Kernel Functions Derived from Fuzzy Clustering and Their Application to Kernel Fuzzy <i>c</i>-Means

Jeongsik Hwang; Sadaaki Miyamoto

doi:10.20965/jaciii.2011.p0090

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JACIII Vol.15 No.1 pp. 90-94

doi: 10.20965/jaciii.2011.p0090

(2011)

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Kernel Functions Derived from Fuzzy Clustering and Their Application to Kernel Fuzzy c-Means

Jeongsik Hwang and Sadaaki Miyamoto

Department of Risk Engineering, School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Ibaraki 305-8573, Japan

Received:

January 18, 2010

Accepted:

April 22, 2010

Published:

January 20, 2011

Keywords:

positive definite kernel, completely monotone function, fuzzy clustering

Abstract

Among widely used kernel functions, such as support vector machines, in data analysis, the Gaussian kernel is most often used. This kernel arises in entropy-based fuzzy c-means clustering. There is reason, however, to check whether other types of functions used in fuzzy c-means are also kernels. Using completely monotone functions, we show they can be kernels if a regularization constant proposed by Ichihashi is introduced. We also show how these kernel functions are applied to kernel-based fuzzy c-means clustering, which outperform the Gaussian kernel in a typical example.

Cite this article as:

J. Hwang and S. Miyamoto, “Kernel Functions Derived from Fuzzy Clustering and Their Application to Kernel Fuzzy c-Means,” J. Adv. Comput. Intell. Intell. Inform., Vol.15 No.1, pp. 90-94, 2011.

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References

[1] B. Schölkopf and A. Smola, “Learning with Kernels,” MIT Press, 2002.
[2] V. N. Vapnik, “Statistical Learning Theory,” Wiley, New York, 1998.
[3] J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum Press, 1981.
[4] J. C. Bezdek, J. Keller, R. Krishnapuram, and N. R. Pal, “Fuzzy Models and Algorithms for Pattern Recognition and Image Processing,” Kluwer, Boston, 1999.
[5] R. N. Davé and R. Krishnapuram, “Robust clustering methods: a unified view,” IEEE Trans. Fuzzy Syst., Vol.5, No.2, pp. 270-293, 1997.
[6] J. C. Dunn, “A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters,” J. of Cybernetics, Vol.3, pp. 32-57, 1974.
[7] R. Krishnapuram and J.M. Keller, “A possibilistic approach to clustering,” IEEE Trans. on Fuzzy Systems, Vol.1, pp. 98-110, 1993.
[8] S. Miyamoto, H. Ichihashi, and K. Honda, “Algorithms for Fuzzy Clustering,” Springer, 2008.
[9] I. J. Schönberg, “Metric spaces and completely monotone functions,” Annals of Mathematics, Vol.39, No.4, pp. 811-841, 1938.
[10] R. P. Li and M. Mukaidono, “Gaussian clustering method based on maximum-fuzzy-entropy interpretation,” Fuzzy Sets and Systems, Vol.102, pp. 253-258, 1999.
[11] S. Miyamoto, “Introduction to Cluster Analysis: Theory and Applications of Fuzzy Clustering,” Morikita-Shuppan, Tokyo, 1990. (in Japanese)
[12] S. Miyamoto and M. Mukaidono, “Fuzzy c-means as a regularization and maximum entropy approach,” Proc. of the 7th Int. Fuzzy Systems Association World Congress (IFSA’97), June 25-30, 1997, Prague, Czech, Vol.II, pp. 86-92, 1997.
[13] S. Akaho, “Kernel Data Analysis,” Iwanami-Shoten, Tokyo, 2008. (in Japanese)
[14] O. Chapelle, J. Weston, and B. Schölkopf, “Cluster Kernels for Semi-Supervised Learning,” Advances in Neural Information Processing Systems, Vol.15, pp. 585-592, MIT Press, 2003.
[15] S. Miyamoto and D. Suizu, “Fuzzy c-means clustering using transformations into high dimensional spaces,” Proc. of FSKD’02: 1st Int. Conf. on Fuzzy Systems and Knowledge Discovery, Singapore, Vol.2, pp. 656-660, Nov. 18-22, 2002.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] B. Schölkopf and A. Smola, “Learning with Kernels,” MIT Press, 2002.

[2] [2] V. N. Vapnik, “Statistical Learning Theory,” Wiley, New York, 1998.

[3] [3] J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum Press, 1981.

[4] [4] J. C. Bezdek, J. Keller, R. Krishnapuram, and N. R. Pal, “Fuzzy Models and Algorithms for Pattern Recognition and Image Processing,” Kluwer, Boston, 1999.

[5] [5] R. N. Davé and R. Krishnapuram, “Robust clustering methods: a unified view,” IEEE Trans. Fuzzy Syst., Vol.5, No.2, pp. 270-293, 1997.

[6] [6] J. C. Dunn, “A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters,” J. of Cybernetics, Vol.3, pp. 32-57, 1974.

[7] [7] R. Krishnapuram and J.M. Keller, “A possibilistic approach to clustering,” IEEE Trans. on Fuzzy Systems, Vol.1, pp. 98-110, 1993.

[8] [8] S. Miyamoto, H. Ichihashi, and K. Honda, “Algorithms for Fuzzy Clustering,” Springer, 2008.

[9] [9] I. J. Schönberg, “Metric spaces and completely monotone functions,” Annals of Mathematics, Vol.39, No.4, pp. 811-841, 1938.

[10] [10] R. P. Li and M. Mukaidono, “Gaussian clustering method based on maximum-fuzzy-entropy interpretation,” Fuzzy Sets and Systems, Vol.102, pp. 253-258, 1999.

[11] [11] S. Miyamoto, “Introduction to Cluster Analysis: Theory and Applications of Fuzzy Clustering,” Morikita-Shuppan, Tokyo, 1990. (in Japanese)

[12] [12] S. Miyamoto and M. Mukaidono, “Fuzzy c-means as a regularization and maximum entropy approach,” Proc. of the 7th Int. Fuzzy Systems Association World Congress (IFSA’97), June 25-30, 1997, Prague, Czech, Vol.II, pp. 86-92, 1997.

[13] [13] S. Akaho, “Kernel Data Analysis,” Iwanami-Shoten, Tokyo, 2008. (in Japanese)

[14] [14] O. Chapelle, J. Weston, and B. Schölkopf, “Cluster Kernels for Semi-Supervised Learning,” Advances in Neural Information Processing Systems, Vol.15, pp. 585-592, MIT Press, 2003.

[15] [15] S. Miyamoto and D. Suizu, “Fuzzy c-means clustering using transformations into high dimensional spaces,” Proc. of FSKD’02: 1st Int. Conf. on Fuzzy Systems and Knowledge Discovery, Singapore, Vol.2, pp. 656-660, Nov. 18-22, 2002.