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Rules in fuzzy rule-based models convey essential knowledge about the system under discussion. As such, they capture the essence of relationships occurring among input and output variables. While the quality of such fuzzy models is predominantly related with the accuracy and eventual interpretability of rules (although to a limited extent), the quality of rules being regarded as generic pieces of knowledge has not been studied so far. In this study, we formulate and investigate this problem by looking at the quality of rules, including aspects of (a) stability, (b) generalizability, and (c) conflict. We identify a concept of rule multiplicity, conflict, and study an emergence of rule generalization. A number of pertinent performance indices are developed, and their usage is presented through a series of experimental studies. [ABSTRACT FROM AUTHOR]

This study focuses on two critical design aspects of a progressive fuzzy polynomial neural network (PFPNN): the influence of the gated recurrent unit (GRU) structure and the implementation of fitness-based candidate neuron selection (FCNS) through two probabilistic strategies. The primary objectives are to enhance modeling accuracy and to reduce the computational load associated with nonlinear regression tasks. Compared with the existing fuzzy rule-based modeling architecture, the proposed dynamic model consists of the GRU structure and the hybrid fuzzy polynomial architecture. In the initial two layers of the PFPNN, we introduce three types of polynomial and fuzzy rules into the GRU neurons (GNs) and fuzzy polynomial neurons (FPNs), which can effectively reveal potential complex relationships in the data space. The synergy of the FCNS strategies and the ℓ 2 regularization learning method is to design a progressive regression model adept at melding the GRU structure with a self-organizing architecture. The proposed GRU structure and polynomial-based neurons significantly improve the modeling accuracy for time-series datasets. The rational utilization of FCNS strategies can reinforce the network structure and discover the potential performance of neurons of the network. Furthermore, the inclusion of ℓ 2 norm regularization provides additional stability to the proposed model and mitigates the overfitting issue commonly encountered in many existing learning methods. We validated the proposed neural networks using six time-series, four machine learning, and two real-world datasets. The PFPNN outperformed other models in the comparison studies in 83.3% of the datasets, emphasizing its superiority in terms of developing a stable deep structure from diverse candidate neurons and reducing computational overhead. [ABSTRACT FROM AUTHOR]