Identification of wiener systems with process noise is a nonlinear errors-in-variables problem

This paper considers the identification of stochastic Wiener dynamic systems, that is linear dynamic systems with process noise, where the measurable output signal is a nonlinear function of the output from the linear system corrupted with additive measurement noise. It is shown how stochastic Wiener system identification can be viewed as a particular non-linear model errors-in-variables problem, for which there exists a large literature. We compare the maximum likelihood method with prediction error minimization methods based on the conditional mean predictor for Wiener systems. Related methods have previously been studied in the framework of identification of non-linear error-in-variables models. We extend these results by taking the input signal to the Wiener system into consideration. For example, the input will affect the variance of the prediction errors. Hence, a prediction error method with a variance weighting is derived to obtain more reliable parameter estimates. An advantage with the prediction error method is that for certain special cases we can avoid numerical integration. We also discuss how the unscented transform can be used to obtain an approximate predictor for the prediction error method. The numerical evaluation of these methods is performed on a simple first order FIR system with a cubic nonlinearity, for which some illustrative analytic properties are derived.