Fuzzy c-Means Clustering for Data with Clusterwise Tolerance Based on L2– and L1-Regularization
Yukihiro Hamasuna*,**, Yasunori Endo**, and Sadaaki Miyamoto**
*Research Fellow of the Japan Society for the Promotion of Science
**Department of Risk Engineering, Faculty of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan
Detecting various kinds of cluster shape is an important problem in the field of clustering. In general, it is difficult to obtain clusters with different sizes or shapes by single-objective function. From that sense, we have proposed the concept of clusterwise tolerance and constructed clustering algorithms based on it. In the field of data mining, regularization techniques are used in order to derive significant classifiers. In this paper, we propose another concept of clusterwise tolerance from the viewpoint of regularization. Moreover, we construct clustering algorithms for data with clusterwise tolerance based on L2– and L1-regularization. After that, we describe fuzzy classification functions of proposed algorithms. Finally, we show the effectiveness of proposed algorithms through numerical examples.
-  M. R. Anderberg, “Cluster Analysis for Applications,” Academic Press, New York, 1973.
-  J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum Press, New York, 1981.
-  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.
-  D. E. Gustafson and W. C. Kessel, “Fuzzy clustering with a fuzzy covariance matrix,” IEEE Conf. on Decision and Control including the 17th Symposium on Adaptive Processes, pp. 761-766, 1978.
-  F. Höppne, F. Klawonn, R. Kruse, and T. Runkler, “Fuzzy Cluster Analysis,” John Wiley & Sons, Chichester, 1999.
-  K. Honda and H. Ichihashi, “Linear fuzzy clustering techniques with missing values and their application to local principal component analysis,” IEEE Trans. on Fuzzy Systems, Vol.12, No.2, pp. 183-193, 2004.
-  S. Miyamoto and N. Kurosawa, “Controlling cluster volume sizes in fuzzy c-means clustering,” Proc. of SCIS & ISIS 2004, pp. 1-4, 2004.
-  Y. Hamasuna, Y. Endo, and S. Miyamoto, “On Tolerant Fuzzy c-Means Clustering,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.13, No.4, pp. 421-428, 2009.
-  Y. Hamasuna, Y. Endo, and S. Miyamoto, “On Tolerant Fuzzy c-Means Clustering and Tolerant Possibilistic Clustering,” Soft Computing, Vol.14, No.5, pp. 487-494, 2010.
-  Y. Endo, R. Murata, H. Haruyama, and S. Miyamoto, “Fuzzy cmeans for data with tolerance,” Proc. of Int. Symposium on Nonlinear Theory and Its Applications (Nolta2005), pp. 345-348, 2005.
-  Y. Hasegawa, Y. Endo, Y. Hamasuna, and S. Miyamoto, “Fuzzy c-means for data with tolerance defined as hyper-rectangle,” Proc. of Modeling Decisions for Artificial Intelligence (MDAI2007), pp. 237-248, 2007.
-  Y. Hamasuna, Y. Endo, Y. Hasegawa, and S. Miyamoto, “Two clustering algorithms for data with tolerance based on hard c-means,” Proc of 2007 IEEE Int. Conf. on Fuzzy Systems (FUZZ-IEEE 2007), pp. 688-691, 2007.
-  Y. Hamasuna, Y. Endo, and S. Miyamoto, “Comparison of Tolerant Fuzzy c-Means Clustering with L2- and L1-Regularization,” Proc. of 2009 IEEE Int. Conf. on Granular Computing (GrC2009), pp. 197-202, 2009.
-  Y. Hamasuna, Y. Endo, and S. Miyamoto, “On Tolerant Fuzzy c-Means Clustering with L1-Regularization,” Int. Fuzzy Systems Association European Society for Fuzzy Logic and Technology (IFSA-EUSFLAT2009), pp. 1152-1157, 2009.
-  R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. of the Royal Statistical Society, Series B, Vol.58, No.1, pp. 267-288, 1996.
-  J. Kazama and J. Tsujii, “Evaluation and extension of maximum entropy models with in equality constraints,” Proc. of 2003 Conf. on Empirical Methods in Natural Language Processing (EMNLP2003), pp. 137-144, 2003.
-  G. C. Cawley, N. L. C. Talbot, and M. Girolami, “Sparse multinomial logistic regression via bayesian L1 regularization,” Advances in Neural Information Processing Systems (NIPS2006), 2006.
-  S. Miyamoto, H. Ichihashi, and K. Honda, “Algorithms for Fuzzy Clustering,” Springer, Heidelberg, 2008.
-  S. Miyamoto, “Formulation of fuzzy c-means clustering using calculus of variations and twofold membership clusters,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.12, No.5, pp. 454-460, 2008.
-  P. M. Williams, “Bayesian regularization and pruning using a laplace prior,” Neural Computation, Vol.7, No.1, pp. 117-143, 1995.