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

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Strength of Spatial Correlation and Spatial
Designs Effects on Covariance Estimation

• Kathryn M. Irvine
• Oregon State University
• Alix I. Gitelman
• Sandra E. Thompson

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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
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Talk Outline

• Stream Sulfate Concentration
• Geostatistical Model
• Preliminary Findings
• Simulations
• Results
• Parameter Estimation
• Discussion

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Study Objective

• Model the spatial heterogeneity of stream
  sulfate concentration in streams in the
  Mid-Atlantic U.S.

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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)

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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

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EMAP and NADP locations
EMAP NADP
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Geostatistical Model
(1)

• 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
• b is a vector of unknown regression
  coefficients
• e(s) is the spatial error process

Where D is matrix of pairwise distances,
f is 1/range, t2 is the partial sill s2
is the nugget
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Effective Range

• Definition
• 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.

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Semi-Variogram
Effective Range
272 km
197 km
Partial Sill
Nugget
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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

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Why Are the Estimates Different?Simulation Study

• Strength of Spatial Correlation?
• NuggetSill 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 nuggetsill ratio
• Thompson (2001) estimation for Matern with 20
  and 50 nugget under different spatial designs

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Is the spatial correlation too weak?
Effective Range Values for Simulations
EMAP Estimates Re-Scaled Range Parameter
1.5 Nugget-to-Sill Ratio 0.50
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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)

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Is it the spatial sample design?
Zimmerman (2006) and Thompson (2001)
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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
• n144 and n361 (In-fill Asymptotics)
• 100 realizations per combination
• RandomFields in R
• Estimation using R code (Ver Hoef 2004)

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1.Estimation of Covariance ParametersThe
Effective Range
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Range Parameter 1
Results for Estimation of Effective Range
Range Parameter 3
Estimation Error estimate - truth
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Range Parameter 1
Results for Estimation of Effective Range
Range Parameter 3
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Range Parameter 1
Results for Estimation of Effective Range
Range Parameter 3
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Range Parameter 1
Results for Estimation of Effective Range
Range Parameter 3
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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

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Discussion

• Which estimation method to use?
• Consistency Results t2f
• (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)

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Acknowledgements

• Co-Authors
• Jay Ver Hoef, Alan Herlihy, Andrew Merton, Lisa
  Madsen

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Questions
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Results1. Estimation of Covariance Parameters
2. Estimation of Autocorrelation Function
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Results2. Estimation of Autocorrelation Function
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Estimation of Autocorrelation FunctionCluster
Design
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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 n361

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Wet Atmospheric Sulfate Deposition
http//www.epa.gov/airmarkets/cmap/mapgallery/mg_w
etsulfatephase1.html
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Estimated Auto-correlation Function for ln(SO4-2)
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Sketch of watershed with overlaid landcover map
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2. Estimation of Autocorrelation
FunctionLattice Design
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Estimation of Autocorrelation FunctionLattice
Design
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2. Estimation of Autocorrelation FunctionRandom
Design
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Estimation of Autocorrelation FunctionRandom
Design
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2. Estimation of Autocorrelation
FunctionCluster Design