Automatica (Journal of IFAC)
Sufficient conditions on general fuzzy systems as function approximators
Automatica (Journal of IFAC)
Universal approximation by hierarchical fuzzy system with constraints on the fuzzy rule
Fuzzy Sets and Systems - Fuzzy models
Approximation accuracy analysis of fuzzy systems as function approximators
IEEE Transactions on Fuzzy Systems
Decomposition property of fuzzy systems and its applications
IEEE Transactions on Fuzzy Systems
Fuzzy clustering for symbolic data
IEEE Transactions on Fuzzy Systems
Analysis and design of hierarchical fuzzy systems
IEEE Transactions on Fuzzy Systems
On multistage fuzzy neural network modeling
IEEE Transactions on Fuzzy Systems
A class of hierarchical fuzzy systems with constraints on the fuzzy rules
IEEE Transactions on Fuzzy Systems
Approximation Capabilities of Hierarchical Fuzzy Systems
IEEE Transactions on Fuzzy Systems
A Survey on Analysis and Design of Model-Based Fuzzy Control Systems
IEEE Transactions on Fuzzy Systems
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When applying gradient descent learning methods to hierarchical fuzzy systems, there is great difficulty in handling the intermediate variables introduced by the hierarchical structures, as the intermediate variables may go outside their definition domain that makes gradient descent learning invalid. To overcome this difficulty, this paper proposes a learning scheme that integrates a normalization process for intermediate variables into gradient descent learning. This ensures that gradient descent methods are applicable to, and correctly used for, learning general hierarchical fuzzy systems. Benchmark datasets are used to demonstrate the validity and advantages of the proposed learning scheme over other existing methods in terms of better accuracy, better transparency, and fewer fuzzy rules and parameters.