Taskinen e al. Gene Sel E ol (2017) 49:36
DOI 10.1186/s12711-017-0310-9
RESEARCH ARTICLE
Single-s ep SNP-BLUP wi hon- he- ly
impu ed geno ypes and esidual polygenic
e ec s
Ma i Taskinen* , Esa A. Män ysaa i and Ismo S andén
Abs ac
Backg ound: Single-s ep genomic bes linea unbiased p edic ion (BLUP) e alua ion combines ela ionship in o ma-
ion om pedig ee and genomic ma ke da a. The inclusion o he genomic in o ma ion in o mixed model equa ions
equi es he in e se o he combined ela ionship ma ix
H
, which has a dense ma ix block o geno yped animals.
Me hods: To a oid in e sion o dense ma ices, single-s ep genomic BLUP can be ans o med o single-s ep single
nucleo ide polymo phism BLUP (SNP-BLUP) which ha e obse ed and impu ed ma ke coe icien s. Simple block
LDL ype decomposi ions o he single-s ep ela ionship ma ix
H
we e de i ed o ob ain di e en ypes o linea ly
equi alen single-s ep genomic mixed model equa ions wi h di e en se s o epa ame ized andom e ec s. Fo
non-geno yped animals, he impu ed ma ke coe icien e ms in he single-s ep SNP-BLUP we e calcula ed on- he- ly
du ing he i e a i e solu ion using spa se ma ix decomposi ions wi hou s o ing he impu ed geno ypes. Residual
polygenic e ec s we e added o geno yped animals and ansmi ed o non-geno yped animals using ela ionship
coe icien s ha a e simila o impu ed geno ypes. The ela ionships we e u he o hogonalized o imp o e con e -
gence o i e a i e me hods.
Resul s: All p esen ed single-s ep SNP-BLUP models can be sol ed e icien ly using i e a i e me hods ha ely on
i e a ion on da a and spa se ma ix app oaches. The e iciency, accu acy and i e a ion con e gence o he de i ed
mixed model equa ions we e es ed wi h a small da ase ha included 73,579 animals o which 2885 we e geno yped
wi h 37,526 SNPs.
Conclusions: In e sion o he la ge and dense genomic ela ionship ma ix was a oided in single-s ep e alua ion
by using ully o hogonalized single-s ep SNP-BLUP o mula ions. The numbe o i e a ions un il con e gence was
smalle in single-s ep SNP-BLUP o mula ions han in he o iginal single-s ep GBLUP when he i abili y was low, bu
inc eased abo e ha o he o iginal single-s ep when he i abili y was high.
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Backg ound
The i s model o simul aneously combine genomic
in o ma ion wi h non-geno yped animal in o ma ion
was single-s ep bes linea unbiased p edic ion (BLUP)
[1, 2] o ssGBLUP. When he numbe o geno yped
animals is la ge, ssGBLUP may become compu a ion-
ally in easible o p ac ical pu poses because i equi es
he in e ses o dense ma ices o sizeequal o he num-
be o geno yped animals, pa icula ly he in e se o he
genomic ela ionship ma ix
G−1
g
. In addi ion, ma ix
Gg
can be singula when he numbe o geno yped indi idu-
als exceeds he numbe o ma ke s. Compu a ional chal-
lenges may ha e been a eason o he slow adop ion o
ssGBLUP ins ead o a mul i-s ep app oach. A compu a-
ionally scalable al e na i e, he algo i hm o p o en and
young (APY), has been sugges ed [3]. In APY, a spa se
G−1
APY
app oxima ion o he
G−1
g
ma ix is c ea ed by se -
ing a diagonal ma ix o a g oup o indi iduals. In p ac-
ice, APY has been able o educe compu a ional cos s
signi ican ly when he numbe o geno yped animals is
e y la ge [4, 5]. Howe e , di e en se s o co e animals
Open Access
G
ene ics
S
elec ion
E olu ion
*Co espondence: ma i. askinen@luke. i
Na u al Resou ces Ins i u e Finland (Luke), Mylly ie 1, Jokioinen, Finland
Page 2 o 15
Taskinen e al. Gene Sel E ol (2017) 49:36
in APY will gi e di e en e alua ions, which may a ec
selec ion decisions.
An al e na i e o mula ion called he eina e single-
s ep single nucleo ide polymo phism BLUP (ssSNP-
BLUP) [6] o e comes some o he majo compu a ional
challenges in ssGBLUP. In pa icula , he e is no need o
cons uc o in e he genomic ela ionship ma ix
Gg
.
The o iginal idea in ssSNP-BLUP ci cum en s he p ob-
lems in ssGBLUP by p edic ing o impu ing geno ypes
o non-geno yped animals, and elying on compu a-
ionally less demanding SNP-based p edic ion ins ead o
b eeding alue based p edic ion. An addi ional ad an age
is ha he ma ke e ec solu ions a e easie o use o
in e im p edic ions.
The ssSNP-BLUP has some compu a ional challenges
as well. A simple implemen a ion o ssSNP-BLUP gen-
e a es and s o es geno ypes o all SNPs o all animals.
This will lead o e y la ge disk s o age and as eading
equi emen s ha will p ohibi use o he app oach o
la ge popula ions. An al e na i e is o make he equi ed
geno ype impu a ions “on- he- ly” ins ead o s o ing he
e y la ge amoun o impu ed geno ypes o ile. Fo p ac-
ical pu poses, he on- he- ly impu a ion equi es a as
compu ing app oach o he impu a ion s ep and/o as
con e gence o he i e a i e me hod.
The o iginal desc ip ion o ssSNP-BLUP app oach
p esen s a wide ange o models han ssGBLUP such as
he use o a numbe o di e en Bayesian SNP model o -
mula ions [6]. In ssGBLUP, in con as o ssSNP-BLUP,
i is ypical o include esidual polygenic (RPG) in o ma-
ion o enhance genomic in o ma ion by including ped-
ig ee-based ela ionships in o he genomic ela ionship
ma ix. Thus, he genomic ela ionship ma ix is usually
“adjus ed” wi h pa o he pedig ee ela ionship ma ix
ei he o supply mo e addi i e ela ionship in o ma-
ion o simply o make he genomic ela ionship ma ix
in e ible. In he o iginal ssSNP-BLUP o mula ion, his
in o ma ion has no been included. The compu a ional
challenges o ssGBLUP has led o he in oduc ion o
se e al equi alen models (e.g., [7–9]). Howe e , hese
al e na i e app oaches ha e had poo con e gence by
i e a i e me hods [7, 9]. One eason is ha he co a i-
ance s uc u es ha e a poo e condi ion numbe which is
a a io o he la ges and smalles eigen alues and is used
o measu e nume ical s abili y [9]. Some o he al e na-
i e e sions o ssGBLUP ha e had SNP e ec s. Because
ssSNP-BLUP is equi alen o ssGBLUP and ssGBLUP has
many equi alen o ms, al e na i e o mula ions o mixed
model equa ions (MME) can be de i ed o ssSNP-BLUP
as well.
In his pape , simple block LDL ype decomposi ions
[10] o he ssGBLUP ela ionship ma ix a e de i ed
o ob ain se e al linea ly equi alen MME. This allows
de i a ion and es ing o se e al equi alen ssSNP-BLUP
MME ha a oid making and s o ing he impu ed geno-
ypes. In his pape , an explici impu a ion o geno ypes
is no needed bu ins ead we apply spa se ma ix decom-
posi ions o a ain pedig ee-based eg essions o e alu-
a ions o geno yped animals on non-geno yped animals
using spa se ma ix decomposi ions. Accu acy and i e a-
ion con e gence o he de i ed equi alen MME a e
es ed on a small No dic dai y ca le da ase .
Me hods
Fo any model, an in ini e numbe o equi alen mod-
els exis . In he ollowing, i s we ecall he concep o
equi alen models and how equi alen models can be
made by a aching co a iance in o ma ion o he model
design ma ix. As an example, equi alence o genomic
BLUP (GBLUP) and SNP-BLUP is p esen ed. Then, ssG-
BLUP is ecalled, and i s co a iance s uc u e o mula ed
using LDL decomposi ion. Equi alen MME a e de i ed
whe e in o ma ion om he genomic ela ionship ma ix
a e a ached o he model design ma ix simila ly as
shown o GBLUP and SNP-BLUP, esul ing in ssSNP-
BLUP. Finally, MME ha ing o hogonalized andom
e ec s o diagonal co a iance s uc u es a e de i ed in
o de o imp o e he con e gence in i e a i e me hods.
A small da ase is used o illus a e pe o mance o he
de i ed equi alen models.
Linea ly equi alen models
Two mixed linea e ec s models, i.e.
wi h he same obse a ions
y
bu di e en ixed (
b
and

b
) and andom e ec s (
u
and

u
), esidual e o s (
e
and

e
), model ma ices (
X
,
Z
,

X
, and

Z
) and a iance s uc-
u es (
G
,
R
,

G
, and

R
), a e said o be linea ly equi alen
models [11–13] i he expec ed alues and he a iances
o he obse a ions a e equal. Thus, models (1) and (2)
a e equi alen i :
Sepa a e equa ions o  ixed and andom e ec s
Mixed model(1) can be sol ed om sepa a e equa ions
o ixed and andom e ec s [14] as:
whe e ma ix
V
needs o be in e ible. The size o ma ix
V
is he numbe o obse a ions, which can be e y
(1)
y=Xb +Zu +e,Va (u)=G,Va (e)=R
(2)
y=

X

b+

Z
u+

e,Va (
u)=

G,Va (
e)=

R
(3)

Xb =

X

b,
V=ZGZ′+R=

Z

G

Z′+

R=

V
.
(4)

b
=(X′V−1X)
−1
X′V−1y

u=GZ′V−
1
(y−X

b)
,
Page 3 o 15
Taskinen e al. Gene Sel E ol (2017) 49:36
la ge, and, he e o e, sol ing he mixed model using his
me hod is seldom easible in p ac ice.
Mixed model equa ions
In p ac ice, Eq.(4) can be sol ed by Hende son’s MME
[14] as ollows:
whe e he a iance ma ix o he andom e ec s
G
and he
esidual a iance ma ix
R
need o be in e ible. MME(5)
usually lead o mo e spa se ma ix sys ems han Eq.(4).
Equi alen models byspli ing he a iance ma ix
Suppose he a iance ma ix
G
can be exp essed as a
ma ix p oduc as ollows:
whe e
M
is ec angula and

G
is an in e ible squa e
ma ix. He e, he ma ix

G
could be, o example, an
iden i y ma ix and could ha e di e en dimensions han
he
G
ma ix.
Ma ices
G
and
Z
a e always oge he in Eq.(4). This
allows us o e-pa ame ize he model as:
and he ea e :
whe e he equi alen model has he same esidual a i-
ance ma ix (

R=R
), he new model ma ix

Z=ZM
, and
he andom e ec s

u
a e de ined as:
Now, acco ding o Eq. (3), he quan i ies wi h a ilde
oge he wi h he same obse a ion ec o
y
, he o iginal
ixed e ec s (

b=

b
), and design ma ix (

X=X
) o m a
linea ly equi alen model(2).
Linea ly equi alen MME
O iginal ixed e ec s

b
and he new andom e ec s

u
o
he linea ly equi alen model can be sol ed, simila ly as
in Eq.(4), om:
o om he co esponding MME(5):
(5)

X
′
R
−1
XX
′
R
−1
Z
Z
′
R
−
1XZ
′
R
−
1Z
+
G
−
1

b

u
=
X
′
R
−1
y
Z
′
R
−
1y
,
(6)
G=M

GM′,
(7)
V
=ZGZ
′
+R
=ZM

GM′Z′+R
=

Z

G

Z′+

R=

V,
(8)

u
=M

GM′Z′V−1(y−X

b)
=M

G

Z′
V−
1
(y−X

b)=M
u,
(9)

u=

G

Z
′
V
−
1(y
−
X

b)
.
(10)

b
=(X′

V−1X)
−1
X′

V−1y

u=

G

Z′
V−
1
(y−X

b),
Because o he equi alence

u=M
u
in Eq.(8), he o igi-
nal andom e ec s

u
can be ob ained om he solu ion o
he linea ly equi alen MME (11) by p e-mul iplying he
new andom e ec s

u
wi h ma ix
M
, i.e.
whe e iden i y ma ix
Ib
has dimension o

b
. No e ha
he in e se o he MME ma ix in Eq.(12) is usually no
e alua ed explici ly, bu a he , he co esponding linea
ma ix equa ion is sol ed using ei he di ec o i e a i e
solu ion me hods.
The o iginal MME (5) can, hus, be sol ed om a
modi ied linea ly equi alen MME (11) whe e he num-
be o andom e ec s in

u
could be smalle o la ge han
in

u
, he a iance s uc u e

G
could be easie o ob ain
o in e han he o iginal
G
, o he new ma ix sys em
could be o he wise nume ically mo e e icien .
P esen a ion o GBLUP andSNP‑BLUP
As an example, conside single- ai GBLUP MME
[15]. The a iance ma ix o andom e ec s
u
is based
on a genomic ela ionship ma ix (
Gg
), which desc ibes
he genomic ela ionships be ween indi iduals, i.e.
Va
(u)
=
σ
2
u
G
g
. The genomic ela ionship ma ix is usu-
ally ully dense and inc eases in o de as he numbe
o geno yped animals inc eases. The in e ed genomic
ela ionship ma ix
Gg
−1
is needed in he solu ion o he
GBLUP MME:
whe e a single ai case is assumed o simplici y,
R=
σ
2
eI
, and
=
σ
2
e
σ
2
u
.
The e a e many ways o cons uc
Gg
. Assume he
genomic ela ionship ma ix
Gg
can be exp essed, in sim-
pli ied o m, using (cen e ed and scaled) ma ke ma ix
Zm
[15] so ha :
whe e
and
Im
is an iden i y ma ix o size equal o he numbe
o ma ke s. Now, a linea ly equi alen MME(11), al e -
na i e o GBLUP MME(13), can be de i ed and o iginal
e ec s sol ed simila ly as in(12) so ha :
(11)

X′R−1XX
′R−1

Z

Z′R−
1
X
Z′R−
1

Z+
G−
1

b

u
=
X′R−1y

Z
′
R
−
1y
.
(12)

b

u

=

Ib0
0M

X′R−1XX
′R−1

X

Z′R−1X

Z′R−1

Z+

G−1
−1
X′R−1y

Z′R−1y
,
(13)

b

u
=
X′XX
′Z
Z′XZ
′Z
+
G−1
g−1
X′y
Z′y
,
(14)
Gg=
Z
m
Z
′
m=
M

GM
′,
(15)
M=Zmand

G=Im,
Page 4 o 15
Taskinen e al. Gene Sel E ol (2017) 49:36
whe e

Z=ZZm
and

is he same a iance a io as in
Eq. (13). This equi alen MME sys em, known as he
SNP-BLUP [16], has ma ke s as andom e ec s ins ead
o indi iduals. Random e ec s a e also “o hogonalized”,
and, hus, in e sion o he dense genomic ela ionship
ma ix
Gg
is a oided.
No e ha , i he ma ke e ec s ha e unequal a iances,
he ela ionship ma ix can be build as:
whe e
and he ma ix
B
is a diagonal co a iance ma ix desc ib-
ing he a iances o di e en ma ke e ec s. Now he
solu ion o SNP-BLUP becomes:
Single‑s ep SNP‑BLUP
In ssGBLUP [1, 2], some indi iduals ha e genomic in o -
ma ion while some ha e only pedig ee in o ma ion. The
model o ssGBLUP is a special case o mixed e ec mod-
els, whe e he be ween-animal ela ionships a e modeled
ia he agg ega ed ela ionship ma ix
H
[1, 2]. The ela-
ionships in
H
a e desc ibed by he pedig ee-based ela-
ionship ma ix
A
, and he genomic ela ionship ma ix
among geno yped animals by
Gg
. Rela ionships among
non-geno yped indi iduals a e cons uc ed om he ped-
ig ee bu modi ied acco ding o he ela ionships among
geno yped animals.
Assuming ha he non-geno yped indi iduals a e
deno ed wi h sub- and supe -sc ip s 1 and he geno yped
indi iduals by sub- and supe -sc ip s 2, he pedig ee ela-
ionship ma ix and i s in e se a e he ollowing:
Assume, as be o e, ha he genomic ela ionship ma ix
has he o m
Gg=
Z
m
Z
′
m
. I he geno yped popula ion
con ains iden ical wins, i.e. clones, o i he e a e mo e
indi iduals han ma ke s, he genomic ela ionship
ma ix
Gg
becomes singula and he usual MME(5) can-
no be cons uc ed. Hence,
Gg
is commonly adjus ed by
eg essing i owa ds he pedig ee ela ionship ma ix
A22
wi h:

b

u
=
Ib0
0Zm

X′XX
′

Z

Z′X
Z′
Z+

Im−1
X′y

Z
′
y
,
(16)
Gg=
Z
m
BZ
′
m=
M

GM
′,
(17)
M=Zmand

G=B,

b

u
=
Ib0
0Zm

X′XX
′

Z

Z′X
Z′
Z+

B−
1
−1
X′y

Z
′
y
.
(18)
A=
A11 A12
A21 A22

and A−1
=
A
11
A
12
A21 A22
.
(19)
Gw=wA22 +(1−w)Gg,
whe e
w
is a scala weigh be ween 0 and 1 and can be
in e p e ed as he ela i e weigh on he polygenic e ec
[2, 15].
Single‑s ep ela ionship ma ix
The in e se o he ssGBLUP a iance ma ix
H
is [1, 2]:
Using block LDL decomposi ion, i is equal o:
whe e
I1
and
I2
a e iden i y ma ices o size equal o
he numbe o non-geno yped and geno yped animals,
espec i ely. Fo impu a ion o geno ypes:
is a eg ession p edic ion o an impu a ion ope a o ha
expands he genomic ela ionship in o ma ion om he
geno yped o he non-geno yped indi iduals [2, 6, 17]. By
in e ing he LDL decomposi ion o Eq.(22), he a iance
ma ix
H
has a simila decomposi ion:
By using block ma ix in e sion iden i ies (23) and
o he “impu ed”
A22
ma ix o
Gw
(19)
he a iance ma ix
H
(24) can be al e na i ely exp essed
as:
whe e
Gimp
is an impu ed genomic ela ionship ma ix as
ollows:
No e ha his ope a es on
Gg
ins ead o
Gw
. Also, no e
ha
Gimp
has a size equal o he numbe o all animals.
Geno yped animals ha e obse ed ma ke da a bu non-
geno yped animals ha e impu ed ma ke da a.
(20)
H
−1
=
A−1
+00
0G
w−
1
−
(A
22
)
−
1

(21)
=
A
11
A
12
A
21
A
22
+Gw−
1
−(A22)−
1
.
(22)
H
−1
=
I10
−
A′
imp
I2

A
11
0
0G
w
−
1

I1−Aimp
0I
2
,
(23)
Aimp =
A12(A22)
−
1
=−
(A11)
−1
A
12
(24)
H=
I1Aimp
0I
2

(A11)
−1
0
0G
w
I10
A′
imp
I2
.
(25)
(A
11
)
−1
=A11 −A12(A22)−
1
A21
(26)

Aimp
I2

A22

A′
imp I2
=
A
−
(A11)
−1
0
00
,
(27)
H=
(1
−
w)

(A11)
−1
0
00
+
wA
+
(1
−
w)Gimp
,
(28)
G
imp
=
Aimp
I2

Gg

A′
imp I2
.
Page 5 o 15
Taskinen e al. Gene Sel E ol (2017) 49:36
Equi alen single‑s ep MME
In he ollowing, he single-s ep ela ionship ma ix
H
will be exp essed as six di e en decomposi ions equi a-
len o Eq.(6):
o di e en ma ices
Mi
and

Gi
. All o hese lead o a
linea ly equi alen ssGBLUP MME sys em o new se s
o andom e ec s wi h, po en ially, di e en nume ical
p ope ies. The linea ly equi alen MME o hese modi-
ied se s o andom e ec s a e simila o Eq.(11) and he
o iginal e ec s can be sol ed simila ly as in Eq.(12):
whe e

Zi=ZMi
and a single ai case is assumed.
No e ha in hese equi alen MME, squa e ma ix

Gi
ep esen s he co a iance s uc u e o he epa a-
me ized andom e ec s,
Mi
hasa ow o each o iginal
andom e ec in
u
o change he model ma ix
Z
, and

Zi
is he ede ined model ma ix.
In o de o de i e linea ly equi alen MME o mul iple
ai cases:
whe e
⊗
is he K onecke p oduc , he gene ic (co) a i-
ance ma ix
G0
o size numbe o ai s is assumed o
ha e a decomposi ion:
Fo example, a simple case would ha e
M0
as iden i y
ma ix and

G0
as
G0
. Now he a iance ma ix
G
can be
exp essed using he decomposi ions o he single-s ep
ela ionship ma ix(29) simila ly as in Eq.(6):
whe e
E ec s can hen be sol ed om linea ly equi alen mul i-
ple ai MME (12):
whe e

Zi=Z(M0⊗Mi)
.
(29)
H=Mi

GiM′
i
,i
=
1,
..., 6,

b

u

=

Ib0
0M
i

X′XX
′

Zi

Z′
iX
Z′
i
Zi+


G−
1
i−1
X′y

Z
′
i
y
,
(30)
G=G0⊗H,
(31)
G0=M0

G0M′
0.
(32)
G=G0⊗H=(M0

G0M′
0)⊗(Mi

GiM′
i)
(33)
=(M0⊗Mi)(

G0⊗

Gi)(M0⊗Mi)′
(34)
=M

GM′,
(35)
M=M0⊗Miand

G=

G0⊗

Gi.


b

u

=

Ib0
0M
0⊗Mi

×

X′R−1XX
′R−1

Zi

Z′
iR−1X

Z′
iR−1

Zi+

G−1
0⊗

G−1
i

−1

X′R−1y

Z′
iR−1y
,
Basic equi alen ssGBLUP MME The LDL decom-
posi ion(24) can be used di ec ly o build he i s lin-
ea ly equi alen ssGBLUP MME o his pape using
H=M1

G1M′
1
whe e:
F om he modi ied ela ionship ma ix

G1
(36), i can be
seen ha his basic equi alen ssGBLUP MME has andom
e ec s o non-geno yped animals wi h a iances
(A
11
)
−1
and o geno yped animals wi h a iances o he adjus ed
genomic ela ionship ma ix
Gw
. The numbe o e ec s in
he sys em is, hus, he same as in he o iginal ssGBLUP.
Basic RPG ssSNP‑BLUP MME O he linea ly equi a-
len ssGBLUP MME o he o m(29) can be de i ed as
well. No e ha he adjus ed genomic ela ionship ma ix
Gw
in Eq.(19) can be exp essed by ma ix p oduc s as
ollows:
whe e
Gg=
Z
m
Z
′
m
. The second linea ly equi alen ssGB-
LUP MME can be buil by subs i u ing
Gw
in Eq. (38) o
Eq. (36):
In his o m, we a oid he in e se o
Gg
in he MME (11).
The coe icien s
w
and
(1−w)
we e also spli using squa e
oo s o ma ix
M2
so ha he new a iance ma ix

G2
can be in e ed e en when
w
is 0 o 1. The i s g oup o
new andom e ec s in Eq.(39), o he non-geno yped
animals, is he same as in he i s , basic equi alen ssG-
BLUP in Eq.(36). Howe e , he geno yped animals now
ha e andom e ec s ela ed h ough he a iance ma ix
A22
. E ec s in his second e ec g oup can be seen as
esidual polygenic e ec s ha can desc ibe e ec s ha
he ma ke e ec s a e unable o model [8]. The hi d
g oup o andom e ec s a e he ma ke e ec s as in SNP-
BLUP(15) and so, his decomposi ion(39) can be called
basic RPG ssSNP‑BLUP MME. Compa ed o he o iginal
ssGBLUP, he second equi alen MME has ma ke e ec s
in addi ion o he animal e ec s.
(36)
M
1
=
I1Aimp
0I
2

and

G1
=
(A11)
−1
0
0G
w.
(37)
G
w
=
I2I2

wA22 0
0(1
−
w)Gg

I2
I2

(38)
=
I2Zm

wA22 0
0(1
−
w)Im

I2
Z′
m,
(39)
M
2=
�
I1
√
wAimp
√
1−wAimpZm
0√wI2√1−wZm
�
�
G2=

(A11)−100
0A
22 0
0 0I
m


.

Page 6 o 15
Taskinen e al. Gene Sel E ol (2017) 49:36
Expanded RPG ssSNP‑BLUP MME A hi d linea ly
equi alen MME o o m(29) can be de i ed om he
al e na i e exp ession o ma ix
H
in Eq. (27) and by
spli ing
Gg=
Z
m
Z
′
m
in Eq.(28) as:
He e, ma ices
E1
and
E2
a e ec angula spa se inci-
dence ma ices ha selec he subse s o non-geno yped
and geno yped animals, espec i ely, om he
A
ma ix.
Bo h
E1
and
E2
ha e he same numbe o columns, i.e.
numbe o all animals. Ma ix
E1
has a ow o each non-
geno yped and ma ix
E2
o each geno yped animal co -
esponding o animal’s column in ma ices
Z1
and
Z2
o
Each ow o bo h
E1
and
E2
has only one non-ze o ele-
men , a alue one a he column co esponding o ha
animal’s loca ion among all o he animals. Hence, when
ows and columns o he ma ices o all animals a e in he
same o de as in ma ix
A
in Eq.(18), ma ix

E1
E2

is an
iden i y ma ix o he size o all animals.
The hi d equi alen MME (40) has h ee g oups o
e ec s simila o he second MME(39). The hi d e ec
g oup has, again, he o hogonal ma ke e ec s, and so
his o mula ion is a ssSNP-BLUP as well. The i s e ec
g oup, o he non-geno yped animals has, howe e , a
cons an mul iplie
√1−w
. Also, he second g oup,
ela ed h ough he pedig ee ela ionship ma ix
A
, has
now e ec s o all animals, and no jus o he geno yped
animals. Thus, in his hi d expanded RPG ssSNP‑BLUP
MME, he non-geno yped animals ha e wo se s o an-
dom e ec s.
Special cases o equi alen ssGBLUP MME These h ee
equi alen MME, (36), (39) and (40), will app oach he
usual animal model when
w→1
. A he limi (
w=1
), he
expanded RPG ssSNP-BLUP MME 3 (40) has clea ly he
ecognizable co a iance s uc u e o
A
. The basic equi -
alen ssGBLUP MME1(36) and he basic RPG ssSNP-
BLUP MME 2(39) a e models whe e he geno yped ani-
mals ac as base animals and he non-geno yped animals
a e eg essed on hem.
Fo he o he di ec ion o
w→0
, he basic equi alen
ssGBLUP MME 1(36) con e ges o an al e na i e p es-
en a ion o he s anda d ssGBLUP. Howe e , i di ides
(40)
M
3=
�√
1−wI1
√
wE1
√
1−wAimpZm
0√wE2√1−wZm
�
�
G3=

(A11)−100
0 A0
0 0I
m


.
(41)
Z=[Z1Z2].
he b eeding alues o non-geno yped animals in o
eg essions on geno yped animals and in o non-impu ed
b eeding alues ha a e no condi ional on hem. Simi-
la ly, a he limi
w=0
he basic and he expanded RPG
ssSNP-BLUP MME, 2(39) and 3(40), coincide wi h he
simple ssSNP-BLUP wi hou esidual polygenic e ec s.
Fo example, in he single ai case, he MME coe icien
ma ix o he basic RPG ssSNP-BLUP MME2 (39) is
whe e
W=(Z
1
Aimp +Z
2
)Zm
.
In he case whe e
w=0
, i.e. he e is no adjus men o
he genomic ela ionship ma ix and, he e o e, no esid-
ual polygenic e ec s, he basic and he expanded RPG
ssSNP-BLUP MME, 2 (39) and 3(40), a e essen ially he
same MME as was de i ed by Fe nando e al.[6]. The
main di e ences a e ha hey ha e mo ed he cen e -
ing e m o he ma ke ma ix
Zm
in o an addi ional ixed
e ec , and hey p oposed o sol e he MME using Bayes-
ian eg ession.
E icien implemen a ion
The h ee linea ly equi alen MME based on Eqs.(36),
(39), and (40) con ain in e ed and non-in e ed e ms o
he pedig ee ela ionship ma ix
A
. In an e icien se up
o sol e hese MME, all hese e ms can be exp essed, o
modi ied in o a o m ha can be exp essed, wi h spa se
ma ices o spa se decomposi ions o spa se ma ices.
This is expec ed o gi e h ee e icien implemen a ions
o hese MME. In p ac ice, ma ix equa ions o hese
MME a e assumed o be sol ed i e a i ely by he p econ-
di ioned conjuga e g adien (PCG) algo i hm. Then, only
a ma ix- ec o p oduc o he MME coe icien ma ix
imes a ec o is pe o med once e e y i e a ion.
Spa se ma ices anddecomposi ions
The in e se o he pedig ee ela ionship ma ix
A−1
can
be exp essed e icien ly [18] as:
whe e animals a e so ed in ( e e sed) age o de om he
younges o he oldes using spa se pe mu a ion ma ix
Q′
, so ha ma ix
L=
(I
−
1
2
P)
′
D
1
2
becomes a lowe i-
angula ma ix in
Q′A−1Q=LL′
. The diagonal ma ix
D
has alues
4/(4−k−Fs)
whe e
k
is he numbe o
known pa en s and
Fs
is he sum o pa en inb eeding
coe icien s. In he “pa en al ma ix”
P
on ow
i
, he e a e
1s in columns co esponding o pa en s o animal
i
. The
pa en al ma ix can be in e p e ed, oge he wi h iden i y
(42)




X
′
XX
′
Z1
0X
′
W
Z′
1XZ
′
1Z1+A11
0Z
′
1W
00A22−10
W′XW
′Z10W
′W
+
Im




,
(43)
A
−1
=
Q

I
−
1
2P
′
D

I
−
1
2P

Q′
=
QLL′Q′
,
Page 7 o 15
Taskinen e al. Gene Sel E ol (2017) 49:36
ma ix
I
, as a e y spa se lowe iangula “Cholesky”
ma ix(
L
)[18].
The pedig ee ela ionship ma ix
A
can be exp essed as
he in e se o i s in e se,
and so he subma ices o he pedig ee ela ionship
ma ix and i s in e se(18) can be ob ained by selec ing
he app op ia e ows and columns as ollows:
whe e
i,j=1, 2
. Ma ix- ec o p oduc s
Aijx
and
Aijx
can be e icien ly compu ed using hese decomposi-
ions [19, 20]. The subma ices
Aij
a e e y spa se, so
hey could al e na i ely be exp essed as sepa a e spa se
ma ices.
The in e se o
A−1
o pa icula pa s o i (e.g.
(A
11
)
−1
)
a e, howe e , in gene al non-spa se and, hus, hese
in e se ma ix e ms should ne e be compu ed explic-
i ly. The spa se subma ix
A11
can be exp essed using
spa si y p ese ing Cholesky ac o iza ion so ha :
whe e ma ix
L1
is spa se lowe iangula and
Q1
spa se pe mu a ion ma ix. No e ha ma ix
L1
has o
be compu ed explici ly, as opposed o ma ix
L
in Eqs.
(43) o(46). Ma ix
L1
has some mo e ill-ins compa ed
o
A11
bu is s ill e y spa se and e icien in use. In [4], i
was demons a ed ha he compu a ions emain a o d-
able e en when he da ase size g ows.
Compu a ions in ol ing ma ix in e sions o
A−1
o
pa s o i can be ans o med in o solu ions o spa se
ma ix equa ion sys ems [20]:
whe e
and
1
a e ec o s o app op ia e sizes in o which
he in e se ma ix ope a ions a e pe o med. He e he
backslash (
) is an ope a o indica ing o wa d o back‑
wa d subs i u ions and emphasizes he impo ance o
a oiding in e ing ma ices. In o he wo ds,
L y
is he
solu ion
x
o equa ion
Lx =y
o can be exp essed as
sol ing (
L
,
y
). No e ha he ma ix p oduc s a e ca e ully
(44)
A=(A−
1
)−1
=Q(L′)−
1
L−
1
Q′,
(45)
A
ij
=
EiQ(L′)
−1
L−1Q′E
′
j
(46)
Aij =
EiQLL
′
Q
′
E
′
j,
(47)
A11 =Q1L1L′
1Q′
1,
(48)
A
=Q(L′)
−1
L−1Q′
=
Q

L
′ 
L
(Q
′
)

(49)
(
A11)
−1
1
=
Q1

L
′
1 
L1
(Q
′
1
1)
,
nes ed wi h pa en hesis so ha only ma ix- ec o ope a-
ions a e pe o med.
Fu he mo e, ollowing [4, 9] he in e se o he ma ix
A22
, needed in he in e sion o he second modi ied
ela ionship ma ix

G2
in Eq.(39), can be exp essed e i-
cien ly using block ma ix in e sion iden i y simila o
Eq.(25) as:
whe e all e ms can be compu ed using Eqs. (45) o(49).
On‑ he‑ ly impu a ion ope a ion
The de i ed equi alen o mula ions in Eqs. (36),
(39), and (40) con ain impu a ion ope a o
Aimp =−
(A11)
−1
A
12
(23) in ma ices
Mi
ha a e
needed when ope a ing wi h he modi ied model ma ix

Zi=ZMi
and i s anspose

Z′
i=
M
′
i
Z
′
, and when calcu-
la ing he o iginal andom e ec s in Eq.(12). When he
MME a e sol ed by he PCG i e a ion algo i hm, he
co e o he algo i hm is a mul iplica ion o he so-called
di ec ion ec o
by he le hand side o he MME(11).
In his mul iplica ion, he impu a ion ope a o , as pa o
he MME coe icien ma ix, ope a es ei he wi h a pa
o he ec o pe aining o andom e ec s o he geno-
yped animals (i.e.
Aimp 2
) o o a ec o o he ma ke
e ec s
m
h ough he ma ke ma ix (i.e.
AimpZm m
).
Thus, in he anspose

Z′
i
side, he impu a ion e m ope -
a es on a ec o o size equal o he numbe o non-gen-
o yped animals (i.e.
A′
imp
1
).
In all cases, he size o he ec o e m ha ope a es on
(A
11
)
−1
equals he numbe o non-geno yped animals,
i.e. size o
1
. Fo example, he impu ed genomic ma ke
da a e m,
−(A
11
)
−1A
12
Zm m
, ha expands he genomic
in o ma ion om geno yped o non-geno yped animals,
can be calcula ed using:
and Eq.(49) so ha :
Vec o
1
is calcula ed om Eq.(46), o wi hou con-
s uc ing any o he ma ices by using ules o
A−1
by
pedig ee in o ma ion [18], o , al e na i ely, as a spa se
ma ix- ec o p oduc o sepa a e spa se ma ix
A12
.
No e ha he ac ual impu a ion o he genomic ma ke
in o ma ion is no needed. The impu a ion ope a ion is
pe o med only implici ly, “on- he- ly” du ing he i e a-
i e solu ion wi hou he need o use, o example, disk
(50)
(A22)−
1
=A
22
−A
21
(A
11
)
−1A
12
,
(51)

2=Zm m
(52)
1=A12
2
(53)
−
(A11)
−1
A12Z
m
m=−
Q1

L
′
1 
L1
(Q
′
1
1)
.
Page 8 o 15
Taskinen e al. Gene Sel E ol (2017) 49:36
s o age. In he no mal impu a ion p ocess [6], he ma ke
in o ma ion needs o be calcula ed o housands, o
e en hund eds o housands ma ke ec o s o geno yped
animals, i.e. columns o ma ke ma ix
Zm
. The p edic ed
ma ke da a ma ix con ains eal numbe s and can be
e y la ge, and, hus, akes a lo o ime and disk space o
gene a e and use.
In he on- he- ly impu a ion p ocess o gene ic e ec s,
howe e , impu a ion is an ope a ion on a “p ojec ion
ec o ” o he geno yped animals, i.e. a linea combina-
ion o he ma ke ec o s. I needs o be pe o med only
wice wi hin each i e a ion ound o each ai . Once o
ma ix

Zi
and ano he ime o he anspose

Z′
i
in ma ix
mul iplica ion o MME coe icien ma ix o Eq.(11). The
impu a ion ope a ion is also needed once be o e he i e -
a ion when calcula ing he igh -hand-side o he new
andom e ec s (

Z′
i
R
−
1
y
) in Eq.(11) and once a he end
o he i e a ion in o de o e ie e he o iginal andom
e ec s (

u=Mi

u
) in Eq.(12).
O hogonaliza ion o  andom e ec s
In he SNP-BLUP e sions o he de i ed equi alen
ssGBLUP, in Eqs. (39) and (40), he ma ke e ec s a e
o hogonal, i.e. hei co a iance ma ix is diagonal.
I u ns ou ha he PCG i e a ion numbe s o hese
wo equi alen ssSNP-BLUP MME a e conside ably
la ge han he o iginal ssGBLUP. In Eq. (39), he
RPG e ec s and genomic alues p edic ed by SNPs
ha e colinea i y, and in Eq.(40), he RPG and he ani-
mal e ec s o non-geno yped animals a e di icul o
sepa a e.
The key o main aining good nume ical p ope ies o
he o iginal ssGBLUP seems o be o “o hogonalize” he
emaining new andom e ec s, oo. The emaining a i-
ance s uc u es can be o hogonalized by spli ing he
a iance ma ices and a aching he wo “hal s” in o he
coe icien ma ices
M
as in Eqs.(6) and(29).
The e m
(A
11
)
−1
in Eqs.(39) and (40) can be o hog-
onalized by using he spa se Cholesky ac o iza ion in
Eq.(47) as ollows:
whe e
No e ha he pe mu a ion ope a o
Q1
can be pe o med
ou side he in e se ope a o . Howe e , he e m
A22
in
Eq.(39) seems o be much mo e di icul o decompose.
S ill, i can be exp essed using he spa se decomposi ion
o he ull ma ix
A
in Eq.(44) as:
(54)
(A
11
)−1
=M11

G11M′
11,
(55)
M11 =Q1
(
L′
1
)
−1
and

G11 =I1.
(56)
A22 =E2AE′
2=M22

G22M′
22,
whe e
and
The ec angula ma ix

A
1
2
2
has dimensions numbe o
geno yped indi iduals imes o al numbe o indi iduals,
hence he ade-o he e is ha Eq.(56) will expand he
second andom e ec g oup o Eq.(39) om geno yped
o all indi iduals (size o
I
).
Using Eqs.(54), (56), and (58), he linea ly equi alen
MME (39) and (40) can be “o hogonalized” in o ou h:
and i h
equi alen MME. Bo h o hese linea ly equi alen (29)
ssSNP-BLUPs sha e he same o hogonal a iance
s uc u e:
and, hus, bo h also ha e he same numbe o new an-
dom e ec s: andom e ec s o he geno yped indi idu-
als, wo se s o andom e ec s o he non-geno yped
indi iduals, and andom e ec s o he ma ke s. The di -
e ence in equi alen MME 4 (59) and 5 (60) is on how
hey di ide he RPG on non-geno yped animals.
The ou h equi alen MME (59) can be called o hog‑
onal ssSNP‑BLUP MME, and he i h MME(60), o igi-
na ing om he expanded RPG ssSNP-BLUP (40),
o hogonal expanded ssSNP‑BLUP MME.
Reduc ion o  he numbe o e ec s byusing ances o s
o geno yped animals
Ma ix
A22
, as he co a iance s uc u e o he geno-
yped animals, was epa ame ized in Eq.(56) using he
ull pedig ee ela ionship ma ix
A
. This epa ame iza-
ion inc eases he numbe o co esponding new an-
dom e ec s om geno yped animals o all animals in he
pedig ee. Howe e , compu a ions in ol ing
A22
equi e
only he geno yped indi iduals and hei ances o s.
Thus, o educe he numbe o ex a new e ec s,
A22
can
be exp essed using a smalle pedig ee and ela ionship
(57)
M
22
=

A
1
2
2
and

G22
=
I
,
(58)

A
1
2
k=
E
k
Q(L′)−1,k
=1, 2.
(59)
M
4=


M11 √wAimp
�
A
1
2
2
√
1−wAimpZm
0√w
�
A
1
2
2
√1
−
wZm

,
(60)
M
5=

√
1−wM11 √w
�
A
1
2
1
√
1−wAimpZm
0√w
�
A
1
2
2
√1
−
wZm

,
(61)
�
G
4=
�
G5=


I1
00
0I 0
0 0I
m

,
Page 9 o 15
Taskinen e al. Gene Sel E ol (2017) 49:36
ma ix

A
con aining he geno yped animals and hei
ances o s [4].
Le he in e se o he pedig ee ela ionship ma ix(43)
o his smalle pedig ee be:
whe e

Q
and

L
a e as be o e in Eq.(44) bu in ol e geno-
yped animals and hei ances o s only. Ma ix
A22
in
Eq.(56) can hen be ep esen ed using he smalle pedi-
g ee as:
whe e

E2
selec s he geno yped indi iduals om he smalle
pedig ee, and he size o iden i y ma ix
Iganc
is he num-
be o geno yped animals and hei ances o s.
The six h linea ly equi alen , educed o hogonal
ssSNP‑BLUP MME, can be de i ed om Eqs.(59) and
(61) as:
He e only he non-geno yped ances o s o he geno-
yped ha e wo se s o andom e ec s, all he o he non-
geno yped and all geno yped animals ha e single se s o
e ec s, in addi ion o he ma ke e ec s.
Da a
The de i ed MME we e es ed using a small No dic Red
dai y ca le da ase and a simple model. The small da a-
se and he model we e pa ially chosen in o de o be
able o use di ec spa se ma ix solu ions o he o iginal
ssGBLUP o ob ain accu a e “co ec solu ions”. The da a
we e de eg essed p oo s o milk yield ha we e based on
es ima ed b eeding alues om he No dic p oduc ion
ai e alua ions by NAV (No dic E alua ions, Denma k).
The e we e 73,579 animals in he pedig ee o which 2885
we e geno yped. Geno yped animals oge he wi h hei
ances o s o m a smalle pedig ee o 6833 animals. The
animals had been geno yped wi h he Illumina Bo ine
SNP50 Bead Chip (Illumina, San Diego, USA). The analy-
sis used 37,526 SNPs ha passed quali y con ol. The e
we e 66,426 non-geno yped and 1222 geno yped animals
wi h pheno ypes. Hence, 1663 animals had a geno ype
bu no pheno ype. We conside ed a single ai model
(62)

A−
1
=

Q

L

L′

Q′,
(63)
A22 =

E
2

A

E
′
2=

M
22

G
22

M
′
22,
(64)

M
22
=

E2

Q(

L
′
)
−1
and

G22
=
I
ganc,
(65)
M
6=
�
M11
√
wAimp
�
M22
√
1−wAimpZm
0√w�
M22 √1−wZm
�
�
G6=

I100
0I
ganc 0
0 0I
m


.
and assumed a he i abili y o 0.5. The genomic da a con-
ained one pai o animals wi h iden ical genomic ma ke
da a and a couple o mo e nea iden ical pai s ha led o
p oblems o he in e sion o he genomic ela ionship
ma ix
Gg
wi hou
A22
adjus men , i.e. in he case
w=0
.
Compa ison s a is ics
The o iginal ssGBLUP wi h in e se a iance ma ix
H−1
o Eq.(20) and he six linea ly equi alen o mula ions o
he o m(29), om Eqs.(36), (39), (40), (59), (60), (61),
and (65), we e implemen ed and es ed in an Oc a e[21]
en i onmen . Spa se ma ix ac o iza ions we e based on
CHOLMOD ou ines[22]. Six di e en weigh s
w
we e
es ed: 0.00, 0.01, 0.10, 0.20, 0.30, and 1.00. Because o
he singula i y in he in e se o he genomic ela ion-
ship ma ix
Gg
−1
wi h he es da a, he o iginal ssGB-
LUP ma ix (20) and he i s equi alen o mula ion(36)
we e no calcula ed when
w=0
. The de i ed new o -
mula ions we e compa ed agains he o iginal ssGBLUP,
mainly ocusing on e iciency, accu acy, and numbe o
i e a ions.
E iciency o he de i ed equi alen ssGBLUP o mu-
la ions elies on he spa si y o he pedig ee ela ionship
ma ices and hei decomposi ions. In e se ma ix ope a-
ions o hese spa se ma ices we e ans o med in o sol -
ing spa se lowe iangula ma ix sys ems. The e iciency
o hese sol ing ope a ions depend on he spa si y s uc-
u e o he ma ices, i.e. numbe o non-ze o elemen s.
Accu acy o he o mula ions was es ed by sol ing he
MME wi h he PCG me hod using Oc a e’s PCG ou ine
(pcg) wi h he diagonal o he MME coe icien ma ix
as he p econdi ione , o wi hou p econdi ioning. Con-
e gence ole ance in pcg was ela i e esidual no m.
The ole ance was chosen o be small (
10−12
) so ha all
sol ed e ec s, wi hou doub , con e ged. Accu acies, o
a he he di e ences om he “exac solu ion”, we e cal-
cula ed as ela i e esidual e o s (

ei
) be ween i e a i ely
ob ained MME solu ions (
si
) and he di ec solu ion
(
sdi ec
) o he o iginal ssGBLUP:
whe e subsc ip
i=1, ...,6
is he o mula ion numbe .
Implemen a ions o he de i ed equi alen ssGBLUP
o mula ion we e no ye s eamlined o speed and, hus,
he execu ion imes we e nei he op imal, no compa a-
ble. The pe o mance o he o mula ions is, he e o e,
es ed by compa ing he numbe o i e a ions o he i e a-
i e solu ion. The pu pose was o demons a e ha he
i e a ion coun s a e compa able o hose ob ained by he
o iginal ssGBLUP. Wi h a la ge numbe o geno yped
indi iduals, he in e sion o he genomic ela ionship
ma ix in he o iginal ssGBLUP becomes a bo leneck
(66)

ei
=
�
s
di ec −
s
i�
�sdi ec �,