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

Rule-based systems, particularly those using "IF antecedents THEN consequent" rules, play a crucial role in mathematical modeling and decision-making. This work focuses on a specific type of rule-based system: the Probabilistic-Fuzzy Inference System. Here, antecedents are represented as fuzzy sets, while consequents are modeled as probability distributions using quantile functions. The inference process relies on the $L_{1}$-Fuzzy transform, also known as the Quantile Fuzzy transform. For each fuzzy antecedent, a weighted quantile of order $p$ is computed, and the inverse quantile transform derives the empirical quantile function for any input value. Initially, weighted quantiles are modeled as scalar values. To better capture dependencies between input and output data, we extend this model to a piecewise linear function, developed in a structured manner with a comprehensive computational algorithm. Its effectiveness is demonstrated through an illustrative example.

Rule-based systems, particularly those using "IF antecedents THEN consequent" rules, play a crucial role in mathematical modeling and decision-making. This work focuses on a specific type of rule-based system: the Probabilistic-Fuzzy Inference System. Here, antecedents are represented as fuzzy sets, while consequents are modeled as probability distributions using quantile functions. The inference process relies on the $L_{1}$-Fuzzy transform, also known as the Quantile Fuzzy transform. For each fuzzy antecedent, a weighted quantile of order $p$ is computed, and the inverse quantile transform derives the empirical quantile function for any input value. Initially, weighted quantiles are modeled as scalar values. To better capture dependencies between input and output data, we extend this model to a piecewise linear function, developed in a structured manner with a comprehensive computational algorithm. Its effectiveness is demonstrated through an illustrative example.