Intermediate variable normalization for gradient descent learning for hierarchical fuzzy system

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.