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| On convergence of covariancematrix of empirical Bayes hyper-parameter estimator |
| YueJu1,2,BiqiangMu3,TianshiChen1 |
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| (1 School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen 518172, Guangdong, China
2 Division of Decision and Control Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, 10044 Stockholm, Sweden;3 Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100190, China) |
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| 摘要: |
| Regularized system identification has become the research frontier of system identification in the past decade. One related
core subject is to study the convergence properties of various hyper-parameter estimators as the sample size goes to infinity.
In 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 given.
However, 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 required. In general, the convergence
in distribution of a sequence of random variables does not necessarily guarantee the convergence of its covariance matrix.
Thus, 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 estimator. In 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 distribution. Moreover, we run numerical simulations to demonstrate the efficacy
of our theoretical results. |
| 关键词: Regularized system identification · Hyper-parameter estimator · Empirical Bayes · Convergence of covariance matrix |
| DOI：https://doi.org/10.1007/s11768-024-00211-z |
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| 基金项目:This work was supported in part by the National Natural Science Foundation of China (No. 62273287), by the Shenzhen Science and Technology Innovation Council (Nos. JCYJ20220530143418040, JCY20170411102101881), and the Thousand Youth Talents Plan funded by the central government of China. |
|
| On convergence of covariancematrix of empirical Bayes hyper-parameter estimator |
| Yue Ju1,2,Biqiang Mu3,Tianshi Chen1 |
| (1 School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen 518172, Guangdong, China
2 Division of Decision and Control Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, 10044 Stockholm, Sweden;3 Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100190, China) |
| Abstract: |
| Regularized system identification has become the research frontier of system identification in the past decade. One related
core subject is to study the convergence properties of various hyper-parameter estimators as the sample size goes to infinity.
In 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 given.
However, 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 required. In general, the convergence
in distribution of a sequence of random variables does not necessarily guarantee the convergence of its covariance matrix.
Thus, 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 estimator. In 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 distribution. Moreover, we run numerical simulations to demonstrate the efficacy
of our theoretical results. |
| Key words: Regularized system identification · Hyper-parameter estimator · Empirical Bayes · Convergence of covariance matrix |
