Estimation and Model Selection for Geostatistical Models Kathryn M. Georgitis Alix I. Gitelman Oregon State University Jennifer A. Hoeting Colorado State University

Aquatic Resource Surveys Designs and Models for Aquatic Resource Surveys DAMARS R82-9096-01 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

Talk Outline Stream Sulfate Concentration G.I.S. Data Sources Bayesian Spatial Model Implementation Problems What exactly is the problem? Simulation results

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

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)

Wet Atmospheric Sulfate Deposition http://www.epa.gov/airmarkets/cmap/mapgallery/mg_wetsulfatephase1.html

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

MAHA/MAIA Stream Locations

Map of NADP and MAHA/MAIA Locations

Sketch of watershed with overlaid landcover map

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

Bayesian Geostatistical Model Priors: b~Np(0,h2I) f~Uniform(a,b) 1/t2 ~ Gamma(g,h) 1/s2 ~Gamma(f,l) (Banerjee et al 2004, and GeoBugs documentation)

Semi-Variogram of ln(SO4-2) Range Partial Sill Nugget

Results using Winbugs 4.1 n=644 tried different covariance functions only exponential without a nugget worked computationally intensive 1000 iterations took approx. 2 1/4 hours

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 n=322 SMCMC & Winbugs: f still did not converge and posterior intervals for all parameters dissimilar

Is the problem the prior specification? Investigated sensitivity to priors Original Priors: b~Np(0,h2I) f~Uniform(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)

Is the problem the presence of a nugget? Simulations: RandomFields package in R Using MAHA coordinates (n=322) Constant mean Exponential covariance with and without a nugget Prior Sensitivity (Berger et al. 2001, Gelman 2004)

Posterior Intervals for f Using Different Priors Prior f~Uniform (4,6) Prior f~Uniform (0,100)

Posterior Intervals for Partial Sill Using Different Priors for f Prior f~Uniform (4,6) Prior f~ Uniform (0,100)

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

Is the Spatial Signal too weak? f= 10 and f = 2.5 100 realizations each combination

Simulation Results for f=10 Bias for ML and REML Estimates

Simulation Results for f=10 Bias for ML and REML Estimates

Simulation Results for f=2.5 Bias for ML and REML Estimates

Simulation Results for f=2.5 Bias for ML and REML Estimates

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

Simulation Results for f=2.5 Bias for ML and REML Estimates

Simulation Results for f=2.5 Bias for ML and REML Estimates

Simulation Results for f=10 Bias for ML and REML Estimates

Results from SMCMC and Winbugs