On Objective-Based Rough Hard and Fuzzy c-Means Clustering
Naohiko Kinoshita* and Yasunori Endo**
*Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan
**Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan
Clustering is one of the most popular unsupervised classification methods. In this paper, we focus on rough clustering methods based on rough-set representation. Rough k-Means (RKM) is one of the rough clustering method proposed by Lingras et al. Outputs of many clustering algorithms, including RKM depend strongly on initial values, so we must evaluate the validity of outputs. In the case of objectivebased clustering algorithms, the objective function is handled as the measure. It is difficult, however to evaluate the output in RKM, which is not objective-based. To solve this problem, we propose new objective-based rough clustering algorithms and verify theirs usefulness through numerical examples.
-  J. MacQueen, “Some methods for classification and analysis of multivariate observations,” Proc. of the 5th Berkeley Symp. on Mathematical Statistics and Probability, Vol.1, pp. 281-297, 1967.
-  J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum Press, New York, 1981.
-  L. A. Zadeh, “Fuzzy Sets,” Information and Control, Vol.8, pp. 338-353, 1965.
-  P. Lingras and G. Peters, “Rough clustering,” WIREs Data Mining and Knowledge Discovery, Vol.1, Issue.1, pp. 64-72, 2011.
-  Z. Pawlak, “Rough Sets,” Int. J. of Computer and Information Sciences, Vol.11, No.5, pp. 341-356, 1982.
-  P. Lingras and C. West, “Interval Set Clustering of Web Users with Rough K-Means,” J. of Intelligent Information Systems, Vol.23, No.1, pp. 5-16, 2004.
-  P. Maji and S. K. Pal, “Rough Set based Generalized Fuzzy CMeans Algorithm and Quantitative Indices,” IEEE Trans. on Systems, Man, and Cybernetics – Part B: Cybernetics, Vol.37, No.6, 2007.
-  G. Peters, “Outliers in Rough k-Means Clustering,” Proc. PReMI 2005, LNCS 3776, pp. 702-707, 2005.
-  S. Mitra, H. Banka, and W. Pedrycz, “Rough-Fuzzy Collaborative Clustering,” IEEE Trans. Systems, Man and Cybernetics Part B, Vol.36, No.4, pp. 795-805, 2006.
-  S. Mitra, “An evolutionary rough partitive clustering,” Pattern Recognition Letters, Vol.25, No.12, pp. 1439-1449, 2004.
-  S. Mitra and B. Barman, “Rough-Fuzzy Clustering: An Application to Medical Imagery,” Proc. RKST 2008, LNAI 5009, pp. 300-307. 2008.
-  S. Hirano and S. Tsumoto, “An Indiscernibility-Based Clustering Method with Iterative Refinement of Equivalence Relations -Rough Clustering-,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.7, No.2, pp. 169-177, 2003.
-  Y. Endo, A. Heki, and Y. Hamasuna, “Non Metric Model Based on Rough Set Representation,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.17, No.4, pp. 540-551, 2013.