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Analysis of temperature and q-parameter dependencies of FCM with Tsallis entropy maximization

This study is dealing with the fuzzy c-means clustering combined with entropy maximization and deterministic annealing. The Tsallis entropy is the q-parameter extension of the Shannon entropy. By maximizing the Tsallis entropy within the framework of fuzzy c-means, a statistical mechanical membership function is obtained, which can be applicable to fuzzy clustering. The major issue of this method is that how should an appropriate q value for a data distribution be determined is unclear. Accordingly, in the previous study, we investigated the relationship between q and the annealing temperature by comparing the area of the membership function quantitatively. The result showed that q and the temperature affect the membership function's area almost inversely. In this article, this relationship is studied more analytically by integrating the membership function directly, and it is approximately confirmed that q and the temperature affect the shape of the membership function inversely. Based on this relationship, a q-incrementation algorithm is developed. An experiment is performed, and it is confirmed that the algorithm works properly. However, its performance is inferior to that of the conventional annealing method.