Fuzzy-Probabilistic Inference System; Quantile Fuzzy Transform; Weighted Quantile Regression; Uniform Fuzzy Partition;Data-Driven Fuzzy Partition

This paper focuses on fuzzy--probabilistic IF--THEN rule-based systems, where antecedents encode fuzzy information and consequents represent probability distributions of the output variable. By combining both types of uncertainty within a unified framework, this approach is effective for time series analysis and forecasting.Given a fuzzy covering of the input universe and an output random variable defined on a probability space, the rules state that if the input belongs to a given fuzzy set, then the output is described by a corresponding quantile function. In practice, uniform or generalized fuzzy partitions are typically constructed by shifting equidistant fuzzy sets along the domain axis. The consequent quantile functions are estimated from data as weighted quantiles, where the weights are given by the membership degrees of input values. These weighted quantiles are obtained by minimizing an asymmetric absolute loss functional. The inference mechanism then evaluates the output quantile at a given input as a normalized weighted average of the rule-wise quantile functions.Although fuzzy--probabilistic inference systems have demonstrated effectiveness in various applications, the construction of an appropriate fuzzy partition remains challenging. Uniform partitions are simple but fail to capture complex structures hidden in the data. This motivates the question of whether a data-driven fuzzy partition can better reflect local behaviour under a well-defined criterion. In this paper, we introduce three algorithmic methods for designing non-uniform, data-dependent fuzzy partitions, while a detailed theoretical analysis is left for future work.

This paper focuses on fuzzy--probabilistic IF--THEN rule-based systems, where antecedents encode fuzzy information and consequents represent probability distributions of the output variable. By combining both types of uncertainty within a unified framework, this approach is effective for time series analysis and forecasting.Given a fuzzy covering of the input universe and an output random variable defined on a probability space, the rules state that if the input belongs to a given fuzzy set, then the output is described by a corresponding quantile function. In practice, uniform or generalized fuzzy partitions are typically constructed by shifting equidistant fuzzy sets along the domain axis. The consequent quantile functions are estimated from data as weighted quantiles, where the weights are given by the membership degrees of input values. These weighted quantiles are obtained by minimizing an asymmetric absolute loss functional. The inference mechanism then evaluates the output quantile at a given input as a normalized weighted average of the rule-wise quantile functions.Although fuzzy--probabilistic inference systems have demonstrated effectiveness in various applications, the construction of an appropriate fuzzy partition remains challenging. Uniform partitions are simple but fail to capture complex structures hidden in the data. This motivates the question of whether a data-driven fuzzy partition can better reflect local behaviour under a well-defined criterion. In this paper, we introduce three algorithmic methods for designing non-uniform, data-dependent fuzzy partitions, while a detailed theoretical analysis is left for future work.