Quantification of the Variance of Estimated Transfer Functions in the Presence of Undermodeling

Abstract

We study the effect of undermodeling on the parameter variance for prediction error time-domain identification with linear model structures. We restrict our consideration to linear time-invariant discrete time single input single output systems. We examine the asymptotic expression for the variance as the number of data tends to infinity. This quantity is known to depend in general on the fourth order statistical properties of the applied input. However, we establish a sufficient condition under which the asymptotic variance is a function of the input power spectrum only. For this case we deliver exact expressions. We show that for a stochastic input the undermodeling contributes to the parameter variance due to the correlation between the prediction errors and its gradients. As an additional contribution we investigate the parameter variance under the assumptions of the stochastic embedding procedure. We show by means of a counterexample that in the framework of stochastic embedding the parameter variance is not necessarily monotonous with respect to the input power spectrum.