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

Genomic selection, a hot topic in genetics, consists in predicting breeding values of selection candidates, using a large number of genetic markers , due to the recent progress in molecular biology. One of the most popular method chosen by geneticists is Ridge regression. In this context, we focus on some predictive aspects of Ridge regression and present theoretical results regarding the accuracy criteria, i.e., the correlation between predicted value and true value. We show the influence of the 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 also given, and we prove that the convergence to an optimal accuracy highly depends on a weighted projection of the signal on each subspace. We discuss also on how to improve the prediction. Last, illustrations on simulated and real data are proposed.