model;
{
##First difference CRW model
#	Created by Ian Jonsen
#		10 March, 2004
#	Last modified 29 October, 2004
pi <- 3.141592653589
pi2 <- 2*pi
npi <- pi*-1
Omega[1,1] <- 1
Omega[1,2] <- 0
Omega[2,1] <- 0
Omega[2,2] <- 1
## priors on process uncertainty
iSigma[1:2,1:2] ~ dwish(Omega[,],2)	
Sigma[1:2,1:2] <- inverse(iSigma[,])
## Priors for first location
for(k in 1:2){
	x[1,k] ~ dt(y[1,k], itau2[1,k], nu[1,k])
	}
## Assume simple random walk to estimate 2nd regular position
x[2,1:2] ~ dmnorm(x[1,], iSigma[,])
theta ~ dunif(npi,pi)	## prior for theta (mean turn angle)
gamma ~ dbeta(1,1)	## prior for gamma (persistence)
corr ~ dunif(0, 10)
## Transition equation
for(t in 2:(RegN-1)){
	## Build transition matrix for rotational component	
	T[t,1,1] <- cos(theta)
	T[t,1,2] <- -sin(theta)
	T[t,2,1] <- sin(theta)
	T[t,2,2] <- cos(theta)
	for(k in 1:2){
		Tdx[t,k] <- T[t,k,1] * (x[t,1] - x[t-1,1]) + T[t,k,2] * (x[t,2] - x[t-1,2])	## matrix multiplication
		x.mn[t,k] <- x[t,k] + gamma * Tdx[t,k]	## predict next location (no process error)
		}
	x[t+1,1:2] ~ dmnorm(x.mn[t,], iSigma[,])	## predict next location (with process error)
	}
##	Measurement equation
for(t in 2:RegN){				## loops over regular time intervals (t)
	for(i in idx[t-1]:(idx[t]-1)){		## loops over observed locations within interval t
		for(k in 1:2){
			itau2c[i,k] <- itau2[i,k] * corr
			zhat[i,k] <- (1 - j[i]) * x[t-1,k] + j[i]*x[t,k]	## interpolate irregularly observed locations
			y[i,k] ~ dt(zhat[i,k], itau2c[i,k], nu[i,k])	## robust measurement equation
			}
		}
	}	
}