Abstract
This paper is dealing with the fuzzy clustering method which combines the deterministic annealing (DA) approach with an entropy, especially the Shannon entropy and the Tsallis entropy. By maximizing the Shannon entropy, the fuzzy entropy, or the Tsallis entropy within the framework of the fuzzy c-means (FCM) method, membership functions similar to the statistical mechanical distribution functions are obtained. We examine characteristics of these entropy-based membership functions from the statistical mechanical point of view. After that, both the Shannon- and Tsallis-entropy-based FCMs are formulated as DA clustering using the very fast annealing (VFA) method as a cooling schedule. Experimental results indicate that the Tsallis-entropy-based FCM is stable with very fast deterministic annealing and suitable for this annealing process.
Highlights
Statistical mechanics investigates the macroscopic properties of a physical system consisting of several elements
There exists a strong relationship between the membership functions of the fuzzy c-means (FCM) clustering [3] with the maximum entropy or entropy regularization methods [4, 5] and the statistical mechanical distribution function
FCM regularized with the Shannon entropy gives a membership function similar to the Boltzmann distribution function [1, 4], and FCM regularized with the fuzzy entropy [6] gives a membership function similar to the Fermi-Dirac distribution function [7]
Summary
Statistical mechanics investigates the macroscopic properties of a physical system consisting of several elements. FCM regularized with the Shannon entropy gives a membership function similar to the Boltzmann (or Gibbs) distribution function [1, 4], and FCM regularized with the fuzzy entropy [6] gives a membership function similar to the Fermi-Dirac distribution function [7] These membership functions are suitable for the annealing methods because they contain a parameter corresponding to the system temperature. The membership function which takes the familiar form of the statistical mechanical distribution function is derived by maximizing the Shannon and fuzzy entropy within the framework of FCM. By introducing VFA to DA, we formulate the Shannon- and Tsallis-entropy based FCMs as very fast DA clustering, to examine their reliabilities. Experiments are performed on the numerical and iris data [13], and the obtained results indicate that Tsallisentropy-based FCM clustering is suitable for very fast DA clustering because of its shape of the membership function
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