Probabilistic-fuzzy, IF-THEN rules, Weighted quantiles, Inverse quantile, fuzzy transform, Quantile estimation, Quantile loss

We focus on the inference process in a probabilistic-fuzzy IF-THEN rule system, where antecedents are modeled as fuzzy sets and consequents as weighted quantile functions, offering insights into the probability distribution of response data. While the weighted arithmetic mean has been the conventional inference method, other suitable approaches exist. This work compares it with three alternatives: the weighted geometric mean, weighted $L_1$ minimization, and a mixture distribution-based method. Through experiments on both simulated and real-world data, we evaluate their performance using standard statistical measures.

We focus on the inference process in a probabilistic-fuzzy IF-THEN rule system, where antecedents are modeled as fuzzy sets and consequents as weighted quantile functions, offering insights into the probability distribution of response data. While the weighted arithmetic mean has been the conventional inference method, other suitable approaches exist. This work compares it with three alternatives: the weighted geometric mean, weighted $L_1$ minimization, and a mixture distribution-based method. Through experiments on both simulated and real-world data, we evaluate their performance using standard statistical measures.