R/gaussian.CARar2.R

In duncanplee/CARBayesST: Spatio-Temporal Generalised Linear Mixed Models for Areal Unit Data

gaussian.CARar2 <- function(formula, data=NULL, W, burnin, n.sample, thin=1,  prior.mean.beta=NULL, prior.var.beta=NULL, prior.nu2=NULL, prior.tau2=NULL, rho.S=NULL, rho.T=NULL, verbose=TRUE)
{
##############################################
#### Format the arguments and check for errors
##############################################
#### Verbose
a <- common.verbose(verbose)  
  
  
#### Frame object
frame.results <- common.frame(formula, data, "gaussian")
N.all <- frame.results$n
p <- frame.results$p
X <- frame.results$X
X.standardised <- frame.results$X.standardised
X.sd <- frame.results$X.sd
X.mean <- frame.results$X.mean
X.indicator <- frame.results$X.indicator 
offset <- frame.results$offset
Y <- frame.results$Y
which.miss <- frame.results$which.miss
n.miss <- frame.results$n.miss  
Y.DA <- Y      
  
  
#### Check on the rho arguments
  if(is.null(rho.S))
  {
  rho <- runif(1)
  fix.rho.S <- FALSE   
  }else
  {
  rho <- rho.S
  fix.rho.S <- TRUE
  }
  if(!is.numeric(rho)) stop("rho.S is fixed but is not numeric.", call.=FALSE)  
  if(rho<0 ) stop("rho.S is outside the range [0, 1].", call.=FALSE)  
  if(rho>1 ) stop("rho.S is outside the range [0, 1].", call.=FALSE)    

  if(is.null(rho.T))
  {
  alpha <- c(runif(1), runif(1))
  fix.rho.T <- FALSE   
  }else
  {
  alpha <- rho.T
  fix.rho.T <- TRUE
  }
  if(!is.numeric(alpha)) stop("rho.T is fixed but is not numeric.", call.=FALSE)  
  if(length(alpha)!=2) stop("rho.T is fixed but is not of length 2.", call.=FALSE)  

  
#### CAR quantities
W.quants <- common.Wcheckformat.leroux(W)
K <- W.quants$n
N <- N.all / K
W <- W.quants$W
W.triplet <- W.quants$W.triplet
W.n.triplet <- W.quants$n.triplet
W.triplet.sum <- W.quants$W.triplet.sum
n.neighbours <- W.quants$n.neighbours 
W.begfin <- W.quants$W.begfin
  
  
#### Priors
  if(is.null(prior.mean.beta)) prior.mean.beta <- rep(0, p)
  if(is.null(prior.var.beta)) prior.var.beta <- rep(100000, p)
  if(is.null(prior.tau2)) prior.tau2 <- c(1, 0.01)
  if(is.null(prior.nu2)) prior.nu2 <- c(1, 0.01)
prior.beta.check(prior.mean.beta, prior.var.beta, p)
prior.var.check(prior.tau2)
prior.var.check(prior.nu2)
  
  
#### MCMC quantities - burnin, n.sample, thin
common.burnin.nsample.thin.check(burnin, n.sample, thin)
  
  
  
#############################
#### Initial parameter values
#############################
mod.glm <- glm(Y~X.standardised-1, offset=offset)
beta.mean <- mod.glm$coefficients
beta.sd <- sqrt(diag(summary(mod.glm)$cov.scaled))
beta <- rnorm(n=length(beta.mean), mean=beta.mean, sd=beta.sd)
  
res.temp <- Y - X.standardised %*% beta - offset
res.sd <- sd(res.temp, na.rm=TRUE)/5
phi <- rnorm(n=N.all, mean=0, sd = res.sd)
tau2 <- var(phi)/10
nu2 <- runif(1, 0, res.sd)
  
  
#### Matrix versions of quantites
offset.mat <- matrix(offset, nrow=K, ncol=N, byrow=FALSE) 
regression.mat <- matrix(X.standardised %*% beta, nrow=K, ncol=N, byrow=FALSE)
phi.mat <- matrix(phi, nrow=K, ncol=N, byrow=FALSE)   
fitted <- as.numeric(offset.mat + regression.mat + phi.mat)
  
  

###############################    
#### Set up the MCMC quantities    
###############################
#### Matrices to store samples
n.keep <- floor((n.sample - burnin)/thin)
samples.beta <- array(NA, c(n.keep, p))
samples.phi <- array(NA, c(n.keep, N.all))
samples.tau2 <- array(NA, c(n.keep, 1))
samples.nu2 <- array(NA, c(n.keep, 1))
  if(!fix.rho.S) samples.rho <- array(NA, c(n.keep, 1))
  if(!fix.rho.T) samples.alpha <- array(NA, c(n.keep, 2))
samples.fitted <- array(NA, c(n.keep, N.all))
samples.loglike <- array(NA, c(n.keep, N.all))
  if(n.miss>0) samples.Y <- array(NA, c(n.keep, n.miss))
  
  
#### Specify the Metropolis quantities
accept <- rep(0,2)
proposal.sd.rho <- 0.05
tau2.shape <- prior.tau2[1] + N.all/2
nu2.shape <- prior.nu2[1] + N.all/2        

  
  
#############################
#### Specify spatial elements
#############################
#### Spatial determinant
  if(!fix.rho.S) 
  {
  Wstar <- diag(apply(W,1,sum)) - W
  Wstar.eigen <- eigen(Wstar)
  Wstar.val <- Wstar.eigen$values
  det.Q.W <-  0.5 * sum(log((rho * Wstar.val + (1-rho))))     
  }else
  {}
  
  
#### Beta update quantities
data.precision.beta <- t(X.standardised) %*% X.standardised
  if(length(prior.var.beta)==1)
  {
  prior.precision.beta <- 1 / prior.var.beta
  }else
  {
  prior.precision.beta <- solve(diag(prior.var.beta))
  }
  
  
#### Check for islands
W.list<- mat2listw(W)
W.nb <- W.list$neighbours
W.islands <- n.comp.nb(W.nb)
islands <- W.islands$comp.id
n.islands <- max(W.islands$nc)
  if(rho==1 & alpha[1]==2 & alpha[2]==-1) 
  {
  tau2.shape <- prior.tau2[1] + prior.tau2[1] + ((N-2) * (K-n.islands))/2
  }else if(rho==1)
  {
  tau2.shape <- prior.tau2[1] + prior.tau2[1] + (N * (K-n.islands))/2        
  }else if(alpha[1]==2 & alpha[2]==-1)
  {
  tau2.shape <- prior.tau2[1] + prior.tau2[1] + ((N-2) * K)/2          
  }else
  {}

  
  
###########################
#### Run the Bayesian model
###########################
#### Start timer
  if(verbose)
  {
  cat("Generating", n.keep, "post burnin and thinned (if requested) samples.\n", sep = " ")
  progressBar <- txtProgressBar(style = 3)
  percentage.points<-round((1:100/100)*n.sample)
  }else
  {
  percentage.points<-round((1:100/100)*n.sample)     
  }
  
  
#### Create the MCMC samples     
  for(j in 1:n.sample)
  {
  ####################################
  ## Sample from Y - data augmentation
  ####################################
    if(n.miss>0)
    {
    Y.DA[which.miss==0] <- rnorm(n=n.miss, mean=fitted[which.miss==0], sd=sqrt(nu2))    
    }else
    {}
  Y.DA.mat <- matrix(Y.DA, nrow=K, ncol=N, byrow=FALSE)
    
    
    
    ##################
    ## Sample from nu2
    ##################
    nu2.offset <- as.numeric(Y.DA.mat - offset.mat - regression.mat - phi.mat)
    nu2.scale <- prior.nu2[2]  + sum(nu2.offset^2)/2
    nu2 <- 1 / rgamma(1, nu2.shape, scale=(1/nu2.scale)) 
    
    
    
    ####################
    ## Sample from beta
    ####################
    fc.precision <- prior.precision.beta + data.precision.beta / nu2
    fc.var <- solve(fc.precision)
    beta.offset <- as.numeric(Y.DA.mat - offset.mat - phi.mat)
    beta.offset2 <- t(X.standardised) %*% beta.offset / nu2 + prior.precision.beta %*% prior.mean.beta
    fc.mean <- fc.var %*% beta.offset2
    chol.var <- t(chol(fc.var))
    beta <- fc.mean + chol.var %*% rnorm(p)        
    regression.mat <- matrix(X.standardised %*% beta, nrow=K, ncol=N, byrow=FALSE)  
    
    
    
    ####################
    ## Sample from phi
    ####################
    phi.offset <- Y.DA.mat - offset.mat - regression.mat
    den.offset <- rho * W.triplet.sum + 1 - rho
    phi.temp <- gaussianar2carupdate(W.triplet, W.begfin, W.triplet.sum,  K, N, phi.mat, tau2, nu2, alpha[1], alpha[2], rho, phi.offset, den.offset)      
    phi <- as.numeric(phi.temp)  - mean(as.numeric(phi.temp))
    phi.mat <- matrix(phi, nrow=K, ncol=N, byrow=FALSE)
    
    
    
    
    ####################
    ## Sample from alpha
    ####################
      if(!fix.rho.T)
      {
      #### Construct the quadratic forms
      temp2 <- alphaquadformcompute(W.triplet, W.triplet.sum, W.n.triplet,  K, N, phi.mat, rho, tau2)
      
      #### Construct the precision matrix
      alpha.prec <- array(c(temp2[[1]], temp2[[3]], temp2[[3]], temp2[[2]]), c(2,2))
      alpha.var <- solve(alpha.prec)
      
      #### Construct the mean vector
      U2 <- (temp2[[1]] * temp2[[5]] - temp2[[3]] * temp2[[4]]) / (temp2[[2]] * temp2[[1]] - temp2[[3]]^2) 
      U1 <- (1 / temp2[[3]]) * (temp2[[5]] - temp2[[2]] * U2)
      alpha.mean <- c(U1, U2)
      alpha <- mvrnorm(n=1, mu=alpha.mean, Sigma=alpha.var)
      }else
      {}

    
    
    ####################
    ## Samples from tau2
    ####################
    temp3 <- tauquadformcomputear2(W.triplet, W.triplet.sum, W.n.triplet,  K, N, phi.mat, rho, alpha[1], alpha[2])
    tau2.scale <- temp3 + prior.tau2[2] 
    tau2 <- 1 / rgamma(1, tau2.shape, scale=(1/tau2.scale))  

    
    
    ##################
    ## Sample from rho
    ##################
      if(!fix.rho.S)
      {
      proposal.rho <- rtruncnorm(n=1, a=0, b=1, mean=rho, sd=proposal.sd.rho)
      temp4 <- tauquadformcomputear2(W.triplet, W.triplet.sum, W.n.triplet,  K, N, phi.mat, proposal.rho, alpha[1], alpha[2])
      det.Q.W.proposal <- 0.5 * sum(log((proposal.rho * Wstar.val + (1-proposal.rho))))
      logprob.current <- N * det.Q.W - temp3 / tau2
      logprob.proposal <- N * det.Q.W.proposal - temp4 / tau2
      hastings <- log(dtruncnorm(x=rho, a=0, b=1, mean=proposal.rho, sd=proposal.sd.rho)) - log(dtruncnorm(x=proposal.rho, a=0, b=1, mean=rho, sd=proposal.sd.rho)) 
      prob <- exp(logprob.proposal - logprob.current + hastings)
      
        if(prob > runif(1))
        {
        rho <- proposal.rho
        det.Q.W <- det.Q.W.proposal
        accept[1] <- accept[1] + 1           
        }else
        {}              
      accept[2] <- accept[2] + 1       
      }else
      {}
    

    
    #########################
    ## Calculate the deviance
    #########################
    fitted <- as.numeric(offset.mat + regression.mat + phi.mat)
    loglike <- dnorm(Y, mean = fitted, sd = rep(sqrt(nu2),N.all), log=TRUE)
    
    
    
    ###################
    ## Save the results
    ###################
      if(j > burnin & (j-burnin)%%thin==0)
      {
      ele <- (j - burnin) / thin
      samples.beta[ele, ] <- beta
      samples.phi[ele, ] <- as.numeric(phi)
        if(!fix.rho.S) samples.rho[ele, ] <- rho
        if(!fix.rho.T) samples.alpha[ele, ] <- alpha
      samples.tau2[ele, ] <- tau2
      samples.nu2[ele, ] <- nu2
      samples.fitted[ele, ] <- fitted
      samples.loglike[ele, ] <- loglike
        if(n.miss>0) samples.Y[ele, ] <- Y.DA[which.miss==0]
      }else
      {}
    
    
    
    ########################################
    ## Self tune the acceptance probabilties
    ########################################
      if(ceiling(j/100)==floor(j/100) & j < burnin)
      {
        if(!fix.rho.S) proposal.sd.rho <- common.accceptrates2(accept[1:2], proposal.sd.rho, 40, 50, 0.5)
      accept <- rep(0,2)
      }else
      {}
    
    
    
    ################################       
    ## print progress to the console
    ################################
      if(j %in% percentage.points & verbose)
      {
      setTxtProgressBar(progressBar, j/n.sample)
      }
}
  
  
#### end timer
  if(verbose)
  {
  cat("\nSummarising results.")
  close(progressBar)
  }else
  {}
  
  
  
###################################
#### Summarise and save the results 
###################################
#### Compute the acceptance rates
  if(!fix.rho.S)
  {
  accept.rho <- 100 * accept[1] / accept[2]
  }else
  {
  accept.rho <- NA    
  }
accept.phi <- 100
accept.beta <- 100
accept.final <- c(accept.beta, accept.phi, accept.rho, 100)
names(accept.final) <- c("beta", "phi", "rho.S", "rho.T")
  
  
#### Compute the fitted deviance
mean.beta <- apply(samples.beta,2,mean)
regression.mat <- matrix(X.standardised %*% mean.beta, nrow=K, ncol=N, byrow=FALSE)   
mean.phi <- matrix(apply(samples.phi, 2, mean), nrow=K, ncol=N)
fitted.mean <- as.numeric(offset.mat + mean.phi + regression.mat)
nu2.mean <- mean(samples.nu2)
deviance.fitted <- -2 * sum(dnorm(Y, mean = fitted.mean, sd = rep(sqrt(nu2.mean),N.all), log = TRUE), na.rm=TRUE)
  
  
#### Model fit criteria
modelfit <- common.modelfit(samples.loglike, deviance.fitted)
  
  
#### Create the fitted values and residuals
fitted.values <- apply(samples.fitted, 2, mean)
response.residuals <- as.numeric(Y) - fitted.values
pearson.residuals <- response.residuals /sqrt(nu2.mean)
residuals <- data.frame(response=response.residuals, pearson=pearson.residuals)
  
  
#### Transform the parameters back to the origianl covariate scale.
samples.beta.orig <- common.betatransform(samples.beta, X.indicator, X.mean, X.sd, p, FALSE)
  

#### Create a summary object
samples.beta.orig <- mcmc(samples.beta.orig)
summary.beta <- t(apply(samples.beta.orig, 2, quantile, c(0.5, 0.025, 0.975))) 
summary.beta <- cbind(summary.beta, rep(n.keep, p), rep(accept.beta,p), effectiveSize(samples.beta.orig), geweke.diag(samples.beta.orig)$z)
rownames(summary.beta) <- colnames(X)
colnames(summary.beta) <- c("Median", "2.5%", "97.5%", "n.sample", "% accept", "n.effective", "Geweke.diag")

summary.hyper <- array(NA, c(5, 7))    
rownames(summary.hyper) <- c("tau2", "nu2", "rho.S", "rho1.T", "rho2.T")     
summary.hyper[1,1:3] <- quantile(samples.tau2, c(0.5, 0.025, 0.975))
summary.hyper[2,1:3] <- quantile(samples.nu2, c(0.5, 0.025, 0.975))
summary.hyper[1, 4:7] <- c(n.keep, 100, effectiveSize(mcmc(samples.tau2)), geweke.diag(mcmc(samples.tau2))$z)    
summary.hyper[2, 4:7] <- c(n.keep, 100, effectiveSize(mcmc(samples.nu2)), geweke.diag(mcmc(samples.nu2))$z) 
  if(!fix.rho.S)
  {
  summary.hyper[3, 1:3] <- quantile(samples.rho, c(0.5, 0.025, 0.975))
  summary.hyper[3, 4:7] <- c(n.keep, accept.rho, effectiveSize(samples.rho), geweke.diag(samples.rho)$z)
  }else
  {
  summary.hyper[3, 1:3] <- c(rho, rho, rho)
  summary.hyper[3, 4:7] <- rep(NA, 4)
  }
if(!fix.rho.T)
{
  summary.hyper[4, 1:3] <- quantile(samples.alpha[ ,1], c(0.5, 0.025, 0.975))
  summary.hyper[4, 4:7] <- c(n.keep, 100, effectiveSize(mcmc(samples.alpha[ ,1])), geweke.diag(mcmc(samples.alpha[ ,1]))$z)
  summary.hyper[5, 1:3] <- quantile(samples.alpha[ ,2], c(0.5, 0.025, 0.975))
  summary.hyper[5, 4:7] <- c(n.keep, 100, effectiveSize(mcmc(samples.alpha[ ,2])), geweke.diag(mcmc(samples.alpha[ ,2]))$z)  
}else
{
  summary.hyper[4, 1:3] <- c(alpha[1], alpha[1], alpha[1])
  summary.hyper[4, 4:7] <- rep(NA, 4)
  summary.hyper[5, 1:3] <- c(alpha[2], alpha[2], alpha[2])
  summary.hyper[5, 4:7] <- rep(NA, 4)
}   

summary.results <- rbind(summary.beta, summary.hyper)
summary.results[ , 1:3] <- round(summary.results[ , 1:3], 4)
summary.results[ , 4:7] <- round(summary.results[ , 4:7], 1)

  
#### Compile and return the results
#### Harmonise samples in case of them not being generated
  if(fix.rho.S & fix.rho.T)
  {
  samples.rhoext <- NA
  }else if(fix.rho.S & !fix.rho.T)
  {
  samples.rhoext <- samples.alpha
  names(samples.rhoext) <- c("rho1.T", "rho2.T")
  }else if(!fix.rho.S & fix.rho.T)
  {
  samples.rhoext <- samples.rho  
  names(samples.rhoext) <- "rho.S"
  }else
  {
  samples.rhoext <- cbind(samples.rho, samples.alpha)
  colnames(samples.rhoext) <- c("rho.S", "rho1.T", "rho2.T")
  }
  if(n.miss==0) samples.Y = NA
  
samples <- list(beta=mcmc(samples.beta.orig), phi=mcmc(samples.phi),  rho=mcmc(samples.rhoext), tau2=mcmc(samples.tau2), nu2=mcmc(samples.nu2), fitted=mcmc(samples.fitted), Y=mcmc(samples.Y))
model.string <- c("Likelihood model - Gaussian (identity link function)", "\nLatent structure model - Autoregressive order 2  CAR model\n")
results <- list(summary.results=summary.results, samples=samples, fitted.values=fitted.values, residuals=residuals, modelfit=modelfit, accept=accept.final, localised.structure=NULL, formula=formula, model=model.string,  X=X)
class(results) <- "CARBayesST"
  
#### Finish by stating the time taken 
  if(verbose)
  {
  b<-proc.time()
  cat("Finished in ", round(b[3]-a[3], 1), "seconds.\n")
  }else
  {}
return(results)
}