Covariance structure regularization via Frobenius-norm discrepancy

In many practical problems, the underlying structure of an estimated covariance matrix is usually blurred due to random noise, particularly when the dimension of the matrix is high. Hence, it is necessary to filter the random noise or regularize the available covariance matrix in certain senses, so that the covariance structure becomes clear. In this paper, we propose a new method for regularizing the covariance structure of a given covariance matrix. By choosing an optimal structure from an available class of covariance structures, the regularization is made in terms of minimizing the discrepancy, defined by Frobenius-norm, between the given covariance matrix and the class of covariance structures. A range of potential candidate structures, including the order-1 moving average structure, compound symmetry structure, order-1 autoregressive structure, order-1 autoregressive moving average structure, are considered. Simulation studies show that the proposed new approach is reliable in regularization of covariance structures. The proposed approach is also applied to real data analysis in signal processing, showing the usefulness of the proposed approach in practice.