Paper:

# Power-Regularized Fuzzy *c*-Means Clustering with a Fuzzification Parameter Less Than One

## Yuchi Kanzawa

Shibaura Institute of Technology

3-7-5 Toyosu, Koto, Tokyo 135-8548, Japan

The present study proposes two types of power-regularized fuzzy *c*-means (pFCM) clustering algorithms with a fuzzification parameter less than one, which supplements previous work on pFCM with a fuzzification parameter greater than one. Both the proposed methods are essentially identical to each other, but not when fuzzification parameter values are specified. Theoretical discussion reveals the property of the proposed methods, and some numerical results substantiate the property of the proposed methods and show that the proposed methods outperform two conventional methods from an accuracy point of view.

- [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. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum Press, New York, 1981.
- [3] S. Miyamoto and M. Mukaidono, “Fuzzy c-Means as a Regularization and Maximum Entropy Approach,” Proc. 7th Int. Fuzzy Systems Association World Congress (IFSA’97), Vol.2, pp. 86-92, 1997.
- [4] S. Miyamoto and K. Umayahara, “Fuzzy Clustering by Quadratic Regularization,” Proc. 1998 IEEE Int. Conf. Fuzzy Syst., pp. 1394-1399, 1998.
- [5] Y. Kanzawa, “Generalization of Quadratic Regularized and Standard Fuzzy c-Means Clustering with respect to Regularization of Hard c-Means,” LNCS, Vol.8234, pp. 152-165, 2013.
- [6] S. Miyamoto and K. Umayahara, “Methods in Hard and Fuzzy Clustering,” in: Z.-Q. Liu and S. Miyamoto (Eds.), Soft Computing and Human-Centered Machines, Springer-Verlag Tokyo, 2000.
- [7] C. L. Blake and C. J. Merz, UCI repository of machine learning databases, a huge collection of artificial and real-world data sets, 1998. Available from: http://archive.ics.uci.edu/ml/ [Accessed January 3, 2016]
- [8] L. Hubert and P. Arabie, “Comparing Partitions,” J. of Classification, Vol.2, pp. 193-218, 1985.
- [9] I. S. Dhillon and D. S. Modha, “Concept Decompositions for Large Sparse Text Data Using Clustering,” Machine Learning, Vol.42, pp. 143-175, 2001.
- [10] S. Miyamoto and K. Mizutani, “Fuzzy Multiset Model and Methods of Nonlinear Document Clustering for Information Retrieval,” LNCS, Vol.3131, pp. 273-283, 2004.
- [11] K. Mizutani, R. Inokuchi, and S. Miyamoto, “Algorithms of Nonlinear Document Clustering based on Fuzzy Set Model,” Int. J. of Intelligent Systems, Vol.23, No.2, pp. 176-198, 2008.
- [12] Y. Kanzawa, “On Kernelization for a Maximizing Model of Bezdek-like Spherical Fuzzy c-means Clustering,” LNCS, Vol.8825, pp. 108-121, 2014.
- [13] Y. Kanzawa, “On Kernelization for a Maximizing Model of Bezdek-like Spherical Fuzzy c-means Clustering,” LNCS, Vol.8825, pp. 108-121, 2014.
- [14] Y. Kanzawa, “A Maximizing Model of Bezdek-like Spherical Fuzzy c-Means,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.19, No.5, pp. 662-669, 2015.
- [15] Y. Kanzawa, “A Maximizing Model of Spherical Bezdek-type Fuzzy Multi-medoids Clustering,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.19, No.6, pp. 738-746, 2015.
- [16] 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.
- [17] R. J. Hathaway and J. C. Bezdek, “NERF C-means: Non-Euclidean Relational Fuzzy Clustering,” Pattern Recognition, Vol.27, pp. 429-437, 1994.
- [18] Y. Kanzawa, “Entropy-regularized Fuzzy Clustering for non-Euclidean Relational Data and for Indefinite Kernel Data,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.16, No.7, pp.784-792, 2012.
- [19] Y. Kanzawa, “Relational Fuzzy c-means and Kernel Fuzzy c-means Using an Object-wise beta-spread Transformation,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.17, No.4, pp. 511-519, 2013.
- [20] Y. Kanzawa, “Relational Fuzzy c-lines Clustering Derived from Kernelization of Fuzzy c-lines,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.18, No.2, pp. 175-181, 2014.
- [21] 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.
- [22] K. Honda, S. Oshio, and A. Notsu, “FCM-type fuzzy co-clustering by K-L information regularization,” Proc. of 2014 IEEE Int. Conf. on Fuzzy Systems, pp. 2505-2510, 2014.
- [23] K. Honda, S. Oshio, and A. Notsu, “Item Membership Fuzzification in Fuzzy Co-clustering Based on Multinomial Mixture Concept,” Proc. of 2014 IEEE Int. Conf. on Granular Computing, pp. 94-99, 2014.
- [24] Y. Kanzawa, “Fuzzy Co-Clustering Algorithms Based on Fuzzy Relational Clustering and TIBA Imputation,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.18, No.2, pp. 182-189, 2014.
- [25] Y. Kanzawa, “On Possibilistic Clustering Methods Based on Shannon/Tsallis-Entropy for Spherical Data and Categorical Multivariate Data,” LNCS, Vol.9321, pp. 125-138, 2015.
- [26] Y. Kanzawa, “Bezdek-type Fuzzified Co-Clustering Algorithm,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.19, No.6, pp. 852-860, 2015.

*c*-Means Clustering with a Fuzzification Parameter Less Than One,”

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.20, No.4, pp. 561-570, 2016

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.20, No.4, pp. 561-570, 2016