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On convergence of covariance matrix of empirical Bayes hyper-parameter estimator

Control Theory and Technology 2024年第2期22卷 149-162页

作者： Yue Ju Biqiang Mu Tianshi Chen School of Data Science and Shenzhen Research Institute of Big Data The Chinese University of Hong KongShenzhenShenzhen518172GuangdongChina Division of Decision and Control Systems School of Electrical Engineering and Computer ScienceKTH Royal Institute of Technology10044StockholmSweden Key Laboratory of Systems and Control Institute of Systems ScienceAcademy of Mathematics and System ScienceChinese Academy of SciencesBeijing100190China

Regularized system identification has become the research frontier of system identification in the past *** related core subject is to study the convergence properties of various hyper-parameter estimators as the sample size goes to *** this paper,we consider one commonly used hyper-parameter estimator,the empirical Bayes(EB).Its convergence in distribution has been studied,and the explicit expression of the covariance matrix of its limiting distribution has been ***,what we are truly interested in are factors contained in the covariance matrix of the EB hyper-parameter estimator,and then,the convergence of its covariance matrix to that of its limiting distribution is *** general,the convergence in distribution of a sequence of random variables does not necessarily guarantee the convergence of its covariance ***,the derivation of such convergence is a necessary complement to our theoretical analysis about factors that influence the convergence properties of the EB hyper-parameter *** this paper,we consider the regularized finite impulse response(FIR)model estimation with deterministic inputs,and show that the covariance matrix of the EB hyper-parameter estimator converges to that of its limiting ***,we run numerical simulations to demonstrate the efficacy of ourtheoretical results.

关键词： Regularized system identification hyper-parameter estimator Empirical Bayes Convergence of covariance matrix

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On Convergence in Distribution of Stein's Unbiased Risk hyper-parameter estimator for Regularized System Identification 41

On Convergence in Distribution of Stein's Unbiased Risk Hype...

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第41届中国控制会议

作者： Yue Ju Tianshi Chen Biqiang Mu Lennart Ljung School of Data Science and Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Key Laboratory of Systems and Control Institute of Systems ScienceAcademy of Mathematics and System ScienceChinese Academy of Sciences Division of Automatic Control Department of Electrical EngineeringLink?ping University

ISBN: (数字)9789887581536

ISBN: (纸本)9781665482561

Asymptotic theory for the regularized system identification has received increasing interests in recent *** this paper,for the finite impulse response(FIR) model and filtered white noise inputs,we show the convergence in distribution of the Stein's unbiased risk estimator(SURE) based hyper-parameter estimator and find factors that influence its convergence *** particular,we consider the ridge regression case to obtain closed-form expressions of the limit of the regression matrix and the variance of the limiting distribution of the SURE based hyper-parameter estimator,and then demonstrate their relation numerically.

关键词： Asymptotic theory Stein's unbiased risk estimator hyper-parameter estimator Regularized system identification

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On Asymptotic Distribution of Generalized Cross Validation hyper-parameter estimator for Regularized System Identification 60

On Asymptotic Distribution of Generalized Cross Validation H...

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60th IEEE Conference on Decision and Control (CDC)

作者： Ju, Yue Chen, Tianshi Mu, Biqiang Ljung, Lennart Chinese Univ Hong Kong Sch Data Sci Shenzhen 518172 Peoples R China Chinese Univ Hong Kong Shenzhen Res Inst Big Data Shenzhen 518172 Peoples R China Chinese Acad Sci Acad Math & Syst Sci Inst Syst Sci Key Lab Syst & Control Beijing 100190 Peoples R China Linkoping Univ Dept Elect Engn Div Automat Control SE-58183 Linkoping Sweden

ISBN: (纸本)9781665436595

Asymptotic theory is one of the core subjects in system identification theory and often used to assess properties of model estimators. In this paper, we focus on the asymptotic theory for the kernel-based regularized system identification and study the convergence in distribution of the generalized cross validation (GCV) based hyper-parameter estimator. It is shown that the difference between the GCV based hyper-parameter estimator and the optimal hyper-parameter estimator that minimizes the mean square error scaled by 1/root N converges in distribution to a zero mean Gaussian distribution, where N is the sample size and an expression of covariance matrix is obtained. In particular, for the ridge regression case, a closed-form expression of the variance is obtained and shows the influence of the limit of the regression matrix on the asymptotic distribution. For illustration, Monte Carlo numerical simulations are run to test our theoretical results.

关键词： Regularized system identification Asymptotic distribution Generalized Cross Validation hyper-parameter estimator

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A Family of hyper-parameter estimators for Regularized Linear System Identification 62

A Family of Hyper-parameter Estimators for Regularized Linea...

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62nd IEEE Conference on Decision and Control (CDC)

作者： Zhang, Meng Chen, Tianshi Mu, Biqiang Chinese Acad Sci Acad Math & Syst Sci Inst Syst Sci Key Lab Syst & Control CAS Beijing 100190 Peoples R China Chinese Univ Hong Kong Sch Data Sci Shenzhen 518172 Peoples R China Chinese Univ Hong Kong Shenzhen Res Inst Big Data Shenzhen 518172 Peoples R China

ISBN: (纸本)9798350301243

hyper-parameter estimation is one of the fundamental issues for kernel-based regularized system identification methods. Empirical Bayes (EB) estimator and Stein's unbiased risk estimator (SURE) are two popular hyper-parameter estimators, but they both have advantages and disadvantages. Specifically, EB is not asymptotically optimal in the mean squared error (MSE) sense but SURE is, while SURE is more sensitive to ill-conditioned regression matrix but EB is more robust. In this paper, to find a better estimator by combining their strength and mitigating their weakness, we propose a family of hyper-parameter estimators by linking EB and SURE estimators together through an index. The finite sample and asymptotic properties of this family of estimators have been established. The Monte Carlo simulation results show that there does exist a 'middle' hyper-parameter estimator in this family that is superior to the EB and SURE.

关键词： Linear system identification Kernel-based regularization method hyper-parameter estimator empirical Bayes estimator Stein's unbiased risk estimator

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