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Spatial and Spatial and Spatio-temporal modeling of Spatio-temporal modeling of the abundance of spawning the abundance of spawning ccohooho salmon salmon

on the Oregon coaston the Oregon coast

R82-9096-01

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 SalmonCoho Salmon

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OverviewOverview

• 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 colorationwww.zoology.ubc.ca/ ~keeley/coho4.jpg

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IntroductionIntroduction

• 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 colorationwww.zoology.ubc.ca/ ~keeley/coho4.jpg

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IntroductionIntroduction

• 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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IntroductionIntroduction

• 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 colorationwww.zoology.ubc.ca/ ~keeley/coho4.jpg

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DataData• 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 UnitsCoho 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 CoastNorth Coast

Mid-CoastMid-Coast

Mid-South Mid-South

CoastCoastUmpquaUmpqua

Coho Population UnitsCoho Population Units

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Spawner abundance has generally been lower in the south/north and higher in the mid-coast of Oregon

Max=270 Max=326 Max=1659

Radius of circle are prop. to observed count

+ zero counts

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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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• Diggle et.al. (1998)

• We fitted a separate model for each year (1998-2003)

• Notation

: observed count of adult spawners

at Coho population , 1, ,

and spatial sampling site , 1, ,

ji

ji j

y

j j P

i n

s

Spatial Analysis Spatial Analysis --Individual Year Analysis

Poisson Generalized Linear Geostatistical ModelPoisson Generalized Linear Geostatistical Model

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• Model for the dataModel for the data:

Yji| λji ~ independent Poisson(lji λji)

• Model for the log of the Poisson rateModel for the log of the Poisson rate

Spatial Analysis Spatial Analysis - - Individual Year Analysis

Poisson Generalized Linear Geostatistical ModelPoisson Generalized Linear Geostatistical Model

non-spatial random effect

length of the site ssjjii (kms)

density of spawning Coho at site s sjji i (counts per km)

spatially correlated random effect

log( )ji j ji jiZ

Dependence among observations is induced by the random spatial process ji

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Spatial Analysis (cont.)Spatial Analysis (cont.)

• Assume

is a correlation parameter

Sites from different coho populations are assumed

independent

21, , MVN( , ) 1, ,

jj n z jZ Z j P Z 0 R = ~

2 ~ MVN ( , ) 1, ,jj n j P η 0 I

( , ) Corr ,

( , ) exp with ( , ) || ||

ji ji ji ji

jj ji ji

Z Z Z Z

d i id i i

R

s s

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Model FittingModel Fitting

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

where

2 2 2

2 2 2

, , , , | | , ,

| , ( ) ( ) ( )

z

z z

p p p

p

λ Z μ y y λ λ Z μ

Z μ

, |

exp λ μ Z η

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• 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 , z

2, , 2

and each of the elements of Z, and λ to generate a Markov chain.

Individual Year Analysis (cont.)Individual Year Analysis (cont.)

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PredictionPrediction

• To obtain predictions at the prediction grid locations of

the spatial component, ZZp

the density of returning

adult coho, p

we sample from their complete conditional distributions:

• p(ZZp |Z, , , zz22)

• p(p |ZZp, , , 22 )

Prediction Grid

Center of prediction grid box denoted by +

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GoalsGoals

• 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 colorationwww.zoology.ubc.ca/ ~keeley/coho4.jpg

Spatio-Temporal ModelSpatio-Temporal Model

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Radius of circle are prop. to observed count

+ zero counts

Counts of returning adult Coho salmonCounts of returning adult Coho salmon1998-20031998-2003

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Yearly Counts by Coho Population (18)Yearly Counts by Coho Population (18)

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• Wikle (2003)

• Notation

• Concern about the short time series available

: observed count of spawning Coho salmon at year ,

coho population and site

for 1,..., (1998, , 2003);

1,..., (1, ,18);

1,...,

tji

ji

tj

y t

j

t T

j P

i n

s

Spatio-Temporal Analysis Spatio-Temporal Analysis Poisson Generalized Linear Geostatistical ModelPoisson Generalized Linear Geostatistical Model

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Spatio-Temporal ModelSpatio-Temporal ModelData ModelData Model

• Model for the data:

Ytij| λtij ~ independent Poisson(ltjiλtij)

• Dependence among observations is induced by the random spatio-temporal process

density of spawning coho at site at time tjis

tji

length of the site ssjji (kms) at time t

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Process ModelProcess Model

log tj i tji tj tji k Z

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

Spatio-temporal dynamic process that accounts for the spatial variation of the coho spawners over time

is an m1 vector representation of the gridded Z-process at the prediction locations

1 18( , , )t t t Z Z Z

1

0

0 1

tji

jm

k

sampled site

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Process Model (cont.)Process Model (cont.)

Assumptions:

2 iid (0, )t j i N hh s· :

, 1, : autoregressive model (STARMA)jtj t j tj W aZ H Z γ

Space-time autoregressive moving average

log tji tji tj tj i k Z

2(0, )jtj mγ MVN Igs:

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• 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

Nearest neighbors model for HNearest neighbors model for HWW,,aa

tjZ

, 1,jtj t j tj W aZ H Z γ

1, 11

2

,

1,

1,3

( , ) ( , ) ( , )

( , )

( , )

( , )

tj t j t j

t j

t

j

j

W x yx y x y x x y

x y y

x x y

a

a

a

Z Z Z

Z

Z

14 , ( , )

( , )

t j

tj

x y x

x y

a

Z

3

4 2

1

:( ,

:upper

:left :right

:lower

)

a

a aw x y

a

1 2 3 4 1 2 3 4( , , , ) do not vary spatially. a a a a a a a a a a

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Model: Model:

• : autoregressive spatial process in the population j

• j is the mean for the autoregresive process W in the Coho Population j

•

•

2MVN( , )j j w j W 1 R

* *

( , ) Corr ,

, exp with , || ||

ji ji ji ji

jj ji ji

W W W W

d i id i i

R

s s

* *, grid locations in

Coho Population

ji ji

j

s s2~ (0, )aa N

, 1jt t t W aZ H Z γ

neighbor coefficients

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• We placed prior to the parameters

• Compute the joint posterior distribution given the data

ImplementationImplementation

2 2 20, , , , ,w α Z

2 2 21 0 1 1

2

1 1 1

2 21,

1 1

2 2 2

, , , , , , , , , , , , ,

| | ,

| , , , | , , ( )

( ) ( ) (

tj

T T w T

nT P

tji tji tji tj

t j i

T P

tj t j j j j w

t j

w

p a

p p

p a p p a

λ λ Z Z Z W μ y y

y λ λ Z

Z Z W W α

, , |

) ( ) α

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ImplementationImplementation

• 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.

2 2 2, , , , , , ,t wa Z W μ

,t λ

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ImplementationImplementation

• 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

2 2 2, , ,w a

0, , , , 1, ,t t t Tα W Z Z λ

5

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Posterior Mean of lambda Posterior Mean of lambda 1998-20001998-2000

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Posterior Mean of lambda 200Posterior Mean of lambda 2001-20031-2003

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Posterior Predictive Posterior Predictive Mean of lambda(2004)Mean of lambda(2004)

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Posterior Predictive Posterior Predictive Mean of lambda(2004)Mean of lambda(2004)

Posterior Mean of lambdaPosterior Mean of lambda2001-20032001-2003

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CommentsComments

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

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Posterior histograms Posterior histograms of of σσ22

ηη, , σσ22γγ, , σσ22

ww , and , and

the neighbor the neighbor coefficient “a”coefficient “a”

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Coho Population UnitsCoho 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

Coho Population UnitsCoho Population Units

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Posterior Sum(lamda[s(ji)]), i=1,Posterior Sum(lamda[s(ji)]), i=1,…n(tj)…n(tj)