Elsevier

Pattern Recognition

Volume 45, Issue 3, March 2012, Pages 1092-1103
Pattern Recognition

Frameworks for multivariate m-mediods based modeling and classification in Euclidean and general feature spaces

https://doi.org/10.1016/j.patcog.2011.08.021Get rights and content

Abstract

This paper presents an extension of m-mediods based modeling technique to cater for multimodal distributions of sample within a pattern. The classification of new samples and anomaly detection is performed using a novel classification algorithm which can handle patterns with underlying multivariate probability distributions. We have proposed two frameworks, namely MMC-ES and MMC-GFS, to enable our proposed multivarite m-mediods based modeling and classification approach workable for any feature space with a computable distance metric. MMC-ES framework is specialized for finite dimensional features in Euclidean space whereas MMC-GFS works on any feature space with a computable distance metric. Experimental results using simulated and complex real life dataset show that multivariate m-mediods based frameworks are effective and give superior performance than competitive modeling and classification techniques especially when the patterns exhibit multivariate probability density functions.

Highlights

► Extension of m-mediods based modeling technique to cater for classes with multimodal PDF. ► A soft classification and anomaly detection adaptive to multimodal distribution of sample. ► Two frameworks are proposed to enable working of proposed classifier for any feature space. ► Proposed MMC-ES is a specialized framework tuned for feature space with computable mean. ► Proposed MMC-GFS framework is applicable to any feature space with computable similarity measure.

Introduction

In recent research, there has been a growth of research attention aimed at the development of sophisticated approaches for pattern modeling and data classification. Detecting anomalous events is an important ability of any good classification system. Classification of unseen samples and anomaly detection require building models of normality. Once the models of normal classes are learnt, these can then be used for classifying new unseen trajectory data as normal (i.e. belonging to one of the modeled classes) or anomalous (not lying in the normality region of modeled classes).

Various machine learning techniques have been proposed for modeling of normal patterns and performing classification using the generated model of normality. Statistical approaches dealing with classification and anomaly detection are based on approximating the density of training data and rejecting test patterns that fall in regions of low density. Khalid and Naftel [9] and Hu et al. [7], [8] models normal motion patterns by estimating single multivariate Gaussian for each class. Khalid and Naftel performs classification using Mahalanobis classifier and anomaly detection using Hotelling's test. In [7], the probability of a sample belonging to each pattern is calculated and the sample is classified to the pattern with the highest probability. However, if the probability of association of sample to the closest pattern is less then a threshold, the sample is deemed anomalous. Approaches using GMM to model normality distribution have also been proposed [1], [2], [3]. Various techniques [4], [5], [6] based on hidden Markov models (HMM) have also been presented for modeling and classification of temporal data. Owen and Hunter [10] use Self-Organizing Feature Maps (SOFM) to learn normal patterns. While classifying unseen samples, if the distance of the sample to its allocated class exceeds a threshold value, the trajectory is identified as anomalous. Marsland et al. [11] propose a novelty filter referred to as Grow When Required (GWR) network that uses SOM, based on habituation, to learn the environment and to discover novel features. The proposed approach is suitable for online use. GWR can add and delete nodes whenever the network in its current state does not sufficiently match the evolving pattern.

Approaches using support vector machines (SVM) have also been proposed [12], [13], [14]. These approaches are based on the principal of separating data belonging to different classes by identifying an optimal hyper plane between them. SVM based approaches involve computation of pairwise distances and time-consuming optimizations. Zhang et al. [12] propose a hybrid approach using SVM and nearest neighbor classifier for content based image recognition with the multiclass setting. Rasch et al. [13] perform classification and anomaly detection using one-class SVM. Various Mahalanobis distance metric learning approaches have also been applied for data clustering and classification [15], [16], [17], [18], [19]. Certain discriminative learning methods such as Fisher Discriminative Learning (FDA) [20] have been proposed for improving the performance of classifier. FDA is a supervised dimensionality reduction mechanism which computes a transformation matrix to maximize inter-class and minimize intra-class scatter. FDA caters for the global distribution of the pattern and gives poor performance in the presence of multivariate distribution of samples within a pattern. An extension of FDA, referred to as local FDA (LFDA) [21] has been proposed to work in the presence of multivariate distribution of samples within a pattern by preserving the local structure of data. LFDA based supervised dimensionality reduction has been combined with well known classifiers such as GMM to improve classifier's performance [22]. However, LFDA can only be applied in feature spaces with a calculable mean and is not applicable to general feature spaces.

In our previous work [28], we have proposed m-mediods based Modeling and Classification approach for features in Euclidean Space (MC-ES). m-mediods based approach models a pattern by a set of cluster centers of mutually disjunctive sub-classes (referred to as mediods) within the pattern. The modeling mechanism is influenced by HSACT-LVQ based clustering mechanism as proposed in our previous work [9]. It has been shown that hierarchical semi-agglomerative approach using a neural network, such as HSACT-LVQ, outperforms hard clustering techniques such as k-Means. k-Means is initialized with the number of cluster centers that are equivalent to the expected number of groupings in the dataset. The cluster centers itself is normally initialized to a randomly picked sample from dataset. This type of hard clustering does not guarantee that the network will identify and distinguish all major groupings. k-Means may organize the cluster centers to represent variations within one major grouping of the data by allocating more than one cluster center to that group. This may be caused by the initialization of more than one cluster center close to or within the distribution of a single pattern. Allocation of more than one output neuron to a pattern will result in having a single cluster center representing multiple patterns. k-Means, therefore, produce poor clustering and classification results due to poor initialization. On the other hand, our HSACT-LVQ algorithm avoids this problem by initializing itself with greater number of cluster centers than the number of groupings to be identified in the dataset. Finer clusters are then merged, based on their similarities, to generate coarse clusters representing the desired number of sub-classes (mediods). The modeling technique, proposed in [28] referred to as m-mediods modeling, models the class containing n members with m-mediods known a-priori. Once the m-mediods model for all the classes have been learnt, the MC-ES approach performs classification of new samples and anomaly detection by checking the closeness of said samples to the models of different classes using hierarchical classifier. The anomaly detection module required specification of threshold which is used globally for all the patterns. However, this approach had unaddressed issues like manual specification of threshold for anomaly detection, identification of appropriate value of threshold for anomaly detection and anomaly detection of patterns with different scale and orientation which is used globally for all the patterns. These issues are addressed by a localized m-mediods based approach (LMC-ES) as proposed in [23] which enables us to automatically select a local significance parameter for each pattern taking into consideration the distribution of individual patterns.

LMC-ES can effectively handle patterns with different orientation and scale and has been shown to give superior performance than competitors including GMM, HMM and SVM based classifiers. However, there are still open issues (i) Modeling, classification and anomaly detection in the presence of multivariate distribution of samples within a pattern, (ii) Soft classification in the presence of multimodal pattern distribution to minimize misclassification, (iii) Modeling and classification in feature spaces for which we can not compute mean.

The contribution of this paper is to present an extension of m-mediods based modeling approach, wherein the multimodal distribution of samples in each pattern is represented using multivariate m-mediods. An approach for multivariate model-based classification and anomaly detection is also presented. The proposed mechanism is based on a soft classification approach which enables the proposed multivariate classifier to adapt to the multimodal distribution of samples within different patterns. We have proposed two frameworks for multivariate m-mediods based modeling and classification applicable to two different feature spaces:

  • 1.

    Finite dimensional features in Euclidean space

  • 2.

    General feature spaces with a computable pairwise similarity measure

This enables our multivariate m-mediods based approach to be used for classification and anomaly detection in any feature space with a given distance function.

The remainder of the paper is organized as follows. In Section 2, an overview of the general working of proposed multivariate m-mediods based modeling and classification approach is presented. Section 3 presents a framework of multivariate modeling and classification for finite dimensional features in Euclidean space with a calculable mean. In Section 4, a modification of multivariate modeling and classification framework to operate in any feature space with a computable similarity function is presented. Comparative evaluation of proposed multivariate m-mediods and previously proposed localized m-mediods [23] based frameworks is presented in Section 5. Experiments have been performed to show the effectiveness of proposed system for modeling, classification and anomaly detection in the presence of multimodal distribution of samples within a pattern, as compared to competitors. These experiments are reported in Section 6. The last section summarizes the paper.

Section snippets

Overview of our classification approach

Classification and anomaly detection in the presence of multivariate distributions of sample within a pattern is a challenging task. Fig. 1 gives an overview of our general multivariate modeling and classification framework to effectively cope with this challenge. The proposed classifier, like any other classifier is composed of two main modules: construction of multivariate m-mediods based model to cater for variation in distribution of samples belonging to a particular pattern and using the

Multivariate modeling and classification for finite dimensional features in Euclidean space (MMC-ES)

In this section, we present a framework for multivariate modeling and classification using m-mediods approach that is applicable to features in Euclidean space with calculable mean.

Multivariate modeling and classification for general feature spaces with a computable pairwise similarity measure (MMC-GFS)

The framework of multivariate m-mediods based modeling and classification, as presented in Section 3, works only with feature spaces with calculable mean. However, for complex feature spaces, it is not always possible to calculate a mean. This section provides a modified multivariate m-mediod based framework for any feature space, given that there is a computable pairwise similarity measure.

Relative merits of proposed modeling and classification algorithms

In this section, we provide a comparative evaluation of the proposed multivariate m-mediods and localized m-mediods [23] based frameworks (LMC-ES) for modeling, classification and anomaly detection. These frameworks can be characterized in terms of the following attributes:

  • Ability to deal with multimodal distribution within a pattern

  • Time complexity of generating m-mediods based model of known patterns

  • Time complexity of classification and anomaly detection using learned models of normality

Experimental results

In this section, we present some results to analyze the performance of the proposed multivariate m-mediods based modeling, classification and anomaly detection as compared to competitive techniques.

Discussion and conclusions

In this paper, we have presented an extension of localized m-mediods based modeling technique to cater for multimodal distribution of samples within a pattern. The strength of the proposed approach is its ability to model complex patterns without imposing any restriction on the distribution of samples within a given pattern. Once the multivariate m-mediods model for all the classes have been learnt, the classification of new trajectories and anomaly detection is then performed using a proposed

Shehzad Khalid graduated from Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Pakistan, in 2000. He received the M. Sc. degree from National University of Science and Technology, Pakistan in 2003 and the Ph.D. degree from the University of Manchester, UK, in 2009. He is currently an Associate professor at the Bahria University of Management and Computer Sciences, Pakistan. His research interests includes: dimensionality reduction, indexing and retrieval, profiling and

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    Shehzad Khalid graduated from Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Pakistan, in 2000. He received the M. Sc. degree from National University of Science and Technology, Pakistan in 2003 and the Ph.D. degree from the University of Manchester, UK, in 2009. He is currently an Associate professor at the Bahria University of Management and Computer Sciences, Pakistan. His research interests includes: dimensionality reduction, indexing and retrieval, profiling and classification, trajectory-based motion learning profiling and classification, computer vision, machine learning.

    Shahid Razzaq received the BS degree in Computer System Engineering from Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Pakistan in 2000 and an MS degree in Computer Science from University of Washington in 2008. He worked at Microsoft Corporation from 2001 to 2008 at the Redmond campus near Seattle, USA. He is currently a lecturer in the Department of Computing at the School of Electrical Engineering and Computer Science (SEECS), a constituent of the National University of Sciences and Technology, Islamabad, Pakistan. His research interests include machine learning and pattern recognition.

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