We present a new methodology to approximate a trapezoidal fuzzy numbers set by using smoothing radial basis functions (RBFs). The methodology uses different error and similarity indices to determine and compare the accuracy of the approximation of the given trapezoidal fuzzy data. For the proposed approximation method a fuzzy radial basis functions type are defined, called fuzzy smoothing radial basis functions under tension. The computation of one of these approximation functions from a given trapezoidal fuzzy data set is described and some convergence results are proved. Finally, some examples in two-dimensions are given to compare the behavior of the presented method by using the proposed error and similarity indices for different configurations of the fuzzy smoothing radial basis functions under tension.

Smoothing b-splines constitute a powerful and popular methodology for performing nonparametric regression with high accuracy. It is well known that the placement of the knots in spline smoothing approximation has an important and considerable effect on the behavior of the final approximation. For this purpose, in this paper a novel methodology is presented for optimal placement and selections of knots, in order to approximate or fit curves to data, using smoothing splines. A new method based on improved clustering algorithm is used to optimally select a reduced number of knots for constructing the base of the b-spline, while ensuring the best accuracy.