Phase Transitions in Fuzzy Clustering Based on Fuzzy Entropy
Makoto Yasuda*, Takeshi Furuhashi**, and Shigeru Okuma***
*Dept. of Electrical and Computer Engineering, Gifu National College of Technology, Shinsei-cho, Motosu-gun, Gifu 501-0495, Japan
**Dept. of Information Engineering, Mie University, 1515 Kamihama-cho, Tsu 514-8507, Japan
***Dept. of Electrical Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
We studied the statistical mechanical characteristics of fuzzy clustering regularized with fuzzy entropy. We obtained Fermi-Dirac distribution as a membership function by regularizing the fuzzy c-means with fuzzy entropy. We then formulated it as direct annealing clustering, and determined the meanings of the Fermi-Dirac function and fuzzy entropy from the statistical mechanical point of view, and showed that this fuzzy clustering is a part of Fermi-Dirac statistics. We also derived the critical temperature at which phase transition occurs in this fuzzy clustering. Then, with a combination of cluster divisions by phase transitions and an adequate division termination condition, we derived fuzzy clustering that automatically determined the number of clusters, as verified by numerical experiments.