Reduction of fuzzy rule base via singular value decomposition

This paper introduces a singular value-based method for reducing a given fu zzy rule set. The method conducts singular value decomposition of the rule consequents and generates certain linear combinations of the original membe rship functions to form new ones for the reduced set. The present work char acterizes membership functions by the conditions of sum normalization (SN), nonnegativeness (NN), and normality (NO). Algorithms to preserve the SN an d NN conditions in the new membership functions are presented. Preservation of the NO condition relates to a high-dimensional convex hull problem and is not always feasible in which case a dosed-to-NO solution may be sought. The proposed method is applicable regardless of the adopted inference parad igms. With product-sum-gravity inference and singleton support fuzzy rule b ase, output errors between the full and reduced fuzzy set are bounded by th e sum of the discarded singular values. The present work discusses three sp ecific applications of fuzzy reduction: fuzzy rule base with singleton supp ort, fuzzy rule base dth nonsingleton support (which includes the case of m issing rules), and the Takagi-Sugeno-Kang (TSK) model. Numerical examples a re presented to illustrate the reduction process.