On the efficient low cost procedure for estimation of high-dimensional prediction error covariance matrices

A simple, efficient algorithm is proposed for estimating the prediction error covariance matrix which plays the key role for successful state estimation in very high dimensional systems. The main results are obtained by introducing the hypothesis on the separability of vertical and horizontal structure of the error covariance matrix and its parameterization. A new parameter optimization problem is formulated which is closely related to the Nearest Kronecker Problem (NKP). This allows to estimate optimally the unknown parameters of the structured parametrized ECM as well as to approach numerically the solution of the traditional NKP in a simple and efficient way. The algorithm for the state estimation will be detailed. The results from experiments on parameter and state estimation problems, for both moderate and high dimensional numerical models, demonstrate a high effectiveness of the proposed filtering approach.