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Fuzzy c-means for data with tolerance by introducing penalty term

In this paper, two new clustering algorithms are proposed for data with some errors. In any of these algorithms, the error is interpreted as one of decision variables - called ldquotolerancerdquo - of a certain optimization problem like the previously proposed algorithm, but the tolerance in new methods is determined by the new introduced penalty term of it in the corresponding objective function. Through some numerical experiments, the difference between our methods and the previously proposed one is discussed.