ABSTRACT
Most real-world classes are fuzzy rather than crisp. It is, therefore, natural to consider fuzzy set theory as a useful tool to deal with classification problems. The use of fuzzy sets in clustering was first suggested by Bellman, Kalaba and Zadeh [1]. Negoita used a separation theorem for fuzzy sets to describe a cluster-based information retrieval system [34]. Ruspini introduced a notion of fuzzy partition to describe the cluster structure of a data set and suggested an algorithm to compute the optimum fuzzy partition [37], [38]. Dunn generalized the minimum-variance clustering procedure to a Fuzzy ISODATA clustering technique [29]. Bezdek generalized Dunn's approach to obtain an infinite family of algorithms known as the Fuzzy n-Means (FNM) algorithms [2], [3J. A good collection of works on fuzzy clustering can be found in [12J. In this chapter, some algorithms to detect the cluster substructure of a fuzzy class will be considered. The algorithms are based on the minimization of a criterion function. The criterion functions are obtained using the local distance with respect to a fuzzy class [19], [20], [22]. In order to correctly detect the unequal size clusters, some adaptive
280 CHAPTER9
distances will be adopted. The mean and the variance of a fuzzy class will be used for data normalization and for the definition of a local metric. The cluster substructure of a fuzzy class C may be detected by considering the local metric induced by C [22], [25]. The search for the cluster substructure of a fuzzy class appears necessary in different circumstances such as the following:
( i) Inadequacy of one-level data classification when the data knowledge requires a more refined analysis.
