Non-Extensive Thermostatistics and Extreme Physical Information for Fuzzy Clustering

Clustering is a widely used knowledge discovery technique. It is used to reveal structures in data that can be extremely useful to the analyst. Partitional clustering attempts to subdivide the data set into subsets or clusters, which are pairwise disjoint, all non empty, and produce the original data set via union. Fuzzy algorithms have been widely studied and applied in this area. In this paper, we focus on fuzzy objective function models whose aim is to assign the data to clusters so that a given objective function is optimized. They can be divided into two categories: probabilistic and possibilistic approaches, each one of them has its own formal and interpretational advantages that are fixed by the specific problem under consideration. We propose a new approach in fuzzy clustering and show how it can be used to obtain a systematic method deriving objective functions. This approach is based on a unifying principle of physics, that of extreme physical information (EPI) defined by Frieden [11], and is inspired by the work of Frieden and A. Plastino, A.R. Plastino and Miller [33], who extend the principle of extremal information in the framework of the non-extensive thermostatistics. The information in question is the trace of the Fisher information matrix for the estimation procedure; this information is shown to be a physical measure of disorder. Then, we show how, with the help of EPI, one can propose an unification/extension of the probabilistic and possibilistic approaches.