Analysis of Temperature andq-Parameter Dependency of FCM with Tsallis Entropy Maximization

Abstract

The Tsallis entropy is aq-parameter extension of the Shannon entropy. By maximizing it within the framework of fuzzyc-means, statistical mechanical membership functions can be derived. We propose a clustering algorithm that includes the membership function and deterministic annealing. One of the major issues for this method is the determination of an appropriate values forqand an initial annealing temperature for a given data distribution. Accordingly, in our previous study, we investigated the relationship betweenqand the annealing temperature. We quantitatively compared the area of the membership function for various values ofqand for various temperatures. The results showed that the effect ofqon the area was nearly the inverse of that of the temperature. In this paper, we analytically investigate this relationship by directly integrating the membership function, and the inversely proportional relationship betweenqand the temperature is approximately confirmed. Based on this relationship, aq-incrementation deterministic annealing fuzzyc-means (FCM) algorithm is developed. Experiments are performed, and it is confirmed that the algorithm works properly. However, it is also confirmed that differences in the shape of the membership function of the annealing method and that of theq-incrementation method are remained.