Paper:

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

## Makoto Yasuda

National Institute of Technology, Gifu College

2236-2 Kamimakuwa, Motosu, Gifu 501-0495, Japan

*c*-means clustering, Tsallis entropy, entropy maximization, deterministic annealing,

*q*-incrementation

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.

*q*-Parameter Dependency of FCM with Tsallis Entropy Maximization,”

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.22 No.5, pp. 666-673, 2018.

- [1] K. Rose, E. Gurewitz, and G. Fox, “A deterministic annealing approach to clustering,” Pattern Recognition Letters, Vol.11, No.9, pp. 589-594, 1990.
- [2] K. Rose, “Deterministic annealing for clustering, compression, classification, regression, and related optimization problems,” Proc. of the IEEE, Vol.86, No.11, pp. 2210-2239, 1998.
- [3] S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi, “Optimization by simulated annealing,” Science, Vol.220, No.4598, pp. 671-680, 1983.
- [4] C. Tsallis, “Possible generalization of Boltzmann-Gibbs statistics,” J. of Statistical Phys., Vol.52, No.1-2, pp.479-487, 1988.
- [5] S. Abe and Y. Okamoto (eds.), “Nonextensive Statistical Mechanics and Its Applications,” Springer, 2001.
- [6] M. Gell-Mann and C. Tsallis (eds.), “Nonextensive Entropy – Interdisciplinary Applications,” Oxford University Press, 2004.
- [7] C. F. L. Lima, F. M. de Assis, and C. P. de Souza, “A comparative study of use of Shannon, Rényi and Tsallis entropy for attribute selecting in network intrusion detection,” Proc. of IEEE Int. Workshop on Measurements and Networking, pp. 77-82, 2011.
- [8] W. Wei, X. Lin, and G. Zhang, “Fast image segmentation based on two-dimensional minimum Tsallis-cross entropy,” Proc. of Int. Conf. on Image Analysis and Signal Processing, pp. 332-335, 2010.
- [9] I. Andricioaei and J. E. Straub, “Generalized simulated annealing algorithms using Tsallis statistics: Application to conformational optimization of a tetrapeptide,” Physical Review E, Vol.53, No.4, pp. 3055-3058, 1996.
- [10] W. Guo and S. Cui, “A Q-EM based simulated annealing algorithm for finite mixture estimation,” IEEE Int. Conf. on Acoustics, Speech and Signal Processing, pp. 1113-1116, 2007.
- [11] K. Honda and H. Ichihashi, “A regularization approach to fuzzy clustering with nonlinear membership weights,” J. Adv. Comput. Intell. Intell. Inform., Vol.11, No.1, pp. 28-34, 2007.
- [12] Y. Kanzawa, “Entropy-regularized fuzzy clustering for non-euclidean relational data and indefinite kernel data,” J. Adv. Comput. Intell. Intell. Inform., Vol.16, No.7, pp. 784-792, 2012.
- [13] M. Ménard, V. Courboulay, and P.-A. Dardignac, “Possibilistic and probabilistic fuzzy clustering: unification within the framework of the non-extensive thermostatistics,” Pattern Recognition, Vol.36, No.6, pp. 1325-1342, 2003.
- [14] M. Ménard, P.-A. Dardignac, and C. C. Chibelushi, “Non-extensive thermostatistics and extreme physical information for fuzzy clustering,” Int. J. of Computational Cognition, Vol.2, No.4, pp. 1-63, 2004.
- [15] M. Yasuda, “Entropy maximization and very fast deterministic annealing approach to fuzzy c-means clustering,” Proc. of the Joint 5th Int. Conf. on Soft Computing and 11th Int. Symp. on Advanced Intelligent Systems, SU-B1-3, pp. 1515-1520, 2010.
- [16] M. Yasuda, “Deterministic annealing approach to fuzzy c-means clustering based on entropy maximization,” Advances in Fuzzy Systems, Vol.2011, 960635, 2011.
- [17] M. Yasuda and Y. Orito, “Multi-
*q*extension of Tsallis entropy based fuzzy*c*-means clustering,” J. Adv. Comput. Intell. Intell. Inform., Vol.18, No.3, pp. 289-296, 2014. - [18] M. Yasuda, “Quantitative analyses and development of a
*q*-incrementation algorithm for FCM with Tsallis entropy maximization,” Advances in Fuzzy Systems, Vol.2015, 404510, 2015. - [19] UCI Machine Learning Repository: Iris Data Set, http://archive.ics.uci.edu/ml/datasets/Iris/ [accessed July 1, 1998]
- [20] UCI Machine Learning Repository: Wine Quality Data Set, http://archive.ics.uci.edu/ml/datasets/Wine+Quality/ [accessed October 7, 2009]
- [21] L. E. Reichl, “A Modern Course in Statistical Physics,” John Wiley & Sons, 1998.
- [22] D. Zwillinger, I. S. Gradshteyn and I. M. Ryzhik. (eds.), “Table of Integrals, Series, and Products,” Academic Press, 2014.
- [23] P. Cortez, A. Cerdeira, F. Almeida, T. Matos, and J. Reis, “Modeling wine preferences by data mining from physicochemical properties,” Decision Support Systems, Vol.47, Issue 4, pp. 547-553, 2009.

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