Spatial%20Prediction%20of%20Coho%20Salmon%20Counts%20on%20Stream%20Networks

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Spatial Prediction of Coho Salmon Counts on
Stream Networks
Dan Dalthorp Lisa Madsen Oregon State
University September 8, 2005
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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.

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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
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Coho Salmon Data

• 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.

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Coastal Stream Networks and Sampling Locations
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GEEs for Spatially Correlated Data

• Liang and Zegers (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
  Zegers 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

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The Latent Process Model
Suppose
The latent process allows for overdispersion
and spatial correlation in .
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The Marginal Model
These assumptions imply
For now, we assume a simple constant-mean model
and a one-parameter exponential correlation
function
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Estimating the Model Parameters
To estimate parameters solve
estimating equations
where
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Iterative Modified Scoring Algorithm
Step 0 Calculate initial estimates
Step 1 Update .
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Step 2 Update .
Step 3 Update .
Iterate steps 1, 2, and 3 until convergence.
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Assessing Model FitEstimating the Mean
 
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Assessing Model Fit Estimating the Variance
 
 
 
 
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Assessing Model Fit Estimating the Range
(Euclidean Distance)
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Assessing Model Fit Estimating the Range
(Stream Distance)
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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)
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Covariance model _
Euclidean Stream 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 _
Null Euclidean Stream
distance 1998 14.72 13.25 14.00 1999
20.58 19.75 21.17 2000 20.05
19.83 19.74 2001 48.69 34.38
37.75 2002 98.53 97.04 97.35 2003
60.49 60.92 58.61
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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 lt Eid) Year
Odds 1998 256152 1999 267132
2000 266171 2001 197198 2002
266171 2003 222197 Total
14741021
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Simulations For each year, 8 scenarios that
mimic the sample means, variances, and ranges
from the data were simulated.
Mean and variance constant 1. Euclidean spatial
correlation 2. Stream network spatial
correlation Mean varies randomly by stream
network variance 3.66 m 1.741 3. Euclidean
spatial correlation long range 4. Euclidean
spatial correlation medium range 5. Euclidean
spatial correlation short range 6. Stream
network spatial correlation long range 7.
Stream network spatial correlation medium
range 8. Stream network spatial correlation
short range
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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)
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Use Euclidean distance or stream distance in
covariance model? Evaluation of predictions via
two measures
where
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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.
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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