Analysis of Temperature and q-Parameter Dependency of FCM with Tsallis Entropy Maximization

摘要

The Tsallis entropy is a q-parameter extension of the Shannon entropy. By maximizing it within the framework of fuzzy c-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 for q and an initial annealing temperature for a given data distribution. Accordingly, in our previous study, we investigated the relationship between q and the annealing temperature. We quantitatively compared the area of the membership function for various values of q and for various temperatures. The results showed that the effect of q on 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 between q and the temperature is approximately confirmed. Based on this relationship, a q-incrementation deterministic annealing fuzzy c-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 the q-incrementation method are remained.