A recursive method of approximation of the inverse of genomic relationship matrix

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A recursive method of

approximation of the inverse of

genomic relationship matrix

P. Faux *,1, I. Misztal2, N. Gengler 1,3

1_{University of Liège, Gembloux Agro-Bio Tech, Belgium} 2_{University of Georgia, Animal and Dairy Science, Athens GA}

3_{National Fund for Scientific Research, Brussels, Belgium}

2011 ADSA-ASAS Joint Annual Meeting New Orleans, LA, July 10-14

* Supported by the National

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Introduction

• _{Inverse of genomic relationship matrix (G)}

useful in different genomic evaluations:

single-step, multiple-steps

• _{Inversion becomes time-consuming when}

number of genotyped animals >50,000

• _{A good approximation of G inverse will}

become useful

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Methods: Approximated

inverse of G

• _{Decomposition of A}-1:

A

-1

=(T

-1

)’ . D

-1

. T

-1

• _{We assume the following model for G}-1:

G

-1

**=(T*)’ . (D**

-1

**)* . T* + E**

Or,

(G

-1

**)=(T)’ . (D**

-1

)* . T*

• _{2 approximations have to be done: T* and (D}-1)*

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Methods: Case of A

-1

• _{Animal a, of sire s and dam d:}

2011 ADSA-ASAS Joint Annual Meeting, New Orleans, July 10-14

s ... d ... a ... s ... d ... a ... 1 1 1 1 -.5 -.5 1 1 A T-1 s ... d ... a ... s ... d ... a ... -.5 -.5



A

x

= -

b

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Methods: Case of G

-1

s ... d ... a ... s ... d ... a ... 1 1 1 1 1 1 G T* s ... d ... a ... s ... d ... a ...

 = -

x

b

A

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Methods: Case of G

-1

s j d ... a ... s j d ... a ... 1 1 1 1 1 1 G T* s j d ... a ... s j d ... a ...

 = -

x

b

A

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Methods: Approximation of

D

-1

(G

-1

**)=(T)’ . (D**

-1

)* . T*



**D = T* . G**

**. (T*)’**

• _{D close to diagonal  2 options:}

1. Invert only diagonal elements  (D-1)* is a diagonal

matrix

2. Apply the same rules as for approximation of G-1 but

for D-1  (D-1)* is a sparse matrix

(D

-1

**)*=(T**

2

*)’ . (D

2-1

)

* . T

2

*

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Methods: Recursive

approximation

• _{Same recursion may be repeated n times on the} “remaining D”

(G

-1

)*

n

=(T*

1

)’ . ... . (T*

n

)’ . (D

n-1

)* . T*

n

. ... . T*

1 • _{At each round, new threshold p}

• _{2 computational bottle-necks are shown in this} equation

• _{Algorithm may be implemented in different ways}

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Examples: Material

• _{Dairy bulls data set:}

– _{1,718 genotyped bulls}

– _{38,416 SNP}

• _{Single-step genomic evaluations on Final Score, each} evaluation uses a different (G-1)*

• _{Construction of G:}

– _G=ZZ’/d

– Both diagonals and off-diagonals of G centered on A₂₂

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Examples: Quality of

approximation

• _{Mean square difference (MSD) between G}-1 and (G-1)*

• MSD between G-1 _{and A}

22-1: 81.49*10-4

Round p MSD 1 0.21 66.53 * 10-4 2 0.017 34.87 * 10-4 3 0.009 16.81 * 10-4 4 0.005 5.97 * 10-4 5 0.003 1.82 * 10-4

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Examples: Incidence on

evaluation

• _{Correlations r between EBV for genotyped animals}

• r between G-1 and A

22-1: 0.76

Round p r 1 0.21 0.79 2 0.017 0.91 3 0.009 0.97 4 0.005 0.99 5 0.003 1.00

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Examples: Sparsity

• Percentage of zeros in lower off-diagonal part of T_f

Round p % of 0 1 0.21 92.35 2 0.017 10.78 3 0.009 2.28 4 0.005 0.86 5 0.003 0.49

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Examples: Others results

• Use of this algorithm to compute (A

₂₂-1

)*

• Closeness with A

₂₂-1

: from 2*10

-4

to 3*10

-5

• _{No incidence on evaluations after 3 rounds}

• _{93% of 0 after 1 round, ... up to ~80%}

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Discussion

• _{Consumption of time}

• _{Savings on both bottle-necks}

• _{Might be interesting in some cases}

• _{Construction of consistent relationships and}

covariances

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Acknowledgements

• _{Animal and Dairy Science (ADS) Department of} University of Georgia (UGA), for hosting and

advising (Mrs H. Wang, Dr S. Tsuruta, Dr I. Aguilar) • _{Animal Breeding and Genetics Group of Animal}

Science Unit of Gembloux Agro-Bio Tech of University of Liège (ULg – Gx ABT)

• _{Holstein Association USA Inc. for providing data}

A recursive method of approximation of the inverse of genomic relationship matrix