25

1

By

Jay M. Ver Hoef

National Marine Mammal Lab

7600 Sand Point Way, NE

Seattle, WA 98115

jay.verhoef@

noaa.gov

By

Jay M. Ver Hoef

National Marine Mammal Lab

7600 Sand Point Way, NE

Seattle, WA 98115

jay.verhoef@

noaa.gov

An Introduction to Statistical

Models for Spatial Data in Ecology

An Introduction to Statistical

Models for Spatial Data in Ecology

2

What do Statisticians Do?

What do Statisticians Do?

3

What is a Model?

What is a Model?

What does it look

like?

How does it work?

Rep

rese

nta

tion

al

Fu

ncti

ona

l

4

Reality

“A

ll m

od

els

are

wro

ng

. W

e m

ake

ten

tati

ve a

ssu

mp

tion

s a

bou

t th

e re

al

wo

rld

wh

ich

we

kno

w a

re f

als

e b

ut

wh

ich

we

bel

ieve

ma

y b

e u

sefu

l.”

-G

eorg

e B

ox

19

76

What are Spatial Statistics

What are Spatial Statistics

Data Model

Representational

What does it look

like?

Points

Lines

Polygons

Probability Model

Lack of Fit

What don’t we

understand?

θ n,

θ s, θ

r

Scientific Model

Functional

How does it work?

Modeling

Modeling

Modeling

Inference

Inference

Inference

020

40

60

80

100

0246810

x

y

26

5

Notation

Notation

Z(s

)D

•D

is t

he

spat

ial

dom

ain

or

area

of

inte

rest

•s

con

tain

s th

e sp

atia

l

coord

inat

es

•Z

is a

val

ue

loca

ted

at

the

spat

ial

coord

inat

es

6

Types of Spatial Data

Types of Spatial Data

{Z

(s):

s∈

D}

•G

eost

ati

stic

al

Da

ta:

Zra

ndo

m;

Dfi

xed

, in

fin

ite,

co

nti

nu

ou

s

•L

att

ice

Da

ta:

Z r

and

om

; D

fix

ed,

fin

ite,

(ir

)reg

ula

rgri

d

•P

oin

t P

att

ern

Da

ta:

Z ≡

1;

D r

and

om

, fi

nit

e

Z(s

)D

7

Examples of GeostatisticalData

Examples of GeostatisticalData

Ozo

ne

Pre

dic

tion

sA

ver

age

Sn

ow

Dep

th

8

Examples of Lattice Data

Examples of Lattice Data 2

312

4

55144

51344

31523

52312

Transform

ed SIDS rates

Plots in a Designed

Experiment

27

9

Examples of Point Patterns

Examples of Point Patterns

Arctic Caribou Calving

Locations

x

yLansing W

oods Hickory

Locations

10

Statistical Models

Statistical Models

)(

)var

( ,

θΣ εεX

βzz

=+

=

un

ob

serv

ed

ob

serv

ed

Prediction

Estimation

Linear Model

) ,

,(

θ εX

zzg

un

ob

serv

ed

ob

serv

ed=

Nonlinear Model

11

•Description of data

•Property of a stochastic process

•Model for a stochastic process

•Statistic

•Function in Fourier analysis

Five Meanings of

Autocorrelation

Five Meanings of

Autocorrelation

12

Four Meanings of Autocorrelation

Four Meanings of Autocorrelation

02

04

06

08

01

00

0246810

x

y

02

04

06

08

01

00

02468

x

y

Independent Errors

AutocorrelatedErrors

Iε 21

0)

var(

,σ

β=

++

=i

ii

εxβ

y

Σε =+

+=

)var

( ,

10

ii

i

εxβ

yβ

01

02

03

04

05

0

-2024

Dis

tance

Empirical Covariance

02

04

06

08

01

00

010203040

Dis

tance

Autocovariance

Autocorrelation Model

Autocorrelation Statistic

AutocorrelatedData 1

2

34

28

13

Autocorrelation Models

Autocorrelation Models

14

Autocorrelation

Autocorrelation

15

Try it! F9

Try it! F9

16

0

Z(1

) ~

N(µ µµµ

,1)

12

34

5

0

12

34

5

Z(i

) ~

N( µ µµµ

,1),

i.i

.d.

=

11

1

11

1

11

1

L

MO

MM

LL Σ

=

10

0

01

0

00

1

L

MO

MM

LL ΣWhy Spatial Statistics?

Why Spatial Statistics?

Z(i

> 1

) =

Z(1

)

29

17

Fits vs. Prediction

Fits vs. Prediction

02

04

06

08

010

0

0246810

x

y

02

04

06

08

0100

02468

x

y

Independent Errors

AutocorrelatedErrors

FitPrediction

Fit = Prediction

Variances different in both cases)

()

var

( ,

θΣ εε

X

β

zz=

+=

uno

bse

rved

ob

serv

ed

18

Estimation and Prediction

Estimation and Prediction )

()

var

( ,

θΣ εεX

βzz

=+

=

unobse

rved

obse

rved

Prediction

Estimation

•Mapping

•Sampling

•Regression

•Designed Experiments

19

=

nn

nn

n

σσ

σ

σσ

σ

σσ

σ

L

MO

MM

LL

21

21

22

21

11

21

1 ΣWhy Do W

e Need

Autocorrelation Models?

Why Do W

e Need

Autocorrelation Models?

)(

)var

( ,

θΣ εεX

βzz

=+

=

uno

bse

rved

ob

serv

ed

20

•Weighted Least Squares

•Generalized Least Squares

•Maximum Likelihood

•Restricted Maximum Likelihood

•Bayes(M

arkov Chain Monte Carlo MCMC)

Estimation?!

Estimation?!

)(

)var

( ,

θΣ εεX

βzz

=+

=

unobse

rved

obse

rved

Leave it to the Statisticians!

30

21

Pitfalls: Valid Autocorrelation

Models

Pitfalls: Valid Autocorrelation

Models

22

Stream Netw

ork Models

Stream Netw

ork Models

SO4Concentration

SO4Concentration

23

Pitfalls: Valid Models for

Stream Netw

orks

Pitfalls: Valid Models for

Stream Netw

orks

Not to scale. All lengths = 1

24

Prediction -Mapping

Prediction -Mapping

Prediction Map

Standard Error Map

)(

)v

ar(

,

θΣ εεX

βzz

=+

=

un

ob

serv

ed

ob

serv

ed

31

25

Mapping -Quantiles

Mapping -Quantiles

26

QuantileMaps

QuantileMaps

27

Probability Maps

Probability Maps

Pro

b=

0.1

23

Pre

dic

tion

= 0

.080

Pre

dic

tion

= 0

.091

Pro

b=

0.0

80

Pre

dic

tion

= 0

.114

Pro

b=

0.6

24

Pre

dic

tion

= 0

.114

Pro

b=

0.5

82

28

•W

hip

tail

Liz

ard

•1

48

loca

tions

in S

outh

ern

Cali

forn

ia

•M

easu

red

the

aver

age

num

ber

cau

ght

in t

rap

s o

ver

80 –

90

tra

pp

ing

ev

ents

in o

ne

yea

r

•D

ata

log-t

ransf

orm

ed,

one

outl

ier

rem

oved

Spatial Regression

Spatial Regression

32

29

•T

her

e w

ere

37

ex

pla

nat

ory

var

iab

les

in 5

bro

ad c

ateg

ori

es:

veg

etat

ion

lay

ers,

veg

etat

ion

typ

es,

top

ogra

ph

ic p

osi

tion

, so

il

typ

es, an

d a

nt

abu

nd

ance

Whiptail Lizard Example

Whiptail Lizard Example

30

California Lizard Data

California Lizard Data

-4.9

053 t

o -

4.4

023

-4.4

023 t

o -

3.8

992

-3.8

992 t

o -

3.3

962

-3.3

962 t

o -

2.8

931

-2.8

931 t

o -

2.3

901

-2.3

901 t

o -

1.8

87

-1.8

87 t

o -

1.3

839

-1.3

839 t

o -

0.8

809

-0.8

809 t

o -

0.3

778

-0.3

778 to 0

.1251

-5-4-3-2-10

Bo

xp

lot

of

raw

data

log(Hyper) abundance

His

tog

ram

of

raw

data

log

(Hyp

er)

a

bund

anc

e

Frequency

-5-4

-3-2

-10

0510152025

)(

)var(

,

θΣ εεX

βzz

=+

=

uno

bse

rved

obse

rved

•Ant abundance

•Percent sandy soil

•Spherical autocorrelation

•Isotropic

Estimation: REML

followed by GLS

31

Exploratory Data Analysis on

Residuals

Exploratory Data Analysis on

Residuals

2/1

2/1

)

1(

ˆ

:R

esi

du

al

d

Stu

den

tize

−−

−−

′′

=−

−

ΣX

X)Σ

XX

(ΣH

1

ii

ii

hM

SE

zµ

-4-3-2-1012

Bo

xp

lot o

f stu

de

nti

ze

d r

esid

ua

ls

log(Hyper) abundance

His

tog

ram

of

stu

de

nti

ze

d r

esid

ua

ls

log

(Hyp

er)

ab

und

ance

Frequency

-4-2

02

0102030405060

32

Model Diagnostics

Model Diagnostics

•L

ikel

ihood

Dis

tan

ce

•C

ook’s

D a

nd

Lev

erag

e

•C

ovar

ian

ce T

race

Based on deleting observations

Mostly for outlier detection

()

)(

)var(

,1

θΣ εεX

βz

=+

=−

obse

rved

Co

ok

sD

Frequency

0.0

0.1

0.2

0.3

0.4

0.5

050100150

33

33

Remove Outlier and Re-fit

Lizard Data

Remove Outlier and Re-fit

Lizard Data

His

tog

ram

of

stu

de

nti

ze

d r

esid

ua

ls

log

(Hyp

er)

ab

und

ance

Frequency

-2-1

01

2

051015202530

-2-1012

Bo

xp

lot o

f s

tud

en

tize

d r

es

idu

als

log(Hyper) abundance

Co

oksD

Frequency

0.0

00.0

20.0

40

.06

0.0

80.1

0

020406080100120

34

Cross-validation

Cross-validation

-5-4

-3-2

-10

-5-4-3-2-10

Cro

ss

-va

lid

ati

on

Sc

att

er

Plo

t

Me

asure

d V

alu

e

Predicted Value

35

Directional Autocorrelation

Directional Autocorrelation

5 Parameters

Isotropy vs.

Anisotropy

36

Cross-validation

Cross-validation

∑

∑∑

=

=

=

<

−

−−

n iii

i

n i

ii

n iii

i

ZzZ

I

zZ

n

ZzZ

n

1

1

2

1

96

.1

)ˆ

r(av

)ˆ

(

:C

over

ag

e

Inte

rval

P

redic

tion

)ˆ

(1

:E

rror

P

redic

tion

S

quare

dM

ean

Root

)ˆ

r(av

)ˆ

(1

:B

ias

ed

Sta

ndard

iz)

()

var

( ,

θΣ εεX

βz

=+

=

−

i

io

bse

rved

Z•Spherical autocorrelation

•Isotropic

•Spherical autocorrelation

•Anisotropic

34

37

Model Selection

Model Selection

•AIC

•AICc

•BIC

•etc.!

-2*loglikelihood+

(Penalty for number of parameters)

Choose the model with the Minimum of these:

Be careful! Some software uses

2*loglikelihood–(Penalty for number of parameters),

in which case you choose the m

axim

um.

Can also use RMSPE and other criteria. W

hy not?

38

Model Selection

Model Selection

39

Final Fitted Model

Final Fitted Model )

()

var

( ,

θΣ εεX

βzz

=+

=

un

ob

serv

ed

ob

serv

ed

40

Spatial Regression

References

Spatial Regression

References

•V

er H

oef

, J.

M.

19

93

. U

niv

ersa

l kri

gin

gfo

r ec

olo

gic

al d

ata.

Pag

es

447

–4

53

in G

oo

dch

ild

, M

.F.,

Par

ks,

B.,

and

Ste

yae

rt,

L.T

. (e

ds.

)

En

vir

on

men

tal

Mo

deli

ng

wit

h G

IS,

Oxfo

rd U

niv

ersi

ty P

ress

, 4

88

p.

•V

er H

oef

, J.

M.,

Cre

ssie

, N

., F

isher

, R

.N.,

and

Cas

e, T

.J.

20

01

.

Unce

rtai

nty

and

sp

atia

l li

nea

r m

od

els

for

eco

logic

al d

ata.

Pag

es2

14

–

23

7 i

n H

un

saker,

C.T

., G

oo

dchil

d,

M.F

.. F

ried

l, M

.A.,

and

Cas

e, T

.J.

(ed

s.),

Sp

ati

al

Un

cert

ain

ty f

or

Eco

log

y:

Imp

lica

tio

ns

for

Rem

ote

Sen

sin

g a

nd

GIS

Ap

pli

ca

tio

ns

Sp

rin

ger

-Ver

lag.

•M

aier

, J.

A.K

., V

er H

oef

, J.

M.,

McG

uir

e, A

.D.,

Bo

wyer

, R

.T.,

Sap

erst

ein,

L.

and

Mai

er,

H.A

. 2

00

6.

Dis

trib

uti

on a

nd

den

sity

of

mo

ose

in r

elat

ion t

o l

and

scap

e ch

arac

teri

stic

s: E

ffec

ts o

f sc

ale.

In

pre

ss, C

an

ad

ian

Jo

urn

al

of

Fo

rest

Rese

arc

h.

35

41

Glades in Ozarks

Glades in Ozarks

42

Glades in Ozarks

Glades in Ozarks

43

+6

5

+6

4

− −−−5

3

− −−−3

2

01

Eff

ect

Tre

atm

en

t

Add T

rts

Est

imat

e

2312

4

55144

51344

31523

52312

Designed Experiment

Designed Experiment

44

Estimation and Prediction

Estimation and Prediction )

()

var

( ,

θΣ εεX

βzz

=+

=

unobse

rved

obse

rved

Prediction

Estimation

•Mapping

•Sampling

•Regression

•Designed Experiments

36

45

Linear Models

Linear Models

ε

X

β

zs

s+

=+

=or

)

()

(i

ji

Zε

τ0ε =

)(

E

Cova

ria

nce

Mod

els

Iε 2 )var

(σ

=In

dep

end

ence

Mo

del

s

Σ ε =)

var

(G

eost

ati

stic

al

Mod

els

Exp

on

enti

al,

Sp

her

ica

l, e

tc.

Σ ε =)

var

(L

att

ice

Mod

els

CA

R,

SA

R,

etc.

46

Estimating Treatm

ent Effects

Estimating Treatm

ent Effects

2312

4

55144

51344

31523

52312

47

Designed Experiment

Experiment

Designed Experiment

Experiment

+6

5

+6

4

− −−−5

3

− −−−3

2

01

Eff

ect

Tre

atm

en

t

1600

tim

es

Est

imat

e

2312

4

55144

51344

31523

52312

48

Designed Experiment Results

Designed Experiment Results

37

49

2312

4

55144

51344

31523

52312

Plots in a

Designed

Experiment

•V

er H

oef

, J.

M.

and

Cre

ssie

, N

. 2

00

1.

Sp

atia

l st

atis

tics

: A

nal

ysi

s o

f fi

eld

exp

erim

ents

. In

Sch

einer,

S.M

. an

d

Gu

revit

ch,

J. (

eds.

), D

esi

gn

an

d

An

aly

sis

of

Eco

log

ical

Exp

eri

men

ts,

Seco

nd

Ed

itio

n,

Oxfo

rd U

niv

ersi

ty

Pre

ss,

p. 2

89

-307

.

•L

enart

, E

.A.,

Bo

wyer

, R

.T.,

Ver

Ho

ef,

J.,

and

Ru

ess

, R

.W. 2

00

2.

Cli

mat

e ch

ange

and

car

ibo

u:

effe

cts

of

sum

mer

wea

ther

on f

ora

ge.

C

an

adia

n

Jo

urn

al

of

Zo

olo

gy

80:

66

4 –

67

8.

Designed Experiments

References

Designed Experiments

References

50

Spatial Sampling

Spatial Sampling

Moose

Su

rvey

Sou

th o

f

Fa

irb

an

ks

~ 4

50

0 m

i2

51

Sources of Randomness

Sources of Randomness

0.0

0.2

0.4

0.6

0.8

1.0

x

0

20

40

value

0.0

0.2

0.4

0.6

0.8

1.0

x

0

20

40

value

Fix

ed

Pa

tte

rn,

Ra

nd

om

Sa

mp

les

Fix

ed

Pa

tte

rn,

Ra

nd

om

Sa

mp

les

Ra

nd

om

Pa

tte

rn,

Fix

ed

Sam

ple

s

Ra

nd

om

Pa

tte

rn,

Fix

ed

Sam

ple

s

0.0

0.2

0.4

0.6

0.8

1.0

x

0

20

40

value

0.0

0.2

0.4

0.6

0.8

1.0

x

0

20

40

value

52

Source of Randomness

Source of Randomness

)1)

(exp(

)

cos(

)

cos(

)si

n(

)si

n(

)(

22

11

22

11

−+

+

++

=

xx

x

xx

xz

ec

cc

c

ss

ss

αβ

αβ

α

βα

βαFix

ed

Pa

tte

rn,

Ra

nd

om

Sa

mp

les

Fix

ed

Pa

tte

rn,

Ra

nd

om

Sa

mp

les

),

0(~

)(

);

()

()

(2

1σ

εε

ρN

xx

xz

xz

ii

ii

+=

−

Ra

nd

om

Pa

tte

rn,

Fix

ed

Sa

mp

les

Ra

nd

om

Pa

tte

rn,

Fix

ed

Sa

mp

les

38

53

Sampling and Geostatistics

Sampling and Geostatistics

54

∫=

Ad

zs

s)(τ

Ad

zA

/)

(∫

=s

sα

Tota

lM

ean

Infinite Population Parameters

Infinite Population Parameters

)(

)var(

,

θΣ εεX

βzz

=+

=

uno

bse

rved

ob

serv

ed

55

0.1

0.3

0.5

0.7

0.9

x

0.1

0.3

0.5

0.7

0.9

y 0.1

0.3

0.5

0.7

0.9

x

0.1

0.3

0.5

0.7

0.9

y

0.1

0.3

0.5

0.7

0.9

x

0.1

0.3

0.5

0.7

0.9

y 0.1

0.3

0.5

0.7

0.9

x

0.1

0.3

0.5

0.7

0.9

y

Fix

ed P

att

ern

, R

an

do

m S

am

ple

s

Simulation Study

Simulation Study

56

Simulation Results

Simulation Results

39

57

Sampling for Finite Populations

Sampling for Finite Populations

58

∑=

=N i

iz

1)

(sτ

∑=

=N i

iz

N1

)(

)/

1(s

α

Finite Population Parameters

Finite Population Parameters

)(

)var(

,

θΣ εεX

βzz

=+

=

uno

bse

rved

ob

serv

ed

59

05

10

x

05

10

15

20

y

12

611

811

12

10

910

11

10

10

912

11

11

11

11

98

10

12

10

11

11

11

10

711

8

10

10

10

10

10

10

12

11

911

11

11

10

99

10

12

12

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911

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10

11

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11

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711

710

811

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611

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77

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10

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99

11

10

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78

11

11

Simulation Study

Simulation Study F

ixed

Pop

ula

tion

,

N =

20

0

Ra

nd

om

Sam

ple

,

n=

10

0

Nu

mb

er o

f

pla

nt

spec

ies

in

70

x 7

0 c

m

plo

ts

60

Simulation Results

Simulation Results

40

61

Simulation Study

Simulation Study

05

10

15

x

05

10

15

y

05

10

15

x

05

10

15

y

62

Simulation Results

Simulation Results

63

##F

air

ban

ks

Fair

ban

ks

Real Example –Moose Survey

Real Example –Moose Survey

64

9 “

Hig

hs,

”

52

Sa

mp

led

33

8 “

Lo

ws,

”

34

Sa

mp

led

64

Conducting the Survey

41

65

Conducting the Survey

66

01

02

03

04

05

001234

Dis

tan

ce

(k

m)

Semivariogram

01

02

03

04

05

00123

Dis

tan

ce

(k

m)

Semivariogram

Hig

h S

tra

tum

Low

Str

atu

m

Modeling Covariance

Modeling Covariance

67

##Fair

ban

ks

Fair

ban

ks

Results

Results

SR

S 985

) ˆ(

115

35

ˆ

=

= τ

τ se

FP

BK 9

78

) ˆ(

113

27

ˆ

=

= τ

τ seTota

l A

rea

Sm

all

Area

FP

BK

153

) ˆ(

143

7ˆ

=

= τ

τ se SR

S (

13

H, 4

L)

227

) ˆ(

153

5ˆ

=

= τ

τ se

68

Summary

Summary

•GeostatisticalMethods for Sampling are

often more precise

•GeostatisticalMethods for Sampling

allow small area estimation

•GeostatisticalMethods for Sampling do

not require randomized designs

•GeostatisticalMethods require modeling

42

69

•V

er H

oef

, J.

M.

20

01

. P

red

icti

ng f

init

e p

op

ula

tions

fro

m s

pat

iall

y

corr

elat

ed d

ata.

20

00

Pro

ceed

ing

s o

f th

e S

ecti

on

on

Sta

tist

ics

an

d t

he

En

vir

on

men

t o

f th

e A

meri

ca

n S

tati

stic

al

Ass

ocia

tio

n,

pgs.

93

-98.

Geostatisticsand Sampling

References

Geostatisticsand Sampling

References

•V

er H

oef

, J.

M.

20

02

.

Sam

pli

ng a

nd

geo

stat

isti

cs

for

spat

ial

dat

a.

Eco

scie

nce

9:

15

2 –

16

1.

•V

er H

oef

, J.

M.

20

06

.

Sp

atia

l M

etho

ds

for

Plo

t-

bas

ed S

amp

ling o

f W

ild

life

Po

pu

lati

ons.

In

pre

ss,

En

vir

on

men

tal

an

d

Eco

log

ica

l S

tati

stic

s70

Reality

“A

ll m

od

els

are

wro

ng

. W

e m

ake

ten

tati

ve a

ssu

mp

tion

s a

bou

t th

e re

al

wo

rld

wh

ich

we

kno

w a

re f

als

e b

ut

wh

ich

we

bel

ieve

ma

y b

e u

sefu

l.”

-G

eorg

e B

ox

19

76

Good Science is a Team Effort

Good Science is a Team Effort

Data Model

Representational

What does it look

like?

Points

Lines

Polygons

Probability Model

Lack of Fit

What don’t we

understand?

θ n,

θ s, θ

r

Scientific Model

Functional

How does it work?

Modeling

Modeling

Modeling

Inference

Inference

Inference

020

40

60

80

100

0246810

x

y