Modeling Space-Time Variation in the Satellite-Derived ... · Modeling Space-Time Variation in the...

Modeling Space-Time Variation in the Satellite-DerivedChlorophyll Index Over Southern India

Petruta C. Caragea

Iowa State UniversityDepartment of Statistics

SAMSI-SAVI Workshop on Environmental StatisticsMarch 5, 2013

P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 1 / 49

The team:

• Maggie Johnson, PhD student (Department of Statistics, Iowa StateUniversity)

• Dan Fortin, PhD student (Department of Statistics, Iowa StateUniversity)

• Pete Atkinson (Department of Geography, University ofSouthampton)

• Jeganathan Chockalingam (Department of Remote Sensing, BirlaInstitute of Technology, India)

• Wendy Meiring (Department of Statistics and Applied Probability,University of California, Santa Barbara)


Why this study

• Climate influences vegetation growth: an increase in the global meantemperature between 1982 and 1999 → increase in global netvegetation productivity (6%)

• Modeling (in space and time) key phenological variables ⇒information on the effects of climate change on vegetation

• Effects of climate change on vegetation phenology arespecies-dependent

• Ideally, construct a global-scale picture

• Satellite-derived vegetation indices: indirect estimates of vegetationphenological events through repeat coverage over the globe

Extract phenological variables to study the seasonal pattern of naturalvegetation and crops at regional to global scales


Phenological variables

• Peak of“greenness”

• Time of onset of“greenness”(start of spring)

• Time of end ofsenescence (endof fall)

• Duration of thegrowing season

• Rate of “greenup”

• Rate ofsenescence


The Data: Chlorophill Index over Southern India

• Data retrieved fromMEdium ResolutionImaging Spectrometer(MERIS) TerrestrialChlorophyll Index(MTCI)

• Consists of 46observations every year(8-day composite)

• Spans 5 years,2003-2007

• Aggregated to 4.6 kmspatial resolution


Methodology: Atkinson, Jeganathan, Dash, (09-11)


Challenges

• Massive spatial data set

• Data spans geographically diverse regions (nonstationary spatialprocesses)

• No ground (“truth”) validation available

• Land use information “aggregated” for each pixel (location)

• Data available at a different spatial resolution after 2007

• Many missing values

• No unique definition (rule) for identifying key phenological variables


Temporal distribution of the Chlorophill Index (MTCI)

Coastal Vegetation Tropical Moist Deciduous

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

2003

2004

2005

2006

2007

2008

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)M

TC

I

2003

2004

2005

2006

2007

2008

Irrigated Agriculture Tropical Evergreen

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

2003

2004

2005

2006

2007

2008

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

2003

2004

2005

2006

2007

2008

Phenological Goalsrelated to

• Model (describe)complex temporaldependence

• Focus is onestimation,assessinguncertainty,characterizingchange, identifyingphenological keyevents, and NOTprediction


Spatial distribution of the Chlorophill Index (MTCI)

January April

10 15 20 25 30 35 40

1520

2530

3540

4550

1

2

3

4

5

10 15 20 25 30 35 4015

2025

3035

4045

50

1

2

3

4

5

August October

10 15 20 25 30 35 40

1520

2530

3540

4550

1

2

3

4

5

10 15 20 25 30 35 40

1520

2530

3540

4550

1

2

3

4

5

Phenological Goalsrelated to

• Model (describe)complex spatialdependence

• Focus is onestimation,assessinguncertainty,characterizingchange, and NOTprediction


Temporal Context

Tropical Evergreen

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

2003

2004

2005

2006

2007

2008


Temporal Context: A simple idea: fit Fourier regression.

Tropical Evergreen

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

2003

2004

2005

2006

2007

2008

ObservationsFourier Fits


Temporal Context: Identify Key Variables.

Tropical Evergreen

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

● ● ● ● ●

● ● ● ● ●

2003

2004

2005

2006

2007

2008



Temporal Context: The Key Variables and the Data.

Tropical Evergreen

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

● ● ● ● ●

● ● ● ● ●

2003

2004

2005

2006

2007

2008

ObservationsFourier Fits 1.

01.

52.

02.

53.

03.

54.

0

Time (years)

MT

CI

● ● ● ● ●

● ● ● ● ●

2003

2004

2005

2006

2007

2008

Observations


Temporal Context: Another example

Coastal Vegetation

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

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2003

2004

2005

2006

2007

2008


01.

52.

02.

53.

03.

54.

0

Time (years)

MT

CI

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2003

2004

2005

2006

2007

2008

Observations


Temporal Context: Conclusions so far1.

01.

52.

02.

53.

03.

54.

0

Time (years)

MT

CI

● ● ● ● ●

● ● ● ● ●

2003

2004

2005

2006

2007

2008


01.

52.

02.

53.

03.

54.

0

Time (years)

MT

CI

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2003

2004

2005

2006

2007

2008


We need

• a smooth curve to identify Key variables (extract “signal”)

• a flexible model to accommodate changes in magnitude, duration, etc.

• to account for temporal dependence


Temporal Context: Include ARMA structure (here AR(4))

Coastal Vegetation

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●●

● ●

●

●

●●

●●

●

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●

2003

2004

2005

2006

2007

2008

ObservationsTS Fits

Conclusions:

• use of additional smoothing from ARMA forecast to identify featuresof interest

• finding the “best” ARMA fit for each location IS difficult

• requires no missing values


Temporal Context: Time Series Dynamic Linear Models.Set-up.

For each location, we have random variables {Yt : t = 1, . . . ,T}

Local Level Model

Yt = µt + εt , with εt ∼ NID(0, σ2ε )

µt+1 = µt + ηt , with ηt ∼ NID(0, σ2η)

with disturbances εt , ηs are independent for all t, s and

µ1 ∼ N (α, c)


Temporal Context: TS Dynamic Linear Models. Set-up.

Local Level Model and Seasonality (Fourier representation)

Yt = µt + εt + γt , with εt ∼ NID(0, σ2ε )

µt+1 = µt + ηt , with ηt ∼ NID(0, σ2η)

Choose period s = 46 (one year) and define

γt =

[s/2]∑j=1

γjt

γj ,t+1 = γjt cosλj + γ∗jt sinλj + ωjt ,

γ∗j ,t+1 = −γjt sinλj + γ∗jt cosλj + ω∗jt ,

with

ωjt , ω∗jt ∼ NID(0, σ2ω), and λj =

2π j

s.


Temporal Context: Dynamic Linear Models. Estimation.

• Estimate unknown parameters σ2ε , σ2η and σ2ω (maximum likelihoodestimation)

• Run the Kalman smoother on the estimated model to obtain the timeseries of the smoothed state estimates

• Obtain standard errors of the estimates and residuals, test forappropriateness of the model

• Computations performed in R using package dlm

(Petris and Petrone, 2009)


Temporal Context: TS Dynamic Linear Model Results.

Coastal Vegetation

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

2003

2004

2005

2006

2007

2008

ObservationsFourier FitsDLM Fits


Temporal Context: TS Dynamic Linear Model Results.

Coastal Vegetation

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

2003

2004

2005

2006

2007

2008

Observations



Coastal Vegetation

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●

●

●

●

●

●

●

2003

2004

2005

2006

2007

2008

Observations 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

● ●

●

●

●

●

2003

2004

2005

2006

2007

2008

Observations

Based on Fourier Regression Based on DLM



Irrigated Agriculture

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●

●

●

●

●

●

●

2003

2004

2005

2006

2007

2008

Observations 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●

●

●● ●

2003

2004

2005

2006

2007

2008

Observations

Based on Fourier Regression Based on DLM


Temporal Context: Conclusions so far

Coastal Vegetation Tropical Evergreen Agriculture

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

● ●●

2003

2004

2005

2006

2007

2008

Observations 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

● ●

●

●

●

●

2003

2004

2005

2006

2007

2008

Observations 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●

●

●● ●

2003

2004

2005

2006

2007

2008

Observations

• DLM approach superior over Fourier Regression

• Extend this to Multivariate Local Level Models (Seemingly UnrelatedTime Series)


Functional Data Analysis Approach (Dan Fortin)

For a fixed location s, χ(s; t) is a random trajectory with representation

χ(s; t) = µ(t) + ε(s; t)

• Use a finite basis representation of the trajectories using basisfunctions that represent the type of temporal variation observed inthe data.

• Use finite basis consisting of eigenfunctions of the temporalcovariance function, i.e. Principal Component Functions.

• Assume ε(s; t) takes values in a reproducing kernel Hilbert space offunctions H ⇒ construct a non-parametric estimator of the temporalcovariance function (use a regularized estimator which penalizessmoothness through the RKHS norm). Closed form estimates of theeigenfunctions ψk(t) are derived from this estimator.


Functional Data Analysis Approach in Practice

Tropical Evergreen

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

2003

2004

2005

2006

2007

2008

ObservationsFunctional 1.

01.

52.

02.

53.

03.

54.

0

Time (years)

MT

CI

● ●

● ● ●

2003

2004

2005

2006

2007

2008

Observations


Functional Data Analysis Approach in Practice

Coastal Vegetation

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

2003

2004

2005

2006

2007

2008

ObservationsFunctional 1.

01.

52.

02.

53.

03.

54.

0

Time (years)

MT

CI

●●

● ● ●

2003

2004

2005

2006

2007

2008

Observations

Pros and Cons

• Provides a streamlined approach to identify phenological indicators

• Computationally intensive

• Ignores spatial dependence— but possible to extend: use a MRFapproach for the coefficients of the basis functions.



January April

10 15 20 25 30 35 40

2025

3035

4045

50

1

2

3

4

5

10 15 20 25 30 35 40

2025

3035

4045

50

1

2

3

4

5

August October

10 15 20 25 30 35 40

2025

3035

4045

50

1

2

3

4

5

10 15 20 25 30 35 40

2025

3035

4045

50

1

2

3

4

5


Spatial distribution of Fourier Regression Coefficients?

β1 β3

0

10

20

30

40

50

0 10 20 30 40 50South

West

−0.50

−0.25

0.00

0.25

0.50value

0

10

20

30

40

50

0 10 20 30 40 50South

West

−0.10.00.10.20.30.4

value

β5 β7

0

10

20

30

40

50

0 10 20 30 40 50South

West

−0.1

0.0

0.1

0.2

value

0

10

20

30

40

50

0 10 20 30 40 50South

West

−0.10−0.050.000.050.10

value

Maggie Johnson

Suggests a possible approach based on a hierarchical model with FourierRegression Coefficients vary with elevation, land use, etc.



January April

10 15 20 25 30 35 40

2025

3035

4045

50

1

2

3

4

5

10 15 20 25 30 35 40

2025

3035

4045

50

1

2

3

4

5

August October

10 15 20 25 30 35 40

2025

3035

4045

50

1

2

3

4

5

10 15 20 25 30 35 40

2025

3035

4045

50

1

2

3

4

5


Spatial Context

• Spatial domain D, locations {si : i = 1, . . . , n}, random variables{Y (si ) : i = 1, . . . , n}, neighborhood Ni

• Markov property [Y (si )|{y(sj) : j 6= i}] = [Y (si )|y(Ni )]; i = 1, . . . , n

One parameter exponential families

• Conditional distribution

fi (y(si )|y(Ni )) = exp [Ai (y(Ni ))y(si )− Bi (y(Ni )) + C (y(si ))]

• Natural parameter function

Ai (y(Ni )) = τ−1(κi ) + γ1

m

∑sj∈Ni

{y(sj)− κj}

Gaussian Models:

Ai (y(Ni )) = κi/σ2 + γ

1

m

∑sj∈Ni

{y(sj)− κj}


Modeling strategy

Ideally,

• Estimate model parameters (κi , σ2 and γ)

• Use exact likelihood function (available for Gaussian MRFs, butimpractical due to large size of the data)

• Alternatively, use Besag’s pseudo-likelihood

• Inference

Model choice is complicated by dimensionality of the data

• Which covariates to use, if necessary?

• What type of spatial dependence?

Need a simple diagnostic tool to guide model choice!


The S-value (Kaiser and Caragea, 2009)

Model-Based Exploratory Statistic

• Exploratory quantity, directly tied to the structure of Markov randomfield models

• Crude estimator of γ

Standard bound

• | γ |< γsb ensures κ ≈ E{Y (si )}• γsb available for exponential family models

• For Gaussian models: γsb = 1σ2

Uses:

1 S/γsb is a measure of strength of dependence

2 if S >> γsb then κ 6= E{Y (si )} in model• directional dependencies• non-constant mean


Uses of the S-value: Detecting strength of dependence

Gaussian model with constant mean κ = 10 and σ2 = 130×30 regular lattice

Case 1: γ = 0.10

• κ = 9.946• S = 0.093

ML Estimates:

• κ = 9.947• γ = 0.112

Case 2: γ = 0.75

• κ = 10.084• S = 0.765

ML Estimates:

• κ = 10.067• γ = 0.830

-1.0 -0.5 0.0 0.5 1.0

-0.4

0.00.20.4

D

r

S-value= 0.093

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.0

0.0

1.0

D

rS-value= 0.765


Uses of the S-value: Detecting directional dependence

Directional Gaussian: κ = 10, σ2 = 1 and γ1 = 0.10 and γ2 = 0.75

Unidirectional

• S = 0.833

ML Estimates:• κ = 9.877

• γ = 0.807

Directional

• S1 = 0.082 S2 = 0.727

ML Estimates:• κ = 9.870

• γ1 = 0.036

• γ2 = 0.787

κ = 9.871

5 10 15 20 25 30

510

1520

2530

X

Y

-1.5 -0.5 0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

D

r

S-value= 0.833

-2 -1 0 1 2

-0.6

-0.2

0.0

0.2

0.4

D

r

S-value= 0.082

-2 -1 0 1 2

-1.5

-0.5

0.51.01.5

D

r

S-value= 0.727


Uses of the S-value: Detecting spatial trend

Data generated with trend: κi = 0.30(ui + vi );unidirectional dependence: γ = 0.10

Const. mean

S = 0.975

With trend

S = 0.234orS = 0.114

5 10 15 20 25 30

510

1520

2530

X

Y

-5 0 5

-8-6

-4-2

02

46

D

r

S-value= 0.975

-0.5 0.0 0.5 1.0

-0.5

0.0

0.5

D

r

S-value= 0.234

-0.5 0.0 0.5 1.0

-0.5

0.0

0.5

D

r

S-value= 0.114


S-value in practiceInvestigate appropriate spatial conditionally specified models for MTCI

January April

10 15 20 25 30 35 40

1520

2530

3540

4550

1.0

1.5

2.0

2.5

3.0

3.5

4.0

10 15 20 25 30 35 40

1520

2530

3540

4550

1.0

1.5

2.0

2.5

3.0

3.5

4.0

August October

10 15 20 25 30 35 40

1520

2530

3540

4550

1.0

1.5

2.0

2.5

3.0

3.5

4.0

10 15 20 25 30 35 40

1520

2530

3540

4550

1.0

1.5

2.0

2.5

3.0

3.5

4.0

January April

●

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●

●

−0.4 −0.2 0.0 0.2 0.4 0.6

−0.

4−

0.2

0.0

0.2

0.4

D

r

S−value= 0.973

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●

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−0.5 0.0 0.5 1.0

−0.

50.

00.

51.

0

D

r

S−value= 1.007

August October

●

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●

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●

●

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

D

r

S−value= 1.007

●

●

●

●

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●

●

●

●

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

D

r

S−value= 1.006


S-value for MTCI

• Neighborhood: 4 nearest neighbors

Calculated S-values for each time point

0.75

0.80

0.85

0.90

0.95

1.00

1.05

Time (years)

S−

valu

e

2003

2004

2005

2006

2007

2008

Cst. Mean, Unidir.

0.75

0.80

0.85

0.90

0.95

1.00

1.05

Time (years)

S−

valu

e

2003

2004

2005

2006

2007

2008

Cst. Mean, Dir. (NS,EW)

0.75

0.80

0.85

0.90

0.95

1.00

1.05

Time (years)

S−

valu

e

2003

2004

2005

2006

2007

2008

Trend, Unidir.


Reasons for a Global structure (trend)?

Land Use Elevation

8

10

12

76 78 80lon

lat


Reasons for a Global structure (trend)?

Land Use Elevation

10 15 20 25 30 35 40

1520

2530

3540

4550

10

20

30

40

10 15 20 25 30 35 40

1520

2530

3540

4550

0

500

1000

1500

2000

.... and possibly many others, such as slope, aspect (temporally stable)temperature, moisture, etc. (temporally variable).


S-values after accounting for Elevation and Land Use0.

750.

800.

850.

900.

951.

001.

05

Time (years)

S−

valu

e

2003

2004

2005

2006

2007

2008

No Trend

0.75

0.80

0.85

0.90

0.95

1.00

1.05

Time (years)

S−

valu

e

2003

2004

2005

2006

2007

2008

Trend (Land Cover)

Recommended model

Trend (Land Cover and Elevation as possible predictors) and unidirectionaldependence for each time point.


Phenological Variable Identification for Tropical Evergreen

Fourier

(no dep)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●●

●

●●

●

2003

2004

2005

2006

2007

2008

Observations 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

● ●●

2003

2004

2005

2006

2007

2008

Observations

DLM

(Cst.Mean)

DLM

(Trend)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

● ●

● ●

●

2003

2004

2005

2006

2007

2008

Observations 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

● ●

● ●

●

2003

2004

2005

2006

2007

2008

Observations

DLM

(Trend +

Spatial)


Phenological Variable Identification for Coastal Vegetation

Fourier

(no dep)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●

●

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●

●

●

●

2003

2004

2005

2006

2007

2008

Observations 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

● ●

●

●

●

●

2003

2004

2005

2006

2007

2008

Observations

DLM

(Cst.Mean)

DLM

(Trend)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

● ●

● ●

●

2003

2004

2005

2006

2007

2008

Observations 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

● ●

● ●

●

2003

2004

2005

2006

2007

2008

Observations

DLM

(Trend +

Spatial)


Phenological Variable Identification for Agriculture

Fourier

(no dep)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●

●

●

●

●

●

●

2003

2004

2005

2006

2007

2008

Observations 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●

●

●● ●

2003

2004

2005

2006

2007

2008

Observations

DLM

(Cst.Mean)

DLM

(Trend)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●

●

● ●

●

● ●

2003

2004

2005

2006

2007

2008

Observations 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●

● ●● ●

●

2003

2004

2005

2006

2007

2008

Observations

DLM

(Trend +

Spatial)


...and sometimes there is just too much noise!

Fourier

(no dep)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●

●

●

●

●

●

●

2003

2004

2005

2006

2007

2008

Observations 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●

●

●●

2003

2004

2005

2006

2007

2008

Observations

DLM

(Cst.Mean)

DLM

(Trend)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●

●

●●

2003

2004

2005

2006

2007

2008

Observations 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (years)

MT

CI

●

●

●

●

●●

2003

2004

2005

2006

2007

2008

Observations

DLM

(Trend +

Spatial)


Spatio-Temporal Context: Conclusions so far

S-values

• Guide model formulation• strength of dependence• type of dependence (e.g. directional)• aptness of mean structure

• Connected to Model form• interpretation within distributional families• direct connection to dependence parameter(s)

Temporal Dynamics

DLM based “signals” are useful with variable identification


Into the Future: connect temporal trends in spatialdependence with natural fluctuations of the vegetationover time

0.80

0.90

Time (years)

S−

valu

e

2003

2004

2005

2006

2007

2008

2.0

2.4

2.8

Time (years)

(Spa

tial)

Mea

n

2003

2004

2005

2006

2007

2008

0.1

0.3

Time (years)

(Spa

tial)

Var

ianc

e

2003

2004

2005

2006

2007

2008


Into the Future: Spatio-temporal models

• Temporal integration via the neighborhood structure

Y Y

Y

Y

Y

Y Y

Y

Y

Y

Time i ((ηηi)) Time i+1 ((ηηi++1))

ηη

• Spatial dependence temporally varying (i.e. ηt)

• Either spatial or temporal dependence as functions of additional(temporally varying) covariates

• Fully Bayesian analysis

• Consider distributions with heavier tails for the temporal model

• Hierarchical spatio-temporal models

• Combine information on different scales (resolutions)

• Assesment!!!P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 48 / 49

References cited and Aknowledgements

From the existing literature:• Jeganathan, C., Dash, J. and Atkinson, P.M. (2010) Mapping

phenology of natural vegetation in India using remote sensing derivedchlorophyll index. International Journal of Remote Sensing.

• Kaiser, M.S., Caragea, P.C. (2009). Exploring dependence with dataon spatial lattices. Biometrics.

Acknowlegements• Maggie Johnson and Dan Fortin (ISU)

• Pete Atkinson (U. of Southampton)

• Jeganathan Chockalingam (Birla Institute of Technology, India)

• Wendy Meiring (University of California, Santa Barbara)

Thank you!!!


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