gaussian.MVlerouxCAR <- function(formula, data=NULL,  W, burnin, n.sample, thin=1, prior.mean.beta=NULL, prior.var.beta=NULL, prior.nu2=NULL, prior.Sigma.df=NULL, prior.Sigma.scale=NULL, rho=NULL, MALA=TRUE, verbose=TRUE)
{
##############################################
#### Format the arguments and check for errors
##############################################
#### Verbose
a <- common.verbose(verbose)
    
    
#### Frame object
frame.results <- common.frame(formula, data, "gaussian")
K <- 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  
J <- ncol(Y)
N.all <- K * J
Y.DA <- Y


#### Create a missing list
    if(n.miss>0)
    {
    miss.locator <- array(NA, c(n.miss, 2))
    colnames(miss.locator) <- c("row", "column")
    locations <- which(t(which.miss)==0)
    miss.locator[ ,1] <- ceiling(locations/J)
    miss.locator[ ,2] <- locations - (miss.locator[ ,1]-1) * J
    }else
    {}


#### Check on MALA argument
    if(length(MALA)!=1) stop("MALA is not length 1.", call.=FALSE)
    if(!is.logical(MALA)) stop("MALA is not logical.", call.=FALSE)  


#### W matrix
    if(!is.matrix(W)) stop("W is not a matrix.", call.=FALSE)
    if(nrow(W)!= K) stop("The number of data points divided by the number of rows in W is not a whole number.", call.=FALSE)
    
    
    
#### rho
    if(is.null(rho))
    {
    rho <- runif(1)
    fix.rho <- FALSE   
    }else
    {
    fix.rho <- TRUE    
    }
    if(!is.numeric(rho) ) stop("rho is fixed but is not numeric.", call.=FALSE)  
    if(rho<0 ) stop("rho is outside the range [0, 1].", call.=FALSE)  
    if(rho>1 ) stop("rho is outside the range [0, 1].", call.=FALSE)      
   

 
#### 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.Sigma.df)) prior.Sigma.df <- J+1
    if(is.null(prior.Sigma.scale)) prior.Sigma.scale <- diag(rep(1,J)) / 1000
    if(is.null(prior.nu2)) prior.nu2 <- c(1, 0.01)
common.prior.beta.check(prior.mean.beta, prior.var.beta, p)
common.prior.varmat.check(prior.Sigma.scale, J)  
common.prior.var.check(prior.nu2)     
 
   
#### MCMC quantities - burnin, n.sample, thin
common.burnin.nsample.thin.check(burnin, n.sample, thin)  
    
    
    
#############################
#### Initial parameter values
#############################
beta <- array(NA, c(p, J))
nu2 <- rep(NA, J)
    for(i in 1:J)
    {
    mod.glm <- lm(Y[ ,i]~X.standardised-1, offset=offset[ ,i])
    beta.mean <- mod.glm$coefficients
    beta.sd <- sqrt(diag(summary(mod.glm)$cov.unscaled)) * summary(mod.glm)$sigma
    beta[ ,i] <- rnorm(n=p, mean=beta.mean, sd=beta.sd)
    nu2[i] <- runif(1, var(mod.glm$residuals)*0.5, var(mod.glm$residuals)*2)
    }
    
res.temp <- Y - X.standardised %*% beta - offset
res.sd <- sd(res.temp, na.rm=TRUE)/5
phi.vec <- rnorm(n=N.all, mean=0, sd=res.sd)
phi <- matrix(phi.vec, nrow=K, byrow=TRUE)
Sigma <- cov(phi)
Sigma.inv <- solve(Sigma)
regression <- X.standardised %*% beta
fitted <- regression + phi + offset

    
    
###############################    
#### Set up the MCMC quantities    
###############################
#### Matrices to store samples    
n.keep <- floor((n.sample - burnin)/thin)
samples.beta <- array(NA, c(n.keep, J*p))
samples.nu2 <- array(NA, c(n.keep, J))
samples.phi <- array(NA, c(n.keep, N.all))
samples.Sigma <- array(NA, c(n.keep, J, J))
    if(!fix.rho) samples.rho <- array(NA, c(n.keep, 1))
samples.loglike <- array(NA, c(n.keep, N.all))
samples.fitted <- array(NA, c(n.keep, N.all))
    if(n.miss>0) samples.Y <- array(NA, c(n.keep, n.miss))
    
    
#### Metropolis quantities
accept <- rep(0,4)
proposal.sd.phi <- 0.1
proposal.sd.rho <- 0.02
Sigma.post.df <- prior.Sigma.df + K  
nu2.posterior.shape <- prior.nu2[1] + 0.5 * K


##################################
#### Set up the spatial quantities
##################################
#### CAR quantities
W.quants <- common.Wcheckformat(W)
W <- W.quants$W
W.triplet <- W.quants$W.triplet
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
Wstar <- diag(apply(W,1,sum)) - W
Q <- rho * Wstar + diag(rep(1-rho,K))
    
    
#### Create the determinant     
    if(!fix.rho)
    {
    Wstar.eigen <- eigen(Wstar)
    Wstar.val <- Wstar.eigen$values
    det.Q <- sum(log((rho * Wstar.val + (1-rho))))    
    }else
    {} 
    
    
#### Check for islands
W.list<- mat2listw(W)
W.nb <- W.list$neighbours
W.islands <- n.comp.nb(W.nb)
islands <- W.islands$comp.id
islands.all <- rep(islands,J)
n.islands <- max(W.islands$nc)
    if(rho==1) Sigma.post.df <- prior.Sigma.df + K - n.islands   
    
    
#### Specify vector variants
Y.vec <- as.numeric(t(Y))


#### Beta update quantities
data.precision <- 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))
    }



###########################
#### 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)
        {
        nu.mat <- matrix(rep(sqrt(nu2), K), nrow=K, byrow=T)
        Y.DA[miss.locator] <- rnorm(n=n.miss, mean=fitted[miss.locator], sd=nu.mat[miss.locator])    
        }else
        {}
        
        
        
    ###################
    ## Sample from beta
    ###################
        for(r in 1:J)
        {
        fc.precision <- prior.precision.beta + data.precision / nu2[r]
        fc.var <- solve(fc.precision)    
        fc.temp1 <- t(((Y.DA[, r] - phi[ , r] - offset[ , r]) %*%  X.standardised) / nu2[r]) + prior.precision.beta %*% prior.mean.beta  
        fc.mean <- fc.var %*% fc.temp1
        chol.var <- t(chol(fc.var))
        beta[ ,r] <- fc.mean + chol.var %*% rnorm(p)   
        }
    regression <- X.standardised %*% beta    

        
    
        ##################
        ## Sample from nu2
        ##################
        fitted.current <- regression + phi + offset
        nu2.posterior.scale <- prior.nu2[2] + 0.5 * apply((Y.DA - fitted.current)^2, 2, sum)
        nu2 <- 1 / rgamma(J, nu2.posterior.shape, scale=(1/nu2.posterior.scale))

        
        
        ##################
        ## Sample from phi
        ##################
        den.offset <- rho * W.triplet.sum + 1 - rho
        phi.offset <- Y.DA - regression - offset
            if(MALA)
            {
            temp1 <- gaussianmcarupdateRW(W.triplet, W.begfin, K, J, phi, phi.offset, den.offset, Sigma.inv, rho, nu2, proposal.sd.phi)      
            }else
            {
            temp1 <- gaussianmcarupdateMALA(W.triplet, W.begfin, K, J, phi, phi.offset, den.offset, Sigma.inv, rho, nu2, proposal.sd.phi)      
            }
        phi <- temp1[[1]]
            for(r in 1:J)
            {
            phi[ ,r] <- phi[ ,r] - mean(phi[ ,r])    
            }
        accept[1] <- accept[1] + temp1[[2]]
        accept[2] <- accept[2] + K    
        
        
        
        ####################
        ## Sample from Sigma
        ####################
        Sigma.post.scale <- t(phi) %*% Q %*% phi + prior.Sigma.scale
        Sigma <- riwish(Sigma.post.df, Sigma.post.scale)
        Sigma.inv <- solve(Sigma)
        
        
        
        ##################
        ## Sample from rho
        ##################
        if(!fix.rho)
        {
            ## Propose a new value
            proposal.rho <- rtruncnorm(n=1, a=0, b=1, mean=rho, sd=proposal.sd.rho)
            Q.prop <- proposal.rho * Wstar + diag(rep(1-proposal.rho), K)
            det.Q.prop <-  sum(log((proposal.rho * Wstar.val + (1-proposal.rho))))    
             
            ## Compute the acceptance rate
            logprob.current <- 0.5 * J * det.Q - 0.5 * sum(diag(t(phi) %*% Q %*% phi %*% Sigma.inv))
            logprob.proposal <- 0.5 * J * det.Q.prop - 0.5 * sum(diag(t(phi) %*% Q.prop %*% phi %*% Sigma.inv))
            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 <- det.Q.prop
                Q <- Q.prop
                accept[3] <- accept[3] + 1           
            }else
            {}              
            accept[4] <- accept[4] + 1       
        }else
        {}
        
        
        
        #########################
        ## Calculate the deviance
        #########################
        fitted <- regression + phi + offset
        loglike <- dnorm(x=as.numeric(t(Y)), mean=as.numeric(t(fitted)), sd=rep(sqrt(nu2), K), log=TRUE)

        
        ###################
        ## Save the results
        ###################
        if(j > burnin & (j-burnin)%%thin==0)
        {
            ele <- (j - burnin) / thin
            samples.beta[ele, ] <- as.numeric(beta)
            samples.nu2[ele, ] <- nu2
            samples.phi[ele, ] <- as.numeric(t(phi))
            samples.Sigma[ele, , ] <- Sigma
            if(!fix.rho) samples.rho[ele, ] <- rho
            samples.loglike[ele, ] <- loglike
            samples.fitted[ele, ] <- as.numeric(t(fitted))
            if(n.miss>0) samples.Y[ele, ] <- Y.DA[miss.locator]
        }else
        {}
        
        
        
        ########################################
        ## Self tune the acceptance probabilties
        ########################################
        if(ceiling(j/100)==floor(j/100) & j < burnin)
        {
        proposal.sd.phi <- common.accceptrates1(accept[1:2], proposal.sd.phi, 40, 50)
            if(!fix.rho)
            {
                proposal.sd.rho <- common.accceptrates2(accept[3:4], proposal.sd.rho, 40, 50, 0.5)
            }
            accept <- c(0,0,0,0)
         }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
accept.phi <- 100 * accept[1] / accept[2]
    if(!fix.rho)
    {
    accept.rho <- 100 * accept[3] / accept[4]
    }else
    {
    accept.rho <- NA    
    }
accept.final <- c(100, accept.phi, rep(100, 2), accept.rho)
names(accept.final) <- c("beta", "phi", "nu2", "Sigma", "rho")


    
#### Compute the fitted deviance
mean.beta <- matrix(apply(samples.beta, 2, mean), nrow=p, ncol=J, byrow=F)
mean.phi <- matrix(apply(samples.phi, 2, mean), nrow=K, ncol=J, byrow=T)
fitted.mean <- X.standardised %*% mean.beta + mean.phi + offset
nu2.mean <- apply(samples.nu2,2,mean)
deviance.fitted <- -2 * sum(dnorm(as.numeric(t(Y)), mean = as.numeric(t(fitted.mean)), sd = sqrt(nu2.mean[rep(1:J,K)]), log = TRUE), na.rm=TRUE)

    
#### Model fit criteria
modelfit <- common.modelfit(samples.loglike, deviance.fitted)


#### transform the parameters back to the origianl covariate scale.
samples.beta.orig <- samples.beta
    for(r in 1:J)
    {
    samples.beta.orig[ ,((r-1)*p+1):(r*p)] <- common.betatransform(samples.beta[ ,((r-1)*p+1):(r*p)], 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(100, (J*p)), effectiveSize(samples.beta.orig), geweke.diag(samples.beta.orig)$z)
col.name <- rep(NA, p*(J-1))

if(is.null(colnames(Y)))
{
    for(r in 1:J)
    {
        col.name[((r-1)*p+1):(r*p)] <- paste("Variable ", r,  " - ", colnames(X), sep="")   
    }
}else
{
    for(r in 1:J)
    {
        col.name[((r-1)*p+1):(r*p)] <- paste(colnames(Y)[r],  " - ", colnames(X), sep="")   
    }
}
rownames(summary.beta) <- col.name
colnames(summary.beta) <- c("Median", "2.5%", "97.5%", "n.sample", "% accept", "n.effective", "Geweke.diag")
    
    summary.hyper <- array(NA, c((2*J+1) ,7))
    ## nu2
    summary.hyper[1:J, 1:3] <-t(apply(samples.nu2, 2, quantile, c(0.5, 0.025, 0.975))) 
    summary.hyper[1:J, 4] <- rep(n.keep, J)
    summary.hyper[1:J, 5] <- rep(100, J)
    summary.hyper[1:J, 6] <- apply(samples.nu2, 2, effectiveSize)
    summary.hyper[1:J, 7] <- geweke.diag(samples.nu2)$z
    
    ## Sigma
    summary.hyper[(J+1):(2*J), 1] <- diag(apply(samples.Sigma, c(2,3), quantile, c(0.5)))
    summary.hyper[(J+1):(2*J), 2] <- diag(apply(samples.Sigma, c(2,3), quantile, c(0.025)))
    summary.hyper[(J+1):(2*J), 3] <- diag(apply(samples.Sigma, c(2,3), quantile, c(0.975)))
    summary.hyper[(J+1):(2*J), 4] <- rep(n.keep, J)
    summary.hyper[(J+1):(2*J), 5] <- rep(100, J)
    summary.hyper[(J+1):(2*J), 6] <- diag(apply(samples.Sigma, c(2,3), effectiveSize))
    for(r in 1:J)
    {
        summary.hyper[(r+J), 7] <- geweke.diag(samples.Sigma[ ,r,r])$z    
    }
    
    if(!fix.rho)
    {
        summary.hyper[(2*J+1), 1:3] <- quantile(samples.rho, c(0.5, 0.025, 0.975))
        summary.hyper[(2*J+1), 4:7] <- c(n.keep, accept.rho, effectiveSize(samples.rho), geweke.diag(samples.rho)$z)
    }else
    {
        summary.hyper[(2*J+1), 1:3] <- c(rho, rho, rho)
        summary.hyper[(2*J+1), 4:7] <- rep(NA, 4)
    }
    
    summary.results <- rbind(summary.beta, summary.hyper)
    rownames(summary.results)[((J*p)+1): nrow(summary.results)] <- c(paste(rep("nu2.",J), 1:J, sep=""), paste(rep("Sigma",J), 1:J, 1:J, sep=""), "rho")
    summary.results[ , 1:3] <- round(summary.results[ , 1:3], 4)
    summary.results[ , 4:7] <- round(summary.results[ , 4:7], 1)
    
    
#### Create the fitted values and residuals
fitted.values <- matrix(apply(samples.fitted, 2, mean), nrow=K, ncol=J, byrow=T)
response.residuals <- Y - fitted.values
nu.mat <- matrix(rep(sqrt(nu2.mean), K), nrow=K, byrow=T)
pearson.residuals <- response.residuals / nu.mat
residuals <- list(response=response.residuals, pearson=pearson.residuals)



#### Compile and return the results
model.string <- c("Likelihood model - Gaussian (identity link function)", "\nRandom effects model - Leroux MCAR\n")
    if(fix.rho) samples.rho=NA
    if(n.miss==0) samples.Y = NA
samples <- list(beta=samples.beta.orig, phi=mcmc(samples.phi), Sigma=samples.Sigma, nu2=mcmc(samples.nu2), rho=mcmc(samples.rho), fitted=mcmc(samples.fitted), Y=mcmc(samples.Y))
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) <- "CARBayes"
    
    
#### 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)
}