ON EXISTENCE AND STRONG CONSISTENCY OF A CLASS OF FUZZYC-MEANS CLUSTERING PROCEDURES
Miin‐Shen Yang, Kai Yu · Cybernetics & Systems · 1992
In this paper the existence and strong consistency of a class of fuzzy c-means (FCM) clustering procedures are established. Suppose that the data set is a simple random sample of observations from a probability distribution, assuming that the second moment of the distribution is finite. Then the solution to the FCM clustering procedures shall exist. The FCM cluster centers and the FCM membership functions will have the property of strong consistency. That is, the sample FCM cluster centers and the population FCM cluster centers will be close to each other with probability one, and also the sample FCM membership functions and the population FCM membership functions will be close to each other with probability one when the sample size increases to infinity.