Evidential clustering of large dissimilarity data

Abstract

In evidential clustering, the membership of objects to clusters is considered to be uncertain and is represented by Dempster-Shafer mass functions, forming a credal partition. The EVCLUS algorithm constructs a credal partition in such a way that larger dissimilarities between objects correspond to higher degrees of conflict between the associated mass functions. In this paper, we present several improvements to EVCLUS, making it applicable to very large dissimilarity data. First, the gradient-based optimization procedure in the original EVCLUS algorithm is replaced by a much faster iterative row-wise quadratic programming method. Secondly, we show that EVCLUS can be provided with only a random sample of the dissimilarities, reducing the time and space complexity from quadratic to roughly linear. Finally, we introduce a two-step approach to construct credal partitions assigning masses to selected pairs of clusters, making the algorithm outputs more informative than those of the original EVCLUS, while remaining manageable for large numbers of clusters.

Evidential Clustering: A Review

“…Three main algorithms have been proposed to generate credal partitions: the Evidential c-means (ECM) [15,16], EK-NNclus [8], and EVCLUS [9,10]. These algorithms are described in the next sections.…”

“…The EVCLUS algorithm [9,10] applies some ideas from Multidimensional Scaling (MDS) [4] to clustering. Let D = (d ij ) be an n×n dissimilarity matrix, where d ij denotes the dissimilarity between objects o i and o j .…”

“…, m n ) from D, we assume that the plausibility pl ij that two objects o i and o j belong to the same class is a decreasing function of the dissimilarity d ij : the more similar are two objects, the more plausible it is that they belong to the same cluster. Now, it can be shown [10] that the plausibility pl ij is equal to 1 − κ ij , where κ ij is the degree of conflict between m i and m j . The credal partition M should thus be determined in such a way that similar objects o i and o j have mass functions m i and m j with low degree of conflict, whereas highly dissimilar objects are assigned highly conflicting mass functions.…”

“…This can be achieved by minimizing the discrepancy between the pairwise degrees of conflict and the dissimilarities, up to some increasing transformation. In [10], we proposed to minimize the following stress function,…”

“…A heuristic for fixing γ is described in [10]. The stress function (23) by the Iterative Row-wise Quadratic Programming (IRQP) [10,21].…”