Quantile and Expectile Smoothing based on L1-norm and L2-norm F-transforms

The fuzzy transform (F-transform), introduced by I. Perfilieva, is a powerful tool for the construction of fuzzy approximation models; it is based on generalized fuzzy partitions and it is obtained by minimizing a quadratic (L₂-norm) functional. In this paper we describe an analogous construction by minimizing an L₁-norm functional, so obtaining the L₁-norm F-transform, which is again a general approximation tool.

The L₁-norm and L₂-norm settings are then used to construct two types of fuzzy-valued of F-transforms, by defining expectile (L₂-norm) and quantile (L₁-norm) extensions of the transforms. This allows to model an observed time series in terms of fuzzy-valued functions, whose level-cuts can be interpreted in the setting of expectile and quantile regression. The proposed methodology is illustrated on some financial daily time series.