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Dimensionality of genomic information and APY inverse of genomic relationship matrix
The objective of this study was to evaluate by simulation the dimensionality of genomic information in closed populations and its effect on genomic predictions using regular or sparse inverses of the genomic relationship matrices (GRM). Six datasets were simulated, representing populations with effective population sizes (Ne) approximately 20, 40, 80, 120, 160 and 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 Morgan, with 49,980 evenly allocated biallelic SNP markers and a total of 4,980 biallelic and randomly distributed QTL affected the trait. The GRM was constructed for each population and analyzed for distribution of eigenvalues. The number of the largest eigenvalues explaining 90, 95, 98 or 99 percent of variation in GRM ranged from 814, 1611, 3701, 6253 (Ne≈20) to 5512, 9245, 15483, 20786 (Ne≈200), respectively. Genomic EBV (GEBV) were computed by single-step genomic BLUP (ssGBLUP) using either a direct inverse of GRM or a sparse inverse with the algorithm for proven and young (APY) that is based on recursion on a random subset of animals, where subset sizes were set to number of the largest eigenvalues explaining 90, 95, 98 or 99 percent of variation in GRM. APY inverse has approximately a linear cost as opposed to cubic for the regular inverse. Accuracies of GEBV for the last generation with APY inverse peaked at EIG98 and were slightly lower with EIG95, EIG99 or the direct inverse. In a situation with large number of SNP markers and genotyped animals, dimensionality of the SNP genomic information defined by the eigenvalues of GRM is approximately a linear function of effective population size, where most information is contained in about NeL largest eigenvalues, with no information beyond 4NeL. Genomic predictions with APY sparse inverse of GRM are more accurate and computationally inexpensive compared to regular inverse.
Keywords:
genomic relationship matrix, genomic recursion, single-step genomic BLUP