Variational Bayes for a Mixed Stochastic/Deterministic Fuzzy Filter

This study, under the variational Bayes (VB) framework, infers the parameters of a Takagi-Sugeno fuzzy filter having deterministic antecedents and stochastic consequents. The aim of this study is to take advantages of the VB framework to design fuzzy-filtering algorithms, which include an automated regularization, incorporation of statistical noise models, and model-comparison capability. The VB method can be easily applied to the linear-in-parameters models. This paper applies the VB method to the nonlinear fuzzy filters without using Taylor expansion for a linear approximation of some nonlinear function. It is assumed that the nonlinear parameters (i.e., antecedents) of the fuzzy filter are deterministic, while linear parameters are stochastic. The VB algorithm, by maximizing a strict lower bound on the data evidence, makes the approximate posterior of linear parameters as close to the true posterior as possible. The nonlinear deterministic parameters are tuned in a way to further increase the lower bound on data evidence. The VB paradigm can be used to design an algorithm that automatically selects the most-suitable fuzzy filter out of the considered finite set of fuzzy filters. This is done by fitting the observed data as a stochastic combination of the different Takagi-Sugeno fuzzy filters such that the individual filters compete with one another to model the data.