## 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

suppressMessages(library(Rcpp))
suppressMessages(library(inline))
suppressMessages(library(compiler))
suppressMessages(library(rbenchmark))


## Actual joint density -- the code which follow implements
## a Gibbs sampler to draw from the following joint density f(x,y)
fun <- function(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 <- 0
    for (i in 1:N) {
        for (j in 1:thin) {
            x <- rgamma(1,3,y*y+4)
            y <- rnorm(1,1/(x+1),1/sqrt(2*(x+1)))
        }
        mat[i,] <- c(x,y)
    }
    mat
}

## 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;  // just to be explicit
'

gslgibbscode <- '
  int N = as<int>(ns);
  int thin = as<int>(thns);
  int i, j;
  gsl_rng *r = gsl_rng_alloc(gsl_rng_mt19937);
  double x=0, y=0;
  NumericMatrix mat(N, 2);
  for (i=0; i<N; i++) {
    for (j=0; j<thin; j++) {
      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));
    }
    mat(i,0) = x;
    mat(i,1) = y;
  }
  gsl_rng_free(r);

  return mat;           // Return to 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))
    x <- seq(0,4,0.01)
    y <- seq(-2,4,0.01)
    z <- outer(x,y,fun)
    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
thn <- 200
res <- benchmark(Rgibbs(N, thn),
                 RCgibbs(N, thn),
                 RcppGibbs(N, thn),
                 GSLGibbs(N, thn),
                 columns=c("test", "replications", "elapsed",
                           "relative", "user.self", "sys.self"),
                 order="relative",
                 replications=10)
print(res)


## And we are done