An efficient genomic prediction method without the direct inverse of the genomic relationship matrix

GBLUP, the most widely used genomic prediction (GP) method, consumes large and increasing amounts of computational resources as the training population size increases due to the inverse of the genomic relationship matrix (GRM). Therefore, in this study, we developed a new genomic prediction method (RHEPCG) that avoids the direct inverse of the GRM by combining randomized Haseman-Elston (HE) regression (RHE-reg) and a preconditioned conjugate gradient (PCG). The simulation results demonstrate that RHEPCG, in most cases, not only achieves similar predictive accuracy with GBLUP but also significantly reduces computational time. As for the real data, RHEPCG shows similar or better predictive accuracy for seven traits of the Arabidopsis thaliana F2 population and four traits of the Sorghum bicolor RIL population compared with GBLUP. This indicates that RHEPCG is a practical alternative to GBLUP and has better computational efficiency.

Estimation of individual animal SNP-BLUP reliability using full Monte Carlo sampling

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Computing time for the full Monte Carlo (MC)-SNP-BLUP was less than for the exact genomic BLUP.

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The full MC-SNP-BLUP better approximated estimated breeding value reliability than an incomplete MCbased SNP-BLUP approach when the residual polygenic effect was high.

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The higher the exact reliability, the smaller the inflation.

Calculation of individual animal reliability of estimated genomic breeding value by SNP-BLUP requires inversion of the mixed model equations (MME). When the SNP-BLUP model includes a residual polygenic (RPG) effect, the size of the MME will be at least the number of genotyped animals (n) plus the number of SNP markers (m). Inversion of the MME in SNP-BLUP involves computations proportional to the cube of the MME size; that is, (n + m)³, which can present a considerable computational burden. We introduce a full Monte Carlo (MC) sampling-based method for approximating reliability in the SNP-BLUP model and compare its performance to the genomic BLUP (GBLUP) model. The performance of the full MC approach was evaluated using 2 data sets, including 19,757 and 222,619 genotyped animals selected from populations with 231,186 and 13.35 million pedigree animals, respectively. Genotypes were available in the data sets for 11,729 and 50,240 SNP markers. An advantage of the full MC approximation method was its low computational demand. A drawback was its tendency to overestimate reliability for animals with low reliability, especially when the weight of the RPG effect was high. The overestimation can be lessened by increasing the number of MC samples.

Updating genetic relationship matrices and their inverses: a methodology note

Most of the calculations for genetic relationship matrices and their inverses utilized in recent (t + 1) evaluations for populations of interest can be avoided by updating these matrices at t evaluations for only new animals. This study describes and develops existing methods for updating pedigree-based and marker-based relationship matrices and their inverses. Updating some of the matrices could benefit from parallel computing.

An Efficient Method to Calculate Genomic Prediction Accuracy for New Individuals

Diagonal elements of the coefficient matrix are necessary to calculate the genomic prediction accuracy. Here an improved methodology is described, to update the inverse of the coefficient matrix (C) for new individuals with a genotype, with and without phenotypes. Computational performance is significantly improved by re-using parts of the coefficient matrix inverse calculations that do not change from one animal to another, in combination with updated calculations for those that do change. This method expedites calculation of accuracy for new individuals with genotypes, without re-doing the whole population, by using the previously calculated matrices.

A method to approximate the inverse of a part of the additive relationship matrix

Single-step genomic predictions need the inverse of the part of the additive relationship matrix between genotyped animals (A22 ). Gains in computing time are feasible with an algorithm that sets up the sparsity pattern of A22-1 (SP algorithm) using pedigree searches, when A22-1 is close to sparse. The objective of this study is to present a modification of the SP algorithm (RSP algorithm) and to assess its use in approximating A22-1 when the actual A22-1 is dense. The RSP algorithm sets up a restricted sparsity pattern of A22-1 by limiting the pedigree search to a maximum number of searched branches. We have tested its use on four different simulated genotyped populations, from 10 000 to 75 000 genotyped animals. Accuracy of approximation is tested by replacing the actual A22-1 by its approximation in an equivalent mixed model including only genotyped animals. Results show that limiting the pedigree search to four branches is enough to provide accurate approximations of A22-1, which contain approximately 80% of zeros. Computing approximations is not expensive in time but may require a great amount of memory (at maximum, approximately 81 min and approximately 55 Gb of RAM for 75 000 genotyped animals using parallel processing on four threads).

Using recursion to compute the inverse of the genomic relationship matrix

Computing the inverse of the genomic relationship matrix using recursion was investigated. A traditional algorithm to invert the numerator relationship matrix is based on the observation that the conditional expectation for an additive effect of 1 animal given the effects of all other animals depends on the effects of its sire and dam only, each with a coefficient of 0.5. With genomic relationships, such an expectation depends on all other genotyped animals, and the coefficients do not have any set value. For each animal, the coefficients plus the conditional variance can be called a genomic recursion. If such recursions are known, the mixed model equations can be solved without explicitly creating the inverse of the genomic relationship matrix. Several algorithms were developed to create genomic recursions. In an algorithm with sequential updates, genomic recursions are created animal by animal. That algorithm can also be used to update a known inverse of a genomic relationship matrix for additional genotypes. In an algorithm with forward updates, a newly computed recursion is immediately applied to update recursions for remaining animals. The computing costs for both algorithms depend on the sparsity pattern of the genomic recursions, but are lower or equal than for regular inversion. An algorithm for proven and young animals assumes that the genomic recursions for young animals contain coefficients only for proven animals. Such an algorithm generates exact genomic EBV in genomic BLUP and is an approximation in single-step genomic BLUP. That algorithm has a cubic cost for the number of proven animals and a linear cost for the number of young animals. The genomic recursions can provide new insight into genomic evaluation and possibly reduce costs of genetic predictions with extremely large numbers of genotypes.

Inversion of a part of the numerator relationship matrix using pedigree information

Background

In recent theoretical developments, the information available (e.g. genotypes) divides the original population into two groups: animals with this information (selected animals) and animals without this information (excluded animals). These developments require inversion of the part of the pedigree-based numerator relationship matrix that describes the genetic covariance between selected animals (A₂₂). Our main objective was to propose and evaluate methodology that takes advantage of any potential sparsity in the inverse of A₂₂ in order to reduce the computing time required for its inversion. This potential sparsity is brought out by searching the pedigree for dependencies between the selected animals. Jointly, we expected distant ancestors to provide relationship ties that increase the density of matrix A₂₂ but that their effect on A22-1

might be minor. This hypothesis was also tested.

Methods

The inverse of A₂₂ can be computed from the inverse of the triangular factor (T^-1) obtained by Cholesky root-free decomposition of A₂₂. We propose an algorithm that sets up the sparsity pattern of T^-1 using pedigree information. This algorithm provides positions of the elements of T^-1 worth to be computed (i.e. different from zero). A recursive computation of A22-1 is then achieved with or without information on the sparsity pattern and time required for each computation was recorded. For three numbers of selected animals (4000; 8000 and 12 000), A₂₂ was computed using different pedigree extractions and the closeness of the resulting A22-1 to the inverse computed using the fully extracted pedigree was measured by an appropriate norm.

Results

The use of prior information on the sparsity of T^-1 decreased the computing time for inversion by a factor of 1.73 on average. Computational issues and practical uses of the different algorithms were discussed. Cases involving more than 12 000 selected animals were considered. Inclusion of 10 generations was determined to be sufficient when computing A₂₂.

Conclusions

Depending on the size and structure of the selected sub-population, gains in time to compute A22-1 are possible and these gains may increase as the number of selected animals increases. Given the sequential nature of most computational steps, the proposed algorithm can benefit from optimization and may be convenient for genomic evaluations.