G-BLUP without inverting the genomic relationship matrix
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1 G-BLUP without inverting the genomic relationship matrix Per Madsen 1 and Jørgen Ødegård 2 1 Center for Quantitative Genetics and Genomics Department of Molecular Biology and Genetics, Aarhus University 2 AquaGen AS Genetics and breeding, Ås, Norway August 22, 2013
2 Outline
3 nterbull 2013 Traditional BLUP EBV s are obtained by solving Henderson s MME. [ X R 1 X X R 1 Z Z R 1 X Z R 1 Z + G 1 [ ˆβ [ X R 1 y Z R 1 y The coefficient matrix is symmetric and involves the inverse of a relationship matrix (G 1 )
4 nterbull 2013 Traditional BLUP EBV s are obtained by solving Henderson s MME. [ X R 1 X X R 1 Z Z R 1 X Z R 1 Z + G 1 [ ˆβ [ X R 1 y Z R 1 y The coefficient matrix is symmetric and involves the inverse of a relationship matrix (G 1 ) In a pedigree based additive model: G 1 (G 0 A) 1 G0 1 A 1
5 Traditional BLUP EBV s are obtained by solving Henderson s MME. [ X R 1 X X R 1 Z Z R 1 X Z R 1 Z + G 1 [ ˆβ [ X R 1 y Z R 1 y The coefficient matrix is symmetric and involves the inverse of a relationship matrix (G 1 ) In a pedigree based additive model: G 1 (G 0 A) 1 G0 1 A 1 In G-BLUP: G 1 (G 0 GSNP ) 1
6 Alternative formulation of MME As shown by Henderson (1984), the MME can be rearranged into an unsymmetric system by multiplying the random part with G G 0 G [ X R 1 X X R 1 Z GZ R 1 X G(Z R 1 Z + G 1 ) [ ˆβ [ X R 1 y GZ R 1 y Interbull 2013
7 Alternative formulation of MME As shown by Henderson (1984), the MME can be rearranged into an unsymmetric system by multiplying the random part with G G 0 G Simplification: [ X R 1 X X R 1 Z GZ R 1 X G(Z R 1 Z + G 1 ) [ ˆβ [ X R 1 y GZ R 1 y [ X R 1 X X R 1 Z GZ R 1 X GZ R 1 Z + I [ ˆβ [ X R 1 y GZ R 1 y
8 Alternative formulation of MME As shown by Henderson (1984), the MME can be rearranged into an unsymmetric system by multiplying the random part with G G 0 G Simplification: [ X R 1 X X R 1 Z GZ R 1 X G(Z R 1 Z + G 1 ) [ ˆβ [ X R 1 y GZ R 1 y [ X R 1 X X R 1 Z GZ R 1 X GZ R 1 Z + I G 1 disappear [ ˆβ [ X R 1 y GZ R 1 y
9 Alternative formulation of MME As shown by Henderson (1984), the MME can be rearranged into an unsymmetric system by multiplying the random part with G G 0 G Simplification: [ X R 1 X X R 1 Z GZ R 1 X G(Z R 1 Z + G 1 ) [ ˆβ [ X R 1 y GZ R 1 y [ X R 1 X X R 1 Z GZ R 1 X GZ R 1 Z + I G 1 disappear G do not need to be positive definite [ ˆβ [ X R 1 y GZ R 1 y
10 Solving strategy Due to the multiplication by G, the Genomic part of the unsymmetric MME, and will typically be the major part of the system
11 Solving strategy Due to the multiplication by G, the Genomic part of the unsymmetric MME, and will typically be the major part of the system Strategy for solving the unsymmetric MME 1: Setup LHS and RHS 2: if enough memory to hold an factorize the system then 3: Solve by direct methods (Can be performed by mulitcore LAPACK subroutines) 4: else 5: Use iterative methods (Gauss-Seidel, PCG, MINRES,...) 6: end if
12 Iterative solving strategy Rearranging the unsymmetric system as: [ X R 1 X 0 0 GZ R 1 Z + I [ ˆβ [ X R 1 y X R 1 Z GZ R 1 y GZ R 1 X ˆβ
13 Iterative solving strategy Rearranging the unsymmetric system as: [ X R 1 X 0 0 GZ R 1 Z + I [ ˆβ [ X R 1 y X R 1 Z GZ R 1 y GZ R 1 X ˆβ Simplification: [ X R 1 X 0 0 GZ R 1 Z + I [ ˆβ [ X R 1 (y Z) GZ R 1 (y X ˆβ)
14 Solving strategy Pseudo code for IOD solver for unsymmetric MME cd0 Initiate 0 (can be a null vector) while cd > ε do y i y Zu i 1 ˆ Compute β i by solve X R 1 X ˆβ i X R 1 y i (direct or iteratively) y i y X ˆβ i Compute i by solve (GZ R 1 Z + I )i GZ R 1 y i (iteratively) cd ui 1 u i u i end while
15 Status for implementation in the DMU-package Direct solver for unsymmetric MME implemented in DMU4
16 Status for implementation in the DMU-package Direct solver for unsymmetric MME implemented in DMU4 IOD solver for unsymmetric MME implemented in DMU5
17 s Solvers tested on the NAV G-BLUP model for Nordic Red Cattle Data G matrix DRP s (protein) for 3662 bulls 5287 animals
18 s Solvers tested on the NAV G-BLUP model for Nordic Red Cattle Data G matrix DRP s (protein) for 3662 bulls 5287 animals IOD solver # of iterations Symmetric MME 60 Global 173 Non-genomic part 173 Genomic part 570
19 s Solvers tested on the NAV G-BLUP model for Nordic Red Cattle Data G matrix DRP s (protein) for 3662 bulls 5287 animals IOD solver # of iterations Symmetric MME 60 Global 173 Non-genomic part 173 Genomic part 570 Solutions are identical
20 Advantages of using unsymmetric MME for G-BLUP: Inversion of the genomic relationship matrix is avoided
21 Advantages of using unsymmetric MME for G-BLUP: Inversion of the genomic relationship matrix is avoided Dimensions that can be handled increases by an order of magnitude
22 Advantages of using unsymmetric MME for G-BLUP: Inversion of the genomic relationship matrix is avoided Dimensions that can be handled increases by an order of magnitude Further improvements
23 Advantages of using unsymmetric MME for G-BLUP: Inversion of the genomic relationship matrix is avoided Dimensions that can be handled increases by an order of magnitude Further improvements As number of typed animals increases, it might be feasible to form elements in G as they are needed using massive parallel computation (GPU s?)
24 Advantages of using unsymmetric MME for G-BLUP: Inversion of the genomic relationship matrix is avoided Dimensions that can be handled increases by an order of magnitude Further improvements As number of typed animals increases, it might be feasible to form elements in G as they are needed using massive parallel computation (GPU s?) Replacing G by H (the One-Step relationship matrix) See EAAP presentation by Ødegård et. al
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