N A R X Models: Optimal Parametric Approximation of Nonparametric Estimators

we have that Bayesian regression, a nonparametric identification technique with several appealing features, can be applied to the identification of NARX (nonlinear ARX) models. However, its computational complexity scales as O(N 3) where N is the data set size. In order to reduce complexity, the challenge is to obtain fixed-order parametric models capable of approximating accurately the nonparametric Bayes estimate avoiding its explicit computation. In this work we derive, optimal finitedimensional approximations of complexity O(N 2) focusing on their use in the parametric identification of NARX models. K e y w o r d s : NARX models, nonparametric identification, parametric identification, Bayesian estimation, neural networks, Gaussian processes. 1 I n t r o d u c t i o n In recent years a large stream of research focused on the black-box identification of nonlinear systems via NARX (Nonlinear Auto Regressive eXogenous) models of the type Yi = f ( Y i 1 , Y i 2 , . . . , Y i -ny , u i 1 , . . . , Ui-n~,) ~6i, (1) where i = 1 ,2 , . . . N, ui and yi denote the scalar input and output, respectively, the integers ny, nu are the maximum lags of past outputs and inputs entering the model, and the additive measurement errors ei are uncorrelated zero-mean Gaussian random variables with Var[ci] 0 .2. The success of NARX models is due both to their capability of capturing nonlinear dynamics and the availability of identification algorithms with a reasonable computational cost [2], [4]. Concerning the generality of such models, in [14] it is shown, under mild assumptions, that any finite-dimensional nonlinear system admits an input-output NARX representation, at least locally.