F-transform with parametric generalized fuzzy partitions

The fuzzy transform (F-transform) has some interesting properties that allow its use in curve fitting and smoothing. We suggest the following: (a) a generalization of fuzzy partitions for the F-transform to control and improve the smoothing effect and (b) a parametrization of the fuzzy numbers constituting the basic functions related to the fuzzy partitions. An optimization criterion is then suggested to obtain the F-transform with best approximation properties, by estimating the basic functions from a family of parametric fuzzy numbers. The generalized fuzzy partitions are characterized by the number of subintervals (the bandwidth of the partition) forming the core of each basic function and we suggest two procedures for bandwidth selection: one based on taut-string variation and a second based on generalized cross validation. Some illustrative examples are included.