A neurofuzzy network knowledge extraction and extended Gram-Schmidt algorithm for model subspace decomposition

This paper introduces a new neurofuzzy model construction and parameter estimation algorithm from observed finite data sets, based on a Takagi-Sugeno (T-S) inference mechanism and a new extended Gram-Schmidt orthogonal decomposition algorithm, for the modeling of a priori unknown dynamical systems in the form of a set of fuzzy rules. The paper introduces a one to one mapping between a fuzzy rule-base and a model matrix feature subspace. Hence, rule-based knowledge can be extracted to enhance model transparency. Model transparency is explored by the derivation of an equivalence between an A-optimality experimental design criterion of the weighting matrix and the average model output sensitivity to the fuzzy rule. The A-optimality experimental design criterion of the weighting matrices of fuzzy rules is used to construct an initial model rule-base. An extended Gram-Schmidt algorithm is then developed to estimate the parameter vector for each rule. This new algorithm decomposes the model rule-bases via an orthogonal subspace decomposition approach, so as to enhance model transparency with the capability of interpreting the derived rule-base energy level.