Fuzzy Partition Entropies and Entropy Constrained Fuzzy Clustering Algorithms

Three families of fuzzy partition entropies are introduced and their relationship with Bezdek's partition entropy is investigated and it is found that more-over, entropy constrained algorithms satisfy a basic requirement for clustering algorithms; they are invariant under uniform scaling of the feature vectors.

Abstract

This article introduces three families of fuzzy partition entropies and investigates their relationship with Bezdek's partition entropy. Bezdek's partition entropy is interpreted as the limit of partition entropies of order a, partition entropies of order β, and R-norm partition entropies introduced in this study. The proposed partition entropies are used to formulate clustering as a constrained minimization problem. This formulation results in entropy constrained fuzzy clustering algorithms as the proposed partition entropies approach Bezdek's partition entropy. The algorithms resulting from the proposed formulation allow the transition from a maximally fuzzy to crisp partitions according to a procedure that resembles an “annealing” process. More-over, entropy constrained algorithms satisfy a basic requirement for clustering algorithms; they are invariant under uniform scaling of the feature vectors.