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 Genotypes soon available from
BFGL:
 50,000 SNPs / animal
 3,000 animals, many more possible
 Need efficient computing algorithms
 Traditional PTAs available from AIPL:
 PTAs combine phenotypes and pedigree
 SNP effects evaluated in second step using deregressed PTAs weighted by
reliability

3

 Simulate SNPs and QTLs
 Compare SNP numbers, size of QTLs
 Calculate genomic EBVs
 Use selection index, G instead of A
 Use iteration on data for SNP effects
 Form haplotypes from genotypes?
 Not tested yet, SNP regression used

4

 Save memory by processing each chromosome separately
 3,000 Holstein bulls to genotype
 17,000 ancestors in pedigree file
 1 billion (20,000 x 50,000 SNPs) genotypes simulated per replicate
 Only 150 million (3,000 x 50,000) genotypes stored for evaluation

5

 Selection index equations for EBV
 u^ = Cov(u,y) Var(y)^{1} (y – Xb)
 u^ = Z Z’ [Z Z’ + R]^{1} (y – Xb)
 R has diagonals = (1 / Reliability)  1
 BLUP equations for marker effects, sum to get EBV
 u^ = Z [Z’R^{1}Z + I k]^{1} Z’R^{1}(y – Xb)
 k = var(u) / var(m)

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 Simple trick to reduce time from quadratic to linear with # SNPs
 Sum coefficients x solutions once
 Sum – diagonal = 3
offdiagonals
 Janss and de Jong, 1999 conference
 Rediscovered by Legarra and Misztal
 Elements of Z are –p and (1 – p), where p is frequency of 2^{nd}
allele

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 Inversion including G matrix
 Animals x markers to hold genotypes
 Animals^{2} to hold elements of G
 <1 Gbyte for 50,000 SNPs, 3000 bulls
 Iteration on genotype data
 Markers + animals
 <.1 Gbyte for 50,000 SNPs, 3000 bulls
 Little memory required for either

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 Inversion including G matrix
 Animals^{2} x markers to form G matrix
 Animals^{3} to invert selection index
 10 hours for 3000 bulls, 50,000 SNPs
 Iteration on genotype data
 Markers x animals x iterations
 16 hours for 1000 iterations
 .997 correlation with inversion

11

 Jacobi iteration
 Use previous round coefficients x solutions
 Adaptive underrelaxation
 Increase relax if convergence improving
 Decrease relax (each round) if diverging
 Solution convergence reasonable
 SD of change < .0001 after 350 rounds
 SD of change < .000001 after 1700 rounds

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 Daughter equivalents
 DE_{Total} = DE_{PA} + DE_{Prog} + DE_{YD}
+ DE_{G}
 DE_{G} is additional DE from genotype
 REL = DE_{total} / (DE_{Total} + k)
 Gains in reliability
 DE_{G} could be about 15 for Net Merit
 More for traits with low heritability
 Less for traits with high heritability

14

 Predictions from 50,000 SNPs using:
 Selection index equations, or
 Iteration on genotype data
 Predictions correlated by up to .9999
 Linear and nonlinear costs OK
 Convergence within 200 to 2500 rounds
 Nonlinear regression improved reliabilities
 Real data predictions available soon
