Some Properties of a Recursive Procedure for High Dimensional Parameter Estimation in Linear Model with Regularization
ABSTRACT
Theoretical results related to properties of a regularized recursive algorithm for estimation of a high dimensional vector of parameters are presented and proved. The recursive character of the procedure is proposed to overcome the difficulties with high dimension of the observation vector in computation of a statistical regularized estimator. As to deal with high dimension of the vector of unknown parameters, the regularization is introduced by specifying a priori non-negative covariance structure for the vector of estimated parameters. Numerical example with Monte-Carlo simulation for a low-dimensional system as well as the state/parameter estimation in a very high dimensional oceanic model is presented to demonstrate the efficiency of the proposed approach.
Theoretical results related to properties of a regularized recursive algorithm for estimation of a high dimensional vector of parameters are presented and proved. The recursive character of the procedure is proposed to overcome the difficulties with high dimension of the observation vector in computation of a statistical regularized estimator. As to deal with high dimension of the vector of unknown parameters, the regularization is introduced by specifying a priori non-negative covariance structure for the vector of estimated parameters. Numerical example with Monte-Carlo simulation for a low-dimensional system as well as the state/parameter estimation in a very high dimensional oceanic model is presented to demonstrate the efficiency of the proposed approach.
KEYWORDS
Linear Model, Regularization, Recursive Algorithm, Non-Negative Covariance Structure, Eigenvalue Decomposition
Linear Model, Regularization, Recursive Algorithm, Non-Negative Covariance Structure, Eigenvalue Decomposition
Cite this paper
Hoang, H. and Baraille, R. (2014) Some Properties of a Recursive Procedure for High Dimensional Parameter Estimation in Linear Model with Regularization. Open Journal of Statistics, 4, 921-932. doi: 10.4236/ojs.2014.411087.
Hoang, H. and Baraille, R. (2014) Some Properties of a Recursive Procedure for High Dimensional Parameter Estimation in Linear Model with Regularization. Open Journal of Statistics, 4, 921-932. doi: 10.4236/ojs.2014.411087.
References
[1] Hoang, H.S. and Baraille, R. (2013) A Regularized Estimator for Linear Regression Model with Possibly Singular Covariance. IEEE Transactions on Automatic Control, 58, 236-241.
http://dx.doi.org/10.1109/TAC.2012.2203552 [2] Daley, R. (1991) Atmospheric Data Analysis. Cambridge University Press, New York. [3] Albert, A. (1972) Regression and the Moore-Penrose Pseudo-Inverse. Academy Press, New York. [4] Hoang, H.S. and Baraille, R. (2014) A Low Cost Filter Design for State and Parameter Estimation in Very High Dimensional Systems. Proceedings of the 19th IFAC Congress, Cape Town, 24-29 August 2014, 3156-3161. [5] Hoang, H.S. and Baraille, R. (2011) Approximate Approach to Linear Filtering Problem with Correlated Noise. Engineering and Technology, 5, 11-23. [6] Golub, G.H. and van Loan, C.F. (1996) Matrix Computations. 3rd Edition, Johns Hopkins University Press, Baltimore. [7] Hoang, H.S., Baraille, R. and Talagrand, O. (2001) On the Design of a Stable Adaptive Filter for High Dimensional Systems. Automatica, 37, 341-359.
http://dx.doi.org/10.1016/S0005-1098(00)00175-8 [8] Hoang, H.S. and Baraille, R. (2011) Prediction Error Sampling Procedure Based on Dominant Schur Decomposition. Application to State Estimation in High Dimensional Oceanic Model. Applied Mathematics and Computation, 218, 3689-3709.
http://dx.doi.org/10.1016/j.amc.2011.09.012 [9] Kalman, R.E. (1960) A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering, 82, 35-45.
http://dx.doi.org/10.1115/1.3662552 [10] Cooper, M. and Haines, K. (1996) Altimetric Assimilation with Water Property Conservation. Journal of Geophysical Research, 101, 1059-1077.
http://dx.doi.org/10.1029/95JC02902 [11] Greenslade, D.J.M. and Young, I.R. (2005) The Impact of Altimeter Sampling Patterns on Estimates of Background Errors in a Global Wave Model. Journal of Atmospheric and Oceanic Technology, 1895-1917. [12] Spall. J.C. (2000) Adaptive Stochastic Approximation by the Simultaneous Perturbation Method. IEEE Transactions on Automatic Control, 45, 1839-1853.
http://dx.doi.org/10.1109/TAC.2000.880982
[1] Hoang, H.S. and Baraille, R. (2013) A Regularized Estimator for Linear Regression Model with Possibly Singular Covariance. IEEE Transactions on Automatic Control, 58, 236-241.
http://dx.doi.org/10.1109/TAC.2012.2203552 [2] Daley, R. (1991) Atmospheric Data Analysis. Cambridge University Press, New York. [3] Albert, A. (1972) Regression and the Moore-Penrose Pseudo-Inverse. Academy Press, New York. [4] Hoang, H.S. and Baraille, R. (2014) A Low Cost Filter Design for State and Parameter Estimation in Very High Dimensional Systems. Proceedings of the 19th IFAC Congress, Cape Town, 24-29 August 2014, 3156-3161. [5] Hoang, H.S. and Baraille, R. (2011) Approximate Approach to Linear Filtering Problem with Correlated Noise. Engineering and Technology, 5, 11-23. [6] Golub, G.H. and van Loan, C.F. (1996) Matrix Computations. 3rd Edition, Johns Hopkins University Press, Baltimore. [7] Hoang, H.S., Baraille, R. and Talagrand, O. (2001) On the Design of a Stable Adaptive Filter for High Dimensional Systems. Automatica, 37, 341-359.
http://dx.doi.org/10.1016/S0005-1098(00)00175-8 [8] Hoang, H.S. and Baraille, R. (2011) Prediction Error Sampling Procedure Based on Dominant Schur Decomposition. Application to State Estimation in High Dimensional Oceanic Model. Applied Mathematics and Computation, 218, 3689-3709.
http://dx.doi.org/10.1016/j.amc.2011.09.012 [9] Kalman, R.E. (1960) A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering, 82, 35-45.
http://dx.doi.org/10.1115/1.3662552 [10] Cooper, M. and Haines, K. (1996) Altimetric Assimilation with Water Property Conservation. Journal of Geophysical Research, 101, 1059-1077.
http://dx.doi.org/10.1029/95JC02902 [11] Greenslade, D.J.M. and Young, I.R. (2005) The Impact of Altimeter Sampling Patterns on Estimates of Background Errors in a Global Wave Model. Journal of Atmospheric and Oceanic Technology, 1895-1917. [12] Spall. J.C. (2000) Adaptive Stochastic Approximation by the Simultaneous Perturbation Method. IEEE Transactions on Automatic Control, 45, 1839-1853.
http://dx.doi.org/10.1109/TAC.2000.880982