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
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.
-  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.
-  J. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum Press, 1981.
-  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.
-  S. Miyamoto and N. Kurosawa, “Controlling Cluster Volume Sizes in Fuzzy c-means Clustering,” Proc. SCIS&ISIS2004, pp. 1-4, 2004.
-  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.
-  S. Miyamoto, H. Ichihashi, and K. Honda, “Algorithms for Fuzzy Clustering,” Springer, 2008.
-  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.
-  K. Honda, S. Oshio, and A. Notsu, “FCM-type Fuzzy Co-Clustering by K-L Information Regularization,” Proc. FUZZ-IEEE2014, pp. 2505-2510, 2014.
-  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.
-  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.
-  Lise’s Inquisitive Students, Machine Learning Research Group @UMD, http://www.cs.umd.edu/sen/lbc-proj/LBC.html [accessed November 1, 2017]
-  V. Tunali, PRETO Data Mining Research, http://www.dataminingresearch.com/ [accessed November 1, 2017]
-  G. Karypis, Karypis Lab, CLUTO – Software for Clustering High-Dimensional Datasets, http://glaros.dtc.umn.edu/ [accessed November 1, 2017]
-  Text REtrieval Conf. (TREC), http://trec.nist.gov [accessed November 1, 2017]
-  TREC: Text REtrieval Conference Relevance Judgments, http://trec.nist.gov/data/qrels_eng/index.html [accessed November 1, 2017]
-  D. Boley et al., “Partitioning-based clustering for web document categorization,” Decision Support Systems, Vol.27, No.3, pp. 329-341, 1999.
-  D. D. Lewis, “Reuters-21578 text categorization test collection distribution 1.0,” http://www.daviddlewis.com/resources/testcollections/reuters21578/ [accessed November 1, 2017]
-  L. Hubert and P. Arabie, “Comparing Partitions,” J. of Classification, Vol.2, pp. 193-218, 1985.
-  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.
-  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.
-  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.
-  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.
-  Y. Kanzawa, “Bezdek-Type Fuzzified Co-Clustering Algorithm,” J. Adv. Comput. Intell. Intell. Inform., Vol.19, No.6, pp. 852-860, 2015.
-  I. S. Dhillon and D. S. Modha, “Concept Decompositions for Large Sparse Text Data Using Clustering,” Machine Learning, Vol.42, pp. 143-175, 2001.
-  S. Miyamoto and K. Mizutani, “Fuzzy Multiset Model and Methods of Nonlinear Document Clustering for Information Retrieval,” LNCS, Vol.3131, pp. 273-283, 2004.
-  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.
-  Y. Kanzawa, “On Kernelization for a Maximizing Model of Bezdek-like Spherical Fuzzy c-means Clustering,” LNCS, Vol.8825, pp. 108-121, 2014.
-  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.
-  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.
-  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.
-  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.
-  R. Krishnapuram and J. M. Keller, “A Possibilistic Approach to Clustering,” IEEE Trans. Fuzzy Syst., Vol.1, pp. 98-110, 1993.
-  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.
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