Strength of Spatial Correlation and Spatial Designs: Effects on Covariance Estimation

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

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


Distance

Au

toco

rre

latio

n

0 2 4 6

0.0

0.2

0.4

0.6

0.8

1.0


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


Lattice Design

Estimation of Autocorrelation FunctionLattice Design


Distance

Au

toco

rre

latio

n

0 2 4 6

0.0

0.2

0.4

0.6

0.8

1.0


Distance

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n

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1.0


Distance

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toco

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latio

n

0 2 4 6

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1.0


Distance

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latio

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Random Design

Estimation of Autocorrelation FunctionRandom Design


Distance

Au

toco

rre

latio

n

0 2 4 6

0.0

0.2

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0.8

1.0


Distance

Au

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Distance

Au

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n

0 2 4 6

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1.0


Distance

Au

toco

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n

0 2 4 6

0.0

0.2

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0.6

0.8

1.0


Cluster Design

Strength of Spatial Correlation and Spatial Designs: Effects on Covariance Estimation

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