In this paper, two new clustering algorithms are proposed for the data with some errors. In any of these algorithms, the error is interpreted as one of decision variables -- called "tolerance" -- of a certain optimization problem like the previously proposed algorithm, but the tolerance is determined based on the opposite criterion to its corresponding previously proposed one. Applying our each algorithm together with its corresponding previously proposed one, a reliability of the clustering result is discussed. Through some numerical experiments, the validity of this paper is discussed.

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.