Distribution Function Estimation in Small Areas for Aquatic Resources Spatial Ensemble Estimates of Temporal Trends in Acid Neutralizing Capacity

1
Distribution Function Estimation in Small Areas
for Aquatic ResourcesSpatial Ensemble Estimates
of Temporal Trendsin Acid Neutralizing Capacity

• Mark Delorey
• F. Jay Breidt
• Colorado State University

2
Project Funding

• The work reported here was developed under the
  STAR Research Assistance Agreement CR-829095
  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.

3
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

4
The Problem

• Evaluation of the Clean Air Act Amendments of
  1990
• examine acid neutralizing capacity (ANC)
• surface waters are acidic if ANC lt 0
• supply of acids from atmospheric deposition
  andwatershed processes exceeds buffering
  capacity
• Temporal trends in ANC within watersheds (8-digit
  HUCs)
• characterize the spatial ensemble of trends
• make a map, construct a histogram, plot an
  empirical distribution function

5
Data Set

• 86 HUCs in Mid-Atlantic Highlands
• ANC in at least two years from 19931998
• HUC-level covariates
• area
• average elevation
• average slope, max slope
• percents agriculture, urban, and forest
• spatial coordinates
• dry acid deposition from NADP

6
Region of Study
7
Locations of Sites
8
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

9
Two Inferential Goals

• Interested in estimating individual HUC-specific
  slopes
• Also interested in ensemblespatially-indexed
  true valuesspatially-indexed estimates
• subgroup analysis what proportion of HUCs have
  ANC increasing over time?
• empirical distribution function (edf)

10
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 EF ei?? is smooth
• Not so appropriate for heteroskedastic
  measurements, explanatory variables, two
  inferential goals

11
Hierarchical Area-Level Model

• Extend model specification by describing
  parameter uncertainty
• Prior specification

12
Bayesian Inference

• Individual estimates use posterior
  meanswhere
• 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)

13
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)

14
Constrained Bayes Estimates

• Compute the scalars
• Form the constrained Bayes (CB) estimates
  aswhere

15
Shrinkage Comparisons for the Slope Ensemble
16
Numerical Illustration

• Compare edfs of estimates to posterior mean of
  F?
• Comparison of ensemble estimates at selected
  quantiles

17
Estimated EDFs of the Slope Ensemble
CB Posterior Mean Bayes
18
Spatial Model

• Letwhere ? is an unknown coefficient vector,
  C (cij) 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

19
Conditional Auto Regressive (CAR) Model

• Let Ah denote a set of neighboring HUCs for HUC h
• The previous formulation is equivalent to
• Cressie and Stern (1991)

20
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

21
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

22
Model Specifications

• Adjacency matrix
• ? diag?hh , h 1,,m nh
  neighbors of HUC h? hk 0, h ? k
• ? can be fixed or random

23
Constrained Bayes with CAR (? fixed)

• If ? is known, Stern and Cressie (1999) show how
  to solve for H1(Y) and H2(Y) under the mean and
  variance constraintsandrespectively, where

24
When ? is Unknown or Random

• We place a uniform prior on ? and minimize the
  Lagrangianwhereto get a system of
  equations that can be used to solve for
• Posterior quantities can be estimated using BUGS
  or other software

25
Spatial Structure
26
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

27
Ongoing Work

• Replace spatial CAR with geostatistical model
  model site responseswhere
• Is CB estimate of rate the same as rate from CB
  estimates?

28
Other Issues

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