Some properties of the fuzzy equivalence matrices

Abstract

ABSTRACT In fuzzy clustering, we always use fuzzy equivalence matrix to find the clustering results. Although fuzzy equivalence matrix is one of the most important tools in fuzzy clustering, it’s hard to judge whether a fuzzy matrix is a fuzzy equivalence matrix directly. In this paper, some simple and useful properties of the fuzzy equivalence matrices have been proposed based on the basic properties proposed in the classi cal articles. Using these properties, the judgment whether a fuzzy matrix is a fuzzy equivalence matrix may become an easy work. Keywords: Fuzzy clustering, fuzzy equivalence matrix 1. INTRODUCTION Given a finite set of objects X, and a measure of similarity for an arbitrary pair of objects, the problem of clustering is to find a partition of X such that the degree of similarity is high for objects within each block (cluster) and low for objects in different blocks. The purpose of clustering is to find a natural classification of given objects on the basis of a conceived concept of similarity. The classification may help to uncover a structure in given data, improve our understanding of a complex system, or suggest a meaningful simplification of a given system [1]. The requirement that given objects be classified into pairwise disjoint subsets (crisp clusters) is overly strong in many practical applications. Fuzzy set theory allows us to weaken it and, hence, make clustering more realistic. So, fuzzy clustering becomes one of the hottest research branches of data mining. One of the approaches to fuzzy clustering is based on fuzzy equivalence relations defined on X. Every fuzzy equivalence relation induces a crisp partition in each of its .-cuts. Using this approach, the fuzzy clustering problem becomes the problem of identifying a fuzzy equivalence relation on X in terms of a conceived concept of similarity. Although this may not be possible to do directly, it is usually easy to determine a fuzzy compatibility relation in terms of an appropriate distance function defined on X. A distance function is always reflexive and symmetric, but it is not transitive. However, once we have a compatibility relation, we can readily obtain a meaningful fuzzy equivalence relation by calculating its transitive closure. The resulting fuzzy equivalence relation offers the user a sequence of crisp partitions on X, which are linearly ordered by refinement ordering and graded by values of . [0, 1]. The user is given a flexibility to choose one of these partitions on the basis of additional criteria [2-4]. But, to calculate the transitive closure of a N×N fuzzy matrix , we need to calculate the 2kth power of the matrix, where k = 1, 2, , N. In the processing, after we calculate the 2k+1th power of the matrix and find that the 2k+1th power of the matrix is equal to the 2kth power of the matrix, the 2kth power of the matrix is the transitive closure of the matrix. In fact, if we can find the 2kth power of the matrix is a fuzzy equivalence matrix directly, we do not need to calculate the 2k+1th power of the matrix. If we can judge a fuzzy matrix is a fuzzy equivalence matrix by means of some simple properties of the equivalence matrix, it may be helpful. In this paper, some properties of the fuzzy equivalence matrix have been proposed and the judgment whether a fuzzy *white70121@163.com