Sign-Perturbed Sums (SPS) with Instrumental Variables for the Identification of ARX Systems - Extended Version

TL;DR

This paper extends the Sign-Perturbed Sums (SPS) method to include instrumental variables, enabling exact finite-sample confidence regions for ARX system identification, applicable to feedback systems, with proven consistency and efficient computation.

Contribution

It introduces a generalized SPS approach using instrumental variables for non-asymptotic confidence regions in ARX system identification, including feedback systems, with theoretical guarantees and practical algorithms.

Findings

Provides exact confidence regions with finite samples

Demonstrates strong consistency of the method

Shows improved computational efficiency in outer-approximation

Abstract

We propose a generalization of the recently developed system identification method called Sign-Perturbed Sums (SPS). The proposed construction is based on the instrumental variables estimate and, unlike the original SPS, it can construct non-asymptotic confidence regions for linear regression models where the regressors contain past values of the output. Hence, it is applicable to ARX systems, as well as systems with feedback. We show that this approach provides regions with exact confidence under weak assumptions, i.e., the true parameter is included in the regions with a (user-chosen) exact probability for any finite sample. The paper also proves the strong consistency of the method and proposes a computationally efficient generalization of the previously proposed ellipsoidal outer-approximation. Finally, the new method is demonstrated through numerical experiments, using both real-world and simulated data.