Distribution Function Estimation in Small Areas for Aquatic Resources Spatial Ensemble Estimates of Temporal Trends in Acid Neutralizing Capacity Mark Delorey F. Jay Breidt Colorado State University This research is funded by U.S.EPA – Science To Achieve Results (STAR) Program Cooperative Agreement # CR

Project Funding The work reported here was developed under the STAR Research Assistance Agreement CR awarded by the U.S. Environmental Protection Agency (EPA) to Colorado State University. This presentation has not been formally reviewed by EPA. The views expressed here are solely those of the presenter and the STARMAP, the Program he represents. EPA does not endorse any products or commercial services mentioned in this presentation.

Outline Statement of the problem: How to get a set of estimates that are good for multiple inferences of acid trends in watersheds? Hierarchical model and Bayesian inference Constrained Bayes estimators -adjusting the variance of the estimators Conditional auto-regressive (CAR) model -introducing spatial correlation Constrained Bayes with CAR Summary

The Problem Evaluation of the Clean Air Act Amendments of examine acid neutralizing capacity (ANC) -surface waters are acidic if ANC < 0 -supply of acids from atmospheric deposition and watershed processes exceeds buffering capacity Temporal trends in ANC within watersheds (8-digit HUC’s) -characterize the spatial ensemble of trends -make a map, construct a histogram, plot an empirical distribution function

Data Set 86 HUC’s in Mid-Atlantic Highlands ANC in at least two years from 1993–1998 HUC-level covariates: -area -average elevation -average slope, max slope -percents agriculture, urban, and forest -spatial coordinates -dry acid deposition from NADP

Region of Study

Locations of Sites

Small Area Estimation Probability sample across region -regional-level inferences are model-free -samples are not sufficiently dense in small watersheds (HUC-8) -need to incorporate auxiliary information through model Two standard types of small area models (Rao, 2003) -area-level: watersheds -unit-level: site within watershed

Two Inferential Goals Interested in estimating individual HUC-specific slopes Also interested in ensemble: spatially-indexed true values: spatially-indexed estimates: -subgroup analysis: what proportion of HUC’s have ANC increasing over time? -“empirical” distribution function (edf):

Deconvolution Approach Treat this as measurement error problem: Deconvolve: -parametric: assume F  in parametric class -semi-parametric: assume F  well-approximated within class (like splines, normal mixtures) -non-parametric: assume E F [e i  ] is smooth Not so appropriate for heteroskedastic measurements, explanatory variables, two inferential goals

Hierarchical Area-Level Model Extend model specification by describing parameter uncertainty: Prior specification:

Bayesian Inference Individual estimates: use posterior means where Do Bayes estimates yield a good ensemble estimate? -use edf of Bayes estimates to estimate F  ? No: Bayes estimates are “over-shrunk” -too little variability to give good representation of edf (Louis 1984, Ghosh 1992)

Adjusted Shrinkage Posterior means not good for both individual and ensemble estimates Improve by reducing shrinkage -sample mean of Bayes estimates already matches posterior mean of -adjust shrinkage so that sample variance of estimates matches posterior variance of true values Louis (1984), Ghosh (1992) Cressie and Stern (1991)

Constrained Bayes Estimates Compute the scalars Form the constrained Bayes (CB) estimates as where

Shrinkage Comparisons for the Slope Ensemble

Numerical Illustration Compare edf’s of estimates to posterior mean of F  : Comparison of ensemble estimates at selected quantiles:

Estimated EDF’s of the Slope Ensemble CB Posterior Mean Bayes

Spatial Model Let where  is an unknown coefficient vector, C = (c ij ) represents the adjacency matrix,  is a parameter measuring spatial dependence,  is a known diagonal matrix of scaling factors for the variance in each HUC, and  is an unknown parameter. Adjacency matrix C can reflect watershed structure

Conditional Auto Regressive (CAR) Model Let A h denote a set of neighboring HUCs for HUC h The previous formulation is equivalent to: Cressie and Stern (1991)

HUC Structure First level (2-digit) divides U.S. into 21 major geographic regions Second level (4-digit) identifies area drained by a river system, closed basin, or coastal drainage area Third level (6-digit) creates accounting units of surface drainage basins or combination of basins Fourth level (8-digit) distinguishes parts of drainage basins and unique hydrologic features

Neighborhood Structure All watersheds within the same HUC-6 region were considered part of same neighborhood No spatial relationship among HUC-4 regions or HUC-2 regions considered at this point

Model Specifications Adjacency matrix:  = diag  hh =, h = 1,…,m; n h = # neighbors of HUC h  hk = 0, h ≠ k  can be fixed or random

Constrained Bayes with CAR (  fixed) If  is known, Stern and Cressie (1999) show how to solve for H 1 (Y) and H 2 (Y) under the mean and variance constraints: and respectively, where

When  is Unknown or Random We place a uniform prior on  and minimize the Lagrangian: where to get a system of equations that can be used to solve for Posterior quantities can be estimated using BUGS or other software

Spatial Structure

Summary In Bayesian context, posterior means are overshrunk; in order to obtain estimates appropriate for ensemble, need to adjust In CAR, if  is known, can find CB estimators following Stern and Cressie (1999); if  is unknown, can still find CB estimators numerically Contour plot indicates that trend slopes of ANC are smoothed and somewhat homogenized within HUC

Ongoing Work Replace spatial CAR with geostatistical model; model site responses where Is CB estimate of rate the same as rate from CB estimates?

Other Issues Restrict to acid-sensitive waters Combine probability and convenience samples Modify spatial structure