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

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 extent of the membership function is determined by a system temperature and a q value. The Tsallis-entropy-based DA-FCM algorithm was developed by combining FCM with the deterministic annealing (DA) method. One of the challenges of this method is to determine an appropriate initial temperature and a q value, according to the data distribution. This is complex, because the center of a cluster is given as a weighted function of the membership function to the power of q or u^q_ik, and it changes its shape by decreasing the temperature or by increasing q. Quantitative relationships between the temperature and q are examined, and the results show that, in order to change u^q_ik 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; the accuracy of classification is shown to be better than that of the conventional method.