Non Metric Model Based on Rough Set Representation
Yasunori Endo*, Ayako Heki**, and Yukihiro Hamasuna***
*Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan
**Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan
***Department of Informatics, Kinki University, 3-4-1 Kowakae, Higashiosaka, Osaka 577-8502, Japan
The non metricmodel is a kind of clustering method in which belongingness or the membership grade of each object in each cluster is calculated directly from dissimilarities between objects and in which cluster centers are not used. The clustering field has recently begun to focus on rough set representation instead of fuzzy set representation. Conventional clustering algorithms classify a set of objects into clusters with clear boundaries, that is, one object must belong to one cluster. Many objects in the real world, however, belong to more than one cluster because cluster boundaries overlap each other. Fuzzy set representation of clusters makes it possible for each object to belong to more than one cluster. The fuzzy degree of membership may, however, be too descriptive for interpreting clustering results. Rough set representation handles such cases. Clustering based on rough sets could provide a solution that is less restrictive than conventional clustering and more descriptive than fuzzy clustering. This paper covers two types of Rough-set-based Non Metric model (RNM). One algorithm is the Roughset-based Hard Non Metric model (RHNM) and the other is the Rough-set-based Fuzzy Non Metric model (RFNM). In both algorithms, clusters are represented by rough sets and each cluster consists of lower and upper approximation. The effectiveness of proposed algorithms is evaluated through numerical examples.
-  J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum, New York, 1981.
-  J. C. Bezdek, J. Keller, R. Krisnapuram, and N. R. Pal, “Fuzzy Models and Algorithms for Pattern Recognition and Image Processing,” The Handbooks of Fuzzy Sets Series, 1999.
-  M. Roubens, “Pattern classification problems and fuzzy sets,” Fuzzy Sets and Systems, Vol.1, pp. 239-253, 1978.
-  J. C. Bezdek, J. W. Davenport, and R. J. Hathaway, “Clustering with the Relational c-Means Algorithms using Different Measures of Pairwise Distance,” Proc. of the 1988 SPIE Technical Symposium on Optics, Electro-Optics, and Sensors, Vol.938, R. D. Juday (Ed.), pp. 330-337, 1988.
-  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.
-  Y. Endo, “On Entropy Based Fuzzy Non Metric Model – Proposal, Kernelization and Pairwise Constraints –,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.16, No.1, pp. 169-173, 2012.
-  P. Lingras and G. Peters, “Rough clustering,” Proc. of the 17th Int. Conf. on Machine Learning (ICML 2000), pp. 1207-1216, 2011.
-  Z. Pawlak, “Rough Sets,” Int. J. of Computer and Information Sciences, Vol.11, No.5, pp. 341-356, 1982.
-  M. Inuiguchi, “Generalizations of Rough Sets: From Crisp to Fuzzy Cases,” Proc. of Rough Sets and Current Trends in Computing, pp. 26-37, 2004.
-  Z. Pawlak, “Rough Classification,” Int. J. of Man-Machine Studies, Vol.20, pp. 469-483, 1984.
-  S. Hirano and S. Tsumoto, “An Indiscernibility-Based Clustering Method with Iterative Refinement of Equivalence Relations,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.7, No.2, pp. 169-177, 2003.
-  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.
-  S. Mitra, H. Banka, and W. Pedrycz, “Rough-Fuzzy Collaborative Clustering,” IEEE Trans. on Systems Man, and Cybernetics, Part B, Cybernetics, Vol.36, No.5, pp. 795-805, 2006.
-  P. Maji and S. K. Pal, “Rough Set Based Generalized Fuzzy CMeans Algorithm and Quantitative Indices,” IEEE Trans. on System, Man and Cybernetics, Part B, Cybernetics, Vol.37, No.6, pp. 1529-1540, 2007.
-  G. Peters, “Rough Clustering and Regression Analysis,” Proc. RSKT’07, LNAI 2007, Vol.4481, pp. 292-299, 2007.
-  S. Mitra and B. Barman, “Rough-Fuzzy Clustering: An Application to Medical Imagery,” Rough Set and Knowledge Technology, LNCS 2008, Vol.5009, pp. 300-307, 2008.
-  S. Miyamoto and M. Mukaidono, “Fuzzy c-Means as a Regularization and Maximum Entropy Approach,” Proc. of the 7th Int. Fuzzy Systems Association World Congress (IFSA’97), Vol.2, pp. 86-92, 1997.
-  S. Miyamoto, K. Umayahara, and M. Mukaidono, “Fuzzy Classification Functions in the Methods of Fuzzy c-Means and Regularization by Entropy,” J. of Japan Society for Fuzzy Theory and Systems Vol.10, No.3, pp. 548-557, 1998.
-  V. N. Vapnik, “Statistical Learning Theory,” Wiley, New York, 1998.
-  V. N. Vapnik, “The nature of Statistical Learning Theory,” 2nd ed., Springer, New York, 2000.
-  Y. Endo, H. Haruyama, and T. Okubo, “On Some Hierarchical Clustering Algorithms Using Kernel Functions,” IEEE Int. Conf. on Fuzzy Systems, #1106, 2004.
-  R. J. Hathaway, J. M. Huband, and J. C. Bezdek, “A Kernelized Non-Euclidean Relational Fuzzy c-Means Algorithm,” Neural, Parallel and Scientific computation, Vol.13, pp. 305-326, 2005.
-  S. Miyamoto, Y. Kawasaki, and K. Sawazaki, “An Explicit Mapping for Fuzzy c-Means Using Kernel Function and Application to Text Analysis,” IFSA/EUSFLAT 2009, 2009.
-  K. Wagstaff and C. Cardie, “Clustering with Instance-level Constraints,” Proc. of the 17th Int. Conf. on Machine Learning, pp. 1103-1110, 2000.
-  K. Wagstaff, C. Cardie, S. Rogers, and S. Schroedl, “Constrained k-means clustering with background knowledge,” Proc. of the 18th Int. Conf. on Machine Learning, pp. 577-584, 2001.