Fuzzy clustering method based on perturbation

Based on the results obtained in the earlier paper [1], the fuzzy clustering method based on perturbation (FCMBP) is further studied and several important results are obtained. First, since the fuzzy similarity matrix equation X2 = X is one of the most important basic components of the FCMBP approach, the structure of the solutions of this matrix equation is investigated. To express the solutions, the concept of fuzzy equivalent standard form is proposed. Second, the existence of the globally and locally optimal fuzzy equivalent matrices, which are nearest to the given fuzzy similarity matrix, is proved. Third, an effective fuzzy clustering method based on the above results is proposed. Finally, two examples, which show that the result obtained by the FCMBP method is more accurate than that obtained by the transitive closure method, are solved.