Interpolation of Fuzzy Data by Using Fuzzy Splines

Abstract

In this paper we define a new set of spline functions called “Fuzzy Splines” to interpolate fuzzy data. Numerical examples will be presented to illustrate the differences between of using our spline and other interpolations that have been studied before.

Hermite and piecewise cubic Hermite interpolation of fuzzy data

“…is a nondecreasing function with respect to r. Similarly, we can prove that (f 1 ) r + φ 1 (ξ 1 ) + (f 2 ) r + φ 2 (ξ 2 ) + (f 1 ) r + ψ 1 (ξ 3 ) + (f 2 ) r + ψ 2 (ξ 4 ) is a nonincreasing function with respect to r for…”

Cardinal Basis Piecewise Hermite Interpolation on Fuzzy Data

“…When ( ) = ( ) = 1− , we will have − fuzzy numbers that involve the triangular fuzzy numbers. For an − fuzzy number = ( , , ), the support is the closed interval [ − , + ] (see, e.g., [6]).…”

“…Theorem 6. Assume that ∈ B and satisfies the piecewise Hermite polynomial cardinal basis function constraints (6). Then, (i) 0 ( ) + +1 0 ( ) ≥ 1, for all ∈ ( , +1 ), = 0, 1, .…”

“…Moreover, Kaleva obtained an interpolating fuzzy cubic spline with the nota-knot condition. Interpolating of fuzzy data was developed to simple Hermite or osculatory interpolation, (3) cubic splines, fuzzy splines, complete splines, and natural splines, respectively, in [4][5][6][7][8] by Abbasbandy et al Later, Lodwick and Santos presented the Lagrange fuzzy interpolating function that loses smoothness at the knots at every -cut; also every -cut ( ̸ = 1) of fuzzy spline with the not--knot boundary conditions of order has discontinuous first derivatives on the knots and based on these interpolants some fuzzy surfaces were constructed [9]. Zeinali et al [10] presented a method of interpolation of fuzzy data by Hermite and piecewise cubic Hermite that was simpler and consistent and also inherited smoothness properties of the generator interpolation.…”

A new approach to universal approximation of fuzzy functions on a discrete set of points

“…Kaleva [3] introduced some properties of Lagrange and cubic spline interpolation. Properties of natural splines and complete splines of odd degrees are introduced in [4,5]. In interpolation problem there are given n + 1 distinct points in R and for each points there is given a fuzzy value in R, we find a fuzzy polynomial of degree at most n which coincides, on these points with given fuzzy values.…”