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

2. Aquatic Resource Surveys
Designs and Models for
Aquatic Resource Surveys
DAMARS R
The research described in this presentation has been funded by the U.S. Environmental Protection Agency through the STAR Cooperative Agreement CR 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

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

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

5. 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)

6. Wet Atmospheric Sulfate Deposition

7. 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

8. MAHA/MAIA Stream Locations

9. Map of NADP and MAHA/MAIA Locations

10. Sketch of watershed with overlaid landcover map

11. 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

12. 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)

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

14. 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

15. 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

16. 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)

17. 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)

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

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

20. 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

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

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

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

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

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

26. 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

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

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

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

30. Results from SMCMC and Winbugs