Paper:

# Fuzzy Clustering Methods for Categorical Multivariate Data Based on *q*-Divergence

## Tadafumi Kondo and Yuchi Kanzawa

Shibaura Institute of Technology

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

*q*-divergence

This paper presents two fuzzy clustering algorithms for categorical multivariate data based on *q*-divergence. First, this study shows that a conventional method for vectorial data can be explained as regularizing another conventional method using *q*-divergence. Second, based on the known results that Kullback-Leibler (KL)-divergence is generalized into the *q*-divergence, and two conventional fuzzy clustering methods for categorical multivariate data adopt KL-divergence, two fuzzy clustering algorithms for categorical multivariate data that are based on *q*-divergence are derived from two optimization problems built by extending the KL-divergence in these conventional methods to the *q*-divergence. Through numerical experiments using real datasets, the proposed methods outperform the conventional methods in term of clustering accuracy.

*q*-Divergence,”

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.22, No.4, pp. 524-536, 2018.

- [1] J. B. MacQueen, “Some Methods for 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, 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 N. Kurosawa, “Controlling Cluster Volume Sizes in Fuzzy
*c*-means Clustering,” Proc. SCIS&ISIS2004, pp. 1-4, 2004. - [5] H. Ichihashi, K. Honda, and N. Tani, “Gaussian Mixture PDF Approximation and Fuzzy
*c*-means Clustering with Entropy Regularization,” Proc. 4th Asian Fuzzy System Symp., pp. 217-221, 2000. - [6] S. Miyamoto, H. Ichihashi, and K. Honda, “Algorithms for Fuzzy Clustering,” Springer, 2008.
- [7] L. Rigouste, O. Cappé, and F. Yvon, “Inference and evaluation of the multinomial mixture model for text clustering,” Information Processing and Management, Vol.43, No.5, pp. 1260-1280, 2007.
- [8] K. Honda, S. Oshio, and A. Notsu, “FCM-type Fuzzy Co-Clustering by K-L Information Regularization,” Proc. FUZZ-IEEE2014, pp. 2505-2510, 2014.
- [9] K. Honda, S. Oshio, and A. Notsu, “Fuzzy Co-Clustering Induced by Multinomial Mixture Models,” J. Adv. Comput. Intell. Intell. Inform., Vol.19, No.6, pp. 717-726, 2015.
- [10] H. Chernoff, “A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on a Sum of Observations,” Ann. Math. Statist., Vol.23, pp. 493-507, 1952.
- [11] Lise’s Inquisitive Students, Machine Learning Research Group @UMD, http://www.cs.umd.edu/sen/lbc-proj/LBC.html [accessed November 1, 2017]
- [12] V. Tunali, PRETO Data Mining Research, http://www.dataminingresearch.com/ [accessed November 1, 2017]
- [13] G. Karypis, Karypis Lab, CLUTO – Software for Clustering High-Dimensional Datasets, http://glaros.dtc.umn.edu/ [accessed November 1, 2017]
- [14] Text REtrieval Conf. (TREC), http://trec.nist.gov [accessed November 1, 2017]
- [15] TREC: Text REtrieval Conference Relevance Judgments, http://trec.nist.gov/data/qrels_eng/index.html [accessed November 1, 2017]
- [16] D. Boley et al., “Partitioning-based clustering for web document categorization,” Decision Support Systems, Vol.27, No.3, pp. 329-341, 1999.
- [17] D. D. Lewis, “Reuters-21578 text categorization test collection distribution 1.0,” http://www.daviddlewis.com/resources/testcollections/reuters21578/ [accessed November 1, 2017]
- [18] L. Hubert and P. Arabie, “Comparing Partitions,” J. of Classification, Vol.2, pp. 193-218, 1985.
- [19] D. Arthur and S. Vassilvitskii, “k-means++: the advantages of careful seeding,” Proc. the 8th Annual ACM-SIAM Symp. on Discrete Algorithms, pp. 1027-1035, 2007.
- [20] C.-H. Oh, K. Honda, and H. Ichihashi, “Fuzzy Clustering for Categorical Multivariate Data,” Proc. Joint 9th IFSA World Congress and 2nd NAFIPS Int. Conf., pp. 2154-2159, 2001.
- [21] K. Kummamuru, A. Dhawale, and R. Krishnapuram, “Fuzzy Co-clustering of Documents and Keywords,” Proc. IEEE Int. Conf. on Fuzzy Sys., Vol.2, pp. 772-777, 2003.
- [22] Y. Kanzawa, “Fuzzy Co-Clustering Algorithms Based on Fuzzy Relational Clustering and TIBA Imputation,” J. Adv. Comput. Intell. Intell. Inform., Vol.18, No.2, pp. 182-189, 2014.
- [23] Y. Kanzawa, “Bezdek-Type Fuzzified Co-Clustering Algorithm,” J. Adv. Comput. Intell. Intell. Inform., Vol.19, No.6, pp. 852-860, 2015.
- [24] I. S. Dhillon and D. S. Modha, “Concept Decompositions for Large Sparse Text Data Using Clustering,” Machine Learning, Vol.42, pp. 143-175, 2001.
- [25] S. Miyamoto and K. Mizutani, “Fuzzy Multiset Model and Methods of Nonlinear Document Clustering for Information Retrieval,” LNCS, Vol.3131, pp. 273-283, 2004.
- [26] K. Mizutani, R. Inokuchi, and S. Miyamoto, “Algorithms of Nonlinear Document Clustering based on Fuzzy Set Model,” Int. J. of Intel. Sys., Vol.23, No.2, pp. 176-198, 2008.
- [27] Y. Kanzawa, “On Kernelization for a Maximizing Model of Bezdek-like Spherical Fuzzy
*c*-means Clustering,” LNCS, Vol.8825, pp. 108-121, 2014. - [28] Y. Kanzawa, “A Maximizing Model of Bezdek-like Spherical Fuzzy
*c*-Means,” Clustering,” J. Adv. Comput. Intell. Intell. Inform., Vol.19, No.5, pp. 662-669, 2015. - [29] Y. Kanzawa, “A Maximizing Model of Spherical Bezdek-type Fuzzy Multi-medoids Clustering,” J. Adv. Comput. Intell. Intell. Inform., Vol.19, No.6, pp. 738-746, 2015.
- [30] A. Cichocki and S. Amari, “Families of Alpha- Beta- and Gamma- Divergences: Flexible and Robust Measures of Similarities,” Entropy, Vol.12 No.6, pp. 1532-1568, 2010.
- [31] A. Cichocki, S. Cruces, and S. Amari, “Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative Matrix Factorization,” Entropy, Vol.13, No.1, pp. 134-170, 2011.
- [32] R. Krishnapuram and J. M. Keller, “A Possibilistic Approach to Clustering,” IEEE Trans. Fuzzy Syst., Vol.1, pp. 98-110, 1993.
- [33] Y. Kanzawa, “On Possibilistic Clustering Methods Based on Shannon/Tsallis-Entropy for Spherical Data and Categorical Multivariate Data,” Lecture Notes in Artificial Intelligence, Vol.9321, pp. 115-128, 2015.

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