Computational Intelligence and Neuroscience

Volume 2018, Article ID 7525786, 9 pages

https://doi.org/10.1155/2018/7525786

## An Improved Kernel Credal Classification Algorithm Based on Regularized Mahalanobis Distance: Application to Microarray Data Analysis

^{1}Department of Physics, Faculty of Sciences, Université Moulay Ismail, Meknes, Morocco^{2}High School of Technology, Université Moulay Ismail, BP 3103 Route d’Agouray, 50006 Toulal, Meknes, Morocco

Correspondence should be addressed to Khawla EL bendadi; moc.liamg@idadneble.k

Received 13 February 2018; Revised 2 May 2018; Accepted 30 May 2018; Published 27 June 2018

Academic Editor: Amparo Alonso-Betanzos

Copyright © 2018 Khawla EL bendadi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Within the kernel methods, an improved kernel credal classification algorithm (KCCR) has been proposed. The KCCR algorithm uses the Euclidean distance in the kernel function. In this article, we propose to replace the Euclidean distance in the kernel with a regularized Mahalanobis metric. The Mahalanobis distance takes into account the dispersion of the data and the correlation between the variables. It differs from Euclidean distance in that it considers the variance and correlation of the dataset. The robustness of the method is tested using synthetic data and a benchmark database. Finally, a set of DNA microarray data from Leukemia dataset was used to show the performance of our method on real-world application.

#### 1. Introduction

Clustering methods are used to classify samples according to specific properties; they consist of organizing a set of objects into clusters, in order to have a high degree of similarity within these clusters. These approaches are applied in a wide variety of areas including data mining, pattern recognition, and computer vision.

Clustering methods were developed by different ways from which we find the hard, fuzzy, and evidential approaches. In the hard clustering [1], clusters are disjointed and does not overlap; we find many algorithms based on kernels which have developed in [2, 3]. On the other hand, in fuzzy clustering [4, 5], a pattern may belong to a number of clusters with a degree of membership.

However, for evidential clustering [6], the concept of metacluster has been introduced to manage the points belonging to overlapping areas.

The theory of belief functions, also known as Dempster-Shafer theory or evidence theory, is an extension of fuzzy approach. It has been shown to be a powerful framework for representing and reasoning with uncertain and imprecise information. The credal classification works under the framework of this theory also called evidential reasoning [7].

A growing number of applications of belief function theory have been reported in unsupervised learning [6, 8, 9], semisupervised learning [10], image processing [11, 12], social network analysis [13], and data analysis of large dissimilarity data [14, 15].

In the literature, there are several methods based on the concept of evidential reasoning as evidential C-means (ECM) [8], Credal C-means (CCM) [16], and credal classification rule (CCR) [17]. In the CCR approach, each pattern is assigned not only to a single cluster, but also to the metacluster. Two varieties of CCR algorithm are developed, one uses Euclidean distance [17] and another version is improved to take into account the data dispersion [18].

However, most of these clustering algorithms present some limits. These methods use Euclidean distance to calculate the similarity between the centers of clusters and data points in the input space. This similarity performs well with hyperspherical data and may not be significant or even deceptive for categorical data type. However, an appropriate clustering method can be introduced to overcome these limitations when the data structure is complex.

Given these limitations, kernel methods have been proposed. Among these methods which have been applied into many learning systems, we find Support Vectors Machine (SVM) [19], Kernel Fuzzy C-means (KFCM) [20], and Kernel Evidential C-means (KECM) [21]. In the same sense, a kernel version of credal classification rule (KCCR) [22, 23] has been recently proposed. KCCR adopts a new kernel metric in the data space to replace the Euclidian distance in CCR with an appropriate kernel function.

When the dissimilarity measure is based on the Euclidean distance, it only characterizes the mean information of a cluster, and therefore it is sensitive to noise and affects cluster divergence. For this, in this paper, we propose a new clustering algorithm which uses the regularized Mahalanobis distance as dissimilarity in the kernel function. On the one hand, the Mahalanobis distance is a useful measure that determines the similarity between datasets. It takes into account the variance and correlation of the data distribution and is often used to detect aberrant data. On the other hand, the computation of the inverse covariance is perhaps impossible or at least unstable. This problem can be classified as inverse, ill-posed, or ill-conditioned problem. This problem is avoided by different methods such as regularization [24, 25].

The article is organized as follows. In the first section, a compendium of credal classification methodology is described, with the CCR algorithm and its kernel variant. An improved version of KCCR method using Mahalanobis distance is detailed in Section 2. In Section 3, the experimental results using simulated data and many examples from the benchmark database will be presented to show the performance of the method. Also, an application on Leukemia dataset is described in this section in order to better evaluate our contribution. The conclusion is given in Section 4.

#### 2. Overview of Credal Classification

The subject of credal classification is exactly to model classification under conditions where not much information is present, by allowing the classifier to produce sets of outcomes rather than single outcomes. The credal classification is based on the theory of belief functions also called evidential reasoning. It allows the objects to belong with different masses of belief, not only to a singleton cluster, but also to a set of clusters named metacluster. The credal classification of the objects is obtained by the combination of the basic belief assignments (BBA) associated with different classes.

In the literature, one can find different approaches, as well as many applications in several fields for the use of credal partition, such as credal C-means (CCM) [16], belief C-means (BCM) [26], and credal classification rule (CCR) [17]. In the next section, brief recall of CCR approach is presented. After that, a kernel credal classification rule (KCCR) is held.

##### 2.1. Credal Classification Rule

The credal classification rule works with credal partition; it calculates the masses of beliefs associated with the specific cluster, the metacluster, and outlier cluster in a simple way.

The interest of the credal classification is clearly evident when the information is insufficient to properly classify an object. For this purpose, the CCR algorithm assigns these objects to the metaclusters (cluster of imprecise objects) computed under the framework of the belief functions. In fact, the credal classification can well reduce errors and correctly capture the imprecision of the classification for dealing with the uncertain data.

A basic belief assignment (BBA) is a function from to , where represent the frame of discernment satisfying and . Two steps are required for the implementation of the CCR approach. First, we start by determining the centers of different cluster types, after we calculate the distance between the objects to be classified and each cluster center in order to construct the BBA.

The computation of the specific cluster center can be established by different methods. As the calculation of the average of the training data or use the centers retrieve by a classification algorithm like FCM or ECM. The determination of the center of the metacluster is considered with the same distance to all the involved centers of the singleton clusters .

The mass of object associated with singleton and metacluster (union from at least two specific clusters) is calculated by the distance between the object and the corresponding cluster (singleton or meta) centers. The mass of belief on specific cluster and metacluster is given by the following [17]:whereThe parameter is an adjustment-weighting factor of the distance between the object and the center of metacluster. The tuning parameter can be fixed to a small value (1 or 2).

The mass of belief for the outlier cluster (all objects very far from all clusters) is controlled by the parameter , according to the following formula:whereIn order to overcome the limitations of CCR method (already cited in the introduction of this paper) with its two variants [17, 18], a kernel-based credal classification rule (KCCR) was proposed to deal with the complex data structure. The CCR computes the similarity between the centers of clusters and data points in the input space using Euclidean distance, contrary to KCCR, which adopts a kernel metric to replace the similarity in CCR.

The limitations of CCR technique are well shown in experimentation 1 (Section 4.1.). In Figure 2, the behavior of the CCR algorithm is presented. It can be inferred that the points whose numbers are “1”, “3”, “4”, and “5” are considered as noise since they are far from the center of the cluster, despite being labelled in a defined cluster. For example, if you take point “1” you see that it belongs to cluster (see Figure 1), and it is considered as noise in Figure 2. It is noted that points “1”, “3”, “4”, and “5” are well classified by using kernel function in the KCCR and MKCCR algorithm (Figure 4).