Fuzzy clustering using deterministic annealing method and its statistical mechanical characteristics

2005 ◽  
Author(s):  
M. Yasuda ◽  
T. Furuhashi ◽  
M. Matsuzaki ◽  
S. Okuma
Keyword(s):  
Fuzzy Clustering ◽  
Mechanical Characteristics ◽  
Deterministic Annealing ◽  
Statistical Mechanical ◽  
Annealing Method

scholarly journals Phase Transitions in Fuzzy Clustering Based on Fuzzy Entropy

2003 ◽  
Vol 7 (3) ◽  
pp. 370-376 ◽  
Author(s):  
Makoto Yasuda ◽  
◽  
Takeshi Furuhashi ◽  
Shigeru Okuma ◽  
◽  
...  
Keyword(s):  
Phase Transitions ◽  
Fuzzy Clustering ◽  
Numerical Experiments ◽  
Mechanical Characteristics ◽  
Point Of View ◽  
Fuzzy Entropy ◽  
Number Of Clusters ◽  
Statistical Mechanical ◽  
Dirac Distribution ◽  
Dirac Function

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.

Analysis of Temperature and q-Parameter Dependency of FCM with Tsallis Entropy Maximization

2018 ◽  
Vol 22 (5) ◽  
pp. 666-673
Author(s):  
Makoto Yasuda ◽  
Keyword(s):  
Membership Function ◽  
Clustering Algorithm ◽  
Annealing Temperature ◽  
Tsallis Entropy ◽  
Deterministic Annealing ◽  
Entropy Maximization ◽  
Statistical Mechanical ◽  
Proportional Relationship ◽  
Parameter Dependency ◽  
The Membership Function

The Tsallis entropy is a q-parameter extension of the Shannon entropy. By maximizing it within the framework of fuzzy c-means, statistical mechanical membership functions can be derived. We propose a clustering algorithm that includes the membership function and deterministic annealing. One of the major issues for this method is the determination of an appropriate values for q and an initial annealing temperature for a given data distribution. Accordingly, in our previous study, we investigated the relationship between q and the annealing temperature. We quantitatively compared the area of the membership function for various values of q and for various temperatures. The results showed that the effect of q on the area was nearly the inverse of that of the temperature. In this paper, we analytically investigate this relationship by directly integrating the membership function, and the inversely proportional relationship between q and the temperature is approximately confirmed. Based on this relationship, a q-incrementation deterministic annealing fuzzy c-means (FCM) algorithm is developed. Experiments are performed, and it is confirmed that the algorithm works properly. However, it is also confirmed that differences in the shape of the membership function of the annealing method and that of the q-incrementation method are remained.

scholarly journals Quantitative Analyses and Development of aq-Incrementation Algorithm for FCM with Tsallis Entropy Maximization

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Makoto Yasuda
Keyword(s):  
Membership Function ◽  
Annealing Temperature ◽  
Data Distribution ◽  
Tsallis Entropy ◽  
Deterministic Annealing ◽  
Entropy Maximization ◽  
Quantitative Analyses ◽  
Statistical Mechanical ◽  
Combinatorial Methods ◽  
The Membership Function

Tsallis entropy is aq-parameter extension of Shannon entropy. By extremizing the Tsallis entropy within the framework of fuzzyc-means clustering (FCM), a membership function similar to the statistical mechanical distribution function is obtained. The Tsallis entropy-based DA-FCM algorithm was developed by combining it with the deterministic annealing (DA) method. One of the challenges of this method is to determine an appropriate initial annealing temperature and aqvalue, according to the data distribution. This is complex, because the membership function changes its shape by decreasing the temperature or by increasingq. Quantitative relationships between the temperature andqare examined, and the results show that, in order to changeuikqequally, inverse changes must be made to the temperature andq. Accordingly, in this paper, we propose and investigate two kinds of combinatorial methods forq-incrementation and the reduction of temperature for use in the Tsallis entropy-based FCM. In the proposed methods,qis defined as a function of the temperature. Experiments are performed using Fisher’s iris dataset, and the proposed methods are confirmed to determine an appropriateqvalue in many cases.

Statistical-mechanical characteristics of dense planar granular systems

2012 ◽  
Vol 14 (2) ◽  
pp. 277-282 ◽  
Author(s):  
Rebecca Hihinashvili ◽  
Raphael Blumenfeld
Keyword(s):  
Mechanical Characteristics ◽  
Granular Systems ◽  
Statistical Mechanical
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