On the accuracy in high-dimensional linear models and its application to genomic selection

Abstract

Genomic selection is today a hot topic in genetics. It consists in predicting breeding values of selection candidates, using the large number of genetic markers now available owing to the recent progress in molecular biology. One of the most popular methods chosen by geneticists is ridge regression. We focus on some predictive aspects of ridge regression and present theoretical results regarding the accuracy criteria, that is, the correlation between predicted value and true value. We show the influence of singular values, the regularization parameter, and the projection of the signal on the space spanned by the rows of the design matrix. Asymptotic results in a high-dimensional framework are given; in particular, we prove that the convergence to optimal accuracy highly depends on a weighted projection of the signal on each subspace. We discuss on how to improve the prediction. Last, illustrations on simulated and real data are proposed.