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

arXiv: https://arxiv.org/abs/1509.04774 · HTML: https://mlsanity.com/paper/1509.04774/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Sep 2015.

## Built on by (most cited first, 5 of 5)

- [Sample Complexity of the Sign-Perturbed Sums Identification Method: Scalar Case](https://mlsanity.com/paper/2401.15792/) (2024)
- [Finite-Sample Identification of Linear Regression Models with Residual-Permuted Sums](https://mlsanity.com/paper/2406.05440/) (2024)
- [Sample Complexity of the Sign-Perturbed Sums Method](https://mlsanity.com/paper/2409.01243/) (2024)
- [Robust Independence Tests with Finite Sample Guarantees for Synchronous Stochastic Linear Systems](https://mlsanity.com/paper/2308.02054/) (2023)
- [Finite Sample Analysis of Distribution-Free Confidence Ellipsoids for Linear Regression](https://mlsanity.com/paper/2409.08801/) (2024)

## Abstract (arXiv)

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.