Algorithms and Performance Analysis for Stochastic Wiener System Identification

TL;DR

This paper derives the Cramér-Rao lower bound and analyzes the statistical performance of system identification methods for stochastic Wiener systems with non-linear sensors, highlighting the impact of non-linearity on accuracy.

Contribution

It provides the first analytical expressions for the CRLB in non-linear Wiener system identification and proposes a ML-inspired method based on Gaussian approximations.

Findings

CRLB expressions quantify the impact of non-linearity on estimation accuracy

Gaussian approximation simplifies analysis and guides method development

Numerical simulations validate theoretical results

Abstract

We analyze the statistical performance of identification of stochastic dynamical systems with non-linear measurement sensors. This includes stochastic Wiener systems, with linear dynamics, process noise and measured by a non-linear sensor with additive measurement noise. There are many possible system identification methods for such systems, including the Maximum Likelihood (ML) method and the Prediction Error Method (PEM). The focus has mostly been on algorithms and implementation, and less is known about the statistical performance and the corresponding Cram\'er-Rao Lower Bound (CRLB) for identification of such non-linear systems. We derive expressions for the CRLB and the asymptotic normalized covariance matrix for certain Gaussian approximations of Wiener systems to show how a non-linear sensor affects the accuracy compared to a corresponding linear sensor. The key idea is to take second order statistics into account by using a common parametrization of the mean and the variance of the output process. This analysis also leads to a ML motivated identification method based on the conditional mean predictor and a Gaussian distribution approximation. The analysis is supported by numerical simulations