Expectile smoothing of time series using F-transform

In this paper, we will illustrate the F-transform based on generalized fuzzy partitions as a tool for expectile smoothing. This allows to represent a time series in terms of a fuzzy-valued function whose level-cuts are modeled by F-transform and estimated by expectile regression. The proposed methodology is illustrated on real economic and …nancial time series. Keywords: Fuzzy Transform, Expectile Smoothing, Fuzzy Time Series 1. F-transform and its properties The fuzzy transform (F-transform) has recently been introduced by I. Per…lieva in [3] (see also [4], [5], [6]) and its properties as a general smoothing tool have been illustrated in [8], [10], [1]. We brie‡y recall the basic de…nitions and properties of the F-transform (see [3], [10]). Given a continuous function f : [a; b] ! R and given a …nite family of fuzzy sets (in particular fuzzy numbers) A = fA1; A2; :::; Ang forming a fuzzy partition of [a; b], the F-transform produces a vector of real numbers F = (F1; F2; :::; Fn) (called the direct F-transform). Each Fk is the minimizer of a weighted squared error between the values f(x) and Fk on the k th subinterval of [a; b]. The direct F-transform F is then used to de…ne the inverse Ftransform function b f : [a; b] ! R and the main result is that b f is an approximating function of f on [a; b]. In the basic setting, each basic function Ak of the fuzzy partition (P;A) has been considered to be zero outside the union of the two adjacent subintervals [xk 1; xk][ [xk; xk+1]; we can generalize (see details in [10]) the concept of a fuzzy partition by taking basic functions that cover more than two consecutive subintervals. Consider an integer r 1 and 2r + 1 consecutive points (and consequently 2r subintervals) of P, xk r; :::; xk; :::; xk+r for all k = 1; 2; :::; n; to complete the notation, we extend the points to x1 r < ::: < x0 < a and b < xn+1 < ::: < xn+r. De…nition 1: ([3], [10]) Let r 1 be a …xed integer number; a fuzzy r-partition of [a; b] is given by a pair (P;A) where P = fa = x1 < x2 < ::: < xn = bg is a decomposition of [a; b], and A is a family of n+ 2r 2 continuous, normal, convex fuzzy numbers A = fA k : [a; b] ! [0; 1]j k = r + 2; :::; n+ r 1g such that a. for k = 1; 2; :::; n, A k is a continuous fuzzy number with A k (xk) = 1 and A (r) k (x) = 0 for x = 2 [xk r; xk+r]; b. for k = 1; 2; :::; n, A k is increasing on [xk r; xk] and decreasing on [xk; xk+r]; c. for k = r + 2; :::; 0, A k is decreasing on [xk; xk+r]; d. for k = n+1; :::n+ r 1, A k is increasing on [xk r; xk]; e. for all x 2 [a; b], the following partition-of-r condition holds n+r 1 P k= r+2 A (r) k (x) = r. The integer r 1 will be called the bandwidth of the partition (P;A). A parametric form of a fuzzy r-partition of [a; b] is obtained by considering n+r 2 shape functions of type L k (x) k = 2; :::; n+ r 1; the basic functions are A (r) k (x) = ><>: L (r) k (x) if x 2 [xk r; xk] 1 L k+r(x) if x 2 [xk; xk+r] 0 otherwise , (1)