This paper provides new clustering algorithms for data with tolerance. Tolerance is understood in a broad sense, e.g., calculation errors and loss of attribute of data. The concept of tolerance is modified by using new concept of tolerance vector. First, the concept is explained and optimization problems of clustering are formulated using the vectors. Second, the problems are solved using Karush-Kuhn-Tucker conditions. Third, the new clustering algorithms are constructed by using the solutions of the problems. Moreover, the effectiveness of proposed algorithms is verified through some numerical examples.

[1] J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum, 1981.
[2] O. Takata and M. Sadaaki, “Fuzzy clustering of Data with Interval Uncertainties,” Journal of Japan Society for Fuzzy Theory and Systems, Vol.12, No.5, pp. 686-695, 2000 (in Japanese).
[3] M. I. Sato and J. Oshima, “On Weighted Principal Component Analysis for Interval-Valued Data and Its Dynamic Feature,” Int. Journal of Innovative Computing, Information and Control, Vol.2, No.1, pp. 69-82, 2006.
[4] Y. Endo and K. Horiuchi, “On Clustering Algorithm for Fuzzy Data,” In Proc. 1997 Int. Symposium on Nonlinear Theory and Its Applications, pp. 381-384, 1997.11.
[5] Y. Endo, “Clustering Algorithm Using Covariance for Fuzzy Data,” In Proc. 1998 Int. Symposium on Nonlinear Theory and Its Applications, pp. 511-514, 1998.9.
[6] Y. Endo, R. Murata, H. Haruyama, and S. Miyamoto, “Fuzzy c-Means for Data with Tolerance,” Proc. 2005 Int. Symposium on Nonlinear Theory and Its Applications, pp. 345-348, Bruges, Belgium, Oct. 19, 2005.
[7] R. Murata, Y. Endo, H. Hideyuki, and S. Miyamoto, “On Fuzzy c-Means for Data with Tolerance,“ Journal of Advanced Computational Intelligence and Intelligent Informatics, Vol.10, No.5, pp. 673-681, 2006.
[8] H. Toyoda, “L1-Norm based Fuzzy Clustering for Data with Tolerance,” Graduation thesis, College of Engineering Systems, University of Tsukuba, 2005 (in Japanese).
[9] 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), June 25-30, 1997, Prague, Chech ,Vol.2, pp. 86-92, 1997.
[10] UCI Machine Learning Databases http://www.ics.uci.edu/ mlearn/databases| verb|/heart-disease/
[11] J. Krzysztof, “L1-norm based fuzzy clustering,” Fuzzy Sets and Systems, Vol.39, pp. 43-50, 1991.
[12] T. Koga, “Clustering Algorithm based on L1-norm space,” Master's thesis, Graduate School of Systems and Information Engineering, University of Tsukuba, 2002 (in Japanese).