Estimation and Model Selection for Geostatistical Models

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Estimation and Model Selection for Geostatistical
Models

• Kathryn M. Georgitis
• Alix I. Gitelman
• Oregon State University
• Jennifer A. Hoeting
• Colorado State University

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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
• G.I.S. Data Sources
• Bayesian Spatial Model
• Implementation Problems
• What exactly is the problem?
• Simulation results

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Original Objective

• Model sulfate concentration in streams in the
  Mid-Atlantic U.S. using a Bayesian
  geostatistical model

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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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Wet Atmospheric Sulfate Deposition
http//www.epa.gov/airmarkets/cmap/mapgallery/mg_w
etsulfatephase1.html
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The Data

• MAHA/MAIA water chemistry data
• 644 stream locations
• Watershed variables
• forest, agriculture, urban, mining
• within ecoregions with high sulfate adsorption
  soils
• National Atmospheric Deposition Program

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MAHA/MAIA Stream Locations
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Map of NADP and MAHA/MAIA Locations
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Sketch of watershed with overlaid landcover map
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Bayesian Geostatistical Model
(1)

• Where Y(s) is observed ln(SO4-2) concentration at
  stream locations
• X(s) is matrix of watershed
  explanatory variables
• b is vector of regression
  coefficients

Where D is matrix of pairwise distances,
f is 1/range, t2 is the partial sill s2 is
the nugget
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Bayesian Geostatistical Model
Priors bNp(0,h2I) fUniform(a,b)
1/t2 Gamma(g,h) 1/s2
Gamma(f,l) (Banerjee et al 2004, and GeoBugs
documentation)
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Semi-Variogram of ln(SO4-2)
Range
Partial Sill
Nugget
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Results using Winbugs 4.1

• n644
• tried different covariance functions
• only exponential without a nugget worked
• computationally intensive
• 1000 iterations took approx. 2 1/4 hours

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New Objective Why is this not working?

• Large N problem?
• Possible solutions
• SMCMC accelerates convergence by
  simultaneously updating multivariate blocks of
  (highly correlated) parameters
• (Sargent et al. 2000, Cowles 2003, Banerjee
  et al 2004 )
• f (1/range) did not converge
• subset data to n322
• SMCMC Winbugs
• f still did not converge and posterior intervals
  for all parameters dissimilar

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Is the problem the prior specification?

• Investigated sensitivity to priors
• Original Priors
• bNp(0,h2I)
• fUniform(a,b)
• 1/t2 Gamma(g,h)
• 1/s2 Gamma(f,l)
• - f Tried Gamma and different Uniform
  distributions (Banerjee et al 2004, Berger et al
  2001)
• Variance components Tried different Gamma
  distributions, half-Cauchy (Gelman 2004)

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Is the problem the presence of a nugget?

• Simulations
• RandomFields package in R
• Using MAHA coordinates (n322)
• Constant mean
• Exponential covariance with and without a nugget
• Prior Sensitivity (Berger et al. 2001, Gelman
  2004)

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Posterior Intervals for f Using Different Priors
Prior fUniform (4,6)
Prior fUniform (0,100)
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Posterior Intervals for Partial SillUsing
Different Priors for f
Prior fUniform (4,6)
Prior f Uniform (0,100)
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Is the Spatial Signal too weak?

• Simulations were using nugget/sill 2/3
• Try using a range of nugget/sill ratios
• Previous research
• Mardia Marshall (1984) spherical with and
  without nugget
• Zimmerman Zimmerman (1991) R.E.M.L vs M.L.E.
  for Exponential without nugget
• Lark (2000) M.O.M. vs M.L.E. for spherical with
  nugget

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Is the Spatial Signal too weak?
f 10 and f 2.5
100 realizations each combination
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Simulation Results for f10Bias for ML and REML
Estimates
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Simulation Results for f10Bias for ML and REML
Estimates
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Simulation Results for f2.5Bias for ML and REML
Estimates
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Simulation Results for f2.5Bias for ML and REML
Estimates
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Conclusions

• Covariance Model Selection Problem
• ML, REML, Bayesian Estimation
• (Harville 1974)
• Infill Asymptotic Properties of M.L.E.
• Ying 1993 Ornstein-Uhlenbeck without nugget
  2-dim.
• lattice design
• Chen et al 2000 Ornstein-Uhlenbeck with nugget
  1-dim.
• Zhang 2004 Exponential without nugget found
  increasing
• range more skewed distributions

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Simulation Results for f2.5Bias for ML and REML
Estimates
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Simulation Results for f2.5Bias for ML and REML
Estimates
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Simulation Results for f10Bias for ML and REML
Estimates
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Results from SMCMC and Winbugs