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 Abstract Data from a surface water monitoring program, the Temporally Integrated Monitoring of Ecosystems (TIME) are used to study trends in acid deposition in surface water. The TIME data consists of a probability sample of lakes and streams. One of the tools used to evaluate characteristics of acidity is the cumulative distribution function of slope trends (acid concentration/year). An understanding of the distribution of these slopes helps evaluate the impact of the to the Clean Air Act Amendments of For example, the proportion of lakes whose acidic neutralizing (ANC) has been decreasing can be estimated. A hierarchical model is constructed to describe these slopes as functions of available auxiliary information; constrained Bayes techniques are used to estimate the ensemble of slope values. Spatial relationships are represented by incorporating a conditional autoregressive model into the constrained Bayes methods. This research is funded by U.S.EPA – Science To Achieve Results (STAR) Program Cooperative Agreements # CR – and # CR – 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): Two Inferential Goals Extend model specification by describing parameter uncertainty: Prior specification: Hierarchical Area-Level Model Bayes estimates (posterior means): where – yield good individual estimates – do not yield a good ensemble estimate – too little variability to give good representation of edf since Bayesian Inference Compute the scalars Form the constrained Bayes (CB) estimates as where This improves the ensemble estimate by reducing shrinkage – sample mean of Bayes estimates already matches posterior mean of – this adjusts shrinkage so that sample variance of estimates matches posterior variance of true values Constrained Bayes Estimates Shrinkage Comparisons for the Slope Ensemble CB Posterior Mean of F  Bayes Estimated EDF’s of the Slope Ensemble 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 Spatial Model Incorporating Spatial 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 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 HUC and Neighborhood Structure Let A h denote a set of neighboring HUCs for HUC h The spatial model is equivalent to (Cressie and Stern, 1991): Adjacency matrix:  = diag(inverse number of neighbors)  hh = 1/m h, h = 1,…,m  hk = 0, h ≠ k 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 We place a uniform prior on  and minimize the Lagrangian: to get a system of equations that can be used to solve for Posterior quantities can be estimated using BUGS or other software Conditional Auto-Regressive with Constrained Bayes Baysian approach Objective 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 HUCs Restrict to acid-sensitive waters Combine probability and convenience samples Other covariates? Modify spatial structure Site-level model? – useful sub-watershed covariates? – spatial scales: HUC to HUC, site to site – more concern with design, normality assumptions Future Work Summary Funding/Disclaimer 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. Evaluation of the Clean Air Act Amendments of 1990 – 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