Reduction of fuzzy rule base via singular value decomposition

This paper introduces a singular value-based method for reducing a given fuzzy rule set. The method conducts singular value decomposition of the rule consequents and generates certain linear combinations of the original membership functions to form new ones for the reduced set. The present work characterizes membership functions by the conditions of sum normalization (SN), nonnegativeness (NN), and normality (NO). Algorithms to preserve the SN and 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 closed-to-NO solution may be sought. The proposed method is applicable regardless of the adopted inference paradigms. With product-sum-gravity inference and singleton support fuzzy rule base, output errors between the full and reduced fuzzy set are bounded by the sum of the discarded singular values. The present work discusses three specific applications of fuzzy reduction: fuzzy rule base with singleton support, fuzzy rule base with nonsingleton support (which includes the case of missing rules), and the Takagi–Sugeno–Kang (TSK) model. Numerical examples are presented to illustrate the reduction process.