Spatial and Spatio-temporal modeling of the abundance of spawning coho salmon on the Oregon coast

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Spatial and Spatio-temporal modeling of the
abundance of spawning coho salmon on the Oregon
coast
Ruben Smith Don L. Stevens Jr. September 11,
2004
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This presentation was supported under STAR
Research Assistance Agreement No. CR82-9096-01
awarded by the U.S. Environmental Protection
Agency to Oregon State University. It has not
been formally reviewed by EPA. The views
expressed in this document are solely those of
authors and EPA does not endorse any products or
commercial services mentioned in this
presentation.
Coho Salmon
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Overview

• Introduction
• Part I Spatial Analysis of the abundance of
  Coho salmon
• Part II Spatio-temporal analysis of the
  abundance of Coho salmon

A male coho salmon with spawning
coloration www.zoology.ubc.ca/ keeley/coho4.jpg
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Introduction

• Coho salmon spend their adult lives at sea and
  return to natal streams along the Oregon Coast to
  spawn
• In 1960s and early 1970s Coho salmon were easily
  available for fishing off Oregon coast
• By late 1970s there were signs that Coho salmon
  stocks have declined in some regions of their
  range
• Coho salmon in Oregon coastal basins are listed
  as threatened under the Endangered Species Act

A male coho salmon with spawning
coloration www.zoology.ubc.ca/ keeley/coho4.jpg
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Introduction

• The Oregon Department of Fisheries and Wildlife
  (ODFW) divides the coastal streams in four
  monitoring areas
• North Coast
• Mid-Coast
• Mid-South Coast,
• Umpqua
• based on genetic variation and life-history
  traits

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Introduction

• Sites are selected using a rotating annual panel
  sample design (Stevens, 1997 Stevens Olsen,
  2000, 2002)
• The sampling unit is about 1-mile long stream
  reach (site)

A male coho salmon with spawning
coloration www.zoology.ubc.ca/ keeley/coho4.jpg
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Data

• ODFW winter spawning Coho surveys
• Visual counts of spawning Coho is used in the
  stream surveys
• Number of surveyed sites
• 420 sites per year.
• 105 per monitoring areas
• Data available for years 1998-2003

http//www.surfingvancouverisland.com/fish/images/
st00b01.jpg
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Coho Population Units
Coho Population Units

• The ODFW identified 33 Coho salmon populations
  units on 4 monitoring areas (North Coast,
  Mid-Coast,Mid-South Coast and Umpqua) based on
• geography
• similarity of habitats
• extinction risk
• potential similarity of life history types

North Coast
Mid-Coast
Mid-South Coast
Umpqua
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Radius of circle are prop. to observed count
zero counts
Max270
Max326
Max1659
Spawner abundance has generally been lower in the
south/north and higher in the mid-coast of Oregon
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• Spatial Analysis Goal
• To generate prediction maps of abundance of
  spawners for each year on the Oregon Coast

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Spatial Analysis -Individual Year
AnalysisPoisson Generalized Linear
Geostatistical Model

• Diggle et.al. (1998)
• We fitted a separate model for each year
  (1998-2003)
• Notation

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Spatial Analysis - Individual Year
AnalysisPoisson Generalized Linear
Geostatistical Model
Dependence among observations is induced by the
random spatial process

• Model for the data
• Yji ?ji independent Poisson(lji ?ji)
• Model for the log of the Poisson rate

length of the site sji (kms)
density of spawning Coho at site sji (counts per
km)
spatially correlated random effect
non-spatial random effect
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Spatial Analysis (cont.)

• Assume
• ? is a correlation parameterSites from
  different coho populations are assumed
  independent

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Model Fitting

• We placed prior on the parameters and compute the
  joint posterior distribution given the data
• where

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Individual Year Analysis (cont.)

• The likelihood is analytically intractable
• We use Markov Chain Monte Carlo (MCMC) methods to
  estimate the parameters
• Gibbs sampler
• Metropolis-Hastings algorithm
• A MATLAB computer program was used to simulate
  realizations from the posterior distributions of
  ?, ?z2,? , ??2 and each of the elements of Z,
  and ? to generate a Markov chain.

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Prediction
Prediction Grid Center of prediction grid box
denoted by

• To obtain predictions at the prediction grid
  locations of
• the spatial component, Zp
• the density of returning adult coho, ?p
• we sample from their complete conditional
  distributions
• p(Zp Z, ?, ? , ?z2)
• p(?p Zp, ?, ?, ??2 )

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Goals
Spatio-Temporal Model

• Predict the spatial abundance of Returning adult
  Coho over time.
• For this presentation we considered only 18 Coho
  populations.

A male coho salmon with spawning
coloration www.zoology.ubc.ca/ keeley/coho4.jpg
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Counts of returning adult Coho salmon 1998-2003
Radius of circle are prop. to observed count
zero counts
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Yearly Counts by Coho Population (18)
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Spatio-Temporal Analysis Poisson Generalized
Linear Geostatistical Model

• Wikle (2003)
• Notation
• Concern about the short time series available

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Spatio-Temporal ModelData Model

• Model for the data
• Ytij ?tij independent Poisson(ltji?tij)
• Dependence among observations is induced by the
  random spatio-temporal process

density of spawning coho at site at time t
length of the site sji (kms) at time t
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Process Model
is an m?1 vector
representation of the gridded Z-process at the
prediction locations
Spatio-temporal dynamic process that accounts for
the spatial variation of the coho spawners over
time
sampled site
known vector that relates sampled locations with
the Z-process. Each sampled site is assigned to
the nearest grid location
spatio-temporal random process that accounts for
observational error and small-scale
spatio-temporal variation
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Process Model (cont.)

• Assumptions

Space-time autoregressive moving average
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Nearest neighbors model for HW,a

• is expressed as linear combination of the
  past value of the process, its four
  nearest-neighbors and an error
• Wj mj x 1 vector is an autoregressive process
  that is allowed to vary spatially

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Model

• 
  autoregressive spatial process in the
  population j
• ?j is the mean for the autoregresive process W
  in the Coho Population j

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Implementation

• We placed prior to the parameters
• Compute the joint posterior distribution given
  the data

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Implementation

• The likelihood is analytically intractable
• We use Markov Chain Monte Carlo (MCMC) methods to
  estimate the parameters
• Gibbs sampler
• Metropolis-Hastings algorithm  
• 2,000 iterations with 1,000 burn-in.

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Implementation

• A MATLAB computer program was used to simulate
  realizations from the posterior distributions of
  
• and each of the elements of
  
• to generate a Markov Chain
• For now, fixed

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Posterior Mean of lambda 1998-2000
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Posterior Mean of lambda 2001-2003
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Posterior Predictive Mean of lambda(2004)
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Posterior Predictive Mean of lambda(2004)
Posterior Mean of lambda 2001-2003
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Comments

• Explore other models consider other lags in time
  (for example three year lag)

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Posterior histograms of s2?, s2?, s2w , and the
neighbor coefficient a
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Coho Population Units
Coho Population Units

• The ODFW identified 33 Coho salmon populations
  units on 4 of the 5 monitoring areas (North
  Coast, Mid-Coast,Mid-South Coast and Umpqua)
  based on
• geography
• similarity of habitats
• extinction risk
• potential similarity of life history types

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Posterior Sum(lamdas(ji)), i1,n(tj)