Covariance-regularized regression and classification for high dimensional problems

We propose covariance-regularized regression, a family of methods for prediction in high dimensional settings that uses a shrunken estimate of the inverse covariance matrix of the features to achieve superior prediction. An estimate of the inverse covariance matrix is obtained by maximizing the log-likelihood of the data, under a multivariate normal model, subject to a penalty; it is then used to estimate coefficients for the regression of the response onto the features. We show that ridge regression, the lasso and the elastic net are special cases of covariance-regularized regression, and we demonstrate that certain previously unexplored forms of covariance-regularized regression can outperform existing methods in a range of situations. The covariance-regularized regression framework is extended to generalized linear models and linear discriminant analysis, and is used to analyse gene expression data sets with multiple class and survival outcomes. Copyright (c) 2009 Royal Statistical Society.