In this paper, two new clustering algorithms based on fuzzy c-means for data with tolerance 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, two objective functions in feature space are shown corresponding to two methods, respectively. Third, 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 original pattern space into higher dimensional feature space than the original one. Last, two iterative algorithms are proposed for the objective functions, respectively.
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