Entropy based Fuzzy c-means Clustering : Analogy with Statistical Mechanics

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  • YASUDA Makoto
    Dept. of Electrical and Computer Engineering, Gifu National College of Technology
  • FURUHASHI Takeshi
    Dept. of Computational Science and Engineering, Nagoya University
  • OKUMA Shigeru
    Dept. of Electrical Engineering, Nagoya University

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  • エントロピーを導入したファジィc-平均法の統計力学的解釈
  • エントロピー オ ドウニュウ シタ ファジィ c ヘイキンホウ ノ トウケイ リキガクテキ カイシャク

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Abstract

In this paper, we summarize the statistical mechanical representation of fuzzy clustering. Then, we give a framework of a possibilistic clustering based on a Bose-Einstein type membership function, and examine its clustering mechanisms. The fuzzy c-means clustering (FCM) method regularized with Shannon entropy gives the Maxwell-Boltzmann (or Gibbs) distribution function as a membership function. Similarly, by introducing fuzzy entropy to the FCM, we obtain the Fermi-Dirac type membership function. In these cases, the constraint that the sum of all particles is fixed is correspondent with the normalization constraint in fuzzy clustering. Furthermore, it is known that the state in which the total number of particles is not conserved exists and written by the Bose-Einstein distribution function. Thus, by the analogy of statistical mechanics, we obtain the Bose-Einstein type membership function without the constraint of normalization and propose a new fuzzy clustering algorithms.

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