Non-asymptotic Confidence Regions for the Transfer Functions of Errors-in-Variables Systems

Finite-sample system identification (FSID) methods provide guaranteed confidence regions for the unknown model parameter of dynamical systems under
mild statistical assumptions
for
a finite number of data points
. In this article, two FSID methods, the leave-out sign-dominant correlation region (LSCR) and sign-perturbed sums (SPS) methods are extended to errors-in-variables (EIV) systems. In EIV systems, both the measured input and the measured output are corrupted by noise, and they are not noise invertible in the sense that the noise signals cannot be recovered from the measured signals given the true system. Hence, standard FSID methods are not applicable. The present article deals with FSID of EIV systems where the input and noise on input are i.i.d. Gaussian processes and the signal-to-noise ratio is known. By utilizing an alternative regression model and swapping the role of the input and the prediction error in the FSID methods, new LSCR and SPS confidence regions are constructed, which include the true model parameter with a guaranteed user-chosen probability. It is shown that the confidence regions are asymptotically included in an
$\epsilon$
-neighborhood of the true parameter. An ellipsoidal approximation which can be computed at low computational cost is proposed for the SPS confidence region. The methods and their properties are illustrated in numerical experiments.