Quantitative analyses and development of a q -incrementation algorithm for FCM with tsallis entropy maximization

Abstract

Tsallis entropy is a q-parameter extension of Shannon entropy. By extremizing the Tsallis entropy within the framework of fuzzy c-means clustering (FCM), a membership function similar to the statistical mechanical distribution function is obtained. The Tsallis entropy-based DA-FCM algorithm was developed by combining it with the deterministic annealing (DA) method. One of the challenges of this method is to determine an appropriate initial annealing temperature and a q value, according to the data distribution. This is complex, because the membership function changes its shape by decreasing the temperature or by increasing q. Quantitative relationships between the temperature and q are examined, and the results showthat, in order to change uikq equally, inverse changes must be made to the temperature and q. Accordingly, in this paper, we propose and investigate two kinds of combinatorial methods for q-incrementation and the reduction of temperature for use in the Tsallis entropy-based FCM. In the proposed methods, q is defined as a function of the temperature. Experiments are performed using Fisher's iris dataset, and the proposed methods are confirmed to determine an appropriate q value in many cases.