Using the Maryland Biological Stream Survey Data to Test Spatial Statistical Models

Using the Maryland Using the Maryland Biological Stream Survey Biological Stream Survey

Data to Test Spatial Data to Test Spatial Statistical ModelsStatistical Models

A Collaborative Approach to A Collaborative Approach to Analyzing Stream Network Analyzing Stream Network

DataDataAndrew A. Merton

OverviewOverview

The material presented here is a subset The material presented here is a subset of the work done by Erin Peterson for of the work done by Erin Peterson for her Ph.D.her Ph.D. Interested in developing geostatistical Interested in developing geostatistical

models for predicting water quality models for predicting water quality characteristics in stream segmentscharacteristics in stream segments

Data: Maryland Biological Stream Survey Data: Maryland Biological Stream Survey (MBSS)(MBSS)

The scope and nature of the problem The scope and nature of the problem requires interdisciplinary collaborationrequires interdisciplinary collaboration

Ecology, geoscience, statistics, others…Ecology, geoscience, statistics, others…

Stream Network DataStream Network Data

The response data is comprised of The response data is comprised of observations observations withinwithin a stream a stream networknetwork What does it mean to be a “neighbor” in What does it mean to be a “neighbor” in

such a framework?such a framework? How does one characterize the distance How does one characterize the distance

between “neighbors”?between “neighbors”? Should distance measures be confined to Should distance measures be confined to

the stream network?the stream network? Does flow (direction) matter?Does flow (direction) matter?

Stream Network DataStream Network Data

Potential explanatory variables are not Potential explanatory variables are not restricted to be within the stream restricted to be within the stream networknetwork Topography, soil type, land usage, etc.Topography, soil type, land usage, etc.

How does one sensibly incorporate How does one sensibly incorporate these explanatory variables into the these explanatory variables into the analysis?analysis? Can we develop tools to aggregate Can we develop tools to aggregate

upstream watershed covariates for upstream watershed covariates for subsequent downstream segments?subsequent downstream segments?

Competing ModelsCompeting Models

Given a collection of competing Given a collection of competing models, how does one select the models, how does one select the “best” model?“best” model? Is one subset of explanatory variables Is one subset of explanatory variables

better or closer to the “true” model?better or closer to the “true” model? Should one assume correlated residuals Should one assume correlated residuals

and, if so, what form should the and, if so, what form should the correlation function take?correlation function take? How does the distance measure impact the How does the distance measure impact the

choice of correlation function?choice of correlation function?

Functional Distances & Functional Distances & Spatial RelationshipsSpatial Relationships

A

B

C

Straight-line Distance (SLD)Is this an appropriate measure of distance?

Influential continuous landscape variables: geology type or acid rain

(As the crow flies…)

Geostatistical models are based on straight-line

distance

A

B

C

Distances and relationships are represented differently depending

on the distance measure


Symmetric Hydrologic Distance (SHD)Hydrologic connectivity

(As the fish swims…)

A

B

C

Distances and relationships are represented differently depending

on the distance measure


Asymmetric Hydrologic Distance (AHD)Longitudinal transport of material

(As the sh*t flows…)

Candidate ModelsCandidate Models

Restrict the model space to general Restrict the model space to general linear modelslinear models

Look at all possible subsets of Look at all possible subsets of explanatory variables explanatory variables XX (Hoeting et al) (Hoeting et al)

Require a correlation structure that can Require a correlation structure that can accommodate the various distance accommodate the various distance measuresmeasures Could assume that the residuals are spatially Could assume that the residuals are spatially

independent, i.e., independent, i.e., S = S = 22II (probably not best) (probably not best) Ver Hoef et al propose a better solutionVer Hoef et al propose a better solution

),(),(~ 2 XNXNZ

Asymmetric Autocovariance Asymmetric Autocovariance Models for Stream NetworksModels for Stream Networks

Weighted asymmetric Weighted asymmetric hydrologic distance (WAHD)hydrologic distance (WAHD)

Developed by Jay Ver Hoef, Developed by Jay Ver Hoef, National Marine Mammal National Marine Mammal Laboratory, SeattleLaboratory, Seattle

Moving average modelsMoving average models

Incorporates flow and uses Incorporates flow and uses hydrologic distancehydrologic distance

Represents discontinuity at Represents discontinuity at confluencesconfluences

Flow

Exponential Correlation StructureExponential Correlation Structure

The exponential correlation function The exponential correlation function can be used for both SLD and SHDcan be used for both SLD and SHD

For AHD, one must multiply For AHD, one must multiply (element- (element-wise) by the weight matrix wise) by the weight matrix AA, i.e., , i.e., ij* = aij ij, hence WAHD

The weights represent the proportion of flow volume that the downstream location receives from the upstream location

Estimating the aij is non-trivial – Need special GIS tools (Theobald et al)

0/exp)1(

01

211,

ijij

ijij

n

jiij hh

h

that such

GIS ToolsGIS ToolsTheobald et al have created automated tools to extract data about hydrologic relationships

between sample points

Visual Basic for Applications programs that:1. Calculate separation distances between sites SLD, SHD, Asymmetric hydrologic distance

(AHD)2. Calculate watershed covariates for each stream

segment Functional Linkage of Watersheds and Streams

(FLoWS)3. Convert GIS data to a format compatible with

statistics software

1 2

3

1 2

3

SLD

1 2

3

SHD AHD

Spatial Weights for WAHDProportional influence: influence of each neighboring sample site on a downstream sample site

•Weighted by catchment area: Surrogate for flow

1. Calculate influence of each upstream segment on segment directly downstream

2. Calculate the proportional influence of one sample site on another• Multiply the edge

proportional influences

3. Output:• n×n weighted incidence

matrix

stream confluencestream segment







matrix

A

BC

DE

F

G

H

survey sitesstream segment







matrix

A

BC

DE

F

G

H

Site PI = B * D * F * G

Parameter EstimationParameter Estimation

Maximize the (profile) likelihood to Maximize the (profile) likelihood to obtain estimates for obtain estimates for , , ,, and and 22

ZXXX 111 ')'()(ˆ

n

XZXZ )()'()(ˆ

12

MLE

s

Profile likelihood:

2log

2

1)ˆlog(

2)2log(

2),ˆ,ˆ;( 22 nnnZprofile

Model SelectionModel Selection

Hoeting et al adapted the Akaike Hoeting et al adapted the Akaike Information Corrected Criterion for Information Corrected Criterion for spatial modelsspatial models AICC estimates the difference between AICC estimates the difference between

the candidate model and the “true” the candidate model and the “true” modelmodel

Select models with small AICCSelect models with small AICC 2

12),,;(2 2

kpn

kpnZprofile AICC

where n is the number of observations, p-1 is the number of covariates, and k is the number of autocorrelation parameters

Spatial Distribution of MBSS Data

N

Summary Statistics for Distance Measures

• Distance measure greatly impacts the number of neighboring sites as well as the median, mean, and maximum separation distance between sites

* Asymmetric hydrologic distance is not weighted here

Summary statistics for distance measures in kilometers using DO (n=826).Distance Measure N Pairs Min Median Mean Max

Straight Line Distance 340725 0.05 101.02 118.16 385.53

Symmetric Hydrologic Distance 62625 0.05 156.29 187.10 611.74

Pure Asymmetric * Hydrologic Distance 1117 0.05 4.49 5.83 27.44

Comparing Distance MeasuresComparing Distance Measures

The “selected” models (one for each The “selected” models (one for each distance measure) were compared by distance measure) were compared by computing the mean square prediction computing the mean square prediction error (MSPE) error (MSPE) GLM: Assumed independent errorsGLM: Assumed independent errors Withheld the same 100 (randomly) Withheld the same 100 (randomly)

selected records from each model fitselected records from each model fit Want MSPE to be smallWant MSPE to be small

p

n

iii

n

ZZMSPE

p

1

2)ˆ(

ANC

0.00

50000.00

100000.00

150000.00

200000.00

250000.00

300000.00

350000.00

GLM SL SH WAH

COND

0.00

5000.00

10000.00

15000.00

20000.00

25000.00

30000.00

35000.00

40000.00

GLM SL SH WAH

DOC

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

GLM SL SH WAH

DO

0.00

0.50

1.00

1.50

2.00

2.50

GLM SL SH WAH

NO3

0.00

0.20

0.40

0.60

0.80

1.00

1.20

GLM SL SH WAH

SO4

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

GLM SL SH WAH

TEMP

6.50

7.00

7.50

8.00

8.50

9.00

GLM SL SH WAH

PHLAB

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

GLM SL SH WAH

MS

PE

GLM

SLD

SHD

WAHD

Comparing Distance MeasuresComparing Distance MeasuresPrediction Performance for Various ResponsesPrediction Performance for Various Responses

Maps of the Relative WeightsMaps of the Relative Weights

Generated maps by kriging Generated maps by kriging (interpolation)(interpolation) Predicted values are linear combinations Predicted values are linear combinations

of the “observed” data, i.e.,of the “observed” data, i.e.,

1

11

1111

11

11111

11211

1111

11

1112

12

)))(()((

)|(

MZ

ZXXXXIXXXX

ZZETTTT

Z1 is the observed data, Z2 is the predicted value, 11 is the correlation matrix for the observed sites, and is the correlation matrix between the prediction site and the observed sites

Relative Weights Used to Make Prediction at Site 465

General Linear Model

Symmetric Hydrologic

Straight-line

Weighted Asymmetric Hydrologic

General Linear Model Straight-line

Symmetric Hydrologic Weighted Asymmetric Hydrologic

Relative Weights Used to Make Prediction at Site 465

Residual Correlations for Site 465

General Linear Model

Symmetric Hydrologic

Straight-line

Weighted Asymmetric Hydrologic

General Linear Model Straight-line

Symmetric Hydrologic Weighted Asymmetric Hydrologic

Residual Correlations for Site 465

Probability-based random survey design• Designed to maximize spatial independence

of survey sites• Does not adequately represent spatial

relationships in stream networks using hydrologic distance measures

Some Comments on the Sampling Design

0 2

244

149133

109

66

38 32

12 7

3519 15 13 6 1 0

0

275

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Fre

quen

cy

Number of Neighboring Sites

244 sites did not have neighbors Sample Size = 881Number of sites with ≥ 1 neighbor: 393Mean number of neighbors per site: 2.81

Conclusions

A collaborative effort enabled the analysis of a complicated problem Ecology – Posed the problem of interest, Ecology – Posed the problem of interest,

provides insight into variable (model) provides insight into variable (model) selectionselection

Geoscience – Development of powerful tools Geoscience – Development of powerful tools based on GISbased on GIS

Statistics – Development of valid covariance Statistics – Development of valid covariance structures, model selection techniquesstructures, model selection techniques

Others – e.g., very understanding (and Others – e.g., very understanding (and sympathetic) spouses…sympathetic) spouses…

Using the Maryland Biological Stream Survey Data to Test Spatial Statistical Models

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