model rwalk_meta; { ## Meta-analysis of individual pathways
## Movement arises from a random walk but the variance depends on an environmental covariate (temperature)
## Observation errors (epsilon_t) are constant among habitats (measurement error)

# Assign hyper-prior distributions for the hyper-parameters of beta
beta.m~dnorm(0,.001)
ibeta.v~dlnorm(0,.001)

# convert ibeta.v to beta.v (Note: Changes the precision to the standard deviation)
beta.v<-1/sqrt(ibeta.v)

#Iterate SSM over multiple pathways
for(k in 1:P){

    ## Process Error (eta_t):
    ## ---------------------
    # eta_t  ~ N(mean = 0, variance = (sigma*exp(-beta*temp))^2 )
    # log(1/(sigma^2))~N(mean = 0, variance = 10^3): Vague prior for the precision.
    isigma[k]~dlnorm(0,.001)

    # convert isigma to sigma (Note: Changes the precision to the standard deviation)
    sigma[k]<-1/sqrt(isigma[k])

    # beta describes how rapidly movement variance declines with increasing temperature
    # beta~N(mean = 0, variance = 1000): This is a vague prior
    beta[k]~dnorm(beta.m,ibeta.v)

    ## Observation error (epsilon_t):
    ## -----------------------------
    # epsilon_t~N(mean = 0, variance = tau^2)
    # log(1/(tau^2))~N(-1.386294, variance= 1): Informative prior for the precision.
    itau[k]~dlnorm(-1.386294,10)

    # convert itau to tau (Note: Changes the precision to the standard deviation)
    tau[k]<-1/sqrt(itau[k])

    ## Initializations:
    ## ---------------
    ## Y ->   True locations (alpha_t)
    ## Y1 -> Observed locations (y_t)

    # In order to get first true location equal to the observed location (i.e. where released) we do the following:
    Y[k,1,1]~dnorm(Y1[k,1,1],100000000)
    Y[k,1,2]~dnorm(Y1[k,1,2],100000000)

    isig.temp[k,1,1]<-0
    isig.temp[k,1,2]<-0

    ## Interate transition equation:
    for(i in 2:N){ # cycles through locations
        for(j in 1:2){  # two dimensions (x & y directions)

            isig.temp[k,i,j]<-(isigma[k]*pow(exp(-1*beta[k]*temp[k,i-1]),-2))
            Y[k,i,j]~dnorm(Y[k,i-1,j],isig.temp[k,i,j])
        }
    }

    ## Iterate measurement equation:
    for(i in 1:N){
        for(j in 1:2){
            Y1[k,i,j]~dnorm(Y[k,i,j],itau[k])
        }
    }
}
}

#These are the data required to fit the above model. The dataset contains the observed locations (Y1) and SST sampled at each location (temp)

Data
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#These are 2 sets of initial values required by the MCMC sampler to run 2 chains simultaneously
Inits
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