Modeling Space-Time Variation in the Satellite-Derived ... · Modeling Space-Time Variation in the...
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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)
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 2 / 49
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
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 3 / 49
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
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 4 / 49
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
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 5 / 49
Methodology: Atkinson, Jeganathan, Dash, (09-11)
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 6 / 49
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
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 7 / 49
Temporal distribution of the Chlorophill Index (MTCI)
Coastal Vegetation Tropical Moist Deciduous
1.0
1.5
2.0
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Time (years)
MT
CI
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Time (years)M
TC
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2004
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Irrigated Agriculture Tropical Evergreen
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Time (years)
MT
CI
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MT
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Phenological Goalsrelated to
• Model (describe)complex temporaldependence
• Focus is onestimation,assessinguncertainty,characterizingchange, identifyingphenological keyevents, and NOTprediction
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 8 / 49
Spatial distribution of the Chlorophill Index (MTCI)
January April
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Phenological Goalsrelated to
• Model (describe)complex spatialdependence
• Focus is onestimation,assessinguncertainty,characterizingchange, and NOTprediction
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 9 / 49
Temporal Context
Tropical Evergreen
1.0
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Time (years)
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P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 10 / 49
Temporal Context: A simple idea: fit Fourier regression.
Tropical Evergreen
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ObservationsFourier Fits
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 11 / 49
Temporal Context: Identify Key Variables.
Tropical Evergreen
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ObservationsFourier Fits
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 12 / 49
Temporal Context: The Key Variables and the Data.
Tropical Evergreen
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ObservationsFourier Fits 1.
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P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 13 / 49
Temporal Context: Another example
Coastal Vegetation
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P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 14 / 49
Temporal Context: Conclusions so far1.
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ObservationsFourier Fits 1.
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ObservationsFourier Fits
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
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 15 / 49
Temporal Context: Include ARMA structure (here AR(4))
Coastal Vegetation
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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
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 16 / 49
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)
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 17 / 49
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.
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 18 / 49
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)
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 19 / 49
Temporal Context: TS Dynamic Linear Model Results.
Coastal Vegetation
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MT
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ObservationsFourier FitsDLM Fits
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 20 / 49
Temporal Context: TS Dynamic Linear Model Results.
Coastal Vegetation
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P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 21 / 49
Temporal Context: The Key Variables and the Data.
Coastal Vegetation
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Observations 1.0
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Observations
Based on Fourier Regression Based on DLM
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 22 / 49
Temporal Context: The Key Variables and the Data.
Irrigated Agriculture
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Based on Fourier Regression Based on DLM
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 23 / 49
Temporal Context: Conclusions so far
Coastal Vegetation Tropical Evergreen Agriculture
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Observations 1.0
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Observations
• DLM approach superior over Fourier Regression
• Extend this to Multivariate Local Level Models (Seemingly UnrelatedTime Series)
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 24 / 49
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.
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 25 / 49
Functional Data Analysis Approach in Practice
Tropical Evergreen
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ObservationsFunctional 1.
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P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 26 / 49
Functional Data Analysis Approach in Practice
Coastal Vegetation
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ObservationsFunctional 1.
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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.
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 27 / 49
Spatial distribution of the Chlorophill Index (MTCI)
January April
10 15 20 25 30 35 40
2025
3035
4045
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10 15 20 25 30 35 40
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P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 28 / 49
Spatial distribution of Fourier Regression Coefficients?
β1 β3
0
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0 10 20 30 40 50South
West
−0.50
−0.25
0.00
0.25
0.50value
0
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40
50
0 10 20 30 40 50South
West
−0.10.00.10.20.30.4
value
β5 β7
0
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0 10 20 30 40 50South
West
−0.1
0.0
0.1
0.2
value
0
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30
40
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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.
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 29 / 49
Spatial distribution of the Chlorophill Index (MTCI)
January April
10 15 20 25 30 35 40
2025
3035
4045
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10 15 20 25 30 35 40
2025
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10 15 20 25 30 35 40
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P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 30 / 49
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}
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 31 / 49
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!
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 32 / 49
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
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 33 / 49
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
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 34 / 49
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
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 35 / 49
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
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 36 / 49
S-value in practiceInvestigate appropriate spatial conditionally specified models for MTCI
January April
10 15 20 25 30 35 40
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4550
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January April
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D
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S−value= 0.973
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D
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S−value= 1.007
August October
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40.
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D
r
S−value= 1.007
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6−
0.4
−0.
20.
00.
20.
40.
6
D
r
S−value= 1.006
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 37 / 49
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.
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 38 / 49
Reasons for a Global structure (trend)?
Land Use Elevation
8
10
12
76 78 80lon
lat
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 39 / 49
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).
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 40 / 49
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.
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 41 / 49
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)
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 42 / 49
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
●
●
●
●
●
●
●
●
●
●
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)
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 43 / 49
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)
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 44 / 49
...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)
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 45 / 49
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
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 46 / 49
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
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 47 / 49
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!!!
P. Caragea (ISU) MTCI in Space and Time SAMSI SAVI Workshop 49 / 49