Distribution Function Estimation in Small Areas for Aquatic Resources

Spatial Ensemble Estimates of Temporal Trendsin Acid Neutralizing Capacity

Mark Delorey

F. Jay Breidt

Colorado State University

This research is funded by

U.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 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

• Examine acid neutralizing capacity• Supply of acids from atmospheric deposition and

watershed processes exceeds buffering capacity• Surface waters are acidic if ANC < 0• 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

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):

m

hh 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

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hhh

e

e

0,N~

ˆ

Hierarchical Area-Level Model

• Extend model specification by describing parameter uncertainty:

• Prior specification:

2 0,NID ,

0,NID ,ˆ

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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

2

2

1

ˆ

ˆVartrˆ

|1

12

2

1

1

CB B B B B Bh h h

1

a a a

ˆHa 1

ˆH

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

Cu

mu

lativ

e P

rob

ab

ility

-1000 -500 0 500 1000 1500

0.0

0.2

0.4

0.6

0.8

1.0

CB

Posterior Mean

Bayes

Conditional Auto Regressive (CAR) 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

1N~,

,N~|ˆ

CIX

D

22 ,,|

,

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:

= Im

fixed at three different levels: 0.09, 0.01, 0

otherwise;0

neighbors and ;centroids 8-HUCbetween distance1 khchk

Constrained Bayes with CAR

• Again, compute H1 and H2:

where

• Then,

111 TT XXXXP

ˆEˆEˆ

ˆVartrˆ

12

211

||

|

PIPI

PI

TTH

H

21

2ˆ

ˆ where,ˆE1ˆE

H

H1a|a|a 1hh

CBh P

Comparison of Estimated EDF’s

Slope in ug / L / year

Cu

mu

lativ

e P

rob

ab

ility

-1000 -500 0 500 1000 1500

0.0

0.2

0.4

0.6

0.8

1.0

CBphi=0.09

Slope in ug / L / year

Cu

mu

lativ

e P

rob

ab

ility

-1000 -500 0 500 1000 1500

0.0

0.2

0.4

0.6

0.8

1.0

CBphi=0.01

Slope in ug / L / year

Cu

mu

lativ

e P

rob

ab

ility

-1000 -500 0 500 1000 1500

0.0

0.2

0.4

0.6

0.8

1.0

CBphi=0

Spatial Structure

0 0

0

0

0

0

0

0

0

0

0 0

0

Summary

• In Bayesian context, posterior means are overshrunk; in order to obtain estimates appropriate for ensemble, need to adjust

• In CAR model, value of does have an effect on edf, but not large effect, possibly due to CB calculation

• Edf of CB estimates in CAR shifts more mass towards positive values

• Contour plot indicates that trend slopes of ANC are smoothed within HUCs

Further Work

• Restrict to acid-sensitive waters• Combine probability and convenience samples• Other covariates?

- deposition maps/trends from CASTNet?• Modify spatial structure• Site-level model?

- useful sub-watershed covariates?- spatial scales: HUC to HUC, site to site- more concern with design, normality assumptions