ON STOCHASTIC CONVERGENCE THEOREMS FOR THE FUZZY<i>C</i>-MEANS CLUSTERING PROCEDURE∗

Abstract

The fuzzy c-means clustering algorithm has been well studied for equal weight distributions on a finite set. Suppose that this situation is generalized to an arbitrary probability distribution on a finite dimensional Euclidean space, assuming that the second moment of the distribution is finite. Now choose ever larger finite random samples from this distribution and compute the standard optimal membership functions for a fuzzy partition into c clusters. Then the convergence of the cluster center points is established in the Hausdorff sense with probability one, provided that there is a unique optimal center point set. These optimal center points are the fixed point of a simple operator, and there is a corresponding iterative algorithm that generalizes the usual procedure.