ON ASYMPTOTIC NORMALITY OF A CLASS OF FUZZY C-MEANS CLUSTERING PROCEDURES

A family of fuzzy c-partitions of a set X in an s-dimensional Euclidean space known as fuzzy c-means (FCM) clustering procedures has been widely used for the problem of clustering the set X of objects into groups of similar individuals. Suppose that the set X is a simple random sample of observations from a population with probability measure P. Now choose ever larger finite random samples from P. We are interested in the stochastically asymptotic properties of the FCM. In this paper, we establish asymptotic normality of the FCM clustering procedures. That is, the asymptotic distribution of the normalized optimal cluster centers is multivariate normal.