Note on orthogonal transformation methods for simplifying fuzzy rule-based models

Simplifying fuzzy rule based models is considered. For the rule selection process, both input space and the output space are considered, where both spaces play important role. The method used is principal component analysis complemented with the OLS method for the selection of the rules. In this selection process, first, the regression matrix of a common radial basis function (RBF) network is considered. The magnitude of the eigenvalues of the RBF matrix of radial basis function network is not central to the selection. The selection is made according to the energy contribution from the graded principal components, where eigenvectors with low eigenvalues may have relatively more energy contribution to the output depending on the model outputs. In parallel to the above gradation process, the influential basis functions are identified, as they are associated with the graded principal components of higher ranks in this very gradation. This approach is extended to normalized RBF matrix for fuzzy systems with singular value decomposition. The comparative results are presented and the implication and importance of the approach is pointed out.