Stochastic Fuzzy Systems
Variational Bayes for a Mixed Stochastic/Deterministic Fuzzy FilterThis study, under variational Bayes (VB) framework, infers the parameters of a TakagiSugeno fuzzy filter having deterministic antecedents and stochastic consequents. The motivation of the study is to take advantages of the VB framework in designing fuzzy filtering algorithms. These advantages include an automated regularization, incorporation of statistical noise models, and model comparison capability. The VB method can be easily applied to the linearinparameters models. This work applies 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 TakagiSugeno fuzzy filters such that the individual filters compete with one another to model the data.

Stationary Fuzzy FokkerPlanck Learning and Stochastic Fuzzy FilteringThe application of nonlinear optimization to the estimation of fuzzy model parameters is well known. For doing reverse of this, the concept of SFFPL (stationary fuzzy FokkerPlanck learning) is introduced i.e. SFFPL applies the fuzzy modeling technique in nonlinear optimization problems. SFFPL is based on the fuzzy approximation of stationary probability density of a stochastic search process associated to the nonlinear optimization problem. A carefully designed algorithm is suggested for SFFPL to locate the optimum point.
We consider the variational Bayes (VB) based inference of a stochastic fuzzy filter whose consequents as well as the antecedents are random variables. The problem of VB inference of stochastic antecedents, due to the nonlinearity of likelihood function, is analytically intractable. The SFFPL algorithm for high dimensional nonlinear optimization, that doesn't require the derivative of the objective function, can be used for numerically solving the stochastic fuzzy filtering problem. 
Related Publications
 M. Kumar, N. Stoll, and R. Stoll, “Variational Bayes for a Mixed Stochastic/Deterministic Fuzzy Filter,” IEEE Transactions on Fuzzy Systems, vol. 18, no. 4, pp. 787801, 2010.
 M. Kumar, N. Stoll, and R. Stoll, “Stationary Fuzzy FokkerPlanck Learning and Stochastic Fuzzy Filtering,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 5, pp. 873889, 2011.