Input Importance Analysis for a Class of Incremental Hierarchical Fuzzy Systems—Theory and Experiments

Hierarchical fuzzy systems (HFSs), as a type of deep-structured fuzzy model, exhibit strong expressive power, excellent scalability, high flexibility, and desirable interpretability, making them highly promising for tackling a variety of complex problems. Despite their widespread applications in various fields such as control, classification, and modeling, very little research has been conducted to analyze how inputs influence the system's behavior. Consequently, this paper presents an in-depth analysis of a class of incremental hierarchical fuzzy systems (IHFSs) from two perspectives of input importance. First, an enhanced mutual information-based input importance ranking method (EMI-IIR) is proposed, enabling a more effective and concise evaluation of the degree to which inputs influence or depend on the ground truth. Next, a comprehensive sensitivity analysis, both at the subsystem level and the entire system level, is performed to illustrate the impact of inputs on the system. Specifically, a novel sensitivity metric, termed average sensitivity (AS), is proposed to simplify sensitivity calculations, thereby significantly reducing computational costs. Several related theorems are provided to theoretically analyze the sensitivity of the system to inputs under specific conditions. Furthermore, an evaluation framework based on the AS, called ASEF, is proposed for assessing the rationality of the constructed IHFS structure, which facilitates the effective design of high-performance systems for modeling tasks. A series of experiments on both synthetic and real-world datasets demonstrate the effectiveness and superiority of the proposed methods. Finally, several potential research directions are suggested to drive further progress.