gaussian.MVCARar2 <- 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.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.MVST(formula, data, "gaussian")
NK <- 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$Y
N.all <- length(Y)
J <- ncol(Y)
which.miss <- frame.results$which.miss
n.miss <- N.all - sum(which.miss)
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(which.miss==0)
miss.locator[ ,1] <- ceiling(locations/J)
miss.locator[ ,2] <- locations - (miss.locator[ ,1]-1) * J
}else
{}
#### W matrix
if(!is.matrix(W)) stop("W is not a matrix.", call.=FALSE)
K <- nrow(W)
N <- NK / K
if(ceiling(N)!= floor(N)) stop("The number of data points in Y divided by the number of rows in W is not a whole number.", call.=FALSE)
#### 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)
#### 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(1/1000,J))
if(is.null(prior.nu2)) prior.nu2 <- c(1, 0.01)
prior.beta.check(prior.mean.beta, prior.var.beta, p)
common.prior.varmat.check(prior.Sigma.scale, J)
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$residuals))
}
res.temp <- Y - X.standardised %*% beta - offset
res.sd <- sd(res.temp, na.rm=TRUE)/5
phi.vec <- rnorm(n=N.all, mean=0, sd=res.sd)
phi <- matrix(phi.vec, ncol=J, byrow=TRUE)
Sigma <- cov(phi)
Sigma.inv <- solve(Sigma)
regression <- X.standardised %*% beta
fitted <- regression + phi + offset
###############################
#### Set up the MCMC quantities
###############################
#### Matrices to store samples
n.keep <- floor((n.sample - burnin)/thin)
samples.beta <- array(NA, c(n.keep, J*p))
samples.nu2 <- array(NA, c(n.keep, J))
samples.phi <- array(NA, c(n.keep, N.all))
samples.Sigma <- array(NA, c(n.keep, J, J))
if(!fix.rho.S) samples.rho <- array(NA, c(n.keep, 1))
if(!fix.rho.T) samples.alpha <- array(NA, c(n.keep, 2))
samples.loglike <- array(NA, c(n.keep, N.all))
samples.fitted <- array(NA, c(n.keep, N.all))
if(n.miss>0) samples.Y <- array(NA, c(n.keep, n.miss))
#### Metropolis quantities
accept <- rep(0,4)
accept.all <- rep(0,4)
proposal.sd.phi <- 0.1
proposal.sd.rho <- 0.02
Sigma.post.df <- prior.Sigma.df + K * N
nu2.posterior.shape <- prior.nu2[1] + 0.5 * K * N
##################################
#### Set up the spatial quantities
##################################
#### CAR quantities
W.quants <- common.Wcheckformat.leroux(W)
W <- W.quants$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$W.begfin
Wstar <- diag(apply(W,1,sum)) - W
Q <- rho * Wstar + diag(rep(1-rho,K))
#### Create the determinant
if(!fix.rho.S)
{
Wstar.eigen <- eigen(Wstar)
Wstar.val <- Wstar.eigen$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
n.islands <- max(W.islands$nc)
if(rho==1 & alpha[1]==2 & alpha[2]==-1)
{
Sigma.post.df <- prior.Sigma.df + ((N-2) * (K-n.islands))/2
}else if(rho==1)
{
Sigma.post.df <- prior.Sigma.df + (N * (K-n.islands))/2
}else if(alpha[1]==2 & alpha[2]==-1)
{
Sigma.post.df <- prior.Sigma.df + ((N-2) * K)/2
}else
{}
#### 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)
{
Y.DA[miss.locator] <- rnorm(n=n.miss, mean=fitted[miss.locator], sd=sqrt(nu2[miss.locator[ ,2]]))
}else
{}
##################
## 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 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 phi
##################
#### Create the offset elements
den.offset <- rho * W.triplet.sum + 1 - rho
phi.offset <- Y.DA - regression - offset
#### Create the random draws to create the proposal distribution
Chol.Sigma <- t(chol(proposal.sd.phi*Sigma))
z.mat <- matrix(rnorm(n=N.all, mean=0, sd=1), nrow=J, ncol=NK)
innovations <- t(Chol.Sigma %*% z.mat)
#### Update the elements of phi
temp1 <- gaussianmvar2carupdateRW(W.triplet, W.begfin, W.triplet.sum, K, N, J, phi, alpha[1], alpha[2], rho, Sigma.inv, nu2, innovations, phi.offset, den.offset)
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] + NK
####################
## Sample from Sigma
####################
Sigma.post.scale <- prior.Sigma.scale + t(phi[1:K, ]) %*% Q %*% phi[1:K, ] + t(phi[(K+1):(2*K), ]) %*% Q %*% phi[(K+1):(2*K), ]
for(t in 3:N)
{
phit <- phi[((t-1)*K+1):(t*K), ]
phitminus1 <- phi[((t-2)*K+1):((t-1)*K), ]
phitminus2 <- phi[((t-3)*K+1):((t-2)*K), ]
temp1 <- phit - alpha[1] * phitminus1 - alpha[2] * phitminus2
Sigma.post.scale <- Sigma.post.scale + t(temp1) %*% Q %*% temp1
}
Sigma <- riwish(Sigma.post.df, Sigma.post.scale)
Sigma.inv <- solve(Sigma)
######################
#### Sample from alpha
######################
if(!fix.rho.T)
{
temp <- MVSTrhoTAR2compute(W.triplet, W.triplet.sum, n.triplet, den.offset, K, N, J, phi, rho, Sigma.inv)
alpha.precision <- matrix(c(temp[[1]], temp[[2]], temp[[2]], temp[[3]]), nrow=2, ncol=2)
alpha.var <- solve(alpha.precision)
alpha.mean <- rep(NA, 2)
alpha.mean[2] <- (temp[[1]] * temp[[5]] - temp[[2]] * temp[[4]]) / (temp[[1]] * temp[[3]] - temp[[2]]^2)
alpha.mean[1] <- (temp[[5]] - temp[[3]] * alpha.mean[2]) / temp[[2]]
alpha <- mvrnorm(n=1, mu=alpha.mean, Sigma=alpha.var)
}else
{}
##################
## Sample from rho
##################
if(!fix.rho.S)
{
## Propose a new value
proposal.rho <- rtruncnorm(n=1, a=0, b=1, mean=rho, sd=proposal.sd.rho)
proposal.Q <- proposal.rho * Wstar + diag(rep(1-proposal.rho), K)
proposal.det.Q <- sum(log((proposal.rho * Wstar.val + (1-proposal.rho))))
proposal.den.offset <- proposal.rho * W.triplet.sum + 1 - proposal.rho
## Compute the quadratic forms based on current and proposed values of rho
temp1.QF <- MVSTrhoSAR2compute(W.triplet, W.triplet.sum, n.triplet, den.offset, K, N, J, phi, rho, alpha[1], alpha[2], Sigma.inv)
temp2.QF <- MVSTrhoSAR2compute(W.triplet, W.triplet.sum, n.triplet, proposal.den.offset, K, N, J, phi, proposal.rho, alpha[1], alpha[2], Sigma.inv)
## Compute the acceptance rate
logprob.current <- 0.5 * J * N * det.Q - 0.5 * temp1.QF
logprob.proposal <- 0.5 * J * N * proposal.det.Q - 0.5 * temp2.QF
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 <- proposal.det.Q
Q <- proposal.Q
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*N), 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.S) samples.rho[ele, ] <- rho
if(!fix.rho.T) samples.alpha[ele, ] <- alpha
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
########################################
k <- j/100
if(ceiling(k)==floor(k))
{
#### Update the proposal sds
proposal.sd.phi <- common.accceptrates1(accept[1:2], proposal.sd.phi, 40, 50)
if(!fix.rho.S)
{
proposal.sd.rho <- common.accceptrates2(accept[3:4], proposal.sd.rho, 40, 50, 0.5)
}
accept.all <- accept.all + accept
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.beta <- 100
accept.phi <- 100 * accept.all[1] / accept.all[2]
if(!fix.rho.S)
{
accept.rho <- 100 * accept.all[3] / accept.all[4]
}else
{
accept.rho <- NA
}
accept.Sigma <- 100
if(!fix.rho.T)
{
accept.alpha <- 100
}else
{
accept.alpha <- NA
}
accept.final <- c(accept.beta, accept.phi, accept.rho, accept.Sigma, accept.alpha)
names(accept.final) <- c("beta", "phi", "rho.S", "Sigma", "rho.T")
#### 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=NK, 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=rep(sqrt(nu2.mean), K*N), 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(accept.beta,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+3) ,7))
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)$z
summary.hyper[(J+1):(2*J), 1] <- diag(apply(samples.Sigma, c(2,3), quantile, c(0.5)))
summary.hyper[(J+1):(2*J), 2] <- diag(apply(samples.Sigma, c(2,3), quantile, c(0.025)))
summary.hyper[(J+1):(2*J), 3] <- diag(apply(samples.Sigma, c(2,3), quantile, c(0.975)))
summary.hyper[(J+1):(2*J), 4] <- rep(n.keep, J)
summary.hyper[(J+1):(2*J), 5] <- rep(100, J)
summary.hyper[(J+1):(2*J), 6] <- diag(apply(samples.Sigma, c(2,3), effectiveSize))
for(r in 1:J)
{
summary.hyper[J+r, 7] <- geweke.diag(samples.Sigma[ ,r,r])$z
}
if(!fix.rho.S)
{
summary.hyper[(2*J+1), 1:3] <- quantile(samples.rho, c(0.5, 0.025, 0.975))
summary.hyper[(2*J+1), 4:5] <- c(n.keep, accept.rho)
summary.hyper[(2*J+1), 6:7] <- c(effectiveSize(samples.rho), geweke.diag(samples.rho)$z)
}else
{
summary.hyper[(2*J+1), 1:3] <- c(rho, rho, rho)
summary.hyper[(2*J+1), 4:5] <- rep(NA, 2)
summary.hyper[(2*J+1), 6:7] <- rep(NA, 2)
}
if(!fix.rho.T)
{
summary.hyper[(2*J+2), 1:3] <- quantile(samples.alpha[ ,1], c(0.5, 0.025, 0.975))
summary.hyper[(2*J+2), 4:5] <- c(n.keep, accept.alpha)
summary.hyper[(2*J+2), 6:7] <- c(effectiveSize(samples.alpha[ ,1]), geweke.diag(samples.alpha[ ,1])$z)
summary.hyper[(2*J+3), 1:3] <- quantile(samples.alpha[ ,2], c(0.5, 0.025, 0.975))
summary.hyper[(2*J+3), 4:5] <- c(n.keep, accept.alpha)
summary.hyper[(2*J+3), 6:7] <- c(effectiveSize(samples.alpha[ ,2]), geweke.diag(samples.alpha[ ,2])$z)
}else
{
summary.hyper[(2*J+2), 1:3] <- c(alpha[1], alpha[1], alpha[1])
summary.hyper[(2*J+2), 4:5] <- rep(NA, 2)
summary.hyper[(2*J+2), 6:7] <- rep(NA, 2)
summary.hyper[(2*J+3), 1:3] <- c(alpha[2], alpha[2], alpha[2])
summary.hyper[(2*J+3), 4:5] <- rep(NA, 2)
summary.hyper[(2*J+3), 6:7] <- rep(NA, 2)
}
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.S", "rho1.T", "rho2.T")
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=NK, ncol=J, byrow=T)
response.residuals <- Y - fitted.values
nu.mat <- matrix(rep(sqrt(nu2.mean), N*K), nrow=N*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 - Multivariate Autoregressive order 2 CAR model\n")
#### 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
colnames(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=samples.beta.orig, phi=mcmc(samples.phi), Sigma=samples.Sigma, nu2=mcmc(samples.nu2), rho=mcmc(samples.rhoext), 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) <- "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)
}
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