Early application of genomic selection relied on SNP estimation with phenotypes or de-regressed proofs (DRP). Chips of 50k SNP seemed sufficient for an accurate estimation of SNP effects. Genomic estimated breeding values (GEBV) were composed of an index with parent average, direct genomic value, and deduction of a parental index to eliminate double counting. Use of SNP selection or weighting increased accuracy with small data sets but had minimal to no impact with large data sets. Efforts to include potentially causative SNP derived from sequence data or high-density chips showed limited or no gain in accuracy. After the implementation of genomic selection, EBV by BLUP became biased because of genomic preselection and DRP computed based on EBV required adjustments, and the creation of DRP for females is hard and subject to double counting. Genomic selection was greatly simplified by single-step genomic BLUP (ssGBLUP). This method based on combining genomic and pedigree relationships automatically creates an index with all sources of information, can use any combination of male and female genotypes, and accounts for preselection. To avoid biases, especially under strong selection, ssGBLUP requires that pedigree and genomic relationships are compatible. Because the inversion of the genomic relationship matrix (G) becomes costly with more than 100k genotyped animals, large data computations in ssGBLUP were solved by exploiting limited dimensionality of genomic data due to limited effective population size. With such dimensionality ranging from 4k in chickens to about 15k in cattle, the inverse of G can be created directly (e.g., by the algorithm for proven and young) at a linear cost. Due to its simplicity and accuracy, ssGBLUP is routinely used for genomic selection by the major chicken, pig, and beef industries. Single step can be used to derive SNP effects for indirect prediction and for genome-wide association studies, including computations of the P-values. Alternative single-step formulations exist that use SNP effects for genotyped or for all animals. Although genomics is the new standard in breeding and genetics, there are still some problems that need to be solved. This involves new validation procedures that are unaffected by selection, parameter estimation that accounts for all the genomic data used in selection, and strategies to address reduction in genetic variances after genomic selection was implemented.

Current status of genomic evaluation.

/pdf/multibreed-genomic-evaluation-for-production-traits-of-dairy-19r6iqin.pdf

Multibreed genomic evaluation for production traits of dairy cattle in the United States using single-step genomic best linear unbiased predictor.

The single-step single nucleotide polymorphism best linear unbiased prediction (ssSNPBLUP) is one of the single-step evaluations that enable a simultaneous analysis of phenotypic and pedigree information of genotyped and non-genotyped animals with a large number of genotypes. The aim of this study was to develop and illustrate several computational strategies to efficiently solve different ssSNPBLUP systems with a large number of genotypes on current computers. The different developed strategies were based on simplified computations of some terms of the preconditioner, and on splitting the coefficient matrix of the different ssSNPBLUP systems into multiple parts to perform its multiplication by a vector more efficiently. Some matrices were computed explicitly and stored in memory (e.g. the inverse of the pedigree relationship matrix), or were stored using a compressed form (e.g. the Plink 1 binary form for the genotype matrix), to permit the use of efficient parallel procedures while limiting the required amount of memory. The developed strategies were tested on a bivariate genetic evaluation for livability of calves for the Netherlands and the Flemish region in Belgium. There were 29,885,286 animals in the pedigree, 25,184,654 calf records, and 131,189 genotyped animals. The ssSNPBLUP system required around 18 GB Random Access Memory and 12 h to be solved with the most performing implementation. Based on our proposed approaches and results, we showed that ssSNPBLUP provides a feasible approach in terms of memory and time requirements to estimate genomic breeding values using current computers.

/pdf/computational-strategies-for-the-preconditioned-conjugate-l24z5wkqak.pdf

Computational strategies for the preconditioned conjugate gradient method applied to ssSNPBLUP, with an application to a multivariate maternal model

The single-step single nucleotide polymorphism best linear unbiased prediction (ssSNPBLUP) method, such as single-step genomic BLUP (ssGBLUP), simultaneously analyses phenotypic, pedigree, and genomic information of genotyped and non-genotyped animals. In contrast to ssGBLUP, SNP effects are fitted explicitly as random effects in the ssSNPBLUP model. Similarly, principal components associated with the genomic information can be fitted explicitly as random effects in a single-step principal component BLUP (ssPCBLUP) model to remove noise in genomic information. Single-step genomic BLUP is solved efficiently by using the preconditioned conjugate gradient (PCG) method. Unfortunately, convergence issues have been reported when solving ssSNPBLUP by using PCG. Poor convergence may be linked with poor spectral condition numbers of the preconditioned coefficient matrices of ssSNPBLUP. These condition numbers, and thus convergence, could be improved through the deflated PCG (DPCG) method, which is a two-level PCG method for ill-conditioned linear systems. Therefore, the first aim of this study was to compare the properties of the preconditioned coefficient matrices of ssGBLUP and ssSNPBLUP, and to document convergence patterns that are obtained with the PCG method. The second aim was to implement and test the efficiency of a DPCG method for solving ssSNPBLUP and ssPCBLUP. For two dairy cattle datasets, the smallest eigenvalues obtained for ssSNPBLUP (ssPCBLUP) and ssGBLUP, both solved with the PCG method, were similar. However, the largest eigenvalues obtained for ssSNPBLUP and ssPCBLUP were larger than those for ssGBLUP, which resulted in larger condition numbers and in slow convergence for both systems solved by the PCG method. Different implementations of the DPCG method led to smaller condition numbers, and faster convergence for ssSNPBLUP and for ssPCBLUP, by deflating the largest unfavourable eigenvalues. Poor convergence of ssSNPBLUP and ssPCBLUP when solved by the PCG method are related to larger eigenvalues and larger condition numbers in comparison to ssGBLUP. These convergence issues were solved with a DPCG method that annihilates the effect of the largest unfavourable eigenvalues of the preconditioned coefficient matrix of ssSNPBLUP and of ssPCBLUP on the convergence of the PCG method. It resulted in a convergence pattern, at least, similar to that of ssGBLUP.

/pdf/deflated-preconditioned-conjugate-gradient-method-for-5yyhcgnuv9.pdf

Deflated preconditioned conjugate gradient method for solving single-step BLUP models efficiently

The number of genotyped animals has increased rapidly creating computational challenges for genomic evaluation. In animal model BLUP, candidate animals without progeny and phenotype do not contribute information to the evaluation and can be discarded. In theory, genotyped candidate animal without progeny can bring information into single-step BLUP (ssGBLUP) and affect the estimation of other breeding values. We studied the effect of including or excluding genomic information of culled bull calves on genomic breeding values (GEBV) from ssGBLUP. In particular, GEBVs of genotyped bulls with daughters and GEBVs of young bulls selected into AI to be progeny tested (test bulls) were studied. The ssGBLUP evaluation was computed using Nordic test day (TD) model and TD data for the Nordic Red Dairy Cattle. The results indicate that genomic information of culled bull calves does not affect the GEBVs of progeny tested reference animals, but if genotypes of the culled bulls are used in the TD ssGBLUP, the genetic trend in the test bulls is considerably higher compared to the situation when genomic information of the culled bull calves is excluded. It seems that by discarding genomic information of culled bull calves without progeny, upward bias of GEBVs of test bulls is reduced.

Reducing bias in the dairy cattle single‐step genomic evaluation by ignoring bulls without progeny

An equivalent computational approach called ssGTBLUP was formulated for the original single-step GBLUP (ssGBLUP). In ssGTBLUP, the genomic relationship matrix has the form G = ZZʹ + C, where the (centered and scaled) Z marker matrix has size n x m (numbers of genotypes and markers), and the C matrix can be easily inverted. The inverse can be written as G-1 = C-1 − TʹT where T is an m by n matrix. When the preconditioned conjugate gradient (PCG) method is used to solve the mixed model equations, a matrix vector product G-1d needs to be computed. In ssGBLUP, this requires n2 multiplications, but in ssGTBLUP, the product TʹTd has 2nm multiplications and C-1d has cn multiplications with the constant c independent of n or m. In an approximate approach called ssGTBLUP(p), the eigendecomposition of ZʹC-1Z is used to reduce the number of rows in the T matrix. Here, p is the percentage of total variance explained by the accepted eigenvalues. The objective of this study was to compare the performance of ssGBLUP, ssGTBLUP, ssGTBLUP(p), and the APY (algorithm for proven and young) method. In APY, the core had 50,000 (APY50K), 30,000 (APY30K), or 10,000 (APY10K) animals. The approaches were tested on the Irish beef carcass conformation genetic evaluation which has a heterogeneous multibreed population. The pedigree had 13.3 million animals. There were m = 54,620 markers available from n = 163,277 genotyped animals. For genotyped animals, the correlations of breeding values between ssGBLUP and ssGTBLUP(p) for the 11 traits in the model ranged from 0.999–1.000 for p = 99, 0.998–1.000 for p = 98, and 0.992–0.998 for p = 95 but were 0.994–1.000 for APY50K, 0.969–0.997 for APY30K, and 0.899–0.967 for APY10K. Computing times per iteration were 4.43, 3.30, 2.69, 2.29, 1.55, 1.76, 1.27, and 0.55 min for ssGBLUP, ssGTBLUP, ssGTBLUP(99), ssGTBLUP(98), ssGTBLUP(95), APY50K, APY30K, and APY10K, respectively. The ssGTBLUP(p) approach allowed a well-defined approximation to ssGBLUP and fast computations

Efficient single-step genomic evaluation for a multibreed beef cattle population having many genotyped animals.

Single-step genomic best linear unbiased prediction (ssGBLUP) models allow the combination of genomic, pedigree, and phenotypic data into a single model, which is computationally challenging for large genotyped populations. In practice, genotypes of animals without their own phenotype and progeny, so-called genotyped selection candidates, can become available after genomic breeding values have been estimated by ssGBLUP. In some breeding programmes, genomic estimated breeding values (GEBV) for these animals should be known shortly after obtaining genotype information but recomputing GEBV using the full ssGBLUP takes too much time. In this study, first we compare two equivalent formulations of ssGBLUP models, i.e. one that is based on the Woodbury matrix identity applied to the inverse of the genomic relationship matrix, and one that is based on marker equations. Second, we present computationally-fast approaches to indirectly compute GEBV for genotyped selection candidates, without the need to do the full ssGBLUP evaluation.The indirect approaches use information from the latest ssGBLUP evaluation and rely on the decomposition of GEBV into its components. The two equivalent ssGBLUP models and indirect approaches were tested on a six-trait calving difficulty model using Irish dairy and beef cattle data that include 2.6 million genotyped animals of which about 500,000 were considered as genotyped selection candidates. When using the same computational approaches, the solving phase of the two equivalent ssGBLUP models showed similar requirements for memory and time per iteration. The computational differences between them were due to the preprocessing phase of the genomic information. Regarding the indirect approaches, compared to GEBV obtained from single-step evaluations including all genotypes, indirect GEBV had correlations higher than 0.99 for all traits while showing little dispersion and level bias.In conclusion, ssGBLUP predictions for the genotyped selection candidates were accurately approximated using the presented indirect approaches, which are more memory efficient and computationally fast, compared to solving a full ssGBLUP evaluation. Thus, indirect approaches can be used even on a weekly basis to estimate GEBV for newly genotyped animals, while the full single-step evaluation is done only a few times within a year. 

https://gsejournal.biomedcentral.com/counter/pdf/10.1186/s12711-023-00808-z

Efficient large-scale single-step evaluations and indirect genomic prediction of genotyped selection candidates

In the last two decades, a number of so-called single-step models have been suggested to incorporate both pedigree and genomic data in statistical breeding value estimation for breeding programs. Yet, steadily growing population sizes and the suboptimal use of computational resources are pushing the practical application of these models to their limits. As an extension to the miraculix package, we have developed tailored solutions for the computation of genotype matrix multiplications, a critical bottleneck when iteratively solving equation systems associated with single-step models. Targeting modern Nvidia® GPUs as well as a broad range of CPU architectures, our implementation significantly accelerates the estimation of breeding values in large population sizes. miraculix is a C/CUDA library released under the Apache 2.0 license. It is freely available at https://github.com/alexfreudenberg/miraculix.

/pdf/accelerated-computations-for-iterative-solver-techniques-in-1lvtp726.pdf

Accelerated computations for iterative-solver techniques in single-step BLUP models

/pdf/increasing-computational-efficiency-in-ssgtblup-by-58gjadipmp.pdf

Increasing computational efficiency in ssGTBLUP by eigendecomposition

Computing time and memory requirements increase with single-step methods to estimate genomic breeding values when the number of genotyped animals increases. Computational costs can be reduced by omitting genotypes of animals without phenotype and progeny, often the candidate animals for selection. Indirect prediction of a candidate animal GEBV can be based on the animal’s genotype and SNP marker solutions (DGV). Alternatively, the sum of DGV and residual polygenic (RPG) effect can be computed, denoted GRV (Genomic and Residual polygenic Value). We applied indirect genomic prediction for a 6 trait calving difficulty evaluation. There were 1.50 million genotyped animals of which 36% were considered candidate animals. Based on our results, DGV showed high accuracy but also bias due to omitting the RPG effect. The GRV prediction had high accuracy and low bias. Computing time was reduced by 33%. 

https://www.wageningenacademic.com/pb-assets/wagen/WCGALP2022/36_006.pdf

360. Indirect genomic prediction reduces computational costs in large-scale single-step evaluations

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