Mark Delorey F. Jay Breidt Colorado State University

Distribution Function Estimation in Small Areas for Aquatic Resources

Spatial Ensemble Estimates of Temporal Trendsin Acid Neutralizing Capacity

Mark DeloreyF. Jay Breidt

Colorado State University

This research is funded byU.S.EPA – Science To AchieveResults (STAR) ProgramCooperativeAgreement

# CR - 829095

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.

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 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

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):

mhh 1

m

hh 1est

m

hh zI

mzF

1

1)(

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

hh

hhh

ee

0,N~

ˆ

Hierarchical Area-Level Model

• Extend model specification by describing parameter uncertainty:

• Prior specification:

2 0,NID ,

0,NID ,ˆ

hhThh

hhhhh ee

x

222, ffff

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)

2

2

ˆ|ˆ1ˆEˆ|E

hh

Thhhhh

Bh x

m

h

m

hh

BBh

1 1

22 ˆ|E

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)

h

Constrained Bayes Estimates

• Compute the scalars

• Form the constrained Bayes (CB) estimates as

where

m

h

BBhH

H

1

22

1

ˆ

ˆVartrˆ

|1

1ˆ

ˆ

1

21

2

HH1a

aa

1

BBh

CBh

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:

m

hh

B zIm

zF1

ˆ|E1)(

Estimated EDF’s of the Slope Ensemble

Slope in ug / L / year

Cum

ulat

ive

Pro

babi

lity

-1000 -500 0 500 1000 1500

0.0

0.2

0.4

0.6

0.8

1.0

CB

Posterior Mean

Bayes

Spatial Model

• Let

where 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

1

2

N~,

,N~|ˆ CIX

D

22 ,,|

,

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)

mh

chkhAk

hhkkhkhkh

,,1

,,N~,| 2

XX

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; nh = # neighbors of HUC h

hk = 0, h ≠ k can be fixed or random

otherwise;0

neighbors and ;centroids 8-HUCbetween distance1 khchk

hn1

hn1

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 constraints:

and

respectively, where

βPIΦPIβYβPβΦβPβ ˆˆΦ

1TΦ

TΦ

1TΦ |E

111 TT XXXXP

βPY|βP ˆ E

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

YβΦXXΦXY,βΦXXΦX

YβPIΦPIβY,βPβΦβPβ

YY,ββΦββ

0

||

|

ˆ

ˆˆ

ˆˆ

1111

*1**1*

1

||

|

TTTTTTT

TTT

T

EE

EE

EEL

111 TT XXXP*

β̂

Spatial Structure

0 0 0

0

0 0

0

0

0

0

0 0 0 0

0

0

0 0

Constrained Bayes with CAR

0 0

0

0

0

0

0 0

0

0

0

0

0

0 0 0

Constrained Bayes

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?

sdssSU

s

sss

tU ti

Tit

ttt

i

θx

ˆ

njissCtfN jit ,,1,;,,~ γsη

Other Issues

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