1 Spatial and Spatio-temporal modeling of the abundance of spawning coho salmon on the Oregon coast R Ruben Smith Don L. Stevens Jr. September 11, 2004

2 This presentation was supported under STAR Research Assistance Agreement No. CR 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

3 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 ~keeley/coho4.jpg

4 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 ~keeley/coho4.jpg

5 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

6 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 ~keeley/coho4.jpg

7 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 :

8 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-SouthCoast Umpqua Coho Population Units

9 Spawner abundance has generally been lower in the south/north and higher in the mid-coast of Oregon Max=270Max=326 Max=1659 Radius of circle are prop. to observed count + zero counts

10 Spatial Analysis Goal  To generate prediction maps of abundance of spawners for each year on the Oregon Coast

11 Diggle et.al. (1998) We fitted a separate model for each year ( ) Notation Spatial Analysis - Poisson Generalized Linear Geostatistical Model Spatial Analysis -Individual Year Analysis Poisson Generalized Linear Geostatistical Model

12 Model for the dataModel for the data: Y j i | λ j i ~ independent Poisson (l j i λ j i ) Model for the log of the Poisson rateModel for the log of the Poisson rate Spatial Analysis - Poisson Generalized Linear Geostatistical Model Spatial Analysis - Individual Year Analysis Poisson Generalized Linear Geostatistical Model non-spatial random effect s ji length of the site s ji (kms) s ji density of spawning Coho at site s ji (counts per km) spatially correlated random effect Dependence among observations is induced by the random spatial process

13 Spatial Analysis (cont.) Assume  is a correlation parameter Sites from different coho populations are assumed independent

14 Model Fitting We placed prior on the parameters and compute the joint posterior distribution given the data where

15 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.)

16 Prediction To obtain predictions at the prediction grid locations of Z  the spatial component, Z p  the density of returning adult coho, p we sample from their complete conditional distributions: Z  z 2p(Z p |Z, , ,  z 2 ) Z   2p( p |Z p, , ,   2 ) Prediction Grid Center of prediction grid box denoted by +

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18 Goals 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 ~keeley/coho4.jpg Spatio-Temporal Model

19 Radius of circle are prop. to observed count + zero counts Counts of returning adult Coho salmon

20 Yearly Counts by Coho Population (18)

21 Wikle (2003) Notation Concern about the short time series available Spatio-Temporal Analysis Poisson Generalized Linear Geostatistical Model

22 Spatio-Temporal Model Data Model Model for the data: Y t ij | λ tij ~ independent Poisson (l tji λ tij ) Dependence among observations is induced by the random spatio-temporal process density of spawning coho at site at time t s j length of the site s ji (kms) at time t

23 Process Model 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 m  1 vector representation of the gridded Z-process at the prediction locations sampled site

24 Process Model (cont.) Assumptions: Space-time autoregressive moving average

25 is expressed as linear combination of the past value of the process, its four nearest-neighbors and an error W j : m j x 1 vector is an autoregressive process that is allowed to vary spatially Nearest neighbors model for H W,a

26 Model: Model: : autoregressive spatial process in the population j  j is the mean for the autoregresive process W in the Coho Population j

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

28 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.

29 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

30 Posterior Mean of lambda

31 Posterior Mean of lambda

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33 Posterior Predictive Mean of lambda(2004)

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

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

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37 Posterior histograms of σ 2 η, σ 2 γ, σ 2 w, and the neighbor coefficient “a”

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42 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 Coho Population Units

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