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## =============================================================================
## A banana-shaped function.
##
## example from Haario et al. 2005
##
## Checked 24-08-09
## =============================================================================
## banana function
Banana <- function (x, a = 1, b = 1) {
y1<-x[1]*a
y2<-x[2]/a -b*(y1^2+a^2)
return(c(y1,y2))
}
## banana matix function
BananaM <- function (X, a=1, b=1) {
y1<-X[,1]*a
y2<-X[,2]/a -b*(y1^2+a^2)
return(cbind(y1,y2))
}
##-----------------------------------------------------------------
## Probability of a multinormally distributed value
##-----------------------------------------------------------------
pmultinorm <- function(value,mean,Cov) {
diff <- value - mean
ex <- -0.5*t(diff) %*% solve(Cov) %*% diff
rdet <- sqrt(det(Cov))
power <- -length(diff)*0.5
return((2.*pi)^power / rdet * exp(ex))
}
## 2-*log sum of squares
BananaSS <- function (p) {
P <- Banana(p)
Cov <- matrix(nr=2,data=c(1,0.9,0.9,1))
-2*sum(log(pmultinorm(P,mean=0,Cov=Cov)))
}
## The Markov chain - simple Metropolis
MCMC <- modMCMC(f=BananaSS, p=c(0,0.5), jump=diag(nrow=2,x=5),niter=1000)
par(mfrow=c(4,2))
plot(MCMC,mfrow=NULL,main="MH")
## The adaptive Metropolis - update proposal every 100 runs
MCMC2 <- modMCMC(f=BananaSS, p=c(0,0.5), jump=diag(nrow=2,x=5),
updatecov=100,niter=1000)
plot(MCMC2,mfrow=NULL,main="AM")
## The Metropolis with delayed rejection
MCMC3 <- modMCMC(f=BananaSS, p=c(0,0.5), jump=diag(nrow=2,x=5),
ntrydr=2,niter=1000)
plot(MCMC3,mfrow=NULL,main="DR")
## The adaptive Metropolis with delayed rejection
MCMC4 <- modMCMC(f=BananaSS, p=c(0,0.5), jump=diag(nrow=2,x=5),
updatecov=100,ntrydr=2,niter=1000)
plot(MCMC4,mfrow=NULL,main="DRAM")
par(mfrow=c(2,2))
plot(MCMC$pars,main="MH")
plot(MCMC2$pars,main="AM")
plot(MCMC3$pars,main="DR")
plot(MCMC4$pars,main="DRAM")
## summaries:
colMeans(BananaM(MCMC4$pars)) # was:c(0,0)
sd(BananaM(MCMC4$pars)) # was:1
cor(BananaM(MCMC4$pars)) # 0.9 off-diagonal
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