On fuzzification and optimization problems of clustering indices

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Results of clustering are qualitatively evaluated by quantities called clustering indices. While many clustering indices are proposed, in B. Desgraupes [Clustering Indices (2016)], Desgraupes reviewed 27 indices, most of which are applicable only to crisp clustering and only one of which is applicable to fuzzy clustering. In the previous article [D. R. Sope and M. Fujio, On a fuzzification of clustering indices, in Proc. Fuzzy System Symposium of the Japan Intelligent Information Fuzzy Society, Vol. 35 (2002), pp. 443–448], the authors gazed on analytic 3 indices among them and modified them to fit fuzzy clustering by regarding the membership degrees as distribution functions of objects over clusters. In this paper, this method called fuzzification, is applied to all the indices of Desgraupes. This investigation also includes optimization of indices. A significant benefit of fuzzy clustering is that the membership degrees allow the optimization problems to be treated as continuous, whereas the ones in the crisp case are discrete, making it generally easier to solve. After giving a precise description of fuzzification which is briefly given in D. R. Sope and M. Fujio [On a fuzzification of clustering indices, in Proc. Fuzzy System Symposium of the Japan Intelligent Information Fuzzy Society, Vol. 35 (2002), pp. 443–448], the authors fuzzify all of 27 indices of Desgraupes and then solve the fuzzy clustering problems having 13 analytic indices among them as the objective functions by the gradient method.