Fuzzy-Probabilistic inference system; Quantile fuzzy transform; Weighted quantile regression; Piecewise linear weighted quantiles

In this work, we consider a particular construction of IF--THEN rules and the associated inference mechanism, which coincide with the so-called quantile fuzzy transform (or $L_{1}$-fuzzy transform). Given a suitable fuzzy partition of the underlying universe and a random variable defined on a probability space, the system is formulated through rules stating that if the input belongs to the $k$-th fuzzy set, then the output is modeled by a corresponding quantile function.The consequent is represented by weighted quantile functions that provide statistical estimates of the output distribution conditioned on the input's membership in the respective fuzzy set. A crucial step in the inference process is the estimation of these quantile functions from data. Traditionally, weighted quantiles are computed via linear programming. We have recently introduced an alternative and computationally efficient method for evaluating weighted quantiles based on the analysis of the right derivative of the associated convex objective function.Although classical weighted quantiles are computationally efficient, they may be inadequate for accurately capturing the local positions of output quantiles over fuzzy inputs. To overcome this limitation, we have extended the weighted quantile approach into a piecewise linear functional form. In this contribution, we propose a slight modification of this construction to enhance its applicability to forecasting tasks. We describe the modified approach, demonstrate its improved inference performance compared to scalar weighted quantiles, and highlight its relevance for forecasting applications.

In this work, we consider a particular construction of IF--THEN rules and the associated inference mechanism, which coincide with the so-called quantile fuzzy transform (or $L_{1}$-fuzzy transform). Given a suitable fuzzy partition of the underlying universe and a random variable defined on a probability space, the system is formulated through rules stating that if the input belongs to the $k$-th fuzzy set, then the output is modeled by a corresponding quantile function.The consequent is represented by weighted quantile functions that provide statistical estimates of the output distribution conditioned on the input's membership in the respective fuzzy set. A crucial step in the inference process is the estimation of these quantile functions from data. Traditionally, weighted quantiles are computed via linear programming. We have recently introduced an alternative and computationally efficient method for evaluating weighted quantiles based on the analysis of the right derivative of the associated convex objective function.Although classical weighted quantiles are computationally efficient, they may be inadequate for accurately capturing the local positions of output quantiles over fuzzy inputs. To overcome this limitation, we have extended the weighted quantile approach into a piecewise linear functional form. In this contribution, we propose a slight modification of this construction to enhance its applicability to forecasting tasks. We describe the modified approach, demonstrate its improved inference performance compared to scalar weighted quantiles, and highlight its relevance for forecasting applications.