A Generalization of Distance Functions for Fuzzy $c$ -Means Clustering With Centroids of Arithmetic Means

Fuzzy c-means (FCM) is a widely used fuzzy clustering method, which allows an object to belong to two or more clusters with a membership grade between zero and one. Despite the considerable efforts made by the clustering community, the common characteristics of distance functions suitable for FCM remain unclear. To fill this crucial void, in this paper, we first provide a generalized definition of distance functions that fit FCM directly. The goal is to provide more flexibility to FCM in the choice of distance functions while preserving the simplicity of FCM by using the centroids of arithmetic means. Indeed, we show that any distance function that fits FCM directly can be derived by a continuously differentiable convex function and, thus, is an instance of the generalized point-to-centroid distance (P2C-D) by definition. In addition, we prove that if the membership grade matrix is nondegenerate, any instance of the P2C-D fits FCM directly. Finally, extensive experiments have been conducted to demonstrate that the P2C-D leads to the global convergence of FCM and that the clustering performances are significantly affected by the choices of distance functions.