New Results in Fuzzy Clustering Based on the Concept of Indistinguishability Relation.

摘要

One of the fundamental problems in clustering is its validation. This issue has been addressed by many researchers in different ways depending on the methods they use to solve the clustering problem, nevertheless the existing approaches are based on the introduction of a validity function which is intented to measure the validity of the clustering obtained by the different methods. In our work we first address this issue and we give a new definition of fuzzy r-cluster that extend Ruspini's definition (Ruspini, 1982). Our definition is based on the new concept of indistinguishability relation (Trillas, 1982) which includes, as particular cases the concepts of similarity relation of Zadeh (1971), probabilistic relation of Menger (1951) and likeness relation of Ruspini (1982). The approach presented here is heavily based on the data themselves. First we construct an indistinguishability relation among a data set (that includes the prototypes or the clusters) based on the degree of membership, to the different clusters, of each element of the data set. Finally, we measure the validity comparing (through a distance measure) the degrees of membership of the data to the clusters, with the degrees or indistinguishability between the data and the prototypes of each cluster. The basic idea is that the smaller the difference between the degrees of membership and the degrees of indistinguishability, the better the clustering. We also suggest that the validity measure obtained can be used to reconsider initial decisions about the election of the prototypes, the number of clusters, etc .. Also since, as we shall see, and as a 'side effect' we get information about the logical and metric properties of the data, we could use such information in order to make a geometrical representation of the data (factor analysis, principal components, etc.). We start giving some definitions and theorems whose proofs are omitted because of space reasons (See Valverde, 1983).