Relational Fuzzy <i>c</i>-Means and Kernel Fuzzy <i>c</i>-Means Using an Object-Wise β-Spread Transformation

Yuchi Kanzawa

doi:10.20965/jaciii.2013.p0511

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JACIII Vol.17 No.4 pp. 511-519

doi: 10.20965/jaciii.2013.p0511

(2013)

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Relational Fuzzy c-Means and Kernel Fuzzy c-Means Using an Object-Wise β-Spread Transformation

Yuchi Kanzawa

Shibaura Institute of Technology, 3-7-5 Toyosu, Koto, Tokyo 135-8548, Japan

Received:

February 28, 2013

Accepted:

April 12, 2013

Published:

July 20, 2013

Keywords:

relational fuzzy clustering, kernel fuzzy clustering, non-Euclidean relational data, indefinite kernel, object-wise β-spread transformation

Abstract

Clustering methods of relational data are often based on the assumption that a given set of relational data is Euclidean, and kernelized clustering methods are often based on the assumption that a given kernel is positive semidefinite. In practice, non-Euclidean relational data and an indefinite kernel may arise, and a β-spread transformation was proposed for such cases, which modified a given set of relational data or a give a kernel Gram matrix such that the modified β value is common to all objects. In this paper, we propose an object-wise β-spread transformation for use in both relational and kernelized fuzzy c-means clustering. The proposed system retains the given data better than conventional methods, and numerical examples show that our method is efficient for both relational and kernel fuzzy c-means.

Cite this article as:

Y. Kanzawa, “Relational Fuzzy c-Means and Kernel Fuzzy c-Means Using an Object-Wise β-Spread Transformation,” J. Adv. Comput. Intell. Intell. Inform., Vol.17 No.4, pp. 511-519, 2013.

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References

[1] J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum, New York, 1981.
[2] 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.
[3] R. J. Hathaway and J. C. Bezdek, “NERF C-means: Non-Euclidean Relational Fuzzy Clustering,” Pattern Recognition, Vol.27, pp. 429-437, 1994.
[4] S. Miyamoto and D. Suizu, “Fuzzy c-Means Clustering Using Kernel Functions in Support Vector Machines,” J. Advanced Computational Intelligence and Intelligent Informatics, Vol.7, No.1, pp. 25-30, 2003.
[5] V. N. Vapnik, “Statistical Learning Theory,” Wiley, New York, 1998.
[6] S.Miyamoto, Y. Kawasaki, and K. Sawazaki, “An Explicit Mapping for Kernel Data Analysis and Application to Text Analysis,” Proc. IFSA-EUSFLAT 2009, pp. 618-623, 2009.
[7] S. Miyamoto and K. Sawazaki, “An Explicit Mapping for Kernel Data Analysis and Application to c-Means Clustering,” Proc. NOLTA 2009, pp. 556-559, 2009.
[8] Y. Kanzawa, Y. Endo, and S. Miyamoto, “Indefinite Kernel Fuzzy c-Means Clustering Algorithms,” Lecture Notes in Computer Science, Vol.6408, pp. 116-128, 2010.
[9] S. Tamura, S. Higuchi, and K. Tanaka, “Pattern Classification Based on Fuzzy Relations,” IEEE Trans. Syst. Man Cybern., Vol.1, No.1, pp. 61-66, 1971.
[10] J. W. Scannell, C. Blakemore, and M. P. Young, “Analysis of Connectivity in the Cat Cerebral Cortex,” J. Neuroscience, Vol.15, No.2, pp. 1463-1483, 1995.
[11] G. Ghosh, A. Strehl, and S. Merugu, “A Consensus Framework for Integrating Distributed Clusterings under Limited Knowledge Sharing,” Proc. NSFWorkshop on Next Generation DataMining, pp. 99-108, 2002.
[12] J. C. Bezdek, J. Keller, R. Krishnapuram, and N.-R. Pal, “Fuzzy Models and Algorithms for Pattern Recognition and Image Processing,” Kluwer Academic Publishing, Boston, 1999.

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] 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.

[3] [3] R. J. Hathaway and J. C. Bezdek, “NERF C-means: Non-Euclidean Relational Fuzzy Clustering,” Pattern Recognition, Vol.27, pp. 429-437, 1994.

[4] [4] S. Miyamoto and D. Suizu, “Fuzzy c-Means Clustering Using Kernel Functions in Support Vector Machines,” J. Advanced Computational Intelligence and Intelligent Informatics, Vol.7, No.1, pp. 25-30, 2003.

[5] [5] V. N. Vapnik, “Statistical Learning Theory,” Wiley, New York, 1998.

[6] [6] S.Miyamoto, Y. Kawasaki, and K. Sawazaki, “An Explicit Mapping for Kernel Data Analysis and Application to Text Analysis,” Proc. IFSA-EUSFLAT 2009, pp. 618-623, 2009.

[7] [7] S. Miyamoto and K. Sawazaki, “An Explicit Mapping for Kernel Data Analysis and Application to c-Means Clustering,” Proc. NOLTA 2009, pp. 556-559, 2009.

[8] [8] Y. Kanzawa, Y. Endo, and S. Miyamoto, “Indefinite Kernel Fuzzy c-Means Clustering Algorithms,” Lecture Notes in Computer Science, Vol.6408, pp. 116-128, 2010.

[9] [9] S. Tamura, S. Higuchi, and K. Tanaka, “Pattern Classification Based on Fuzzy Relations,” IEEE Trans. Syst. Man Cybern., Vol.1, No.1, pp. 61-66, 1971.

[10] [10] J. W. Scannell, C. Blakemore, and M. P. Young, “Analysis of Connectivity in the Cat Cerebral Cortex,” J. Neuroscience, Vol.15, No.2, pp. 1463-1483, 1995.

[11] [11] G. Ghosh, A. Strehl, and S. Merugu, “A Consensus Framework for Integrating Distributed Clusterings under Limited Knowledge Sharing,” Proc. NSFWorkshop on Next Generation DataMining, pp. 99-108, 2002.

[12] [12] J. C. Bezdek, J. Keller, R. Krishnapuram, and N.-R. Pal, “Fuzzy Models and Algorithms for Pattern Recognition and Image Processing,” Kluwer Academic Publishing, Boston, 1999.