Efficient approximations of the inverse of a part of the additive relationship matrix

Abstract Text:

The single-step genomic BLUP requires the inverse of A₂₂, additive relationship matrix gathering relationships between genotyped animals. Inversion of A₂₂ may be time-consuming when the number of genotyped animals increases. The sparsity pattern of its inverse may be computed using pedigree information and may help to reduce computations. However, in some cases, the inverse may be extremely dense. In this study, we propose an alternate implementation that restricts the sparsity pattern of the inverse of A₂₂ in order to approximate its inverse. Such approximations are then computed for different orders of A₂₂ (10,000; 25,000; 50,000 and 75,000). Using the best approximations, reliabilities of breeding values in equivalent model instead of animal model are slightly inferior. Moreover, computing times of approximations are limited (e.g. 4h for an order of 75,000) and may be sped up by a factor of 3 using parallel computing.

Keywords:

Additive relationships

Genomic selection

Matrix inversion