Regret Analysis with Almost Sure Convergence for OBF-ARX Filter
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929492193181696 |
|---|---|
| author | Li, Jiayun Lu, Yiwen Mo, Yilin |
| author_facet | Li, Jiayun Lu, Yiwen Mo, Yilin |
| contents | This paper considers the output prediction problem for an unknown Linear Time-Invariant (LTI) system. In particular, we focus our attention on the OBF-ARX filter, whose transfer function is a linear combination of Orthogonal Basis Functions (OBFs), with the coefficients determined by solving a least-squares regression. We prove that the OBF-ARX filter is an accurate approximation of the Kalman Filter (KF) by quantifying its online performance. Specifically, we analyze the average regret between the OBF-ARX filter and the KF, proving that the average regret over $N$ time steps converges to the asymptotic bias at the speed of $O(N^{-0.5+ε})$ almost surely for all $ε>0$. Then, we establish an upper bound on the asymptotic bias, demonstrating that it decreases exponentially with the number of OBF bases, and the decreasing rate $τ(\boldsymbolλ, \boldsymbolμ)$ explicitly depends on the poles of both the KF and the OBF. Numerical results on diffusion processes validate the derived bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05390 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regret Analysis with Almost Sure Convergence for OBF-ARX Filter Li, Jiayun Lu, Yiwen Mo, Yilin Optimization and Control This paper considers the output prediction problem for an unknown Linear Time-Invariant (LTI) system. In particular, we focus our attention on the OBF-ARX filter, whose transfer function is a linear combination of Orthogonal Basis Functions (OBFs), with the coefficients determined by solving a least-squares regression. We prove that the OBF-ARX filter is an accurate approximation of the Kalman Filter (KF) by quantifying its online performance. Specifically, we analyze the average regret between the OBF-ARX filter and the KF, proving that the average regret over $N$ time steps converges to the asymptotic bias at the speed of $O(N^{-0.5+ε})$ almost surely for all $ε>0$. Then, we establish an upper bound on the asymptotic bias, demonstrating that it decreases exponentially with the number of OBF bases, and the decreasing rate $τ(\boldsymbolλ, \boldsymbolμ)$ explicitly depends on the poles of both the KF and the OBF. Numerical results on diffusion processes validate the derived bounds. |
| title | Regret Analysis with Almost Sure Convergence for OBF-ARX Filter |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2409.05390 |