## Simple Gibbs Sampler Example
## Adapted from Darren Wilkinson's post at
## http://darrenjw.wordpress.com/2010/04/28/mcmc-programming-in-r-python-java-and-c/
##
## Sanjog Misra and Dirk Eddelbuettel, June-July 2011
## Actual joint density -- the code which follow implements
## a Gibbs sampler to draw from the following joint density f(x,y)
x*x * exp(-x*
y*
y -
y*
y + 2*
y - 4*x)
}
## Note that the full conditionals are propotional to
## f(x|y) = (x^2)*exp(-x*(4+y*y)) : a Gamma density kernel
## f(y|x) = exp(-0.5*2*(x+1)*(y^2 - 2*y/(x+1)) : Normal Kernel
## There is a small typo in Darrens code.
## The full conditional for the normal has the wrong variance
## It should be 1/sqrt(2*(x+1)) not 1/sqrt(1+x)
## This we can verify ...
## The actual conditional (say for x=3) can be computed as follows
## First - Construct the Unnormalized Conditional
fy.unnorm <-
function(
y) fun(3,
y)
## Then - Find the appropriate Normalizing Constant
K <- integrate(fy.unnorm,-Inf,Inf)
## Finally - Construct Actual Conditional
fy <-
function(
y) fy.unnorm(
y)/K$val
## Now - The corresponding Normal should be
fy.dnorm <-
function(
y) {
x <- 3
dnorm(
y,1/(1+x),sqrt(1/(2*(1+x))))
}
## and not ...
fy.dnorm.wrong <-
function(
y) {
x <- 3
dnorm(
y,1/(1+x),sqrt(1/((1+x))))
}
if (interactive()) {
## Graphical check
## Actual (gray thick line)
curve(fy,-2,2,
col=
'grey',lwd=5)
## Correct Normal conditional (blue dotted line)
curve(fy.dnorm,-2,2,
col=
'blue',add=T,lty=3)
## Wrong Normal (Red line)
curve(fy.dnorm.wrong,-2,2,
col=
'red',add=T)
}
## Here is the actual Gibbs Sampler
## This is Darren Wilkinsons R code (with the corrected variance)
## But we are returning only his columns 2 and 3 as the 1:N sequence
## is never used below
Rgibbs <-
function(
N,thin) {
mat <- matrix(0,ncol=2,nrow=N)
x <- 0
y <-
rnorm(1,1/(x+1),1/sqrt(2*(x+1)))
}
}
}
## We can also try the R compiler on this R function
RCgibbs <- cmpfun(Rgibbs)
## For example
## mat <- Rgibbs(10000,10); dim(mat)
## would give: [1] 10000 2
## Now for the Rcpp version -- Notice how easy it is to code up!
gibbscode <- '
using namespace Rcpp; // inline does that for us already
// n and thin are SEXPs which the Rcpp::as function maps to C++ vars
int N = as<int>(n);
int thn = as<int>(thin);
int i,j;
NumericMatrix mat(N, 2);
RNGScope scope; // Initialize Random number generator
// The rest of the code follows the R version
double x=0, y=0;
for (i=0; i<N; i++) {
for (j=0; j<thn; j++) {
x = ::Rf_rgamma(3.0,1.0/(y*y+4));
y = ::Rf_rnorm(1.0/(x+1),1.0/sqrt(2*x+2));
}
mat(i,0) = x;
mat(i,1) = y;
}
return mat; // Return to R
'
# Compile and Load
RcppGibbs <- cxxfunction(signature(
n=
"int", thin =
"int"),
gibbscode, plugin="Rcpp")
gslgibbsincl <- '
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
using namespace Rcpp;
'
gslgibbscode <- '
int N = as<int>(ns);
int thin = as<int>(thns);
gsl_rng *r = gsl_rng_alloc(gsl_rng_mt19937);
x = gsl_ran_gamma(r,3.0,1.0/(
y*
y+4));
y = 1.0/(x+1)+gsl_ran_gaussian(r,1.0/sqrt(2*x+2));
}
}
gsl_rng_free(r);
'
## Compile and Load
GSLGibbs <- cxxfunction(signature(ns="int", thns = "int"),
body=gslgibbscode, includes=gslgibbsincl,
plugin="RcppGSL")
## without RcppGSL, using cfunction()
#GSLGibbs <- cfunction(signature(ns="int", thns = "int"),
# body=gslgibbscode, includes=gslgibbsincl,
# Rcpp=TRUE,
# cppargs="-I/usr/include",
# libargs="-lgsl -lgslcblas")
## Now for some tests
## You can try other values if you like
## Note that the total number of interations are N*thin!
Ns <- c(1000,5000,10000,20000)
thins <- c(10,50,100,200)
tim_R <- rep(0,4)
tim_RC <- rep(0,4)
tim_Rgsl <- rep(0,4)
tim_Rcpp <- rep(0,4)
for (i in seq_along(Ns)) {
tim_R[i] <- system.time(mat <- Rgibbs(Ns[i],thins[i]))[3]
tim_RC[i] <- system.time(cmat <- RCgibbs(Ns[i],thins[i]))[3]
tim_Rgsl[i] <- system.time(gslmat <- GSLGibbs(Ns[i],thins[i]))[3]
tim_Rcpp[i] <- system.time(rcppmat <- RcppGibbs(Ns[i],thins[i]))[3]
cat("Replication #", i, "complete \n")
}
## Comparison
speedup <- round(tim_R/tim_Rcpp,2);
speedup2 <- round(tim_R/tim_Rgsl,2);
speedup3 <- round(tim_R/tim_RC,2);
summtab <- round(rbind(tim_R,tim_RC, tim_Rcpp,tim_Rgsl,speedup3,speedup,speedup2),3)
colnames(summtab) <- c("N=1000","N=5000","N=10000","N=20000")
rownames(summtab) <- c("Elasped Time (R)","Elasped Time (RC)","Elapsed Time (Rcpp)", "Elapsed Time (Rgsl)",
"SpeedUp Rcomp.","SpeedUp Rcpp", "SpeedUp GSL")
print(summtab)
## Contour Plots -- based on Darren's example
if (interactive() &&
require(KernSmooth)) {
op <- par(mfrow=c(4,1),mar=c(3,3,3,1))
contour(x,y,z,
main="Contours of actual distribution",xlim=c(0,2), ylim=c(-2,4))
fit <- bkde2D(
as.matrix(
mat),c(0.1,0.1))
contour(drawlabels=T, fit$x1, fit$x2, fit$fhat, xlim=c(0,2), ylim=c(-2,4),
main=paste("Contours of empirical distribution:",round(tim_R[4],2)," seconds"))
fitc <- bkde2D(
as.matrix(rcppmat),c(0.1,0.1))
contour(fitc$x1,fitc$x2,fitc$fhat,xlim=c(0,2), ylim=c(-2,4),
main=paste("Contours of Rcpp based empirical distribution:",round(tim_Rcpp[4],2)," seconds"))
fitg <- bkde2D(
as.matrix(gslmat),c(0.1,0.1))
contour(fitg$x1,fitg$x2,fitg$fhat,xlim=c(0,2), ylim=c(-2,4),
main=paste("Contours of GSL based empirical distribution:",round(tim_Rgsl[4],2)," seconds"))
par(op)
}
## also use rbenchmark package
N <- 20000
res <- benchmark(Rgibbs(N,
thn),
columns=c(
"test",
"replications",
"elapsed",
"relative", "user.self", "sys.self"),
replications=10)
print(res)
## And we are done
1.8.1.2 
