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Spatial Prediction of Coho Salmon Counts on Stream Networks

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Dan Dalthorp

Lisa Madsen

Oregon State University

September 8, 2005

Sponsors

- U.S. EPA STAR grant # CR-829095
- U.S. EPA Program for Cooperative Research on Aquatic
- Indicators at Oregon State University grant # CR-83168201-0.

Outline

• Introduction

(i) Coho salmon data

(ii) GEEs for spatial data

• Latent process model for spatially correlated counts

• Estimation and results

• Cross-validation

• Simulation study

• Conclusions and future research

- Adult Coho salmon counts at selected points in Oregon coastal
- stream networks for 1998 through 2003.
- Euclidean distance between sampled points.
- Stream distance between sampled points.

Coastal Stream Networks and Sampling Locations

- Liang and Zeger’s (1986) pioneering paper in Biometrika
- introduced GEEs for longitudinal data.
- Zeger (1988) developed GEE analysis for a time series of counts
- using a latent process model.
- McShane, Albert, and Palmatier (1997) adapted Zeger’s model and
- analysis to spatially correlated count data.
- Gotway and Stroup (1997) used GEEs to model and predict spatially
- correlated binary and count data.
- Lin and Clayton (2005) develop asymptotic theory for GEE estimators
- of parameters in a spatially correlated logistic regression model

Suppose:

The latent process allows for overdispersion and

spatial correlation in .

The Marginal Model

These assumptions imply:

For now, we assume a simple constant-mean model and

a one-parameter exponential correlation function:

To estimate parameters solve estimating equations:

where

Step 0: Calculate initial estimates

Step 1: Update .

Step 2: Update .

Step 3: Update .

Iterate steps 1, 2, and 3 until convergence.

Year

Sample

Mean

Euclidean

Distance

Stream

Distance

1998

6.2451

6.0

6.4941

1999

9.0025

8.7286

9.0765

2000

11.92

10.898

11.481

2001

31.359

31.597

34.541

2002

46.494

46.782

46.725

2003

44.453

41.005

41.829

Year

Sample

Std. Dev.

Euclidean

Distance

Stream

Distance

1998

222.07

221.59

221.59

1999

443.65

442.61

442.54

2000

384.59

384.75

383.90

2001

2508.6

2502.3

2512.3

2002

9286.6

9265.4

9265.4

2003

3650.2

3653.4

3648.4

Assessing Model Fit – Estimating the Range (Stream Distance)

Cross validation to compare predictions based on three different assumptions about the underlying spatial process:

1. Null model (spatial independence) :

2. Spatial correlation as a function of Euclidean distance (ed):

3. Spatial correlation as a function of stream network distance (id)

Covariance model _

EuclideanStream distance

1998 -0.001 -0.047

1999 0.007 -0.037

2000 0.013 0.011

2001 -0.005 -0.005

2002 -0.008 -0.007

2003 -0.002 0.020

1. Bias? Not an issue...

2. Precision?

Covariance model _

NullEuclideanStream distance

1998 14.7213.25 14.00

1999 20.58 19.7521.17

2000 20.05 19.83 19.74

2001 48.6934.38 37.75

2002 98.5397.04 97.35

2003 60.49 60.92 58.61

Variances of predicteds

Null Euclidean Stream

0.04 10.32 4.95

0.05 12.07 7.68

0.04 11.46 6.40

0.13 38.08 33.36

0.22 15.74 10.65

0.14 24.34 25.99

Odds(|Eed| < |Eid|)

Year Odds

1998 256:152

1999 267:132

2000 266:171

2001 197:198

2002 266:171

2003 222:197

Total 1474:1021

Simulations

For each year, 8 scenarios that mimic the sample means,

variances, and ranges from the data were simulated.

Mean and variance constant

1. Euclideanspatial correlation

2. Stream network spatial correlation

Mean varies randomly by stream network; variance = 3.66 m 1.741

3.Euclideanspatial correlation; long range

4.Euclideanspatial correlation; medium range

5.Euclideanspatial correlation; short range

6.Stream networkspatial correlation; long range

7.Stream networkspatial correlation; medium range

8.Stream networkspatial correlation; short range

Simulation proceedure

1. Simulate vector Z of correlated lognormal-Poissons

to cover all sampling sites (n≈ 400)

2. Estimate parameters (m, s2, range) via latent process

regression from simulated data for a subset of the sampling

sites (blue)

3. Predict Z at the remaining sites (red, m ≈ 400) using:

(Gotway and Stroup 1997)

4. Repeat 100 times for each scenario (8) and year (6)

Use Euclidean distance or stream distance in covariance model?

Evaluation of predictions via two measures:

where:

Summary of Findings

Cross-validations:

1. MSPEs same for Euclidean distance and stream network distance;

2. Errors usually smaller with Euclidean distance;

3. Population spikes more likely to be detected with Euclidean distance.

Simulations:

1. Euclidean spatial process: Euclidean covariance gives smaller MSPE than does

stream network distance covariance;

2. Stream network process: Euclidean covariance model MSPEs comparable to

those of stream distance model EXCEPT when network means varied and range

of correlation was large.

- Future work
- -- Incorporate covariates (with some misaligned data);
- -- Incorporate downstream distances/flow ratios;
- -- Spatio-temporal modeling;
- -- Rank correlations in place of covariances;
- -- Model selection;
- -- Non-random data;