| 1. |
- Djehiche, Boualem, 1962-, et al.
(författare)
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Non asymptotic estimation lower bounds forLTI state space models with Cramér-Rao and van Trees
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We study the estimation problem for linear time-invariant (LTI) state-space models with Gaussian excitation of an unknown covariance. We provide non asymptotic lower bounds for the expected estimation error and the mean square estimation risk of the least square estimator, and the minimax mean square estimation risk. These bounds are sharp with explicit constants when the matrix of the dynamics has no eigenvalues on the unit circle and are rate-optimal when they do. Our results extend and improve existing lower bounds to lower bounds in expectation of the mean square estimation risk and to systems with a general noise covariance. Instrumental to our derivation are new concentration results for rescaled sample covariances and deviation results for the corresponding multiplication processes of the covariates, a differential geometric construction of a prior on the unit operator ball of small Fisher information, and an extension of the Cramér-Rao and van Treesinequalities to matrix-valued estimators.
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| 2. |
- Mazhar, Othmane, Ph.D. student, 1990-
(författare)
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Data driven modeling in the presence of time series structure: : Improved bounds and effective algorithms
- 2022
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of five appended papers devoted to modeling tasks where the desired models are learned from data sets with an underlying time series structure. We develop a statistical methodology for providing efficient estimators and analyzing their non-asymptotic behavior. We further suggest novel algorithmic design techniques for obtaining practical procedures to compute these estimators. Specifically, we study time series models of increasing levels of difficulty. In the first paper, we study change point and clustering systems where the dynamic structure of the time series is entirely encoded in the combinatorial properties of the estimated parameters. We then investigate in the second paper Finite Input Response (FIR) models, which exhibit a time-shifted random design. The obtained results are then generalized in the third paper to linear Hidden Markov models since they are infinite impulse response models with a particular polynomial structure. Finally, in the fourth and fifth papers, we investigate linear time-invariant (LTI) state-space models where the covariates generated along the path of the system are not just dependent but also dependent on the estimated parameter. Hence, the spectral properties of this estimated parameter affect the estimation performance. Throughout this journey, we develop a statistical methodology for deriving statistically efficient estimators. This statistical methodology relays on the idea that efficient estimators should strike a compromise between a signal term and a noise term. The signal term is intimately related to the spectral properties of the design matrix, and the noise term is intimately associated with the covariates multiplication process. To quantify both of these terms and obtain upper bounds for the estimation errors, we develop new concentration and deviation inequalities based on chaining integrals and self-normalized martingale inequalities. We also obtain lower bounds for the estimation errors by extending the Cramér-Rao inequality to a biased estimator and alow-rank Fisher information and provide an information geometric construction of carefully chosen priors on sets of matrices to obtain a van Tree inequality describing the minimax rate for the estimation problem. Finally, on the algorithmic side, we design efficient estimation procedures based on dynamic programming, penalized least squares, and the Ho-Kalman algorithm to take into account the data’s time series structure.
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| 3. |
- Shi, Shengling, et al.
(författare)
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Finite-Sample Analysis of Identification of Switched Linear Systems With Arbitrary or Restricted Switching
- 2022
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Ingår i: IEEE Control Systems Letters. - : IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC. - 2475-1456. ; 7, s. 121-126
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Tidskriftsartikel (refereegranskat)abstract
- For the identification of switched systems with measured states and a measured switching signal, this letter aims to analyze the effect of switching strategies on the estimation error. The data is assumed to be collected from globally asymptotically or marginally stable switched systems under switches that are arbitrary or subject to an average dwell time constraint. Then the switched system is estimated by the least-squares (LS) estimator. To capture the effect of the parameters of the switching strategies on the LS estimation error, finite-sample error bounds are developed in this letter. The obtained error bounds show that the estimation error is logarithmic of the switching parameters when there are only stable modes; however, when there are unstable modes, the estimation error bound can increase linearly as the switching parameter changes. This suggests that in the presence of unstable modes, the switching strategy should be properly designed to avoid the significant increase of the estimation error.
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