On Entropy Based Fuzzy Non Metric Model – Proposal, Kernelization and Pairwise Constraints –

Yasunori Endo

doi:10.20965/jaciii.2012.p0169

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JACIII Vol.16 No.1 pp. 169-173

doi: 10.20965/jaciii.2012.p0169

(2012)

Paper:

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On Entropy Based Fuzzy Non Metric Model – Proposal, Kernelization and Pairwise Constraints –

Yasunori Endo

Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan

Received:

January 15, 2011

Accepted:

October 3, 2011

Published:

January 20, 2012

Keywords:

clustering, fuzzy non metric model, entropy regularization, kernel, pairwise constraint

Abstract

The fuzzy non metric model is a kind of clustering method in which belongingness or the membership grade of each datum to each cluster is calculated directly from dissimilarities between data, and cluster centers are not used. In this paper, we first construct a new fuzzy non metric model with entropy regularization. Second, we kernelize the proposed method by introducing kernel functions. Third, we consider pairwise constraints with the proposed method. We then confirm the above methods through some simple numerical examples.

Cite this article as:

Y. Endo, “On Entropy Based Fuzzy Non Metric Model – Proposal, Kernelization and Pairwise Constraints –,” J. Adv. Comput. Intell. Intell. Inform., Vol.16 No.1, pp. 169-173, 2012.

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References

[1] J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum, New York, 1981.
[2] J. C. Bezdek, J. Keller, R. Krisnapuram, and N. R. Pal, “Fuzzy Models and Algorithms for Pattern Recognition and Image Processing,” The Handbooks of Fuzzy Sets Series, 1999.
[3] M. Roubens, “Pattern classification problems and fuzzy sets,” Fuzzy Sets and Systems, Vol.1, pp. 239-253, 1978.
[4] J. C. Bezdek, J. W. Davenport, and R. J. Hathaway, “Clustering with the Relational c-Means Algorithms using Different Measures of Pairwise Distance,” R. D. Juday (Ed.), Proc. of the 1988 SPIE Technical Symp. on Optics, Electro-Optics, and Sensors, Vol.938, pp. 330-337, 1988.
[5] R. J. Hathaway, J. W. Davenport, and J. C. Bezdek, “Relational Duals of the c-Means Clustering Algorithms,” Pattern Recognition, Vol.22, No.2, pp. 205-212, 1989.
[6] 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), Vol.2, pp. 86-92, 1997.
[7] S. Miyamoto, K. Umayahara, and M. Mukaidono, “Fuzzy Classification Functions in the Methods of Fuzzy c-Means and Regularization by Entropy,” J. of Japan Society for Fuzzy Theory and Systems, Vol.10, No.3, pp. 548-557, 1998.
[8] V. N. Vapnik, “Statistical Learning Theory,” Wiley, New York, 1998.
[9] V. N. Vapnik, “The nature of Statistical Learning Theory,” 2nd ed., Springer, New York, 2000.
[10] Y. Endo, H. Haruyama, and T. Okubo, “On Some Hierarchical Clustering Algorithms Using Kernel Functions,” IEEE Int. Conf. on Fuzzy Systems, #1106, 2004.
[11] R. J. Hathaway, J. M. Huband, and J. C. Bezdek, “A Kernelized Non-Euclidean Relational Fuzzy c-Means Algorithm,” Neural, Parallel and Scientific computation, Vol.13, pp. 305-326, 2005.
[12] S.Miyamoto, Y. Kawasaki, and K. Sawazaki, “An Explicit Mapping for Fuzzy c-Means Using Kernel Function and Application to Text Analysis,” IFSA/EUSFLAT 2009, 2009.
[13] K. Wagstaff and C. Cardie, “Clustering with Instance-level Constraints,” Proc. of the 17th Int. Conf. on Machine Learning, pp. 1103-1110, 2000.
[14] K. Wagstaff, C. Cardie, S. Rogers, and S. Schroedl, “Constrained k-means clustering with background knowledge,” Proc. of the 18th Int. Conf. on Machine Learning, pp. 577-584, 2001.
[15] J. Mercer, “Functions of Positive and Negative Type and Their Connection with the Theory of Integral Equations,” Philosophical Trans. of the Royal Society A, Vol.209, pp. 415-446, 1909.
[16] Y. Endo, R. Murata, H. Haruyama, and S. Miyamoto, “Fuzzy c-Means for Data with Tolerance,” Proc. 2005 Int. Symp. on Nonlinear Theory and Its Applications, pp. 345-348, 2005.
[17] Y. Kanzawa, Y. Endo, and S. Miyamoto, “Entropy Regularized Fuzzy c-Means for Data with Tolerance introducing Penalty Term in Feature Space,” SCIS&ISIS 2008, 2008.

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

[1] [1] J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum, New York, 1981.

[2] [2] J. C. Bezdek, J. Keller, R. Krisnapuram, and N. R. Pal, “Fuzzy Models and Algorithms for Pattern Recognition and Image Processing,” The Handbooks of Fuzzy Sets Series, 1999.

[3] [3] M. Roubens, “Pattern classification problems and fuzzy sets,” Fuzzy Sets and Systems, Vol.1, pp. 239-253, 1978.

[4] [4] J. C. Bezdek, J. W. Davenport, and R. J. Hathaway, “Clustering with the Relational c-Means Algorithms using Different Measures of Pairwise Distance,” R. D. Juday (Ed.), Proc. of the 1988 SPIE Technical Symp. on Optics, Electro-Optics, and Sensors, Vol.938, pp. 330-337, 1988.

[5] [5] R. J. Hathaway, J. W. Davenport, and J. C. Bezdek, “Relational Duals of the c-Means Clustering Algorithms,” Pattern Recognition, Vol.22, No.2, pp. 205-212, 1989.

[6] [6] 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), Vol.2, pp. 86-92, 1997.

[7] [7] S. Miyamoto, K. Umayahara, and M. Mukaidono, “Fuzzy Classification Functions in the Methods of Fuzzy c-Means and Regularization by Entropy,” J. of Japan Society for Fuzzy Theory and Systems, Vol.10, No.3, pp. 548-557, 1998.

[8] [8] V. N. Vapnik, “Statistical Learning Theory,” Wiley, New York, 1998.

[9] [9] V. N. Vapnik, “The nature of Statistical Learning Theory,” 2nd ed., Springer, New York, 2000.

[10] [10] Y. Endo, H. Haruyama, and T. Okubo, “On Some Hierarchical Clustering Algorithms Using Kernel Functions,” IEEE Int. Conf. on Fuzzy Systems, #1106, 2004.

[11] [11] R. J. Hathaway, J. M. Huband, and J. C. Bezdek, “A Kernelized Non-Euclidean Relational Fuzzy c-Means Algorithm,” Neural, Parallel and Scientific computation, Vol.13, pp. 305-326, 2005.

[12] [12] S.Miyamoto, Y. Kawasaki, and K. Sawazaki, “An Explicit Mapping for Fuzzy c-Means Using Kernel Function and Application to Text Analysis,” IFSA/EUSFLAT 2009, 2009.

[13] [13] K. Wagstaff and C. Cardie, “Clustering with Instance-level Constraints,” Proc. of the 17th Int. Conf. on Machine Learning, pp. 1103-1110, 2000.

[14] [14] K. Wagstaff, C. Cardie, S. Rogers, and S. Schroedl, “Constrained k-means clustering with background knowledge,” Proc. of the 18th Int. Conf. on Machine Learning, pp. 577-584, 2001.

[15] [15] J. Mercer, “Functions of Positive and Negative Type and Their Connection with the Theory of Integral Equations,” Philosophical Trans. of the Royal Society A, Vol.209, pp. 415-446, 1909.

[16] [16] Y. Endo, R. Murata, H. Haruyama, and S. Miyamoto, “Fuzzy c-Means for Data with Tolerance,” Proc. 2005 Int. Symp. on Nonlinear Theory and Its Applications, pp. 345-348, 2005.

[17] [17] Y. Kanzawa, Y. Endo, and S. Miyamoto, “Entropy Regularized Fuzzy c-Means for Data with Tolerance introducing Penalty Term in Feature Space,” SCIS&ISIS 2008, 2008.