Strength of Spatial Correlation and Spatial Designs: Effects on Covariance Estimation Kathryn M. Irvine Oregon 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 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 R

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 SO 4 -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 locations EMAP NADP

Geostatistical Model Where Y(s) is a vector of observed ln(SO 4 -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 Where D is matrix of pairwise distances,  is 1/range,   is the partial sill   is the nugget (1)

Effective Range Definition: 1) Distance beyond which the correlation between observations is less than or equal to ) Distance where the semi-variogram reaches 95% of the sill.

Semi-Variogram 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 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 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? 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 = estimate - truth

Range Parameter = 1 Range Parameter = 3 Results for Estimation of Effective Range

Range Parameter = 1 Range Parameter = 3 Results for Estimation of Effective Range

Range Parameter = 1 Range Parameter = 3 Results for Estimation of Effective Range

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

Results 1. Estimation of Covariance Parameters 2. Estimation of Autocorrelation Function

Results: 2. Estimation of Autocorrelation Function

Estimation of Autocorrelation Function Cluster Design

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

Estimated Auto-correlation Function for ln(SO 4 -2 )

Sketch of watershed with overlaid landcover map

2. Estimation of Autocorrelation Function Lattice Design

Estimation of Autocorrelation Function Lattice Design

2. Estimation of Autocorrelation Function Random Design

Estimation of Autocorrelation Function Random Design

2. Estimation of Autocorrelation Function Cluster Design