Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Sp
ace
-tim
em
od
ellin
g:
Aca
sest
ud
y
Joh
an
Lin
dst
rom
1,2
1C
en
tre
for
Ma
the
ma
tica
lSci
en
ces
Lu
nd
Un
ive
rsit
y
2D
ep
art
me
nt
of
Sta
tist
ics
Un
ive
rsit
yo
fW
ash
ing
ton
Dan
ish
Tech
nic
al
Un
ivers
ity
Aq
ua
Ch
arl
ott
en
lun
dO
cto
ber
17,
2012
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y1
/21
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Ove
rvie
w
Lec
ture
1:
Sp
ati
al
mo
dell
ing
Lec
ture
2:
Gau
ssia
nM
ark
ov
Ran
do
mFie
lds
Lec
ture
3:
Sp
ati
o-T
em
po
ral
mo
dell
ing
—A
case
stu
dy
1.
Sp
ati
o-t
em
po
ral
fram
ew
ork
s2.
Exam
ple
so
fSp
ati
o-t
em
po
ral
mo
dell
ing
3.
Mo
dell
ing
NO
xin
Lo
sA
ng
ele
s—
Aca
sest
ud
y
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y2
/21
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Spat
io-t
emp
oral
mod
elli
ng
Sp
ati
o-t
em
po
ral
mo
dels
typ
icall
yfa
llin
too
ne
of
two
main
cate
go
ries:
1.
Sp
ati
al
field
sevo
lvin
gin
tim
e
2.
Sp
ati
all
yvary
ing
tim
ese
ries
Th
em
od
ell
ing
stra
teg
ysh
ou
ldb
eb
ase
do
nth
eavail
ab
led
ata
,th
esc
ien
tifi
cq
uest
ion
an
dco
mp
uta
tio
nal
con
sid
era
tio
ns.
Lo
tso
fre
cen
tw
ork
,le
ssavail
ab
le“o
ff-t
he-s
helf
”m
eth
od
s/p
ack
ag
es.
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y3
/21
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Spat
io-t
emp
oral
mod
elli
ng
—E
xam
ple
s
◮A
class
of
covari
an
cefu
nct
ion
sfo
rsp
ace
-tim
ed
ata
(Gn
eit
ing
,2002;
Fu
en
tes
et
al.
,2008).
◮M
od
ell
ing
of
PM
2.5
usi
ng
ase
ries
of
spati
all
yco
rrela
ted
field
s(P
aci
ore
ket
al.
,2009).
◮A
sep
ara
ble
space
-tim
em
od
el
form
ula
ted
usi
ng
GM
RF:s
(Cam
ele
ttiet
al.
,2012).
◮Ph
ysi
csb
ase
dm
od
ell
ing
of
rain
fall
,u
sed
top
ost
pro
cess
fore
cast
s(S
igri
stet
al.
,2012).
◮D
yn
am
icm
od
el
for
cou
ple
den
vir
on
men
tal
vari
ab
les
(Ip
po
liti
et
al.
,2012). Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y4
/21
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Bac
kgro
un
dL
osA
nge
les
Th
eM
ESA
Air
stu
dy
◮Th
eM
ult
i-Eth
nic
Stu
dy
of
Ath
ero
scle
rosi
s(M
ESA
)is
ala
rge
stu
dy
of
card
iovasc
ula
rd
isease
s.
◮It
foll
ow
sm
ore
than
6000
peo
ple
fro
msi
xco
mm
un
itie
s.◮
Ba
ltim
ore
◮C
hic
ag
o◮
Los
An
gele
s◮
Min
ne
ap
olis
–Sa
int
Pa
ul
◮N
ew
Yo
rk◮
Win
sto
n–Sa
lem
◮M
ESA
Air
isan
EPA
stu
dy
of
ho
wair
po
llu
tio
neff
ect
sca
rdio
vasc
ula
rd
isease
s.
◮Th
ep
rim
ary
po
llu
tan
tsin
the
MESA
Air
stu
dy
are
PM
2.5
an
dN
Ox
(th
isca
se).
◮See
Szp
iro
et
al.
(2010);
Sam
pso
net
al.
(2011);
Lin
dst
rom
et
al.
(2011)
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y5
/21
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Bac
kgro
un
dL
osA
nge
les
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y6
/21
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Bac
kgro
un
dL
osA
nge
les
Ava
ilab
led
ata
—L
osA
nge
les
050100150200250
ME
SA
mo
nit
ori
ng
Date
Location
2000
2002
2004
2006
2008
2010
ME
SA
hom
e
ME
SA
fix
ed
ME
SA
snapshot
AQ
S s
ites
Typ
eo
fsi
tesi
tes
Sta
rtd
ate
En
dd
ate
Nb
r.o
fo
bs.
AQ
S20
1999–01–27
2009–10–07
4180
MESA
fixed
52005–12–07
2009-0
7-0
1399
MESA
ho
me
84
2006–05–24
2008–02–13
155
MESA
snap
sho
t1177
2006–07–05
2007–01–31
449
1Sn
ap
sho
td
ate
s:2006–07–05,
2006–10–25,
an
d2007–01–31
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y7
/21
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Bac
kgro
un
dL
osA
nge
les
Ava
ilab
led
ata
—L
osA
nge
les
0123456
Gle
nd
ora
60
37
00
16
Da
te
NOx (log ppb)
20
00
20
02
20
04
20
06
20
08
20
10
Ob
se
rva
tio
ns
Fitte
d s
mo
oth
tre
nd
log
(Ca
lin
e+
1)
0123456
Ly
nw
oo
d 6
03
71
30
1
Da
te
NOx (log ppb)
20
00
20
02
20
04
20
06
20
08
20
10
0123456
Co
sta
Me
sa
60
59
00
07
Da
te
NOx (log ppb)
20
00
20
02
20
04
20
06
20
08
20
10
0123456
A H
om
e c
los
e t
o L
yn
wo
od
60
37
13
01
Da
te
NOx (log ppb)
20
00
20
02
20
04
20
06
20
08
20
10
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y8
/21
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Com
bin
edm
odel
Spat
io-t
emp
oral
mod
el
We
mo
del
the
log
ari
thm
of
each
2-w
eek
avera
ge
as
y(s,t)=
m∑
i=1
βi(
s)f i(t)+ν(
s,t).
f i(t)
Sm
oo
thte
mp
ora
ltr
en
ds
wit
hf 1(t)≡
1an
df 2(t),...,
f m(t)
mean
zero
.
βi(
s)Sp
ati
all
yvary
ing
coeffi
cien
tsfo
rth
ete
mp
ora
ltr
en
ds.
ν(s,
t)R
esi
du
als
,m
od
ell
ed
as
am
ean
zero
Gau
ssia
nfi
eld
that
isin
dep
en
den
tin
tim
eb
ut
has
spati
al
stru
ctu
re.
Th
esm
oo
thte
mp
ora
ltr
en
ds,
f i(t)
are
com
pu
teu
sin
ga
sin
gu
lar
valu
ed
eco
mp
osi
tio
no
fth
ed
ata
matr
ix,Y
(see
Fu
en
tes
et
al.
,2006,
an
dth
eco
mp
ute
rexerc
ise).
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y9
/21
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Com
bin
edm
odel
Spat
io-t
emp
oral
mod
el(c
ont.
)
βi(
s)∈
N(X
iαi,Σβ
i(θ
B))
Xi
Desi
gn
matr
ices,
that
incl
ud
es
geo
gra
ph
ical
covari
ate
s(d
iffe
ren
tfo
reach
i).
αi
Reg
ress
ion
coeffi
cien
ts.
Σβ
iC
ovari
an
cem
atr
ixd
esc
rib
ing
ad
dit
ion
al
spati
al
dep
en
den
cen
ot
cap
ture
db
yth
eg
eo
gra
ph
ical
covari
ate
s.
θB
Para
mete
rso
fth
eco
vari
an
cem
atr
ices.
ν(s,
t)∈
N(0,Σν(θν))
Σν
Blo
ckd
iag
on
al
covari
an
cem
atr
ixfo
rth
ere
sid
uals
.
θν
Para
mete
rso
fth
ere
sid
ual
covari
an
cem
atr
ix.
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y1
0/2
1
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Com
bin
edm
odel
Com
bin
edm
odel
Wri
tin
gth
em
od
el
on
matr
ixfo
rmw
eo
bta
in
Y=
FB+ν
wh
ere B∈
N
X1
00
0X
20
00
X3
α1
α2
α3
,
Σβ
1(θ
B)
00
0Σβ
2(θ
B)
00
0Σβ
3(θ
B)
an
d
ν∈
N
0,
Σt=
1,ν(θν)
0···
0Σ
t=2,ν(θν)
0
00
. ..
.
Bo
thB
an
dν
are
Gau
ssia
nan
dw
eh
ave
[Y|θ
B,θν,α]∈
N(
FXα,Σν(θν)+
FΣ
B(θ
B)F
⊤)
,
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y1
1/2
1
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Com
bin
edm
odel
Com
bin
edm
odel
(con
t.)
Ou
rm
od
el
isn
ow
[Y|θ
B,θν,α]∈
N(
FXα,Σν(θν)+
FΣ
B(θ
B)F
⊤)
,
an
dp
ara
mete
rsca
nb
eest
imate
db
ym
axim
isin
gth
elo
g-l
ikeli
ho
od
l(θ
B,θν,α|Y
).
Est
imati
on
of
the
ab
ove
mo
del
isco
mp
uta
tio
nall
yexp
en
sive
an
dw
ere
du
ceth
eco
mp
uta
tio
nal
cost
by:
1.
Use
pro
file
likeli
ho
od
tore
du
ceth
en
um
ber
of
para
mete
rsin
the
log
-lik
eli
ho
od
.
2.
Uti
lise
the
blo
ckst
ruct
ure
inΣ
Ban
dΣν,to
red
uce
the
com
pu
tati
on
al
bu
rden
.
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y1
2/2
1
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Sim
pli
fica
tion
sC
omp
uta
tion
alis
sues
Lik
elih
ood
sim
pli
fica
tion
s
◮M
atr
ixalg
eb
raca
nb
eu
sed
to“si
mp
lify
”th
eli
keli
ho
od
(Harv
ille
,1997;
Pete
rsen
an
dPed
ers
en
,2008).
◮A
san
exam
ple
we
stu
dy
the
dete
rmin
an
to
fth
elo
g-l
ikeli
ho
od
log∣ ∣
Σν+
FΣ
BF⊤∣ ∣
=lo
g|Σν|+
log|Σ
B|+
log∣ ∣ ∣Σ
−1
B+
F⊤Σ
−1
νF∣ ∣ ∣
Th
ism
ay
no
tse
em
sim
ple
rb
ut:
1.Σν+
FΣ
BF⊤
isd
en
seN
×N
-matr
ix,an
dco
mp
uti
ng
the
dete
rmin
an
tre
qu
iresO(
N3)
op
era
tio
ns.
2.Σν
an
dΣ
Bare
bo
thb
lock
dia
go
nal,
wit
h“sm
all
”b
lock
s.
3.Σ
−1
B+
F⊤Σ
−1
νF
isa
den
sem
n×
nm
-matr
ix.
Co
mp
uti
ng
the
dete
rmin
an
tre
qu
iresO(
m3n
3)
op
era
tio
ns,
wit
hm
n≪
N.
Wh
ere
:
NTo
tal
nu
mb
er
of
ob
serv
ati
on
s.
nTo
tal
nu
mb
er
of
ob
serv
ed
site
s.
mN
um
ber
of
tem
po
ral
basi
sfu
nct
ion
s(i
ncl
.in
terc
ep
t).
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y1
3/2
1
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Sim
pli
fica
tion
sC
omp
uta
tion
alis
sues
Com
pu
tati
onal
issu
es
1000
2000
3000
4000
5000
Co
mp
ute
r ti
me f
or
evalu
ati
on
of
the p
rofi
le lo
g−
likelih
oo
d
Num
ber
of observ
ations
Time (s)
0.050.5550
Naïv
e, 1 to 2
86 locations
Optim
ised, 1 to 5
0 locations
Optim
ised, 51 to 1
00 locations
Optim
ised, 101 to 2
00 locations
Optim
ised, 201 to 2
86 locations
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y1
4/2
1
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Est
.P
ar.
Pre
dic
ted
aver
age
Val
idat
ion
Est
imat
edp
aram
eter
s
−101234
F1 − Intercept
F2 − Intercept
F3 − Intercept
−0.20.00.20.40.60.81.0
Inte
rce
pt
(F1
)
Dist. to A1
Dist. to road
A1 & A2 in buffer
A3 in buffer
Pop. in 2km buffer
Dist. to coast
Co
va
ria
nc
e p
ara
me
ters
F1 − range
F1 − sill
F2 − range
F2 − sill
F3 − range
F3 − sill
Res. − range
Res. − sill
Res. − nugget
1e−021e−011e+001e+011e+02
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y1
5/2
1
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Est
.P
ar.
Pre
dic
ted
aver
age
Val
idat
ion
Pre
dic
ted
aver
age
NO
xco
nce
ntr
atio
n—
Los
An
gele
s
Joh
anL
ind
stro
m-
joh
anl@
mat
hs.
lth
.se
Cas
est
ud
y1
6/2
1
Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
eren
ces
Est
.P
ar.
Pre
dic
ted
aver
age
Val
idat
ion
Mod
elva
lid
atio
n—
NO
xin
Los
An
gele
s
23456
Gle
nd
ora
60
37
00
16
Da
te
NOx (log ppb)
20
00
20
02
20
04
20
06
20
08
20
10
Ob
se
rva
tio
ns
Pre
dic
tio
ns
95
% C
I
23456
Ly
nw
oo
d 6
03
71
30
1
Da
te
NOx (log ppb)
20
00
20
02
20
04
20
06
20
08
20
10
23456
Co
sta
Me
sa
60
59
00
07
Da
te
NOx (log ppb)
20
00
20
02
20
04
20
06
20
08
20
10
23456
A H
om
e c
los
e t
o L
yn
wo
od
60
37
13
01
Da
te
NOx (log ppb)
20
00
20
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Spat
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oral
Dat
aM
odel
Sim
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lts
Ref
eren
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Est
.P
ar.
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ted
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age
Val
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Spat
io-t
emp
oral
Dat
aM
odel
Sim
pli
fica
tion
sR
esu
lts
Ref
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ind
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Ref
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ind
stro
m-
joh
anl@
mat
hs.
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.se
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est
ud
y2
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1
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Dat
aM
odel
Sim
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fica
tion
sR
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Ref
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anL
ind
stro
m-
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mat
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