Strength of Spatial Correlation
and Spatial Designs: Effects on Covariance Estimation
Kathryn M. IrvineOregon State University
Alix I. Gitelman
Sandra E. Thompson
The research described in this presentation has been funded by the U.S. Environmental Protection Agency through the STAR Cooperative Agreement CR82-9096-01 Program on Designs and Models for Aquatic Resource Surveys at Oregon State University. It has not been subjected to the Agency's review and therefore does not necessarily reflect the views of the Agency, and no official endorsement should be inferred
R82-9096-01
Talk Outline
• Stream Sulfate Concentration– Geostatistical Model– Preliminary Findings
• Simulations
• Results– Parameter Estimation
• Discussion
Study Objective:
Model the spatial heterogeneity of stream sulfate concentration in streams in the Mid-
Atlantic U.S.
Why stream sulfate concentration?
– Indirectly toxic to fish and aquatic biota• Decrease in streamwater pH • Increase in metal concentrations (AL)
– Observed positive spatial relationship with atmospheric SO4
-2 deposition (Kaufmann et al. 1991)
The Data
• EMAP water chemistry data– 322 stream locations
• Watershed variables: – % forest, % agriculture, % urban, % mining
– % within ecoregions with high sulfate adsorption soils
• National Atmospheric Deposition Program
EMAP and NADP locationsMAHA/MAIANADPEMAP
NADP
Geostatistical Model
( ) ( ) ( )Y s X s s Where Y(s) is a vector of observed ln(SO4
-2) concentration at stream locations (s)
X(s) is a matrix of watershed explanatory variables is a vector of unknown regression coefficients
(s) is the spatial error process
( ) ~ (0, )ns N Σ2 2 exp( ) Σ I D
Where D is matrix of pairwise distances, is 1/range,
is the partial sill is the nugget
(1)
Effective RangeDefinition: 1) Distance beyond which the correlation between observations is less than or equal to 0.05.
2) Distance where the semi-variogram reaches 95% of the sill.
2 2
2
1log 0.05
Semi-Variogram
0 100 200 300
km
0.0
0.5
1.0
1.5
Sem
i-Var
iogr
am
EmpiricalMLREML
Nugget
Partial Sill
Effective Range
197 km
272 km
Interpretations of Spatial Covariance Parameters
• Patch Characteristics (Rossi et al. 1992; Robertson and Gross 1994; Dalthorp et al. 2000;
Schwarz et al. 2003 and more)
– Effective Range ~ Size of Patch– Nugget ~ Tightness of Patches
• Sample Design Modifications– Effective Range: Independent Samples– Nugget: Measurement Error
Why Are the Estimates Different?Simulation Study
Strength of Spatial Correlation?
– Nugget:Sill ratio and/or Range Parameter• Mardia & Marshall (1984): measurement error increases
variability of ML estimates of range
• Zimmerman & Zimmerman (1991): REML and ML better when spatial signal weak (short range)
• Lark (2000): ML better compared to MOM when short range and large nugget:sill ratio
• Thompson (2001): estimation for Matern with 20% and 50% nugget under different spatial designs
Is the spatial correlation too weak?
Effective Range Values for Simulations
Nugget-to-Sill RatioRange Parameter 0.10 0.33 0.50 0.67 0.90
1 2.89 2.59 2.30 1.90 0.693 8.67 7.77 6.90 5.70 2.07
EMAP Estimates Re-Scaled:
Range Parameter ~1.5 Nugget-to-Sill Ratio ~0.50
Is it the spatial sample design?
-Cluster design optimal for covariance parameter estimation (Pettitt and McBratney 1993; Muller and Zimmerman 1999; Zhu and Stein 2005; Xia et al. 2006;
Zimmerman 2006; Zhu and Zhang 2006)
Is it the spatial sample design? n=144 Lattice
0 2 4 6 8 10
02
46
81
0
n=361 Lattice
0 2 4 6 8 10
02
46
81
0
n=144 Random
0 2 4 6 8 10
02
46
81
0n=361 Random
0 2 4 6 8 10
02
46
81
0
n=144 Cluster
0 2 4 6 8 10
02
46
81
0
n=361 Cluster
0 2 4 6 8 10
02
46
81
0
Zimmerman (2006) and Thompson (2001)
Simulation Study
• Spatial Designs: Lattice, Random, Cluster
• Range Parameter = 1 and 3• Nugget/Sill Ratio:
0.10, 0.33, 0.50, 0.67, 0.90
• n=144 and n=361 (In-fill Asymptotics)
• 100 realizations per combination• RandomFields in R• Estimation using R code (Ver Hoef 2004)
1.Estimation of Covariance Parameters
The Effective Range
Range Parameter = 1 Range Parameter = 3
Results for Estimation of Effective Range
Estimation Error
Ratio Design Method 10% 50% 90%
0.10 grid ML -0.88 -0.20 0.76
REML -0.82 0.00 1.11
random ML -0.88 -0.25 0.62
REML -0.79 -0.08 0.92
cluster ML -1.01 -0.27 0.94
REML -0.92 -0.11 1.40
0.90 grid ML -359.92 -0.20 0.48
REML -300.49 0.10 502.84
random ML -341.89 -0.16 0.59
REML -295.10 0.06 1390.17
cluster ML -7.21 -0.30 0.65
REML -1.04 -0.02 30.84
Estimation Error Ratio Design Method 10% 50% 90%
0.10 grid ML -4.79 -2.18 2.31
REML -4.40 -0.86 9.89
random ML -4.48 -2.01 3.02
REML -4.03 -0.58 10.89
cluster ML -5.07 -2.45 3.98
REML -4.66 -0.74 12.75
0.90 grid ML -37.75 -1.31 0.77
REML -2.18 -0.10 2464.44
random ML -30.42 -1.34 0.84
REML -2.15 -0.14 726.00
cluster ML -2.53 -1.40 1.63
REML -1.91 0.35 1255.04
Estimation Error = estimate - truth
Range Parameter = 1 Range Parameter = 3
Results for Estimation of Effective Range
Estimation Error
Ratio Design Method 10% 50% 90%
0.10 grid ML -0.88 -0.20 0.76
REML -0.82 0.00 1.11
random ML -0.88 -0.25 0.62
REML -0.79 -0.08 0.92
cluster ML -1.01 -0.27 0.94
REML -0.92 -0.11 1.40
0.90 grid ML -359.92 -0.20 0.48
REML -300.49 0.10 502.84
random ML -341.89 -0.16 0.59
REML -295.10 0.06 1390.17
cluster ML -7.21 -0.30 0.65
REML -1.04 -0.02 30.84
Estimation Error Ratio Design Method 10% 50% 90%
0.10 grid ML -4.79 -2.18 2.31
REML -4.40 -0.86 9.89
random ML -4.48 -2.01 3.02
REML -4.03 -0.58 10.89
cluster ML -5.07 -2.45 3.98
REML -4.66 -0.74 12.75
0.90 grid ML -37.75 -1.31 0.77
REML -2.18 -0.10 2464.44
random ML -30.42 -1.34 0.84
REML -2.15 -0.14 726.00
cluster ML -2.53 -1.40 1.63
REML -1.91 0.35 1255.04
Range Parameter = 1 Range Parameter = 3
Results for Estimation of Effective Range
Estimation Error
Ratio Design Method 10% 50% 90%
0.10 grid ML -0.88 -0.20 0.76
REML -0.82 0.00 1.11
random ML -0.88 -0.25 0.62
REML -0.79 -0.08 0.92
cluster ML -1.01 -0.27 0.94
REML -0.92 -0.11 1.40
0.90 grid ML -359.92 -0.20 0.48
REML -300.49 0.10 502.84
random ML -341.89 -0.16 0.59
REML -295.10 0.06 1390.17
cluster ML -7.21 -0.30 0.65
REML -1.04 -0.02 30.84
Estimation Error Ratio Design Method 10% 50% 90%
0.10 grid ML -4.79 -2.18 2.31
REML -4.40 -0.86 9.89
random ML -4.48 -2.01 3.02
REML -4.03 -0.58 10.89
cluster ML -5.07 -2.45 3.98
REML -4.66 -0.74 12.75
0.90 grid ML -37.75 -1.31 0.77
REML -2.18 -0.10 2464.44
random ML -30.42 -1.34 0.84
REML -2.15 -0.14 726.00
cluster ML -2.53 -1.40 1.63
REML -1.91 0.35 1255.04
Range Parameter = 1 Range Parameter = 3
Results for Estimation of Effective Range
Estimation Error
Ratio Design Method 10% 50% 90%
0.10 grid ML -0.88 -0.20 0.76
REML -0.82 0.00 1.11
random ML -0.88 -0.25 0.62
REML -0.79 -0.08 0.92
cluster ML -1.01 -0.27 0.94
REML -0.92 -0.11 1.40
0.90 grid ML -359.92 -0.20 0.48
REML -300.49 0.10 502.84
random ML -341.89 -0.16 0.59
REML -295.10 0.06 1390.17
cluster ML -7.21 -0.30 0.65
REML -1.04 -0.02 30.84
Estimation Error Ratio Design Method 10% 50% 90%
0.10 grid ML -4.79 -2.18 2.31
REML -4.40 -0.86 9.89
random ML -4.48 -2.01 3.02
REML -4.03 -0.58 10.89
cluster ML -5.07 -2.45 3.98
REML -4.66 -0.74 12.75
0.90 grid ML -37.75 -1.31 0.77
REML -2.18 -0.10 2464.44
random ML -30.42 -1.34 0.84
REML -2.15 -0.14 726.00
cluster ML -2.53 -1.40 1.63
REML -1.91 0.35 1255.04
Summary Covariance Parameter Estimation
• Effective Range :– ML under-estimate the truth
– REML more skewed in 90th percentile (large nugget-to-sill and range parameter)
• Partial Sill:– ML under-estimate the truth
– REML more skewed in 90th percentile
• Nugget:– estimated well; particularly with cluster design
Discussion
– Which estimation method to use?
– Consistency Results: Chen et al. 2000, Zhang and Zimmerman 2005)
– Uncertainty estimates for REML and ML• REML: Increasing Domain (Cressie and Lahiri 1996)
• ML: Increasing Domain and Infill Asymptotics
(Zhang and Zimmerman 2005)
Acknowledgements
• Co-Authors
• Jay Ver Hoef, Alan Herlihy, Andrew Merton, Lisa Madsen
Questions
Results1. Estimation of Covariance Parameters
2. Estimation of Autocorrelation Function
Results:2. Estimation of Autocorrelation Function
Estimation of Autocorrelation FunctionCluster Design
ML for range=1 n=361
Distance
Au
toco
rre
latio
n
0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
REML for range=1 n=361
Distance
Au
toco
rre
latio
n
0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
ML for range=3 n=361
Distance
Au
toco
rre
latio
n
0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
REML for range=3 n=361
Distance
Au
toco
rre
latio
n
0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
Summary: Estimation of Autocorrelation Function
• Overall Patterns:
– ML and REML poor performance with stronger
spatial correlation (larger effective ranges)
– REML large variability
– ML under-estimation
– ‘BEST’ case:
Cluster Design with range parameter = 1 and n=361
Wet Atmospheric Sulfate Deposition
http://www.epa.gov/airmarkets/cmap/mapgallery/mg_wetsulfatephase1.html
Estimated Auto-correlation Function
for ln(SO4-2)
0 100 200 300 400 500
km
0.0
0.2
0.4
0.6
0.8
1.0
Est
imat
ed A
utoc
orre
latio
n F
unct
ion
MLREML
Sketch of watershed with overlaid landcover map
ForestMiningUrban
Agriculture
2. Estimation of Autocorrelation Function
Lattice Design
Estimation of Autocorrelation FunctionLattice Design
ML for range=1 n=361
Distance
Au
toco
rre
latio
n
0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
REML for range=1 n=361
Distance
Au
toco
rre
latio
n
0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
ML for range=3 n=361
Distance
Au
toco
rre
latio
n
0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
REML for range=3 n=361
Distance
Au
toco
rre
latio
n
0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
2. Estimation of Autocorrelation Function
Random Design
Estimation of Autocorrelation FunctionRandom Design
ML for range=1 n=361
Distance
Au
toco
rre
latio
n
0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
REML for range=1 n=361
Distance
Au
toco
rre
latio
n
0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
ML for range=3 n=361
Distance
Au
toco
rre
latio
n
0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
REML for range=3 n=361
Distance
Au
toco
rre
latio
n
0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
2. Estimation of Autocorrelation Function
Cluster Design