Cluster validity for fuzzy clustering algorithms

The proportion exponent is introduced as a measure of the validity of the clustering obtained for a data set using a fuzzy clustering algorithm. It is assumed that the output of an algorithm includes a fuzzy nembership function for each data point. We show how to compute the proportion of possible memberships whose maximum entry exceeds the maximum entry of a given membership function, and use these proportions to define the proportion exponent. Its use as a validity functional is illustrated with four numerical examples and its effectiveness compared to other validity functionals, namely, classification entropy and partition coefficient.