Hadi Hossein-Abad Mahdipour

Many variants of fuzzy c-means (FCM) clustering method are applied to crisp numbers but only a few of them are extended to non-crisp numbers, mainly due to the fact that the latter needs complicated equations and exhausting calculations. Vector form of fuzzy c-means (VFCM), proposed in this paper, simplifies the FCM clustering method applying to non-crisp (symbolic interval and fuzzy) numbers. Indeed, the VFCM method is a simple and general form of FCM that applies the FCM clustering method to various types of numbers (crisp and non-crisp) with different correspondent metrics in a single structure, and without any complex calculations and exhaustive derivations. The VFCM maps the input (crisp or non-crisp) features to crisp ones and then applies the conventional FCM to the input numbers in the resulted crisp features' space. Finally, the resulted crisp prototypes in the mapped space would be influenced by inverse mapping to obtain the main prototypes' parameters in the input features' space. Equations of FCM applied to crisp, symbolic interval and fuzzy numbers (i.e., LR-type, trapezoidal-type, triangular-type and normal-type fuzzy numbers) are obtained in this paper, using the proposed VFCM method. Final resulted equations are the same as derived equations in [7, 9, 10, 13, 18, 19, 30, 38–40] (the FCM clustering method applying to non-crisp numbers directly and without using VFCM), while the VFCM obtains these equations using a single structure for all cases [7, 9, 10, 13, 18, 19, 30, 38–40] without any complex calculations. It is showed that VFCM is able to clustering of normal-type fuzzy numbers, too. Simulation results approve truly of normal-type fuzzy numbers clustering.