Development of the method for dynamic regularization of selected estimates in the correlation matrices of observations

Abstract

The problem of formation of sample estimates of correlation matrices of observations by the «computational stability – consistency» criterion is considered. The problem of zero eigenvalues inherent in the problem of static regularization of sample estimates of correlation matrices is revealed. The solution of this problem by the static regularization method leads to the fact that the sample estimate of the regularized matrix is similar, but not identical to the original one in terms of consistency. Therefore, the problem of investigating the regularization of the sample estimate of the correlation matrix with respect to the solution of inverse problems under a priori uncertainty is actualized. In such a situation, the regularizing parameter of the inverse problem should be updated in real time as the input data arrive. To solve the revealed problem, an alternative method of dynamic regularization is proposed. In the study, the computational stability, convergence and consistency of sample estimates of correlation matrices of observations under a priori uncertainty are analyzed. The optimum function of dynamic regularization of sample estimates of correlation matrices of observations is obtained, the evaluation of which does not require prediction data and additional computing resources to search for the optimum value of the regularization parameter. The numerical results confirming the main findings are presented. The developed method of dynamic regularization of sample estimates of correlation matrices is an alternative to static regularization and allows resolving the «computational stability – consistency» contradiction when forming sample estimates of correlation matrices. Unlike static regularization, the procedure of dynamic regularization unambiguously connects the optimum dynamic regularization function with the matrix dimension and the size of the observed sample, which allows eliminating the problem of choosing the regularization parameter under a priori uncertainty with respect to the input data of the computational problem. In addition, the dynamic regularization method is characterized by simplicity of computational operations in real time in the absence of a priori information. Application of the method of dynamic regularization of sample estimates of correlation matrices extends the capabilities of a wide class of information systems that are designed to solve ill-posed inverse problems under a priori uncertainty