  The estimation of covariance matrices of multiple classes with limited
training data is a difficult problem. The sample covariance matrix (SCM) is
known to perform poorly when the number of variables is large compared to the
available number of samples. In order to reduce the mean squared error (MSE) of
the SCM, regularized (shrinkage) SCM estimators are often used. In this work,
we consider regularized SCM (RSCM) estimators for multiclass problems that
couple together two different target matrices for regularization: the pooled
(average) SCM of the classes and the scaled identity matrix. Regularization
toward the pooled SCM is beneficial when the population covariances are
similar, whereas regularization toward the identity matrix guarantees that the
estimators are positive definite. We derive the MSE optimal tuning parameters
for the estimators as well as propose a method for their estimation under the
assumption that the class populations follow (unspecified) elliptical
distributions with finite fourth-order moments. The MSE performance of the
proposed coupled RSCMs are evaluated with simulations and in a regularized
discriminant analysis (RDA) classification set-up on real data. The results
based on three different real data sets indicate comparable performance to
cross-validation but with a significant speed-up in computation time.
