Covariance Structure Regularization via Frobenius-Norm Discrepancy

In many practical problems, the underlying structure of an estimated co-
variance matrix is usually blurred due to random noise, particularly when
the dimension of the matrix is high. Hence, it is necessary to filter the ran-
dom 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 ma-
trix. 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.