6
Spatio-temporal Data Model Simplifications Results References Space-time modelling: A case study Johan Lindstr ¨ om 1,2 1 Centre for Mathematical Sciences Lund University 2 Department of Statistics University of Washington Danish Technical University Aqua Charlottenlund October 17, 2012 Johan Lindstr¨ om - [email protected] Case study 1/21 Spatio-temporal Data Model Simplifications Results References Overview Lecture 1: Spatial modelling Lecture 2: Gaussian Markov Random Fields Lecture 3: Spatio-Temporal modelling — A case study 1. Spatio-temporal frameworks 2. Examples of Spatio-temporal modelling 3. Modelling NO x in Los Angeles — A case study Johan Lindstr¨ om - [email protected] Case study 2/21 Spatio-temporal Data Model Simplifications Results References Spatio-temporal modelling Spatio-temporal models typically fall into one of two main categories: 1. Spatial fields evolving in time 2. Spatially varying time series The modelling strategy should be based on the available data, the scientific question and computational considerations. Lots of recent work, less available “off-the-shelf” methods/packages. Johan Lindstr¨ om - [email protected] Case study 3/21 Spatio-temporal Data Model Simplifications Results References Spatio-temporal modelling — Examples A class of covariance functions for space-time data (Gneiting, 2002; Fuentes et al., 2008). Modelling of PM 2.5 using a series of spatially correlated fields (Paciorek et al., 2009). A separable space-time model formulated using GMRF:s (Cameletti et al., 2012). Physics based modelling of rainfall, used to postprocess forecasts (Sigrist et al., 2012). Dynamic model for coupled environmental variables (Ippoliti et al., 2012). Johan Lindstr¨ om - [email protected] Case study 4/21

Spatio-temporal modelling Spatio-temporal modelling — … · Spatio-temporal Data Model Simplifications Results References Combined model Spatio-temporal model We model the logarithm

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Spat

io-t

emp

oral

Dat

aM

odel

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pli

fica

tion

sR

esu

lts

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eren

ces

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ace

-tim

em

od

ellin

g:

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sest

ud

y

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an

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dst

rom

1,2

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tre

for

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tica

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nd

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ive

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ive

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ish

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ivers

ity

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2012

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est

ud

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Spat

io-t

emp

oral

Dat

aM

odel

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pli

fica

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esu

lts

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eren

ces

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rvie

w

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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—

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sest

ud

y

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m-

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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

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joh

anl@

mat

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lth

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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

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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

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ori

ng

Date

Location

2000

2002

2004

2006

2008

2010

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e

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ed

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snapshot

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ites

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eo

fsi

tesi

tes

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rtd

ate

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dd

ate

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r.o

fo

bs.

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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

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ap

sho

td

ate

s:2006–07–05,

2006–10–25,

an

d2007–01–31

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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

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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

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20

00

20

02

20

04

20

06

20

08

20

10

Joh

anL

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anl@

mat

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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-

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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···

t=2,ν(θν)

0

00

. ..

.

Bo

thB

an

are

Gau

ssia

nan

dw

eh

ave

[Y|θ

B,θν,α]∈

N(

FXα,Σν(θν)+

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)+

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

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m-

joh

anl@

mat

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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+

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+

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

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

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

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Cas

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4/2

1

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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

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Cas

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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

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Bib

liog

rap

hy

I

Cam

ele

tti,

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Lin

dg

ren

,F.

,Sim

pso

n,

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an

dR

ue,

H.

(2012),

“Sp

ati

o-t

em

po

ral

mo

deli

ng

of

part

icu

late

matt

er

con

cen

trati

on

thro

ug

hth

eSPD

Eap

pro

ach

,”A

StA

Ad

van

ces

inSta

tist

ical

An

aly

sis,

On

lin

e,

1–23.

Fu

en

tes,

M.,

Ch

en

,L.,

an

dD

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,J.

M.(2

008),

“A

class

of

no

nse

para

ble

an

dn

on

stati

on

ary

spati

alte

mp

ora

lco

vari

an

cefu

nct

ion

s,”

En

vir

on

metr

ics,

19,

487–507.

Fu

en

tes,

M.,

Gu

tto

rp,

P.,

an

dSam

pso

n,

P.D

.(2

006),

“U

sin

gtr

an

sfo

rms

toan

aly

ze

space

-tim

ep

roce

sses,

”in

Sta

tist

ical

Meth

od

sfo

rSp

ati

o-T

em

po

ral

Syst

em

s,ed

s.Fin

ken

stad

t,B

.,H

eld

,L.,

an

dIs

ham

,V

.,C

RC

/Ch

ap

man

an

dH

all

,p

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77–150.

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eit

ing

,T.

(2002),

“N

on

sep

ara

ble

,Sta

tio

nary

Co

vari

an

ceFu

nct

ion

sfo

rSp

ace

-Tim

eD

ata

,”J.

Am

er.

Sta

tist

.A

sso

c.,

97,

590–600.

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eren

ces

Bib

liog

rap

hy

II

Harv

ille

,D

.A

.(1

997),

Matr

ixA

lgeb

raFro

ma

Sta

tist

icia

n’s

Pers

pect

ive,

Sp

rin

ger,

1st

ed

.

Ipp

oli

ti,L.,

Vale

nti

ni,

P.,

an

dG

am

erm

an

,D

.(2

012),

“Sp

ace

-tim

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od

ell

ing

of

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ple

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ote

mp

ora

len

vir

on

men

tal

vari

ab

les,

”J.

Ro

y.Sta

tist

.So

c.Ser.

C,

61,

175–200.

Lin

dst

rom

,J.

,Szp

iro

,A

.A

.,Sam

pso

n,

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.,Sh

ep

pard

,L.,

Oro

n,A

.,R

ich

ard

s,M

.,an

dLars

on

,T.

(2011),

“A

Fle

xib

leSp

ati

o-T

em

po

ral

Mo

del

for

Air

Po

llu

tio

n:

All

ow

ing

for

Sp

ati

o-T

em

po

ral

Co

vari

ate

s,”

Tech

.R

ep

.W

ork

ing

Pap

er

370,

UW

Bio

stati

stic

sW

ork

ing

Pap

er

Seri

es.

Paci

ore

k,

C.

P.,

Yan

osk

y,J.

D.,

Pu

ett

,R

.C

.,Lad

en

,F.

,an

dSu

h,

H.

H.

(2009),

“Pra

ctic

al

Larg

e-S

cale

Sp

ati

o-T

em

po

ral

Mo

eli

ng

of

Part

icu

late

Matt

er

Co

nce

ntr

ati

on

s,”

An

n.

Sta

tist

.,3,

370–397.

Pete

rsen

,K

.B

.an

dPed

ers

en

,M

.S.

(2008),

“Th

eM

atr

ixC

oo

kb

oo

k,”

htt

p:/

/matr

ixco

okb

oo

k.c

om

.

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eren

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Bib

liog

rap

hy

III

Sam

pso

n,

P.D

.,Szp

iro

,A

.A

.,Sh

ep

pard

,L.,

Lin

dst

rom

,J.

,an

dK

au

fman

,J.

D.(2

011),

“Pra

gm

ati

cEst

imati

on

of

aSp

ati

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ral

Air

Qu

ali

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od

el

wit

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ula

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on

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rin

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ata

,”A

tmo

.En

vir

on

.,45,

6593–6606.

Sig

rist

,F.

,K

un

sch

,H

.R

.,an

dSta

hel,

W.

A.

(2012),

“A

nSPD

EB

ase

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ral

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del

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Larg

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ata

Sets

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han

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cip

itati

on

Fo

reca

sts,

”Te

ch.

rep

.,Sem

inar

for

Sta

tist

ics,

Dep

art

men

to

fM

ath

em

ati

cs,

ETH

Zu

rich

,Zu

rich

,Sw

itzerl

an

d.

Szp

iro

,A

.,Sam

pso

n,

P.,Sh

ep

pard

,L.,

Lu

mle

y,T.

,A

dar,

S.,

an

dK

au

fman

,J.

(2010),

“Pre

dic

tin

gin

tra-u

rban

vari

ati

on

inair

po

llu

tio

nco

nce

ntr

ati

on

sw

ith

com

ple

xsp

ati

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em

po

ral

dep

en

den

cies,

”En

vir

on

metr

ics,

21,

601–631.

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