Strategies for inversion of the additive relationship matrix among genotyped animals

(1)

Strategies for inversion of

the addi0ve rela0onship matrix

among genotyped animals

P. Faux and N. Gengler

(2)

Introduc0on: The case of A

₂₂

vs. A



A

₂₂

= subpart of A whose inversion is required,

e.g. in ssGBLUP

(3)

Introduc0on: The case of A

₂₂

vs. A



A

₂₂

= subpart of A whose inversion is required,

e.g. in ssGBLUP

A

(4)

Introduc0on: The case of A

₂₂

vs. A



A

₂₂

= subpart of A whose inversion is required,

e.g. in ssGBLUP

A

₂₂

genotyped

non genotyped

ge

no

ty

pe

d

no

n

ge

no

ty

pe

d

(5)

Introduc0on: The case of A

₂₂

vs. A



The inverse of A is computed as a sum of

vector products

(Henderson, 1976)

A

_(i)

!1

=

A

(i!1)

!1

₀

"

0

0 #

$

%

&

'

(

+

!

(i)

!b

_(i)

1 #

$

%

&

'

(

! "

b

(i)

1 #

$%

&

'(

(6)

Introduc0on: The case of A

₂₂

vs. A



The inverse of A is computed as a sum of

vector products

T

_A(i)

!1

=

T

_A(i!1)

!1

₀

!b

"

_(i)

1 #

$

%

&

'

(

A

_(i)

!1

=

A

(i!1)

!1

₀

"

0

0 #

$

%

&

'

(

+

!

(i)

!b

_(i)

1 #

$

%

&

'

(

! "

b

(i)

1 #

$%

&

'(

D

!1

_A(i)

=

D

A(i!1)

!1

₀

"

0 _!

_(i)

#

$

%

&

'

(

A

_(i)

!1

(7)

Introduc0on: The case of A

₂₂

vs. A



A

₂₂

= subpart of A whose inversion is required,

e.g. in ssGBLUP

A

₂₂

genotyped

non genotyped

ge

no

ty

pe

d

no

n

ge

no

ty

pe

d

(8)

Sparsity in the inverse factor of A

₂₂



Example: An animal and its parents

C

(9)

Sparsity in the inverse factor of A

₂₂



Example: An animal and its parents

C

A

B

T

-‐1

_...

_A

_...

_B

_...

_C

_...

...

C

...

≠0

0 ?

(10)

Issues and Objec0ve



How sparse is the inverse of

A

₂₂

?

... How sparse is the inverse factor (

T

-‐1

_{)
of}

_A

22 ?



How a putaXve sparsity could be used in

computaXon of the inverse?



_{Main
objecXve:}

_{To
avoid
useless}

computa0ons

(11)

Sparsity in the inverse factor of A

₂₂



How to deal with more complex cases?



By a

comprehensive search in the pedigree



« SP Algorithm »



Explores pedigree brances and apply simple rules



Uses only pedigree and incidence vector



Returns a symbolic inverse factorizaXon

(12)

Sparsity in the inverse factor of A

₂₂



Some

performances

on diﬀerent sizes of

A

₂₂

:

Processor: Intel Xeon 64-‐bit

64Gb of RAM,

Clock speed: 3.16GHz

(13)

Strategies to take sparsity into account

1. Successive construcXon of the inverse

How to get b?

A

!1

_22(i)

=

A

22(i!1)

!1

₀

"

0

0 #

$

%

&

'

(

+

!

(i)

!b

_(i)

1 #

$

%

&

'

(

! "

b

(i)

1 #

$%

&

'(

1. b

_(i)

= A

_22(i!1)

!1

A

_22(i!1)

(:,1: i !1)

2. A

_22(i!1)

b

_(i)

= A

_22(i!1)

(:,1: i !1)

(14)

Strategies to take sparsity into account

1. RestricXng the product only to elements of b

diﬀerent from 0

b

_(i)

_{= A}

!1

_22(i!1)

_A

22(i!1)

(:,1: i !1) " x = Zy

≠0

=0

=

x

Z

y

x

Z

y

=

0

(15)

Strategies to take sparsity into account

2. Solving a linear system of lower size

A

_22(i!1)

b

_(i)

_{= A}

_22(i!1)

_{(:,1: i !1) " Zx = y}

=0

≠0

=0

=

Z

x

0 y

=

Z

x

y

(16)

Strategies to take sparsity into account

Processor: Intel Xeon 64-‐bit

8Gb of RAM,

Clock speed: 3 GHz

(17)



Order of A

₂₂

= Number of genotyped animals



Depends on the pedigree (depth, lines, ...)

(18)

Strategies to take sparsity into account

3. Storing the inverse of A

₂₂

from Xme to Xme

and updaXng this inverse only for recent

animals

A

_{22(t )}

!1

=

A

22(t!1)

!1

₀

"

0

0 #

$

%

&

'

(

+

!

(i)

!b

_(i)

1 #

$

%

&

'

(

! "

b

(i)

1 #

$%

&

'(

(19)

Strategies to take sparsity into account

A

_22(t)

genotyped

non genotyped

ge

no

ty

pe

d

no

n

ge

no

ty

pe

d

A

_22(t+1)

newly

genotyped

ne

w

ly

ge

no

ty

pe

d

(20)

Strategies to take sparsity into account

A

_22(t)

genotyped

non genotyped

ge

no

ty

pe

d

no

n

ge

no

ty

pe

d

A

_22(t+1)

newly

genotyped

ne

w

ly

ge

no

ty

pe

d

(21)

Take-‐home messages

1.  Sparsity pafern of the inverse of A

₂₂

can be

set up without matrix computaXons, even for

large matrices

(22)

Take-‐home messages

1.  Sparsity pafern of the inverse of A

₂₂

can be

set up without matrix computaXons, even for

large matrices

2.  Using sparsity reduces Xme for inversion, if

that inversion uses the inverse factor

(23)

Take-‐home messages

1.  Sparsity pafern of the inverse of A

₂₂

can be

set up without matrix computaXons, even for

large matrices

2.  Using sparsity reduces Xme for inversion, if

that inversion uses the inverse factor

3.  As the order of A

₂₂

increases, inversion

shrinks to solve mulXple small linear systems

that are idenXﬁed by SP algorithm

(24)

Acknowledgements

• Fonds NaXonal de la Recherche Luxembourg (FNR)

• CONVIS s.c.

• AssociaXon Wallonne de l’Elevage (AWE)