Some Theory for and Applications of Gaussian Markov Random ... · GMRF:s Results References Some...
Transcript of Some Theory for and Applications of Gaussian Markov Random ... · GMRF:s Results References Some...
GM
RF:s
Result
sR
efe
rences
Som
eT
heo
ryfo
ran
dA
pplica
tion
sof
Gau
ssia
nM
arko
vR
andom
Fie
lds
Joh
anLin
dst
rom
1D
avid
Bol
in1
Fin
nLin
dgr
en1
Hav
ard
Rue2
1C
entr
efo
rM
ath
em
ati
calScie
nces
Lund
Univ
ers
ity
&2D
epart
ment
ofM
ath
em
ati
calScie
nces
NT
NU
,Tro
ndheim
Sea
ttle
Feb
ruar
y3,
2009
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Ove
rvie
w
Gau
ssia
nM
arko
vra
ndom
fiel
ds
Bas
ics
Appro
xim
atin
gM
ater
nco
vari
ance
sIN
LA
–Fas
tes
tim
atio
n
Res
ult
sT
he
Sah
elV
eget
ation
Pre
cipitation
Dep
thD
ata
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
Gau
ssia
nM
arko
vR
andom
Fie
lds
(GM
RF:s
)
◮A
Gau
ssia
nM
arko
vra
ndom
fiel
d(G
MR
F)
isa
Gau
ssia
nra
ndom
fiel
dw
ith
aM
arko
vpro
per
ty.
◮T
he
nei
ghbou
rsN
ito
apoi
nt
s iar
eth
epoi
nts
that
inso
me
sense
are
clos
eto
s i.
◮T
he
Gau
ssia
nra
ndom
fiel
dx∈
N( m,Q−1) h
asa
join
tdis
trib
uti
onth
atsa
tisfi
es
p(x
i|{x
j:j6=
i})
=p(x
i|{x
j:j∈N
i}).
◮j
/∈N
i⇐⇒
xi⊥
xj|{ x
k:k
/∈{i
,j}}
⇐⇒
Qi,j=
0.
◮T
he
den
sity
is
p(x
)=
|Q|1
/2
(2
p)n/2exp(−
1 2(x
−
m)⊤ Q(x−m))
◮Fas
tal
gori
thm
sth
atuti
lise
the
spar
sity
ofQ
exis
t(c
-pac
kage
GM
RFlib)
See
Rue
and
Hel
d(2
005)
for
exte
nsi
vedet
ails
onG
MR
F:s
.Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
GM
RF:s
–Sim
plified
expre
ssio
ns
For
mula
ting
the
den
sity
thro
ugh
the
pre
cisi
onm
atri
xin
stea
dof
thro
ugh
the
cova
rian
cem
atri
xsi
mplifies
seve
ralco
mm
onex
pre
ssio
nfo
rG
auss
ian
model
s.
◮C
ondit
ional
expec
tati
on
{xI|x
J,J
6=I}
∈N( m I−Q−1 I,
IQ
I,J(x
J−
m J),Q−1 I,I
).
◮H
irea
rchic
alm
odel
ling
x|θ
∈N( B
θ,Q
−1) ,
θ∈
N( m,Q−1 θ
) ,[ x θ
]∈
N
([ B
m m] ,[Q
QB
B⊤
QB
⊤Q
B+
Qθ
] −1)
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
GM
RF:s
–Sim
plified
expre
ssio
ns
For
mula
ting
the
den
sity
thro
ugh
the
pre
cisi
onm
atri
xin
stea
dof
thro
ugh
the
cova
rian
cem
atri
xsi
mplifies
seve
ralco
mm
onex
pre
ssio
nfo
rG
auss
ian
model
s.
◮C
ondit
ional
expec
tati
on
{xI|x
J,J
6=I}
∈N( m I−Q−1 I,
IQ
I,J(x
J−
m J),Q−1 I,I
).
◮H
irea
rchic
alm
odel
ling
x|θ
∈N( B
θ,Q
−1) ,
θ∈
N( m,Q−1 θ
) ,[ x θ
]∈
N
([ B
m m] ,[Q
QB
B⊤
QB
⊤Q
B+
Qθ
] −1)
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
GM
RF:s
–P
revio
us
lim
itat
ions
◮H
owto
choos
eor
const
ruct
Q?
◮A
GM
RF
mig
ht
be
com
puta
tion
ally
effec
tive
but
itis
diffi
cult
inco
nst
ruct
pre
cisi
onm
atri
ces
that
resu
ltin
reas
onab
leco
vari
ance
funct
ions
for
the
under
lyin
gG
auss
ian
fiel
ds.
◮V
ario
us
ad-h
oc
met
hods
exis
t.A
com
mon
solu
tion
isto
use
asm
allnei
ghbou
rhood
and
let
the
pre
cisi
onbet
wee
ntw
opoi
nts
dep
end
onth
edis
tance
bet
wee
nth
epoi
nts
.
◮R
ue
and
Tje
lmel
and
(200
2)cr
eate
dG
MR
F:s
onre
ctan
gula
rgr
ids
inth
atap
pro
xim
ate
Gau
ssia
nfiel
ds
wit
ha
wid
ecl
ass
ofco
vari
ance
funct
ions.
◮H
avin
gth
efiel
ddefi
ned
only
ona
regu
lar
grid
lead
sto
issu
esw
ith
map
pin
gth
eob
serv
atio
ns
toth
egr
idpoi
nts
.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
GM
RF:s
–P
revio
us
lim
itat
ions
◮H
owto
choos
eor
const
ruct
Q?
◮A
GM
RF
mig
ht
be
com
puta
tion
ally
effec
tive
but
itis
diffi
cult
inco
nst
ruct
pre
cisi
onm
atri
ces
that
resu
ltin
reas
onab
leco
vari
ance
funct
ions
for
the
under
lyin
gG
auss
ian
fiel
ds.
◮V
ario
us
ad-h
oc
met
hods
exis
t.A
com
mon
solu
tion
isto
use
asm
allnei
ghbou
rhood
and
let
the
pre
cisi
onbet
wee
ntw
opoi
nts
dep
end
onth
edis
tance
bet
wee
nth
epoi
nts
.
◮R
ue
and
Tje
lmel
and
(200
2)cr
eate
dG
MR
F:s
onre
ctan
gula
rgr
ids
inth
atap
pro
xim
ate
Gau
ssia
nfiel
ds
wit
ha
wid
ecl
ass
ofco
vari
ance
funct
ions.
◮H
avin
gth
efiel
ddefi
ned
only
ona
regu
lar
grid
lead
sto
issu
esw
ith
map
pin
gth
eob
serv
atio
ns
toth
egr
idpoi
nts
.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
GM
RF:s
–P
revio
us
lim
itat
ions
◮H
owto
choos
eor
const
ruct
Q?
◮A
GM
RF
mig
ht
be
com
puta
tion
ally
effec
tive
but
itis
diffi
cult
inco
nst
ruct
pre
cisi
onm
atri
ces
that
resu
ltin
reas
onab
leco
vari
ance
funct
ions
for
the
under
lyin
gG
auss
ian
fiel
ds.
◮V
ario
us
ad-h
oc
met
hods
exis
t.A
com
mon
solu
tion
isto
use
asm
allnei
ghbou
rhood
and
let
the
pre
cisi
onbet
wee
ntw
opoi
nts
dep
end
onth
edis
tance
bet
wee
nth
epoi
nts
.
◮R
ue
and
Tje
lmel
and
(200
2)cr
eate
dG
MR
F:s
onre
ctan
gula
rgr
ids
inth
atap
pro
xim
ate
Gau
ssia
nfiel
ds
wit
ha
wid
ecl
ass
ofco
vari
ance
funct
ions.
◮H
avin
gth
efiel
ddefi
ned
only
ona
regu
lar
grid
lead
sto
issu
esw
ith
map
pin
gth
eob
serv
atio
ns
toth
egr
idpoi
nts
.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
Mat
ern
cova
rian
ces
(Ber
tilM
ater
n,19
17–2
007)
◮T
he
Mat
ern
cova
rian
cefa
mily
onu∈
Rd:
r(u,v
)=
C(x
(u),
x(v
))=
s221− n G(n)( k‖v−u‖)n K n
(k‖v−u‖)w
ith
scal
e(i
nve
rse
range
)
k>0andshape/
smoot
hnes
s
n>0,an
dK
namodifiedBes
selfu
nct
ion.
◮Fie
lds
wit
hM
ater
nco
vari
ance
sar
eso
luti
ons
toan
SP
DE
Whit
tle
(195
4)bas
edon
the
Lap
laci
an,
D=∇⊤ ∇,( k2 −D) a/2 x
(u)
=
t2 E(u),w
her
e
E(u)isspatialw
hit
enoi
se,an
d
a=n+d/2.◮
Par
amet
erlink:
s2 =t2G(
n) G(a)k2n (4p)d/2Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
SP
DE
issu
es
◮N
on-u
niq
uen
ess:
Ifx(u
)is
aso
luti
onto
the
SP
DE
for
a=2,soisx(u
)+
c·e
xp(ke⊤ u),foran
yunit
lengt
hve
ctor
ean
dan
yco
nst
ant
c.
◮N
on-s
tati
onar
ity:
On
abou
nded
dom
ain,th
eSP
DE
solu
tion
sar
enon
-sta
tion
ary,
unle
ssco
ndit
ioned
onsu
itab
lebou
ndar
ydis
trib
uti
ons.
◮P
ract
ical
solu
tion
toth
enon
-uniq
uen
ess
and
non
-sta
tion
arity:
Zer
o-nor
mal
-der
ivat
ive
(Neu
man
n)
bou
ndar
ies
reduce
the
impac
tof
the
null-s
pac
eso
luti
ons.
◮R
esult
ing
cova
rian
ce,fo
r
W=[0,L]⊂R:
C(x
(u),
x(v
))≈
r M(u
,v)+
r M(u
,−v)+
r M(u
,2L−
v)
=r M
(0,v
−u)+
r M(0
,v+
u)+
r M(0
,2L−
(v+
u))
.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
SP
DE
issu
es
◮N
on-u
niq
uen
ess:
Ifx(u
)is
aso
luti
onto
the
SP
DE
for
a=2,soisx(u
)+
c·e
xp(ke⊤ u),foran
yunit
lengt
hve
ctor
ean
dan
yco
nst
ant
c.
◮N
on-s
tati
onar
ity:
On
abou
nded
dom
ain,th
eSP
DE
solu
tion
sar
enon
-sta
tion
ary,
unle
ssco
ndit
ioned
onsu
itab
lebou
ndar
ydis
trib
uti
ons.
◮P
ract
ical
solu
tion
toth
enon
-uniq
uen
ess
and
non
-sta
tion
arity:
Zer
o-nor
mal
-der
ivat
ive
(Neu
man
n)
bou
ndar
ies
reduce
the
impac
tof
the
null-s
pac
eso
luti
ons.
◮R
esult
ing
cova
rian
ce,fo
r
W=[0,L]⊂R:
C(x
(u),
x(v
))≈
r M(u
,v)+
r M(u
,−v)+
r M(u
,2L−
v)
=r M
(0,v
−u)+
r M(0
,v+
u)+
r M(0
,2L−
(v+
u))
.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
The
finit
eel
emen
tm
ethod
◮A
stoch
asti
cw
eak
form
ula
tion
ofth
eSP
DE
stat
esth
at
[ ⟨
f k,( D−k2 )a/2x⟩]
k=
1,.
..,n
D =[ 〈
f k,W〉] k=
1,.
..,n
for
each
set
ofte
stfu
nct
ions{f k} .
◮W
euse
Nsi
mple
pie
cew
ise
linea
rte
stfu
nct
ions,
equal
toth
epie
cew
ise
linea
rbas
isfu
nct
ions,
x(u
)=∑
j
y k(u)w j,andcom
pute
the
resu
ltin
gdis
trib
uti
ons
by
explici
tly
calc
ula
ting
the
expec
tati
onve
ctor
and
pre
cisi
onm
atri
xfo
rw
(=x).
◮For
a=2,theweak
form
ula
tion
can
be
wri
tten
Kw
=[⟨
f i,( D−k2 )y j⟩ ]
i,j=
1,.
..,N
wD =[ 〈
f k,W〉] k=
1,.
..,N
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
Con
stru
ctio
nof
Q
◮W
ith
the
hel
pof
Gre
en’s
firs
tid
enti
ty,
Ci,j=
〈y i, y j〉,K
i,j=⟨ y i,( k2 −D)y
j⟩=
k2 C i,j+〈∇y i,∇
y j〉,fo
rN
eum
ann
bou
ndar
ies.
◮M
arko
vifi
edLea
stSquar
esan
dG
aler
kin
solu
tion
s:
C=
dia
g(〈
y i,1〉),(“opti
mal
”ap
pro
xim
atio
n)
Q1, k=K,(Lea
stSquar
es)
Q2,k=KC−1 K,
(Gal
erkin
)
Q
a,k=KC−1 Q a
−2,kC−1 K, a=
3,4,
...(G
aler
kin
recu
rsio
n)
Nei
ghbou
rhood
radiu
seq
ual
s
a.◮
Sim
ple
clos
ed-for
mex
pre
ssio
ns
for
the
mat
rix
elem
ents
.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
Appro
xim
atin
gM
ater
nco
vari
ance
s(L
indgr
enan
dR
ue,
2007
)
That
fiel
ds
wit
hM
ater
nco
vari
ance
sar
eso
luti
ons
toan
SP
DE
has
bee
nuse
dto
const
ruct
GM
RF:s
that
appro
xim
ate
fiel
ds
wit
hM
ater
nco
vari
ance
for
a∈Z+ .◮
Wel
l-defi
ned
SP
DE
oncu
rved
man
ifol
ds
(e.g
.a
glob
e).
◮Pos
sible
toin
troduce
bot
han
isot
ropy
and
non
-sta
tion
arity.
◮Spat
ially
osci
llat
ing
fiel
ds
can
be
intr
oduce
dvia
aco
mple
xve
rsio
nof
the
SP
DE
:
(h1
+ih
2−
∇⊤
(H1
+iH
2)∇
)(x1(u
)+
ix2(u
))=
E 1(u)+i E 2(u)
◮T
he
pre
cisi
onm
atri
xof
the
appro
xim
atin
gG
MR
Fis
found
usi
ng
the
finit
eel
emen
tm
ethod
ona
tria
ngu
lati
onof
irre
gula
rly
spac
edpoi
nts
.◮
The
resu
ltin
gG
MR
Fis
defi
ned
onth
epoi
nts
ofth
etr
iangu
lati
on,m
akin
git
suit
able
for
model
ling
fiel
ds
that
are
obse
rved
atir
regu
lar
loca
tion
s.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
Appro
xim
atin
gM
ater
nco
vari
ance
s(L
indgr
enan
dR
ue,
2007
)
That
fiel
ds
wit
hM
ater
nco
vari
ance
sar
eso
luti
ons
toan
SP
DE
has
bee
nuse
dto
const
ruct
GM
RF:s
that
appro
xim
ate
fiel
ds
wit
hM
ater
nco
vari
ance
for
a∈Z+ .◮
Wel
l-defi
ned
SP
DE
oncu
rved
man
ifol
ds
(e.g
.a
glob
e).
◮Pos
sible
toin
troduce
bot
han
isot
ropy
and
non
-sta
tion
arity.
◮Spat
ially
osci
llat
ing
fiel
ds
can
be
intr
oduce
dvia
aco
mple
xve
rsio
nof
the
SP
DE
:
(h1
+ih
2−
∇⊤
(H1
+iH
2)∇
)(x1(u
)+
ix2(u
))=
E 1(u)+i E 2(u)
◮T
he
pre
cisi
onm
atri
xof
the
appro
xim
atin
gG
MR
Fis
found
usi
ng
the
finit
eel
emen
tm
ethod
ona
tria
ngu
lati
onof
irre
gula
rly
spac
edpoi
nts
.◮
The
resu
ltin
gG
MR
Fis
defi
ned
onth
epoi
nts
ofth
etr
iangu
lati
on,m
akin
git
suit
able
for
model
ling
fiel
ds
that
are
obse
rved
atir
regu
lar
loca
tion
s.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
Appro
xim
atin
gM
ater
nco
vari
ance
s(L
indgr
enan
dR
ue,
2007
)
That
fiel
ds
wit
hM
ater
nco
vari
ance
sar
eso
luti
ons
toan
SP
DE
has
bee
nuse
dto
const
ruct
GM
RF:s
that
appro
xim
ate
fiel
ds
wit
hM
ater
nco
vari
ance
for
a∈Z+ .◮
Wel
l-defi
ned
SP
DE
oncu
rved
man
ifol
ds
(e.g
.a
glob
e).
◮Pos
sible
toin
troduce
bot
han
isot
ropy
and
non
-sta
tion
arity.
◮Spat
ially
osci
llat
ing
fiel
ds
can
be
intr
oduce
dvia
aco
mple
xve
rsio
nof
the
SP
DE
:
(h1
+ih
2−
∇⊤
(H1
+iH
2)∇
)(x1(u
)+
ix2(u
))=
E 1(u)+i E 2(u)
◮T
he
pre
cisi
onm
atri
xof
the
appro
xim
atin
gG
MR
Fis
found
usi
ng
the
finit
eel
emen
tm
ethod
ona
tria
ngu
lati
onof
irre
gula
rly
spac
edpoi
nts
.◮
The
resu
ltin
gG
MR
Fis
defi
ned
onth
epoi
nts
ofth
etr
iangu
lati
on,m
akin
git
suit
able
for
model
ling
fiel
ds
that
are
obse
rved
atir
regu
lar
loca
tion
s.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
INLA
–Fas
tes
tim
atio
n(R
ue
and
Mar
tino,
2007
)
◮A
ssum
ean
under
lyin
gG
MR
Fw
ith
(pos
sibly
non
-Gau
ssia
n)
poi
nt
obse
rvat
ions,
i.e.
x∈
N( m(Y),Q(Y)−
1)
p(y
i|x, Y)=p(y i|x i,
Y).◮
Ifth
eob
serv
atio
ns
are
non
-Gau
ssia
nth
elo
g-like
lihood
can
be
appro
xim
ated
wit
ha
Gau
ssia
nby
aTay
lor
expan
sion
ofth
eob
serv
atio
nden
sity
,p(y
i|xi,
Y).◮
Do
num
eric
alop
tim
isat
ion
ofth
elo
g-like
lihood,usi
ng
the
abov
eap
pro
xim
atio
n.
◮O
nce
the
MA
P-e
stim
ator
isfo
und,use
the
Hes
sian
ofth
elo
g-like
lihood
todet
erm
ine
the
like
lyva
riat
ions
ofth
elike
lihood
and
do
num
eric
alin
tegr
atio
n.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
INLA
–Fas
tes
tim
atio
n(R
ue
and
Mar
tino,
2007
)
◮A
ssum
ean
under
lyin
gG
MR
Fw
ith
(pos
sibly
non
-Gau
ssia
n)
poi
nt
obse
rvat
ions,
i.e.
x∈
N( m(Y),Q(Y)−
1)
p(y
i|x, Y)=p(y i|x i,
Y).◮
Ifth
eob
serv
atio
ns
are
non
-Gau
ssia
nth
elo
g-like
lihood
can
be
appro
xim
ated
wit
ha
Gau
ssia
nby
aTay
lor
expan
sion
ofth
eob
serv
atio
nden
sity
,p(y
i|xi,
Y).◮
Do
num
eric
alop
tim
isat
ion
ofth
elo
g-like
lihood,usi
ng
the
abov
eap
pro
xim
atio
n.
◮O
nce
the
MA
P-e
stim
ator
isfo
und,use
the
Hes
sian
ofth
elo
g-like
lihood
todet
erm
ine
the
like
lyva
riat
ions
ofth
elike
lihood
and
do
num
eric
alin
tegr
atio
n.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
INLA
–Fas
tes
tim
atio
n(R
ue
and
Mar
tino,
2007
)
◮A
ssum
ean
under
lyin
gG
MR
Fw
ith
(pos
sibly
non
-Gau
ssia
n)
poi
nt
obse
rvat
ions,
i.e.
x∈
N( m(Y),Q(Y)−
1)
p(y
i|x, Y)=p(y i|x i,
Y).◮
Ifth
eob
serv
atio
ns
are
non
-Gau
ssia
nth
elo
g-like
lihood
can
be
appro
xim
ated
wit
ha
Gau
ssia
nby
aTay
lor
expan
sion
ofth
eob
serv
atio
nden
sity
,p(y
i|xi,
Y).◮
Do
num
eric
alop
tim
isat
ion
ofth
elo
g-like
lihood,usi
ng
the
abov
eap
pro
xim
atio
n.
◮O
nce
the
MA
P-e
stim
ator
isfo
und,use
the
Hes
sian
ofth
elo
g-like
lihood
todet
erm
ine
the
like
lyva
riat
ions
ofth
elike
lihood
and
do
num
eric
alin
tegr
atio
n.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
INLA
–Fas
tes
tim
atio
n(R
ue
and
Mar
tino,
2007
)
◮A
ssum
ean
under
lyin
gG
MR
Fw
ith
(pos
sibly
non
-Gau
ssia
n)
poi
nt
obse
rvat
ions,
i.e.
x∈
N( m(Y),Q(Y)−
1)
p(y
i|x, Y)=p(y i|x i,
Y).◮
Ifth
eob
serv
atio
ns
are
non
-Gau
ssia
nth
elo
g-like
lihood
can
be
appro
xim
ated
wit
ha
Gau
ssia
nby
aTay
lor
expan
sion
ofth
eob
serv
atio
nden
sity
,p(y
i|xi,
Y).◮
Do
num
eric
alop
tim
isat
ion
ofth
elo
g-like
lihood,usi
ng
the
abov
eap
pro
xim
atio
n.
◮O
nce
the
MA
P-e
stim
ator
isfo
und,use
the
Hes
sian
ofth
elo
g-like
lihood
todet
erm
ine
the
like
lyva
riat
ions
ofth
elike
lihood
and
do
num
eric
alin
tegr
atio
n.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
INLA
–Fas
tes
tim
atio
n(c
ont.
)
The
Good:
◮V
ery
fast
com
par
edto
MC
MC
.
◮G
ives
GO
OD
appro
xim
atio
ns
ofth
epos
teri
orden
siti
es,
p(x
i|Y
,Y map),p(x
i|Y
),p( Y i|Y).
The
Bad
:
◮Lim
ited
toa
few
(5-1
0)hyper
par
amet
ers,
Y.◮
Giv
eslim
ited
info
rmat
ion
abou
tin
tera
ctio
nbet
wee
ndiff
eren
tpar
amet
ers,
p(x
i,xj|Y
),p(Y i,Y j|Y).
The
Ugl
y:
◮Lar
gem
emor
yre
quir
men
tsco
mpar
edto
MC
MC
ofa
GM
RF.
This
ism
ainly
due
toth
enum
eric
alin
tegr
atio
n.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
INLA
–Fas
tes
tim
atio
n(c
ont.
)
The
Good:
◮V
ery
fast
com
par
edto
MC
MC
.
◮G
ives
GO
OD
appro
xim
atio
ns
ofth
epos
teri
orden
siti
es,
p(x
i|Y
,Y map),p(x
i|Y
),p( Y i|Y).
The
Bad
:
◮Lim
ited
toa
few
(5-1
0)hyper
par
amet
ers,
Y.◮
Giv
eslim
ited
info
rmat
ion
abou
tin
tera
ctio
nbet
wee
ndiff
eren
tpar
amet
ers,
p(x
i,xj|Y
),p(Y i,Y j|Y).
The
Ugl
y:
◮Lar
gem
emor
yre
quir
men
tsco
mpar
edto
MC
MC
ofa
GM
RF.
This
ism
ainly
due
toth
enum
eric
alin
tegr
atio
n.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Basic
sM
atern
IN
LA
INLA
–Fas
tes
tim
atio
n(c
ont.
)
The
Good:
◮V
ery
fast
com
par
edto
MC
MC
.
◮G
ives
GO
OD
appro
xim
atio
ns
ofth
epos
teri
orden
siti
es,
p(x
i|Y
,Y map),p(x
i|Y
),p( Y i|Y).
The
Bad
:
◮Lim
ited
toa
few
(5-1
0)hyper
par
amet
ers,
Y.◮
Giv
eslim
ited
info
rmat
ion
abou
tin
tera
ctio
nbet
wee
ndiff
eren
tpar
amet
ers,
p(x
i,xj|Y
),p(Y i,Y j|Y).
The
Ugl
y:
◮Lar
gem
emor
yre
quir
men
tsco
mpar
edto
MC
MC
ofa
GM
RF.
This
ism
ainly
due
toth
enum
eric
alin
tegr
atio
n.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Afr
ican
Sah
el
◮T
he
insp
irat
ion
for
the
follow
ing
applica
tion
sw
asan
arti
cle
by
Eklu
ndh
and
Ols
son
(200
3).
◮A
sem
i-ar
idre
gion
dir
ectl
yso
uth
ofth
eSah
ara
des
ert.
◮Sta
rtin
gin
the
late
1960
’s,th
ear
easu
ffer
eddro
ugh
tsfo
rov
ertw
enty
year
s.
◮R
ecen
tst
udie
sin
dic
ate
ave
geta
tion
reco
very
.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Nor
mal
ised
Diff
eren
ceV
eget
atio
nIn
dex
◮D
ata
take
nfr
omth
eN
OA
A/N
ASA
Pat
hfinder
AV
HR
RLan
d(P
AL)
dat
ase
t.
◮N
DV
Iis
calc
ula
ted
usi
ng
sate
llit
em
easu
rem
ents
ofth
ere
flec
tance
from
the
Ear
th’s
surf
ace.
◮T
he
PA
Ldat
ase
tco
nta
ins
36m
easu
rem
ents
per
year
.W
euse
aggr
egat
edye
arly
dat
a.
◮T
he
ques
tion
is“H
asth
eam
ount
ofve
geta
tion
incr
ease
d?”
Fig
ure
:N
DV
Idata
for
the
Sahel1983
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Veg
etat
ion
model
(Bol
inet
al.,
2008
)
◮Spat
ialm
odel
for
mea
sure
dN
DV
I:
Y(s
i,t)
=xt(s
i)+
e it,Yt=
AtX
t+
Et.
wher
ext
are
late
nt
fiel
ds
const
rain
edto
asu
mof
tem
por
altr
ends,
xt(s
i)=
m ∑ j=1
f j(t
)·K
j(s i
)
wher
ef j
are
tem
por
altr
end
funct
ions
wit
hsp
atia
lly
vary
ing
regr
essi
onco
effici
ents
Kj
◮T
he
obse
rvat
ion
mat
rice
sA
tdet
erm
ine
whic
hpoi
nts
are
obse
rved
atea
chti
me
poi
nt
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Reg
ress
ion
model
◮W
ithou
tth
etr
end
const
rain
t,a
sim
ple
pri
orm
odel
for
xt
wou
ldbe
aW
hit
tle
fiel
dw
ith
k=0,withprec
isio
nQ
x.
◮T
he
tren
dre
stri
ctio
npro
vid
esa
nat
ura
lpri
orfo
rK
=[K
⊤ 1,.
..,K
⊤ m]⊤
as
K|t∈N(0,(tQ)
−1)
wher
eQ
=(F
⊤F)⊗
Qx,an
dF
=[f
1,.
..,f
m]
◮T
he
resi
dual
vari
ance
isal
low
edto
vary
acro
sssp
ace,
S e=diag(s2 i)
◮C
olle
ctin
gth
edat
aan
dob
serv
atio
nm
atri
ces
ina
sim
ilar
man
ner
,
Y=
AK
+E
=dia
g(A
t)(
F⊗
I n)K
+E
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Pos
teri
ordis
trib
uti
on
◮T
he
pos
teri
ordis
trib
uti
onfo
rK
give
nth
edat
aY
=[Y
⊤ 0,.
..,Y
⊤ T−
1]⊤
and
the
par
amet
ers
is
(K|Y
,S e,t)∈N(m K|•
,Q−
1K|•
),w
ith m K|•=Q−1 K|•
A⊤
S−1 eY,Q
K|•
=
tQ+A⊤ S−1 eA.◮
The
unknow
npar
amet
ers
(t,s2 1,.
..,
s2 n)
can
be
esti
mat
edusi
ng
anE
Mal
gori
thm
,ex
plo
itin
gth
eG
MR
Fst
ruct
ure
inth
eupdat
ing
equat
ions.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Lin
ear
tren
des
tim
atio
n
◮Slo
pe
(upper
)an
dIn
terc
ept
(low
er)
esti
mat
es.
K2 E
stim
ate
−0.3
−0.2
−0.1
00.1
K1 E
stim
ate
0246810
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Tre
nd
sign
ifica
nce
◮Sig
nifi
cance
esti
mat
esfo
rth
esl
ope
ofth
elinea
rtr
ends
OLS
GM
RF
Sig
nific
ant N
egative
Non−
Sig
nific
ant N
egative
Non−
Sig
nific
ant P
ositiv
eS
ignific
ant P
ositiv
e
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Pre
cipit
atio
n(L
indst
rom
and
Lin
dgr
en,20
08)
00.2
1.8
10
Yea
rly
pre
cipit
atio
n,in
met
res,
for
the
443
stat
ions
that
hav
ere
por
ted
mea
sure
men
tsfo
r19
82.
Dat
afr
omth
eG
lobal
His
tori
cal
Clim
atol
ogy
Net
wor
k.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Irre
gula
rtr
iangu
lati
on
◮W
edo
not
nee
da
den
sere
gula
rgr
idfo
rm
odel
ling
the
pre
cipit
atio
n.
◮T
he
GM
RF
const
ruct
ion
allo
ws
for
irre
gula
rtr
iangu
lati
ons,
whic
hre
duce
sth
eco
mputa
tion
alburd
en.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Spat
io-t
empor
alm
odel
◮G
iven
ala
tent
pre
cipit
atio
nfiel
dx,tr
ansf
orm
edob
serv
atio
ns
are
assu
med
Gau
ssia
n:
Y(s
i,t)|x
∈N
(x(s
i,t)
,
s2 ).◮
We
use
anA
R(1
)-pro
cess
inti
me,
wit
hsp
atia
lly
corr
elat
ednoi
sean
dsp
atia
lly
vary
ing
mea
n:
Xt+
1−
m=a·( Xt−
m) +
h t,(h t| k2 , t)∈N(
0,Q
−1
S),
X1∈
N(m,Q−1 S
/(1−
a2))
,
m=Bθ.an
dw
eca
nw
rite
X∈
N(1
⊗B
θ,( Q
t⊗
Qs
) −1)
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Bas
isfu
nct
ions
for
the
mea
n
lati
tude
transformedprecipitation
812
160
0.751.5
Lef
tLat
itude
ism
odel
led
usi
ng
abro
ken
linea
rtr
end.
Rig
ht
Inad
dit
ion
toth
ebro
ken
tren
d15
B-s
pline
surf
ace
bas
isfu
nct
ions
are
use
d.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Hie
rarc
hic
alm
odel
grap
h
a tb takb k tk2
a
m θQ θ θ
X
a sb s s2Y
Dir
ecte
dac
ycl
icgr
aph
des
crib
ing
the
resu
ltin
ghie
rarc
hic
alm
odel
.T
he
join
tly
Gau
ssia
nfiel
ds
Xan
dθ
can
be
inte
grat
edou
t.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Mar
kov
chai
nM
onte
Car
lo
◮Fir
stw
etr
ansf
orm
the
par
amet
ers
toob
tain
par
amet
ers
valid
onR
:
˜s2 =log(s2 ), ˜k2 =log(k2 ),a
=lo
g(1
+a)
−lo
g(1
−a)
,
˜q=log(q),◮
Con
stru
cta
pro
pos
aldis
trib
uti
onby
usi
ng
ara
ndom
wal
kpro
pos
alon
the
tran
sfor
med
vari
able
spac
e.
◮H
owev
erin
itia
lru
ns
ona
reduce
ddat
aset
sw
ith
coar
ser
tria
ngu
lati
onsh
owed
stro
ng
dep
enden
cies
inth
epos
teri
ordis
trib
uti
on,le
adin
gus
toa
corr
elat
edpro
pos
aldis
trib
uti
on:
˜ y new∈N( ˜ y old,S proposal)
.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Mar
kov
chai
nM
onte
Car
lo
◮Fir
stw
etr
ansf
orm
the
par
amet
ers
toob
tain
par
amet
ers
valid
onR
:
˜s2 =log(s2 ), ˜k2 =log(k2 ),a
=lo
g(1
+a)
−lo
g(1
−a)
,
˜q=log(q),◮
Con
stru
cta
pro
pos
aldis
trib
uti
onby
usi
ng
ara
ndom
wal
kpro
pos
alon
the
tran
sfor
med
vari
able
spac
e.
◮H
owev
erin
itia
lru
ns
ona
reduce
ddat
aset
sw
ith
coar
ser
tria
ngu
lati
onsh
owed
stro
ng
dep
enden
cies
inth
epos
teri
ordis
trib
uti
on,le
adin
gus
toa
corr
elat
edpro
pos
aldis
trib
uti
on:
˜ y new∈N( ˜ y old,S proposal)
.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Mar
kov
chai
nM
onte
Car
lo(c
ont.
)
s
2
a
a
k2 k
2
q
˜s
2
a
a
˜k2 ˜k2
˜qT
wo-
dim
ensi
onal
his
togr
ams
illu
stra
ting
the
dep
enden
cebet
wee
nth
eco
mpon
ents
of(y|Y)before(
left
pan
e)an
daf
ter
(rig
ht
pan
e)th
etr
ansf
orm
atio
n.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Com
puta
tion
alburd
en
◮T
he
dom
inat
ing
cost
for
each
MC
MC
iter
atio
nis
calc
ula
tion
ofth
eC
hol
esky
fact
oris
atio
nof
Q.
◮In
vert
ing
afu
llco
vari
ance
mat
rix
isO( n
3) ,
◮G
iven
asp
atia
lG
MR
Fon
ala
ttic
ew
ith
npoi
nts
the
Chol
esky
fact
oris
O( n
3/2) ,
◮G
iven
asp
atio
-tem
por
alG
MR
Fon
ala
ttic
ew
ith
npoi
nts
the
Chol
esky
fact
oris
O( n
2) ,
How
bad
isth
ead
dit
ional
burd
enfo
rth
ete
mpor
aldep
enden
ce?
nSpat
ial
Spat
io-t
empor
al20
390.
05s
—15
·203
9=
3058
52.
48s
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Com
puta
tion
alburd
en
◮T
he
dom
inat
ing
cost
for
each
MC
MC
iter
atio
nis
calc
ula
tion
ofth
eC
hol
esky
fact
oris
atio
nof
Q.
◮In
vert
ing
afu
llco
vari
ance
mat
rix
isO( n
3) ,
◮G
iven
asp
atia
lG
MR
Fon
ala
ttic
ew
ith
npoi
nts
the
Chol
esky
fact
oris
O( n
3/2) ,
◮G
iven
asp
atio
-tem
por
alG
MR
Fon
ala
ttic
ew
ith
npoi
nts
the
Chol
esky
fact
oris
O( n
2) ,
How
bad
isth
ead
dit
ional
burd
enfo
rth
ete
mpor
aldep
enden
ce?
nSpat
ial
Spat
io-t
empor
al20
390.
05s
—15
·203
9=
3058
52.
48s
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Com
puta
tion
alburd
en
◮T
he
dom
inat
ing
cost
for
each
MC
MC
iter
atio
nis
calc
ula
tion
ofth
eC
hol
esky
fact
oris
atio
nof
Q.
◮In
vert
ing
afu
llco
vari
ance
mat
rix
isO( n
3) ,
◮G
iven
asp
atia
lG
MR
Fon
ala
ttic
ew
ith
npoi
nts
the
Chol
esky
fact
oris
O( n
3/2) ,
◮G
iven
asp
atio
-tem
por
alG
MR
Fon
ala
ttic
ew
ith
npoi
nts
the
Chol
esky
fact
oris
O( n
2) ,
How
bad
isth
ead
dit
ional
burd
enfo
rth
ete
mpor
aldep
enden
ce?
nSpat
ial
Spat
io-t
empor
al20
390.
05s
—15
·203
9=
3058
52.
48s
46.8
s
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Kri
ging
resu
lts
from
MC
MC
sim
ula
tion
of(t,k2 ,a, s2
) 00.2
1.8
10
Inte
rpol
ated
year
lypre
cipit
atio
n,in
met
res,
for
1982
.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Dep
thD
ata
◮W
ehav
em
easu
rem
ents
ofse
abea
ddep
thfo
ran
area
outs
ide
the
Sw
edis
hco
ast.
◮T
he
dat
ahas
bee
nm
odel
led
usi
ng
anunder
lyin
gG
MR
Ffiel
dw
ith
Mat
ern
cova
rian
cean
dunknow
nex
pec
tati
onco
nsi
stin
gof
anar
bit
rary
pla
ne
(e.g
.K
rigi
ng)
.
◮T
he
rough
ly12
’000
mea
sure
men
tlo
cati
ons
hav
ebee
ntr
inag
ula
ted
resu
ltin
gin
agr
idco
nta
inin
g20
’000
poi
nts
.
◮Pos
teri
orden
siti
esfo
rth
eth
ree
hyper
par
amet
ers
ofth
em
odel
and
for
the
under
lyin
gfiel
dw
her
ees
tim
ated
usi
ng
INLA
.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Dep
thD
ata
◮W
ehav
em
easu
rem
ents
ofse
abea
ddep
thfo
ran
area
outs
ide
the
Sw
edis
hco
ast.
◮T
he
dat
ahas
bee
nm
odel
led
usi
ng
anunder
lyin
gG
MR
Ffiel
dw
ith
Mat
ern
cova
rian
cean
dunknow
nex
pec
tati
onco
nsi
stin
gof
anar
bit
rary
pla
ne
(e.g
.K
rigi
ng)
.
◮T
he
rough
ly12
’000
mea
sure
men
tlo
cati
ons
hav
ebee
ntr
inag
ula
ted
resu
ltin
gin
agr
idco
nta
inin
g20
’000
poi
nts
.
◮Pos
teri
orden
siti
esfo
rth
eth
ree
hyper
par
amet
ers
ofth
em
odel
and
for
the
under
lyin
gfiel
dw
her
ees
tim
ated
usi
ng
INLA
.
◮Tot
ales
tim
atio
nti
me
ona
Cor
e2D
uo
des
kto
p:
8m
inute
s.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
The
Sahel
Depth
Data
Dep
thD
ata
–R
esult
s
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s
GM
RF:s
Result
sR
efe
rences
Bib
liog
raphy
Bolin,D
.,Lin
dst
rom
,J.,
Eklu
ndh,L.,
and
Lin
dgre
n,F.(2
008),
“Fast
Est
imation
ofSpatially
Dep
enden
tTem
pora
lV
eget
ation
Tre
nds
usi
ng
Gauss
ian
Mark
ov
Random
Fie
lds,
”Subm
itte
dto
Com
put.
Sta
tist
.and
Data
Anal.
Eklu
ndh,L.and
Ols
son,L.(2
003),
“V
eget
ation
index
tren
ds
for
the
Afr
ican
Sahel
1982-1
999,”
Geo
phys.
Res
.Let
t.,30,1430–1433.
Lin
dgre
n,F.and
Rue,
H.(2
007),
“E
xplici
tco
nst
ruct
ion
ofG
MR
Fappro
xim
ations
togen
eralise
dM
ate
rnFie
lds
on
irre
gula
rgri
ds,
”Tec
h.R
ep.12,C
entr
efo
rM
ath
ematica
lSci
ence
s,Lund
Univ
ersi
ty,Lund,Sw
eden
.
Lin
dst
rom
,J.and
Lin
dgre
n,F.(2
008),
“M
odel
ing
Yea
rly
cum
ula
tive
Pre
cipitation
over
the
Afr
ican
Sahel
usi
ng
aG
auss
ian
Mark
ov
Random
Fie
ld,”
Inpre
para
tion.
Rue,
H.and
Hel
d,L.(2
005),
Gauss
ian
Mark
ov
Random
Fie
lds;
Theo
ryand
Applica
tions,
vol.
104
of
Monogra
phs
on
Sta
tist
ics
and
Applied
Pro
bability,
Chapm
an
&H
all/C
RC
.
Rue,
H.and
Mart
ino,S.(2
007),
“A
ppro
xim
ate
Bayes
ian
infe
rence
for
hie
rarc
hic
al
Gauss
ian
Mark
ov
random
fiel
dm
odel
s,”
J.Sta
tist
.P
lann.and
Infe
rence
,137,
3177–3192.
Rue,
H.and
Tje
lmel
and,H
.(2
002),
“Fitting
Gauss
ian
Mark
ov
Random
Fie
lds
toG
auss
ian
Fie
lds,
”Sca
nd.J.Sta
tist
.,29,31–49.
Whittle,
P.(1
954),
“O
nSta
tionary
Pro
cess
esin
the
Pla
ne,
”B
iom
etri
ka,41,
434–449.
Johan
Lin
dstrom
-jo
hanl@
maths.lth.s
eT
heory
and
Applicatio
ns
ofG
MR
F:s