A Maximizing Model of Bezdek-Like Spherical Fuzzy <i>c</i>-Means

Yuchi Kanzawa

doi:10.20965/jaciii.2015.p0662

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JACIII Vol.19 No.5 pp. 662-669

doi: 10.20965/jaciii.2015.p0662

(2015)

Paper:

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A Maximizing Model of Bezdek-Like Spherical Fuzzy c-Means

Yuchi Kanzawa

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

Received:

January 31, 2015

Accepted:

June 16, 2015

Published:

September 20, 2015

Keywords:

fuzzy c-means, spherical clustering

Abstract

In this study, a maximizing model of Bezdek-like spherical fuzzy c-means clustering is proposed, which is based on the regularization of the maximizing model of spherical hard c-means. Such a maximizing model was unclear in Bezdek-like method, whereas other types of methods have been investigated well both in minimizing and maximizing model. Using theoretical analysis and numerical experiments, the classification rule of the proposed method is shown. Using numerical examples, the proposed method is shown to be valid for document clustering, because documents are represented as spherical data via term document-inverse document frequency weighting and normalization processing.

Cite this article as:

Y. Kanzawa, “A Maximizing Model of Bezdek-Like Spherical Fuzzy c-Means,” J. Adv. Comput. Intell. Intell. Inform., Vol.19 No.5, pp. 662-669, 2015.

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References

[1] J. B. MacQueen, “Some Methods of Classification and Analysis of Multivariate Observations,” Proc. 5th Berkeley Symp. on Math. Stat. and Prob., pp. 281-297, 1967.
[2] J. Dunn, “A Fuzzy Relative of the Isodata Process and Its Use in Detecting Compact, Well-Separated Clusters,” J. of Cybernetics, Vol.3, No.3, pp. 32-57, 1973.
[3] J. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum Press, New York, 1981.
[4] N. R. Pal and J. C. Bezdek, “On Cluster Validity for Fuzzy c-Means Model,” IEEE Trans. Fuzzy Syst., Vol.1, pp. 370-379, 1995.
[5] S. Miyamoto and M. Mukaidono, “Fuzzy c-Means as a Regularization and Maximum Entropy Approach,” Proc. 7^th Int. Fuzzy Systems Association World Congress (IFSA’97), Vol.2, pp. 86-92, 1997.
[6] I. S. Dhillon and D. S. Modha, “Concept Decompositions for Large Sparse Text Data Using Clustering,” Machine Learning, Vol.42, pp. 143-175, 2001.
[7] S. Miyamoto and K. Mizutani, “Fuzzy Multiset Model and Methods of Nonlinear Document Clustering for Information Retrieval,” LNCS, Vol.3131, pp. 273-283, 2004.
[8] S. Miyamoto, H. Ichihashi, and K. Honda, “Algorithms for Fuzzy Clustering,” Springer, 2008.
[9] S. Miyamoto and K. Umayahara, “Methods in Hard and Fuzzy Clustering,” in Z.-Q. Liu, and S. Miyamoto (eds.), Soft Computing and Humancentered Machines, Springer-Verlag Tokyo, (2000).
[10] C. Buchta, M. Kober, I. Feinerer, and K. Hornik, “Spherical k-Means Clustering,” J. of Statistical Software, Vol.50, No.10, 2012.
[11] A. Banerjee, I. S. Dhillon, J. Ghosh, and S. Sra, “Clustering on the Unit Hypersphere using von Mises-Fisher Distributions,” J. of Machine Learning Research, Vol.6, pp. 1345-1382, 2005.
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[13] D. Boley, M. Gini, R. Gross, E. H. Han, K. Hasting, G. Karypis, V. Kumar, B. Mobasher, and J. Moore, “Document Categorization and Query Generation on the World Wide Web using WebACE,” AI Review, Vol.11, pp. 365-391, 1999.
[14] G. Ghosh, A. Strehl, and S. Merugu, “A Consensus Framework for Integrating Distributed Clusterings under Limited Knowledge Sharing,” Proc. NSF Workshop on Next Generation Data Mining, pp. 99-108, 2002.
[15] 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.
[16] M. Roubens, “Pattern Classification Problems and Fuzzy Sets,” Fuzzy Sets and Syst., Vol.1, pp. 239-253, 1978.
[17] M. P. Windham, “Numerical Classification of Proximity Data with Assignment Measures,” J. Classification, Vol.2, pp. 157-172, 1985.
[18] R. Krishnapuram and J. M. Keller, “A Possibilistic Approach to Clustering,” IEEE Trans. on Fuzzy Systems, Vol.1, pp. 98-110, 1993.
[19] C. Oh, K. Honda, and H. Ichihashi, “Fuzzy Clustering for Categorical Multivariate Data,” Proc. IFSA World Congress and 20th NAFIPS Int. Conf., pp. 2154-2159, 2001.
[20] D. Abril, G. Navarro-Arribas, and V. Torra, “Spherical Microaggregation: Anonymizing Sparse Vector Spaces,” Computers & Security, Vol.49, pp. 28-44, 2015.

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

[1] [1] J. B. MacQueen, “Some Methods of Classification and Analysis of Multivariate Observations,” Proc. 5th Berkeley Symp. on Math. Stat. and Prob., pp. 281-297, 1967.

[2] [2] J. Dunn, “A Fuzzy Relative of the Isodata Process and Its Use in Detecting Compact, Well-Separated Clusters,” J. of Cybernetics, Vol.3, No.3, pp. 32-57, 1973.

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

[4] [4] N. R. Pal and J. C. Bezdek, “On Cluster Validity for Fuzzy c-Means Model,” IEEE Trans. Fuzzy Syst., Vol.1, pp. 370-379, 1995.

[5] [5] S. Miyamoto and M. Mukaidono, “Fuzzy c-Means as a Regularization and Maximum Entropy Approach,” Proc. 7^th Int. Fuzzy Systems Association World Congress (IFSA’97), Vol.2, pp. 86-92, 1997.

[6] [6] I. S. Dhillon and D. S. Modha, “Concept Decompositions for Large Sparse Text Data Using Clustering,” Machine Learning, Vol.42, pp. 143-175, 2001.

[7] [7] S. Miyamoto and K. Mizutani, “Fuzzy Multiset Model and Methods of Nonlinear Document Clustering for Information Retrieval,” LNCS, Vol.3131, pp. 273-283, 2004.

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

[9] [9] S. Miyamoto and K. Umayahara, “Methods in Hard and Fuzzy Clustering,” in Z.-Q. Liu, and S. Miyamoto (eds.), Soft Computing and Humancentered Machines, Springer-Verlag Tokyo, (2000).

[10] [10] C. Buchta, M. Kober, I. Feinerer, and K. Hornik, “Spherical k-Means Clustering,” J. of Statistical Software, Vol.50, No.10, 2012.

[11] [11] A. Banerjee, I. S. Dhillon, J. Ghosh, and S. Sra, “Clustering on the Unit Hypersphere using von Mises-Fisher Distributions,” J. of Machine Learning Research, Vol.6, pp. 1345-1382, 2005.

[12] [12] T. Mitchell, “20 Newsgroups.” UCI KDD Archive, available at: http://kdd.ics.uci.edu/databases/20newsgroups/ [Accessed Febrary 14, 2015]

[13] [13] D. Boley, M. Gini, R. Gross, E. H. Han, K. Hasting, G. Karypis, V. Kumar, B. Mobasher, and J. Moore, “Document Categorization and Query Generation on the World Wide Web using WebACE,” AI Review, Vol.11, pp. 365-391, 1999.

[14] [14] G. Ghosh, A. Strehl, and S. Merugu, “A Consensus Framework for Integrating Distributed Clusterings under Limited Knowledge Sharing,” Proc. NSF Workshop on Next Generation Data Mining, pp. 99-108, 2002.

[15] [15] 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.

[16] [16] M. Roubens, “Pattern Classification Problems and Fuzzy Sets,” Fuzzy Sets and Syst., Vol.1, pp. 239-253, 1978.

[17] [17] M. P. Windham, “Numerical Classification of Proximity Data with Assignment Measures,” J. Classification, Vol.2, pp. 157-172, 1985.

[18] [18] R. Krishnapuram and J. M. Keller, “A Possibilistic Approach to Clustering,” IEEE Trans. on Fuzzy Systems, Vol.1, pp. 98-110, 1993.

[19] [19] C. Oh, K. Honda, and H. Ichihashi, “Fuzzy Clustering for Categorical Multivariate Data,” Proc. IFSA World Congress and 20th NAFIPS Int. Conf., pp. 2154-2159, 2001.

[20] [20] D. Abril, G. Navarro-Arribas, and V. Torra, “Spherical Microaggregation: Anonymizing Sparse Vector Spaces,” Computers & Security, Vol.49, pp. 28-44, 2015.