Fuzzy c-Means Algorithms for Data with Tolerance Using Kernel Functions

Yuchi KANZAWA  Yasunori ENDO  Sadaaki MIYAMOTO  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E91-A   No.9   pp.2520-2534
Publication Date: 2008/09/01
Online ISSN: 1745-1337
DOI: 10.1093/ietfec/e91-a.9.2520
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Nonlinear Theory and its Applications)
Category: Soft Computing
fuzzy c-means,  clustering,  tolerance,  kernel functions,  

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In this paper, two new clustering algorithms based on fuzzy c-means for data with tolerance using kernel functions are proposed. Kernel functions which map the data from the original space into higher dimensional feature space are introduced into the proposed algorithms. Nonlinear boundary of clusters can be easily found by using the kernel functions. First, two clustering algorithms for data with tolerance are introduced. One is based on standard method and the other is on entropy-based one. Second, the tolerance in feature space is discussed taking account into soft margin algorithm in Support Vector Machine. Third, two objective functions in feature space are shown corresponding to two methods, respectively. Fourth, Karush-Kuhn-Tucker conditions of two objective functions are considered, respectively, and these conditions are re-expressed with kernel functions as the representation of an inner product for mapping from the original pattern space into a higher dimensional feature space. Fifth, two iterative algorithms are proposed for the objective functions, respectively. Through some numerical experiments, the proposed algorithms are discussed.