Quantile and expectile smoothing based on L1-norm and L2-norm fuzzy 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) error function. In this paper, within the discrete setting, we describe an analogous construction by minimizing an L1-norm error function, 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 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.