bglm (version 1.0)

expmh: Function for Bayesian Estimation in Exponential Distribution Models

Description

In this function the transition kernel proposed by Gamerman (1997) is implemented for a Metropolis Hastings algorithm, in order to sample the posterior distribution of the regression parameters given de data in a exponential distribution model. A normal Prior is assumed for the regression parameters. For now only the log link is implemented.

Usage

"expmh"(y, X, b = rep(0, dim(X)[2]), 
B = diag(rep(100, dim(X)[2])), N = 3000, flag = F,...)
"expmh"(formula, data = list(), ...)

Arguments

formula

an object of class formula: a symbolic description of the model to be fitted.

data

A data frame containing the variables in the model.

...

When using expmh.formula, the formula object encapsulates the arguments y and X of expmh.default, thus ... represents all other arguments needed in expmh.default to be passed to expmh.formula

y

Response variable, vector of positive numbers.

X

Design matrix

b

Mean of the normal prior distribution of the regression parameters.

B

Covariance matrix of the normal prior distribution of the regression parameters.

N

Number mcmc simulations of the posterior distributions of the regression parameters given de data.

flag

Logical, if TRUE iterations and acceptance ratio of the samples is printed to monitor the mcmc progress.

Value

A list with the following objects:

chain

A matrix where mcmc simulations of the posterior distributions of the regression parameters given the data is stored. Rows correspond to mcmc simulation and columns correspond to the regression parameters.

Deviance

a vector with -2*l(y,chain[i,]), where l(.,.) is the log-likelihood of the model.

Accepted_samples

An integer with the number of samples accepted by the M-H algorithm.

Details

See Gamerman (1997) for the details.

References

Gamerman, D. 1997. Sampling from the posterior distribution in generalized linear mixed models. Statistics and Computing, 7, 57-68.

Examples

################
### Simulated data
################
library(coda)
library(car)
n<-500
(beta<-runif(2,-10,10))
###limits of mu
ld<-c(10,70)
###generate X according to ld
xlm<-c((log(ld[1])-beta[1])/beta[2],(log(ld[2])-beta[1])/beta[2])
####design matrix
X<-as.matrix(data.frame(x0=rep(1,n),x1=runif(n,sort(xlm)[1],sort(xlm)[2])))
mu<-sapply(1:dim(X)[1],function(i){exp(X[i,]%*%beta)})
######generate the data according to the model
y<-rexp(n,1/mu)
###fit the model
bres<-expmh(y~X[,2],N=3000)
###compare with true beta
round(data.frame(true.beta=beta,
                 mh.mean=apply(bres$chain,2,mean),
                 mh.sd=apply(bres$chain,2,sd)),5)
#####examine MCMC chains.                 
dev.new(width=9,height=6)
par(mfrow=c(2,3))
plot(as.ts(bres$chain[,1]),cex.main=0.9,main=expression(beta[0]),
     ylab="",xlab="iterations" )
plot(density(bres$chain[,1]),cex.main=0.9, col="red", lwd=2, 
     main=expression(beta[1]) )
autocorr.plot(mcmc(bres$chain[,1]),cex.main=0.9, col="red", lwd=2, 
              main=expression(beta[0]),auto.layout=FALSE ,lag.max=100)
plot(as.ts(bres$chain[,2]),cex.main=0.9,main=expression(beta[1]),
     ylab="",xlab="iterations" )
plot(density(bres$chain[,2]),cex.main=0.9, col="red", lwd=2, 
     main=expression(beta[0]) )
autocorr.plot(mcmc(bres$chain[,2]),cex.main=0.9, col="red", lwd=2,
              main=expression(beta[1]),auto.layout=FALSE ,lag.max=100)