems for pure diffusion part. The latter modified problems are solved easily and directly by the potential-theoretic methods developed by Dayanik and Karatzas (2003) and Dayanik (2008) . A similar sequential plan was followed by Dayanik et al. (2008a) and Sezer (2010) to solve sequential change detection problems, each admitting a one-sided optimal stopping region with only one decision boundary. The multisource Bayesian sequential binary hypothesis testing problem, however, is far more challenging with a two-sided optimal stopping region and two critical decision boundaries, which should be determined simultaneously. Optimal stopping problems for jump-diffusions, which admit two-sided optimal stopping rules, appear also in finance and real-option theory for pricing American-type financial contracts, which, we believe, can be tackled very effectively with the same method of this paper. We refer the reader to Salminen (1985) , Lerche (1997, 2001 ), Christensen and Irle (2011), Cissé et al. (2012) for examples of and additional remarks on problems with two-sided stopping regions. For the general theory of optimal stopping for (jump) diffusions, the books by Shiryaev (1978) , Peskir and Shiryaev (2006) , Øksendal and Sulem (2007) and the references cited therein can be consulted.. In Sect. 5, we show that the general multisource sequential testing problem can be reduced to a simple one where the observations consist of those of only one Brownian motion and only one compound Poisson process (i.e., d = m = 1). Therefore, in the remainder except Sect. 5, we assume that d = m = 1 and drop the superscripts used to identify local characteristics, arrival times, and marks with different Brownian motions and compound Poisson processes.. In Sect. 2, we start with the precise description of the problem and the derivation of an auxiliary optimal stopping problem. Section 3 introduces a key jump operator and successive approximations to the value function of the auxiliary optimal stopping problem, whose solution is explicitly identified, along with the Bayes-optimal decision rule for the Bayesian sequential binary hypothesis testing problem, in Sect. 4. Section 5 shows how to reduce the multisource problem to that of one Brownian motion and one compound Poisson process, the solution of which was already given in Sect. 4. Section 6 concludes with a numerical algorithm to find Bayes ε-optimal decision rules and its illustrations on several numerical examples. Some of the proofs are deferred to the Appendix.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Linear-Time Algorithms for Tree Root Problems Abstract Let T be a tree on a set V of nodes. The p-th power T p of T is the graph on V such that any two nodes u and w of V are adjacent in T p if and only if the distance of u and w in T is at most p. Given an n-node m-edge graph G and a positive integer p, the p-th tree root problem asks for a tree T , if any, such that G = T p . Given an n-node m-edge graph G, the tree root problem asks for a positive integer p and a tree T , if any, such that G = T p . Kearney and Corneil gave the best previously known algorithms for both problems. Their algorithm for the former (respectively, latter) problem runs in O(n 3 ) (respectively, O(n 4 )) time. In this paper, we give O(n + m)-time algorithms for both problems. Let H be a graph on a set V of nodes. The p-th power H p of H is the graph on V such that any two nodes u and w of V are adjacent in H p if and only if the distance of u and w in H is at most p. If G = H p , then we say that graph H is a p-th root of graph G or, equivalently, G is the p-th power of H . Graph roots and graph powers have been extensively studied in the literature [1-6, 12, 15, 16, 18-21, 23, 26, 28, 38, 39, 41, 43, 47, 51] . See [8, Sect. 10.6 ] for a survey. Motwani and Sudan [40] proved that recognizing squares of graphs is NP-complete. Lau [33] showed that squares of bipartite graphs can be recognized in polynomial time and proved the NPcompleteness of recognizing cubes of bipartite graphs. Lau and Corneil [34] also studied the tractability of recognizing powers of proper interval, split, and chordal graphs. Lin and Skiena [35] gave a linear-time algorithm to find square roots of planar graphs.. If G = T p for some tree T and integer p, we say that G is a p-th tree power and call tree T a p-th tree root of G. Given a graph G and a positive integer p, the p-th tree root problem asks for a tree T , if any, with G = T p . Given a graph G, the tree root problem asks for a tree T and an integer p, if any, with G = T p . Various versions of tree-root problems are associated with fundamental issues in distributed computing [32, 36] and phylogeny [7, 13, 43] . Ross and Harary [45] characterized squares of trees and showed that square tree roots, when they exist, are unique up to isomorphism. Lin and Skiena [35] gave a linear-time algorithm to recognize squares of trees. Kearney and Corneil [32] gave the best previously known algorithms for the p-th tree root problem and the tree root problem. Their algorithm for the p-th tree root problem runs in O(n 3 ) time for any n-node m-edge graph, leading to an O(n 4 )-time algorithm for the tree root problem. Gupta and Singh [25] gave a characterization of p-th tree powers and proposed a heuristic algorithm to construct a p-th tree root. Their algorithm runs in O(n 3 ) time, but its correctness is not proved in the paper. It was unknown whether the p-th tree root problem can be solved in o(n 3 ) time [32, 33] . In this paper we improve Kearney and Corneil's result [32] by giving O(m + n)-time algorithms for the tree root problem and the p-th tree root problem for any given p. Our results lead to the first known O(m + n)-time obtainable informationtheoretically optimal 2n + O(log n)-bit succinct encoding for any n-node m-edge tree power. (See, e.g., [14, 27, 31, 37, 42, 46] for results in succinct encodings.). The rest of the paper is organized as follows. Section 2 gives the preliminaries. Section 3 gives our O(m + n)-time algorithm for the 2h-th tree root problem for any given positive integer h. Section 4 gives our O(m + n)-time algorithm for the (2h + 1)-st tree root problems for any given non-negative integer h. Section 5 gives our O(m+n)-time algorithm for the tree root problem. Section 6 concludes the paper. The combination of Theorems 3.1 and 4.3 yields a linear-time algorithm for the p-th tree root problem for any given integer p. Moreover, we show how to compute in linear time the smallest even integer p (Theorem 5.2) and odd integer p (Theorem 5.3), if any, such that a graph admits a p-th tree root. Combining Theorems 3.1, 4.3, 5.2, and 5.3, we have a linear-time algorithm for the tree-root problem. It would be of interest to see if our techniques can be extended to work for larger graph classes.

O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


An integrated approach for structural damage identification using wavelet neuro-fuzzy model Structural damage can be identified by processing structural vibration response signals and excitation data, and thus the suitability of signal processing methods is essential to structural damage identification. To explore an intelligent signal processing method for structural damage identification, the paper integrated wavelet real-time filtering algorithm, Adaptive Neruo-Fuzzy Inference System (ANFIS) and interval modeling technique to process structural response signals and excitation data. With Wavelet Transform (WT) algorithm filtering random noise, ANFIS was found to model the structural behavior properly and interval modeling technique to quantify damage index accurately. The rapid identifications of several unknown damages and small damages indicate the efficiency of this integrated method. The comparison of these results and some other signal processing methods shows that, the proposed method can be used to identify both the time and the location when the structural damage occurs unexpectedly. Vibration based damage identification approach is a common kind of structural damage identification method. In these methods, Wavelet Transformation (WT) method has been widely exploited in treatment of vibration response and construction of damage feature extractor; Adaptive Neuro-Fuzzy Inference System (ANFIS) has been used to indicate damage characteristic. In most studies, WT was used as an effective signal analysis tool and brings damage-incurred singularities into prominence, addressing features of damage and enabling a solution to damage identification (Cao, Cheng, Su, & Xu, 2012) . ANFIS can be fed into extracted features in order to identify the size and position of the damage and to estimate structural operating conditions (Saeed, Galybin, & Popov, 2013) .. One of the difficulties in identification process is the treatment of uncertainty caused by noise and variability, which has been well handled by WT algorithm (Peretto, Sasdelli, & Tinarelli, 2005) , AN-FIS (Ch & Mathur, 2010) or interval modeling technique (Lew, 2011) . WT algorithm and ANFIS may be integrated by interval modeling technique.. The uncertainty of vibration response in helicopter is caused by numerous complex loading conditions, highly flexible uses and strong randomicity in maneuvering, etc. Flap, lag, and torsion motions of rotor system in the non-symmetric flow caused by rotating and sliding lead to complicated aerodynamic coupling load and structural vibration load acting on moving parts and adjacent structures. Furthermore, vibration phenomenon is the intrinsic characteristic of helicopter, which causes 40% of the accidents. Thus, helicopter structural damage identification using vibration response is challenging yet plausible if the uncertainty is carefully processed.. The main task of this paper is to deal with uncertainty using signal processing algorithm in the process of helicopter structural damage identification, and the purpose is to explore an intelligent approach for vibration based structural damage identification using wavelet real-time filtering algorithm, ANFIS and interval modeling technique cooperatively. The integrated approach takes WT method as a real-time filtering algorithm, ANFIS as a structural response behavior modeling tool, and interval modeling technique as a damage feature extractor.. The proposed approach begins by filtering the sensor's output time series with wavelet real-time filtering algorithm. After trained by input-output time series, ANFIS is then used to model the structure and predict the displacement response. The interval modeling technique accepts prediction results as the input and gives uncertainty coordinates as the output. Information fusion is subsequently accomplished and then the damage can be identified after damage index is constructed.. This approach can be used as a structural damage identification method, which provides new ideas for structural modeling and structural damage feature extracting. Furthermore, this study may cast light on parameter identification schemes. In this paper, ANFIS is used for structural modeling. After trained by input-output response data, ANFIS acts as a real structure and outputs structural displacement vector. The structural damage identification method proposed in this paper can realize dynamic identification of unknown damage time and location at 0.01 s after its occurrence. Thus, we can possibly further realize real-time identification of damage if the computer performance meets the requirements. This paper deals with helicopter structural damage identification problem when the frequency and magnitude of harmonic excitation are invariable. In extreme situations like transient state and maneuvering, the excitations acting on helicopter fuselage are not harmonic. Thus, structural damage occurring in these flight conditions can not be identified using the method proposed in this paper. The case study in this paper focuses on a small kind of unmanned helicopter whose flight conditions are relatively simple. Flight states with different harmonic excitations acting on the structure may be further divided into sub-states, within which the excitation is stable harmonic excitation. When the helicopter transforms flight conditions or shifts sub-states among the same flight condition, it is necessary to switch identification method corresponding to the frequency and magnitude of harmonic excitation.. The proposed method can realize rapid damage identification at 0.01 s after its occurrence if the computer performance meets the computing requirements, and this is what makes this paper special. In this paper, all the data during the 11s simulation are computed integrally. So it is difficult to judge whether the computer can complete the calculation of data generated in 0.01 s simulation process within 0.01 s and to predict which kind of computer can identify at 10.01 s the damage occurring at 10 s in real-time.. The identification effect is relatively obvious if simulation analysis is under low noise level, whilst un-conspicuous if under high noise level. The damage status can't be effectively identified at present, and it needs further study. This dynamic identification may also be taken as a kind of real-time approach. Furthermore, as structural damage can be regarded as a dynamic variation of structural parameter, this structural damage identification method may be also exploited as structural parameter identification method. The structural parameter may be updated after structural damage has been identified and located.

O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O B I I O O O B I I I O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O B I I O B I I I O O B I I O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O


Fuzzy human motion analysis: A review Human Motion Analysis (HMA) is currently one of the most popularly active research domains as such significant research interests are motivated by a number of real world applications such as video surveillance, sports analysis, healthcare monitoring and so on. However, most of these real world applications face high levels of uncertainties that can affect the operations of such applications. Hence, the fuzzy set theory has been applied and showed great success in the recent past. In this paper, we aim at reviewing the fuzzy set oriented approaches for HMA, individuating how the fuzzy set may improve the HMA, envisaging and delineating the future perspectives. To the best of our knowledge, there is not found a single survey in the current literature that has discussed and reviewed fuzzy approaches towards the HMA. For ease of understanding, we conceptually classify the human motion into three broad levels: Low-Level (LoL), Mid-Level (MiL), and High-Level (HiL) HMA. Human motion analysis (HMA) refers to the analysis and interpretation of human movements over time. For decades, it has been a popular research topic that crossovers many domains such as biology [1, 2] , psychology [3, 4] , multimedia [5] and so on. In the computer vision domain, HMA has been emerging actively over the years due to the advancement of video camera technologies and the availability of more sophisticated computer vision algorithms in the public domain. Here, the HMA concerns the detection, tracking and recognition of human, and more generally the understanding of human behaviors, from image sequences involving humans. Amongst all, video surveillance is one of the most important real-time applications [6, 7, 8, 9, 10] . For instance, as illustrated in Figure 1 , the Madrid, London and Boston marathon bombing tragedies, happened in 2004, 2005 and 2013 respectively, would not have been worse if an intelligent video surveillance system that capable of automatically detecting abnormal human behavior was installed in the public areas. Apart from that, HMA also contributed in video retrieval [11] , sports analysis [12, 13, 14] , healthcare monitoring [15, 16] , human-computer interaction [17] and so on.. The importance and popularity of the HMA system has led to several surveys in the literature, as indicated in Table 1 . One of the earliest surveys was [18] , focused on various methods employed in the analysis of the human body motion, which is in non-rigid form. [19] gave an overview on the motion extraction methods using the motion capture systems and focused on action recognition, individual body parts recognition, and body configuration estimation. [20] used the same taxonomy as in [19] , but engaging different labels for the three classes, that is further dividing the classes into subclasses yielding a more comprehensive taxonomy. [21] gave an overview on the applications of visual analysis of human movements, and their taxonomy covered the 2D and 3D approaches with and without the explicit shape models.. As the works in this area prosper, public datasets start to gain importance in the vision community to meet different research challenges. The KTH [38] and the Weizmann [39, 40] datasets were the most popular human actions *Chern Hong Lim and Ekta Vats contributed equally to this paper. **Corresponding author. Reviewed the state-of-the-art of "looking at people" focusing on person identification and surveillance monitoring. Human action recognition using multiple views: A comparative perspective on recent developments. Presented a review and comparative study of recent multi-view 2D and 3D approaches for HMA. Fuzzy set theory has been effectively applied in many ways that revealed a number of fuzzy approaches such as FIS, FCM, Fuzzy qualitative reasoning, etc. This paper takes the initiative to review the works that employed these fuzzy approaches in the HMA system which has not been done previously. From the studies, one can notice that the fuzzy approaches are capable of handling the uncertainty that abounded in each level of the HMA system (LoL, MiL, and HiL). The fundamental factors that endowed such capability to the fuzzy approaches include the ability to perform soft labeling and the flexibility to adapt to different uncertainties. However, most of the reported works herein did not utilize the standard HMA datasets as their baseline. Anyway, the current datasets are mostly too ideal to reflect the real world scenarios that is full of uncertainties. The generation of the fuzzy dataset for HMA could be one of the potential future works other than the early event detection and the still image action recognition. After reviewing a number of works using the fuzzy approaches in HMA, we identified some important factors that make fuzzy approaches successful in improving the overall system performance and these will be discussed in this section along with the potential future works.

O B I I O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O