Genomic relationship matrix (GRM) can be inverted by Algorithm for Proven and Young (APY) based on recursion on a random subset of animals. While a regular inverse has a cubic cost, the cost of the APY inverse can be close to linear. Theory for APY assumes that optimal size of the subset (maximizing accuracy of genomic predictions) is due to a limited dimensionality of GRM, which is a function of effective populations size (Ne). The objective of this study was to evaluate these assumptions by simulation. Six populations were simulated with approximate effective population size (Ne) from 20 to 200. Each population consisted of 10 non-overlapping generations, with 25,000 animals per generation and phenotypes available for generations 1 to 9. The last three generations were fully genotyped assuming genome length L=30. The GRM was constructed for each population and analyzed for distribution of eigenvalues. Genomic estimated breeding values (GEBV) were computed by single-step GBLUP using either a direct or APY inverse of GRM. The sizes of the subset in APY were set to the number of the largest eigenvalues explaining x% of variation (EIGx, x=90, 95, 98, 99) in GRM. Accuracies of GEBV for the last generation with APY inverse peaked at EIG98 and were slightly lower with EIG95, EIG99 or the direct inverse. Most information in GRM is contained in about NeL largest eigenvalues, with no information beyond 4NeL. Genomic predictions with APY inverse of GRM are more accurate than by regular inverse.
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