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Title: Collaborative Filtering in a Non-Uniform World: Learning with the Weighted Trace Norm
Abstract: We show that matrix completion with trace-norm regularization can be significantly hurt when entries of the matrix are sampled non-uniformly. We introduce a weighted version of the trace-norm regularizer that works well also with non-uniform sampling. Our experimental results demonstrate that the weighted trace-norm regularization indeed yields significant gains on the (highly non-uniformly sampled) Netflix dataset.
Title: Model-Driven Constraint Programming
Abstract: Constraint programming can definitely be seen as a model-driven paradigm. The users write programs for modeling problems. These programs are mapped to executable models to calculate the solutions. This paper focuses on efficient model management (definition and transformation). From this point of view, we propose to revisit the design of constraint-programming systems. A model-driven architecture is introduced to map solving-independent constraint models to solving-dependent decision models. Several important questions are examined, such as the need for a visual highlevel modeling language, and the quality of metamodeling techniques to implement the transformations. A main result is the s-COMMA platform that efficiently implements the chain from modeling to solving constraint problems
Title: Reconstruction of signals with unknown spectra in information field theory with parameter uncertainty
Abstract: The optimal reconstruction of cosmic metric perturbations and other signals requires knowledge of their power spectra and other parameters. If these are not known a priori, they have to be measured simultaneously from the same data used for the signal reconstruction. We formulate the general problem of signal inference in the presence of unknown parameters within the framework of information field theory. We develop a generic parameter uncertainty renormalized estimation (PURE) technique and address the problem of reconstructing Gaussian signals with unknown power-spectrum with five different approaches: (i) separate maximum-a-posteriori power spectrum measurement and subsequent reconstruction, (ii) maximum-a-posteriori power reconstruction with marginalized power-spectrum, (iii) maximizing the joint posterior of signal and spectrum, (iv) guessing the spectrum from the variance in the Wiener filter map, and (v) renormalization flow analysis of the field theoretical problem providing the PURE filter. In all cases, the reconstruction can be described or approximated as Wiener filter operations with assumed signal spectra derived from the data according to the same recipe, but with differing coefficients. All of these filters, except the renormalized one, exhibit a perception threshold in case of a Jeffreys prior for the unknown spectrum. Data modes, with variance below this threshold do not affect the signal reconstruction at all. Filter (iv) seems to be similar to the so called Karhune-Loeve and Feldman-Kaiser-Peacock estimators for galaxy power spectra used in cosmology, which therefore should also exhibit a marginal perception threshold if correctly implemented. We present statistical performance tests and show that the PURE filter is superior to the others.
Title: Geometric approach to sampling and communication
Abstract: Relationships that exist between the classical, Shannon-type, and geometric-based approaches to sampling are investigated. Some aspects of coding and communication through a Gaussian channel are considered. In particular, a constructive method to determine the quantizing dimension in Zador's theorem is provided. A geometric version of Shannon's Second Theorem is introduced. Applications to Pulse Code Modulation and Vector Quantization of Images are addressed.
Title: Rewriting Constraint Models with Metamodels
Abstract: An important challenge in constraint programming is to rewrite constraint models into executable programs calculat- ing the solutions. This phase of constraint processing may require translations between constraint programming lan- guages, transformations of constraint representations, model optimizations, and tuning of solving strategies. In this paper, we introduce a pivot metamodel describing the common fea- tures of constraint models including different kinds of con- straints, statements like conditionals and loops, and other first-class elements like object classes and predicates. This metamodel is general enough to cope with the constructions of many languages, from object-oriented modeling languages to logic languages, but it is independent from them. The rewriting operations manipulate metamodel instances apart from languages. As a consequence, the rewriting operations apply whatever languages are selected and they are able to manage model semantic information. A bridge is created between the metamodel space and languages using parsing techniques. Tools from the software engineering world can be useful to implement this framework.
Title: Using ATL to define advanced and flexible constraint model transformations
Abstract: Transforming constraint models is an important task in re- cent constraint programming systems. User-understandable models are defined during the modeling phase but rewriting or tuning them is manda- tory to get solving-efficient models. We propose a new architecture al- lowing to define bridges between any (modeling or solver) languages and to implement model optimizations. This architecture follows a model- driven approach where the constraint modeling process is seen as a set of model transformations. Among others, an interesting feature is the def- inition of transformations as concept-oriented rules, i.e. based on types of model elements where the types are organized into a hierarchy called a metamodel.
Title: Convergence of Bayesian Control Rule
Abstract: Recently, new approaches to adaptive control have sought to reformulate the problem as a minimization of a relative entropy criterion to obtain tractable solutions. In particular, it has been shown that minimizing the expected deviation from the causal input-output dependencies of the true plant leads to a new promising stochastic control rule called the Bayesian control rule. This work proves the convergence of the Bayesian control rule under two sufficient assumptions: boundedness, which is an ergodicity condition; and consistency, which is an instantiation of the sure-thing principle.
Title: Structured, sparse regression with application to HIV drug resistance
Abstract: We introduce a new version of forward stepwise regression. Our modification finds solutions to regression problems where the selected predictors appear in a structured pattern, with respect to a predefined distance measure over the candidate predictors. Our method is motivated by the problem of predicting HIV-1 drug resistance from protein sequences. We find that our method improves the interpretability of drug resistance while producing comparable predictive accuracy to standard methods. We also demonstrate our method in a simulation study and present some theoretical results and connections.
Title: A new approach to content-based file type detection
Abstract: File type identification and file type clustering may be difficult tasks that have an increasingly importance in the field of computer and network security. Classical methods of file type detection including considering file extensions and magic bytes can be easily spoofed. Content-based file type detection is a newer way that is taken into account recently. In this paper, a new content-based method for the purpose of file type detection and file type clustering is proposed that is based on the PCA and neural networks. The proposed method has a good accuracy and is fast enough.
Title: A Complete Characterization of Statistical Query Learning with Applications to Evolvability
Abstract: Statistical query (SQ) learning model of Kearns (1993) is a natural restriction of the PAC learning model in which a learning algorithm is allowed to obtain estimates of statistical properties of the examples but cannot see the examples themselves. We describe a new and simple characterization of the query complexity of learning in the SQ learning model. Unlike the previously known bounds on SQ learning our characterization preserves the accuracy and the efficiency of learning. The preservation of accuracy implies that that our characterization gives the first characterization of SQ learning in the agnostic learning framework. The preservation of efficiency is achieved using a new boosting technique and allows us to derive a new approach to the design of evolutionary algorithms in Valiant's (2006) model of evolvability. We use this approach to demonstrate the existence of a large class of monotone evolutionary learning algorithms based on square loss performance estimation. These results differ significantly from the few known evolutionary algorithms and give evidence that evolvability in Valiant's model is a more versatile phenomenon than there had been previous reason to suspect.
Title: Efficiently Discovering Hammock Paths from Induced Similarity Networks
Abstract: Similarity networks are important abstractions in many information management applications such as recommender systems, corpora analysis, and medical informatics. For instance, by inducing similarity networks between movies rated similarly by users, or between documents containing common terms, and or between clinical trials involving the same themes, we can aim to find the global structure of connectivities underlying the data, and use the network as a basis to make connections between seemingly disparate entities. In the above applications, composing similarities between objects of interest finds uses in serendipitous recommendation, in storytelling, and in clinical diagnosis, respectively. We present an algorithmic framework for traversing similarity paths using the notion of `hammock' paths which are generalization of traditional paths. Our framework is exploratory in nature so that, given starting and ending objects of interest, it explores candidate objects for path following, and heuristics to admissibly estimate the potential for paths to lead to a desired destination. We present three diverse applications: exploring movie similarities in the Netflix dataset, exploring abstract similarities across the PubMed corpus, and exploring description similarities in a database of clinical trials. Experimental results demonstrate the potential of our approach for unstructured knowledge discovery in similarity networks.
Title: Message-Passing Algorithms: Reparameterizations and Splittings
Abstract: The max-product algorithm, a local message-passing scheme that attempts to compute the most probable assignment (MAP) of a given probability distribution, has been successfully employed as a method of approximate inference for applications arising in coding theory, computer vision, and machine learning. However, the max-product algorithm is not guaranteed to converge to the MAP assignment, and if it does, is not guaranteed to recover the MAP assignment. Alternative convergent message-passing schemes have been proposed to overcome these difficulties. This work provides a systematic study of such message-passing algorithms that extends the known results by exhibiting new sufficient conditions for convergence to local and/or global optima, providing a combinatorial characterization of these optima based on graph covers, and describing a new convergent and correct message-passing algorithm whose derivation unifies many of the known convergent message-passing algorithms. While convergent and correct message-passing algorithms represent a step forward in the analysis of max-product style message-passing algorithms, the conditions needed to guarantee convergence to a global optimum can be too restrictive in both theory and practice. This limitation of convergent and correct message-passing schemes is characterized by graph covers and illustrated by example.
Title: Coarse-grained modeling of multiscale diffusions: the p-variation estimates
Abstract: We study the problem of estimating parameters of the limiting equation of a multiscale diffusion in the case of averaging and homogenization, given data from the corresponding multiscale system. First, we review some recent results that make use of the maximum likelihood of the limiting equation. In particular, it has been shown that in the averaging case, the MLE will be asymptotically consistent in the limit while in the homogenization case, the MLE will be asymptotically consistent only if we subsample the data. Then, we focus on the problem of estimating the diffusion coefficient. We suggest a novel approach that makes use of the total $p$-variation, as defined in the theory of rough paths and avoids the subsampling step. The method is applied to a multiscale OU process.
Title: Asymptotically Stable Walking of a Five-Link Underactuated 3D Bipedal Robot
Abstract: This paper presents three feedback controllers that achieve an asymptotically stable, periodic, and fast walking gait for a 3D (spatial) bipedal robot consisting of a torso, two legs, and passive (unactuated) point feet. The contact between the robot and the walking surface is assumed to inhibit yaw rotation. The studied robot has 8 DOF in the single support phase and 6 actuators. The interest of studying robots with point feet is that the robot's natural dynamics must be explicitly taken into account to achieve balance while walking. We use an extension of the method of virtual constraints and hybrid zero dynamics, in order to simultaneously compute a periodic orbit and an autonomous feedback controller that realizes the orbit. This method allows the computations to be carried out on a 2-DOF subsystem of the 8-DOF robot model. The stability of the walking gait under closed-loop control is evaluated with the linearization of the restricted Poincar\'e map of the hybrid zero dynamics. Three strategies are explored. The first strategy consists of imposing a stability condition during the search of a periodic gait by optimization. The second strategy uses an event-based controller. In the third approach, the effect of output selection is discussed and a pertinent choice of outputs is proposed, leading to stabilization without the use of a supplemental event-based controller.
Title: Graph Zeta Function in the Bethe Free Energy and Loopy Belief Propagation
Abstract: We propose a new approach to the analysis of Loopy Belief Propagation (LBP) by establishing a formula that connects the Hessian of the Bethe free energy with the edge zeta function. The formula has a number of theoretical implications on LBP. It is applied to give a sufficient condition that the Hessian of the Bethe free energy is positive definite, which shows non-convexity for graphs with multiple cycles. The formula clarifies the relation between the local stability of a fixed point of LBP and local minima of the Bethe free energy. We also propose a new approach to the uniqueness of LBP fixed point, and show various conditions of uniqueness.
Title: High-dimensional variable selection for Cox's proportional hazards model
Abstract: Variable selection in high dimensional space has challenged many contemporary statistical problems from many frontiers of scientific disciplines. Recent technology advance has made it possible to collect a huge amount of covariate information such as microarray, proteomic and SNP data via bioimaging technology while observing survival information on patients in clinical studies. Thus, the same challenge applies to the survival analysis in order to understand the association between genomics information and clinical information about the survival time. In this work, we extend the sure screening procedure Fan and Lv (2008) to Cox's proportional hazards model with an iterative version available. Numerical simulation studies have shown encouraging performance of the proposed method in comparison with other techniques such as LASSO. This demonstrates the utility and versatility of the iterative sure independent screening scheme.
Title: Co-channel Interference Cancellation for Space-Time Coded OFDM Systems Using Adaptive Beamforming and Null Deepening
Abstract: Combined with space-time coding, the orthogonal frequency division multiplexing (OFDM) system explores space diversity. It is a potential scheme to offer spectral efficiency and robust high data rate transmissions over frequency-selective fading channel. However, space-time coding impairs the system ability to suppress interferences as the signals transmitted from two transmit antennas are superposed and interfered at the receiver antennas. In this paper, we developed an adaptive beamforming based on least mean squared error algorithm and null deepening to combat co-channel interference (CCI) for the space-time coded OFDM (STC-OFDM) system. To illustrate the performance of the presented approach, it is compared to the null steering beamformer which requires a prior knowledge of directions of arrival (DOAs). The structure of space-time decoders are preserved although there is the use of beamformers before decoding. By incorporating the proposed beamformer as a CCI canceller in the STC-OFDM systems, the performance improvement is achieved as shown in the simulation results.
Title: Iterative exact global histogram specification and SSIM gradient ascent: a proof of convergence, step size and parameter selection
Abstract: The SSIM-optimized exact global histogram specification (EGHS) is shown to converge in the sense that the first order approximation of the result's quality (i.e., its structural similarity with input) does not decrease in an iteration, when the step size is small. Each iteration is composed of SSIM gradient ascent and basic EGHS with the specified target histogram. Selection of step size and other parameters is also discussed.
Title: Interactive Submodular Set Cover
Abstract: We introduce a natural generalization of submodular set cover and exact active learning with a finite hypothesis class (query learning). We call this new problem interactive submodular set cover. Applications include advertising in social networks with hidden information. We give an approximation guarantee for a novel greedy algorithm and give a hardness of approximation result which matches up to constant factors. We also discuss negative results for simpler approaches and present encouraging early experimental results.
Title: Approximation by log-concave distributions, with applications to regression
Abstract: We study the approximation of arbitrary distributions $P$ on $d$-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback--Leibler-type functional. We show that such an approximation exists if and only if $P$ has finite first moments and is not supported by some hyperplane. Furthermore we show that this approximation depends continuously on $P$ with respect to Mallows distance $D_1(\cdot,\cdot)$. This result implies consistency of the maximum likelihood estimator of a log-concave density under fairly general conditions. It also allows us to prove existence and consistency of estimators in regression models with a response $Y=\mu(X)+\epsilon$, where $X$ and $\epsilon$ are independent, $\mu(\cdot)$ belongs to a certain class of regression functions while $\epsilon$ is a random error with log-concave density and mean zero.
Title: Asymptotic risks of Viterbi segmentation
Abstract: We consider the maximum likelihood (Viterbi) alignment of a hidden Markov model (HMM). In an HMM, the underlying Markov chain is usually hidden and the Viterbi alignment is often used as the estimate of it. This approach will be referred to as the Viterbi segmentation. The goodness of the Viterbi segmentation can be measured by several risks. In this paper, we prove the existence of asymptotic risks. Being independent of data, the asymptotic risks can be considered as the characteristics of the model that illustrate the long-run behavior of the Viterbi segmentation.
Title: Improved EM for Mixture Proportions with Applications to Nonparametric ML Estimation for Censored Data
Abstract: Improved EM strategies, based on the idea of efficient data augmentation (Meng and van Dyk 1997, 1998), are presented for ML estimation of mixture proportions. The resulting algorithms inherit the simplicity, ease of implementation, and monotonic convergence properties of EM, but have considerably improved speed. Because conventional EM tends to be slow when there exists a large overlap between the mixture components, we can improve the speed without sacrificing the simplicity or stability, if we can reformulate the problem so as to reduce the amount of overlap. We propose simple "squeezing" strategies for that purpose. Moreover, for high-dimensional problems, such as computing the nonparametric MLE of the distribution function with censored data, a natural and effective remedy for conventional EM is to add exchange steps (based on improved EM) between adjacent mixture components, where the overlap is most severe. Theoretical considerations show that the resulting EM-type algorithms, when carefully implemented, are globally convergent. Simulated and real data examples show dramatic improvement in speed in realistic situations.
Title: Robust Independent Component Analysis by Iterative Maximization of the Kurtosis Contrast with Algebraic Optimal Step Size
Abstract: Independent component analysis (ICA) aims at decomposing an observed random vector into statistically independent variables. Deflation-based implementations, such as the popular one-unit FastICA algorithm and its variants, extract the independent components one after another. A novel method for deflationary ICA, referred to as RobustICA, is put forward in this paper. This simple technique consists of performing exact line search optimization of the kurtosis contrast function. The step size leading to the global maximum of the contrast along the search direction is found among the roots of a fourth-degree polynomial. This polynomial rooting can be performed algebraically, and thus at low cost, at each iteration. Among other practical benefits, RobustICA can avoid prewhitening and deals with real- and complex-valued mixtures of possibly noncircular sources alike. The absence of prewhitening improves asymptotic performance. The algorithm is robust to local extrema and shows a very high convergence speed in terms of the computational cost required to reach a given source extraction quality, particularly for short data records. These features are demonstrated by a comparative numerical analysis on synthetic data. RobustICA's capabilities in processing real-world data involving noncircular complex strongly super-Gaussian sources are illustrated by the biomedical problem of atrial activity (AA) extraction in atrial fibrillation (AF) electrocardiograms (ECGs), where it outperforms an alternative ICA-based technique.
Title: Plugin procedure in segmentation and application to hyperspectral image segmentation
Abstract: In this article we give our contribution to the problem of segmentation with plug-in procedures. We give general sufficient conditions under which plug in procedure are efficient. We also give an algorithm that satisfy these conditions. We give an application of the used algorithm to hyperspectral images segmentation. Hyperspectral images are images that have both spatial and spectral coherence with thousands of spectral bands on each pixel. In the proposed procedure we combine a reduction dimension technique and a spatial regularisation technique. This regularisation is based on the mixlet modelisation of Kolaczyck and Al.
Title: Estimation for High-Dimensional Linear Mixed-Effects Models Using $\ell_1$-Penalization
Abstract: We propose an $\ell_1$-penalized estimation procedure for high-dimensional linear mixed-effects models. The models are useful whenever there is a grouping structure among high-dimensional observations, i.e. for clustered data. We prove a consistency and an oracle optimality result and we develop an algorithm with provable numerical convergence. Furthermore, we demonstrate the performance of the method on simulated and a real high-dimensional data set.
Title: Non-equilibrium dynamics of stochastic point processes with refractoriness
Abstract: Stochastic point processes with refractoriness appear frequently in the quantitative analysis of physical and biological systems, such as the generation of action potentials by nerve cells, the release and reuptake of vesicles at a synapse, and the counting of particles by detector devices. Here we present an extension of renewal theory to describe ensembles of point processes with time varying input. This is made possible by a representation in terms of occupation numbers of two states: Active and refractory. The dynamics of these occupation numbers follows a distributed delay differential equation. In particular, our theory enables us to uncover the effect of refractoriness on the time-dependent rate of an ensemble of encoding point processes in response to modulation of the input. We present exact solutions that demonstrate generic features, such as stochastic transients and oscillations in the step response as well as resonances, phase jumps and frequency doubling in the transfer of periodic signals. We show that a large class of renewal processes can indeed be regarded as special cases of the model we analyze. Hence our approach represents a widely applicable framework to define and analyze non-stationary renewal processes.
Title: Partition Decoupling for Multi-gene Analysis of Gene Expression Profiling Data
Abstract: We present the extention and application of a new unsupervised statistical learning technique--the Partition Decoupling Method--to gene expression data. Because it has the ability to reveal non-linear and non-convex geometries present in the data, the PDM is an improvement over typical gene expression analysis algorithms, permitting a multi-gene analysis that can reveal phenotypic differences even when the individual genes do not exhibit differential expression. Here, we apply the PDM to publicly-available gene expression data sets, and demonstrate that we are able to identify cell types and treatments with higher accuracy than is obtained through other approaches. By applying it in a pathway-by-pathway fashion, we demonstrate how the PDM may be used to find sets of mechanistically-related genes that discriminate phenotypes.
Title: Supervised Learning of Digital image restoration based on Quantization Nearest Neighbor algorithm
Abstract: In this paper, an algorithm is proposed for Image Restoration. Such algorithm is different from the traditional approaches in this area, by utilizing priors that are learned from similar images. Original images and their degraded versions by the known degradation operators are utilized for designing the Quantization. The code vectors are designed using the blurred images. For each such vector, the high frequency information obtained from the original images is also available. During restoration, the high frequency information of a given degraded image is estimated from its low frequency information based on the artificial noise. For the restoration problem, a number of techniques are designed corresponding to various versions of the blurring function. Given a noisy and blurred image, one of the techniques is chosen based on a similarity measure, therefore providing the identification of the blur. To make the restoration process computationally efficient, the Quantization Nearest Neighborhood approaches are utilized.
Title: Word level Script Identification from Bangla and Devanagri Handwritten Texts mixed with Roman Script
Abstract: India is a multi-lingual country where Roman script is often used alongside different Indic scripts in a text document. To develop a script specific handwritten Optical Character Recognition (OCR) system, it is therefore necessary to identify the scripts of handwritten text correctly. In this paper, we present a system, which automatically separates the scripts of handwritten words from a document, written in Bangla or Devanagri mixed with Roman scripts. In this script separation technique, we first, extract the text lines and words from document pages using a script independent Neighboring Component Analysis technique. Then we have designed a Multi Layer Perceptron (MLP) based classifier for script separation, trained with 8 different wordlevel holistic features. Two equal sized datasets, one with Bangla and Roman scripts and the other with Devanagri and Roman scripts, are prepared for the system evaluation. On respective independent text samples, word-level script identification accuracies of 99.29% and 98.43% are achieved.
Title: A fuzzified BRAIN algorithm for learning DNF from incomplete data
Abstract: Aim of this paper is to address the problem of learning Boolean functions from training data with missing values. We present an extension of the BRAIN algorithm, called U-BRAIN (Uncertainty-managing Batch Relevance-based Artificial INtelligence), conceived for learning DNF Boolean formulas from partial truth tables, possibly with uncertain values or missing bits. Such an algorithm is obtained from BRAIN by introducing fuzzy sets in order to manage uncertainty. In the case where no missing bits are present, the algorithm reduces to the original BRAIN.
Title: Query Learning with Exponential Query Costs
Abstract: In query learning, the goal is to identify an unknown object while minimizing the number of "yes" or "no" questions (queries) posed about that object. A well-studied algorithm for query learning is known as generalized binary search (GBS). We show that GBS is a greedy algorithm to optimize the expected number of queries needed to identify the unknown object. We also generalize GBS in two ways. First, we consider the case where the cost of querying grows exponentially in the number of queries and the goal is to minimize the expected exponential cost. Then, we consider the case where the objects are partitioned into groups, and the objective is to identify only the group to which the object belongs. We derive algorithms to address these issues in a common, information-theoretic framework. In particular, we present an exact formula for the objective function in each case involving Shannon or Renyi entropy, and develop a greedy algorithm for minimizing it. Our algorithms are demonstrated on two applications of query learning, active learning and emergency response.
Title: Handwritten Bangla Basic and Compound character recognition using MLP and SVM classifier
Abstract: A novel approach for recognition of handwritten compound Bangla characters, along with the Basic characters of Bangla alphabet, is presented here. Compared to English like Roman script, one of the major stumbling blocks in Optical Character Recognition (OCR) of handwritten Bangla script is the large number of complex shaped character classes of Bangla alphabet. In addition to 50 basic character classes, there are nearly 160 complex shaped compound character classes in Bangla alphabet. Dealing with such a large varieties of handwritten characters with a suitably designed feature set is a challenging problem. Uncertainty and imprecision are inherent in handwritten script. Moreover, such a large varieties of complex shaped characters, some of which have close resemblance, makes the problem of OCR of handwritten Bangla characters more difficult. Considering the complexity of the problem, the present approach makes an attempt to identify compound character classes from most frequently to less frequently occurred ones, i.e., in order of importance. This is to develop a frame work for incrementally increasing the number of learned classes of compound characters from more frequently occurred ones to less frequently occurred ones along with Basic characters. On experimentation, the technique is observed produce an average recognition rate of 79.25 after three fold cross validation of data with future scope of improvement and extension.
Title: Supervised Classification Performance of Multispectral Images
Abstract: Nowadays government and private agencies use remote sensing imagery for a wide range of applications from military applications to farm development. The images may be a panchromatic, multispectral, hyperspectral or even ultraspectral of terra bytes. Remote sensing image classification is one amongst the most significant application worlds for remote sensing. A few number of image classification algorithms have proved good precision in classifying remote sensing data. But, of late, due to the increasing spatiotemporal dimensions of the remote sensing data, traditional classification algorithms have exposed weaknesses necessitating further research in the field of remote sensing image classification. So an efficient classifier is needed to classify the remote sensing images to extract information. We are experimenting with both supervised and unsupervised classification. Here we compare the different classification methods and their performances. It is found that Mahalanobis classifier performed the best in our classification.