Bayesian zero-order TSK fuzzy system modeling

Abstract

Different from the existing TSK fuzzy system modeling methods, a novel zero-order TSK fuzzy modeling method called Bayesian zero-order TSK fuzzy system (B-ZTSK-FS) is proposed from the perspective of Bayesian inference in this paper. The proposed method B-ZTSK-FS constructs zero-order TSK fuzzy system by using the maximum a posteriori (MAP) framework to maximize the corresponding posteriori probability. First, a joint likelihood model about zero-order TSK fuzzy system is defined to derive a new objective function which can assure that both antecedents and consequents of fuzzy rules rather than only their antecedents of the most existing TSK fuzzy systems become interpretable. The defined likelihood model is composed of three aspects: clustering on the training set for antecedents of fuzzy rules, the least squares (LS) error for consequent parameters of fuzzy rules, and a Dirichlet prior distribution for fuzzy cluster memberships which is considered to not only automatically match the “sum-to-one” constraints on fuzzy cluster memberships, but also make the proposed method B-ZTSK-FS scalable for large-scale datasets by appropriately setting the Dirichlet index. This likelihood model indeed indicates that antecedent and consequent parameters of fuzzy rules can be linguistically interpreted and simultaneously optimized by the proposed method B-ZTSK-FS which is based on the MAP framework with the iterative sampling algorithm, which in fact implies that fuzziness and probability can co-jointly work for TSK fuzzy system modeling in a collaborative rather than repulsive way. Finally, experimental results on 28 synthetic and real-world datasets are reported to demonstrate the effectiveness of the proposed method B-ZTSK-FS in the sense of approximation accuracy, interpretability and scalability.