Abstract
In this paper, we consider a fuzzy c-means (FCM) clustering algorithm combined with the deterministic annealing method and the Tsallis entropy maximization. The Tsallis entropy is a q-parameter extension of the Shannon entropy. By maximizing the Tsallis entropy within the framework of FCM, membership functions similar to statistical mechanical distribution functions can be derived. One of the major considerations when using this method is how to determine appropriate q values and the highest annealing temperature, Thigh , for a given data set. Accordingly, in this paper, a method for determining these values simultaneously without introducing any additional parameters is presented. In our approach, the membership function is approximated by a series of expansion methods and the K-means clustering algorithm is utilized as a preprocessing step to estimate a radius of each data distribution. The results of experiments indicate that the proposed method is effective and both q and Thigh can be determined automatically and algebraically from a given data set.
Highlights
Techniques from statistical mechanics can be used for the investigation of the macroscopic properties of a physical system consisting of many elements
The Tsallis entropy is a q-parameter extension of the Shannon entropy
This membership function has a similar form to the statistical mechanical distribution function, and is suitable for use with annealing methods because it contains a parameter corresponding to the system temperature
Summary
Techniques from statistical mechanics can be used for the investigation of the macroscopic properties of a physical system consisting of many elements. In the field of fuzzy clustering, a membership function was derived by maximizing the Tsallis entropy within the framework of FCM [10] [11] [12]. This membership function has a similar form to the statistical mechanical distribution function, and is suitable for use with annealing methods because it contains a parameter corresponding to the system temperature. We estimate the radius of the distribution of the data set, and apply the approximation method to determine q and Thigh. By applying the variational method, the stationary condition for the Tsallis entropy functional yields the following membership function for Tsallis-FCM [12]:
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