Abstract
The well-known Fuzzy C-Means (FCM) algorithm for data clustering has been extended to Evidential C-Means (ECM) algorithm in order to work in the belief functions framework with credal partitions of the data. Depending on data clustering problems, some barycenters of clusters given by ECM can become very close to each other in some cases, and this can cause serious troubles in the performance of ECM for the data clustering. To circumvent this problem, we introduce the notion of imprecise cluster in this paper. The principle of our approach is to consider that objects lying in the middle of specific classes (clusters) barycenters must be committed with equal belief to each specific cluster instead of belonging to an imprecise meta-cluster as done classically in ECM algorithm. Outliers object far away of the centers of two (or more) specific clusters that are hard to be distinguished, will be committed to the imprecise cluster (a disjunctive meta-cluster) composed by these specific clusters. The new Belief C-Means (BCM) algorithm proposed in this paper follows this very simple principle. In BCM, the mass of belief of specific cluster for each object is computed according to distance between object and the center of the cluster it may belong to. The distances between object and centers of the specific clusters and the distances among these centers will be both taken into account in the determination of the mass of belief of the meta-cluster. We do not use the barycenter of the meta-cluster in BCM algorithm contrariwise to what is done with ECM. In this paper we also present several examples to illustrate the interest of BCM, and to show its main differences with respect to clustering techniques based on FCM and ECM.
Highlights
In the data clustering analysis, the credal partition based on the belief functions theory has been introduced recently in (Denœux and Masson, 2003, 2004; Masson and Denœux, 2004, 2008)
In the new Belief C-Means (BCM) algorithm that we propose in this paper, the mass of belief of the specific cluster for each object is computed from the distance between the object and the center of the cluster, and the mass of belief of a meta-cluster is computed both from the distances between object and prototypes of the involved specific clusters, and the distances among these prototypes
BCM is an extension of Fuzzy C-Means (FCM) and an alternative of Evidential C-Means (ECM)
Summary
In the data clustering analysis, the credal partition based on the belief functions theory has been introduced recently in (Denœux and Masson, 2003, 2004; Masson and Denœux, 2004, 2008). In ECM algorithm, the barycenter of a meta-cluster is obtained in averaging the centers of the specific clusters involved in the meta-cluster it is related with It implies that the objects lying in the middle of the several specific clusters will be considered to belong to the meta-cluster represented by the union (disjunction) of these specific clusters. In ECM, the mass of belief for associating the object xi with an element Aj of 2X denoted by mij , mxi ðAjÞ, is determined from the distance dij between xi and the prototype vector vj of the element. Mij is obtained by the minimization of the following objective function under a constraint to obtain the best credal partitioning problem (see Masson and Denœux, 2008 for justifications and details):.
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