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Finite Time Identification in Unstable Linear Systems

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  • Mohamad Kazem Shirani Faradonbeh
  • Ambuj Tewari
  • George Michailidis

Abstract

Identification of the parameters of stable linear dynamical systems is a well-studied problem in the literature, both in the low and high-dimensional settings. However, there are hardly any results for the unstable case, especially regarding finite time bounds. For this setting, classical results on least-squares estimation of the dynamics parameters are not applicable and therefore new concepts and technical approaches need to be developed to address the issue. Unstable linear systems arise in key real applications in control theory, econometrics, and finance. This study establishes finite time bounds for the identification error of the least-squares estimates for a fairly large class of heavy-tailed noise distributions, and transition matrices of such systems. The results relate the time length (samples) required for estimation to a function of the problem dimension and key characteristics of the true underlying transition matrix and the noise distribution. To establish them, appropriate concentration inequalities for random matrices and for sequences of martingale differences are leveraged.

Suggested Citation

  • Mohamad Kazem Shirani Faradonbeh & Ambuj Tewari & George Michailidis, 2017. "Finite Time Identification in Unstable Linear Systems," Papers 1710.01852, arXiv.org, revised Jun 2018.
  • Handle: RePEc:arx:papers:1710.01852
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    1. Nielsen, Bent, 2005. "Strong Consistency Results For Least Squares Estimators In General Vector Autoregressions With Deterministic Terms," Econometric Theory, Cambridge University Press, vol. 21(03), pages 534-561, June.
    2. Nielsen, Bent, 2010. "Analysis Of Coexplosive Processes," Econometric Theory, Cambridge University Press, vol. 26(03), pages 882-915, June.
    3. Katarina Juselius & Zorica Mladenovic, 2002. "High Inflation, Hyperinflation and Explosive Roots: The Case of Yugoslavia," Discussion Papers 02-23, University of Copenhagen. Department of Economics.
    4. Lai, T. L. & Wei, C. Z., 1983. "Asymptotic properties of general autoregressive models and strong consistency of least-squares estimates of their parameters," Journal of Multivariate Analysis, Elsevier, vol. 13(1), pages 1-23, March.
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