Covariance Matrix Estimation With Heterogeneous Samples

We consider the problem of estimating the covariance matrix M_p of an observation vector, using heterogeneous training samples, i.e., samples whose covariance matrices are not exactly M_p. More precisely, we assume that the training samples can be clustered into K groups, each one containing L_k, snapshots sharing the same covariance matrix M_k. Furthermore, a Bayesian approach is proposed in which the matrices M_k. are assumed to be random with some prior distribution. We consider two different assumptions for M_p. In a fully Bayesian framework, M_p is assumed to be random with a given prior distribution. Under this assumption, we derive the minimum mean-square error (MMSE) estimator of M_p which is implemented using a Gibbs-sampling strategy. Moreover, a simpler scheme based on a weighted sample covariance matrix (SCM) is also considered. The weights minimizing the mean square error (MSE) of the estimated covariance matrix are derived. Furthermore, we consider estimators based on colored or diagonal loading of the weighted SCM, and we determine theoretically the optimal level of loading. Finally, in order to relax the a priori assumptions about the covariance matrix M_p, the second part of the paper assumes that this matrix is deterministic and derives its maximum-likelihood estimator. Numerical simulations are presented to illustrate the performance of the different estimation schemes.