On the accuracy in high dimensional linear models under imperfect linkage disequilibrium

Genomic selection (GS) consists in predicting breeding values of selection candidates, using a large number of genetic markers. An important question in GS is the determination of the number of markers required for a good prediction. Many studies show that it becomes useless to consider too many markers. In contrast, for some species, the number of markers remains too small to cover the huge genome size. Under such sparse genetic map, it is likely to observe some imperfect linkage disequilibrium: the alleles at a gene location and at a marker located nearby vary. In this context, we tackle here the problem of imperfect linkage disequilibrium in the Ridge regression framework. We present theoretical results regarding the accuracy criteria, i.e., the correlation between predicted value and true value. We show the influence of the projection of the causal regression function (i.e. at genes) on the space spanned by the columns of the design matrix (i.e. at markers). Asymptotic results, in a high dimensional framework, are given, and we prove that the convergence to an optimal accuracy depends on a few limiting factors. This study generalizes our recent results (Rabier et al. (2018)) obtained under perfect linkage disequi-librium. Last, illustrations on simulated and real data are proposed.