Shrinkage Approach for Spatiotemporal EEG Covariance Matrix Estimation

The characterization of the background activity in electroencephalography (EEG) is of interest in many problems, such as in the study of the brain rhythms and in the solution of the inverse problem for source localization. In most cases the background activity is modeled as a random process, and a basic characterization is done via the second order moments of the process, i.e., the spatiotemporal covariance. The general spatiotemporal covariance matrix of the background activity in EEG is extremely large. To reduce its dimensionality it is generally decomposed as a Kronecker product of a spatial and a temporal covariance matrices. They are generally estimated from the data using sample estimators, which have numerical and statistical problems when the number of trials is small. We present a shrinkage estimator for both EEG spatial and temporal covariance matrices of the background activity. We show that this estimator outperforms the commonly used ones when the quantity of available data is low. We find sufficient conditions for the consistency of the shrinkage estimator and present some results concerning its numerical stability. We compare several shrinkage approaches and show how to improve the estimator by incorporating known structure in the covariance matrix based on background activity models. Results using simulated and real EEG data support our approach.