## Benchmark Model 1
## Modified from Parnell et al. (2013)

model {
  # target data likelihood
  for(i in 1:N) {
    Y[i,1:J] ~ dmnorm(mu[i,],OmegaresZ)
    mu[i,1:J] <- p[i,1:K]%*%s[1:K,1:J,i]
    }


  # s - sources
  for(i in 1:N) {
    for(k in 1:K) {
     s[k,1:J,i] ~ dmnorm(muS[,k],OmegaS[,,k])
    }
  }

  # s contraints
   for(i in 1:N) {
      for(k in 1:K) {
         for (j in 1:J) {
            positive[k,j,i] ~ dinterval(s[k,j,i], zero)
      }
   }
   }
   
  
  # p - proportions
  for(i in 1:N) {
    p[i,1:K] <- expphiV[i,1:K]/sum(expphiV[i,1:K])
    phiV[i,1:K] <- phi[i,1:(K-1)]%*%tV
    for(k in 1:K) {
      expphiV[i,k] <- exp(phiV[i,k])
    }
  }

  # phi
  for(i in 1:N) {
    for(k in 1:(K-1)) {
      phi[i,k] ~ dnorm(muphi[k],tauphi[k])
    }
    philast[i] <- -sum(phi[i,1:(K-1)])
  }

  # hyperparameters for phi prior
  for(k in 1:(K-1)) {
     muphi[k] ~ dnorm(0,1)
     tauphi[k] ~ dgamma(2,1)
    sigma2phi[k] <- 1/tauphi[k]
  }

  # Prior on OmegaresZ
  # combined precision matrix of residual + measurement error
  OmegaresZ ~ dwish(Rres,kres)
  SigmaresZ <- inverse(OmegaresZ)

}