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