190 lines
5.7 KiB
R
190 lines
5.7 KiB
R
## Simple Gibbs Sampler Example
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## Adapted from Darren Wilkinson's post at
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## http://darrenjw.wordpress.com/2010/04/28/mcmc-programming-in-r-python-java-and-c/
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##
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## Sanjog Misra and Dirk Eddelbuettel, June-July 2011
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## Updated by Dirk Eddelbuettel, Mar 2020
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suppressMessages({
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library(Rcpp)
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library(rbenchmark)
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})
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## Actual joint density -- the code which follow implements
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## a Gibbs sampler to draw from the following joint density f(x,y)
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fun <- function(x,y) {
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x*x * exp(-x*y*y - y*y + 2*y - 4*x)
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}
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## Note that the full conditionals are propotional to
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## f(x|y) = (x^2)*exp(-x*(4+y*y)) : a Gamma density kernel
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## f(y|x) = exp(-0.5*2*(x+1)*(y^2 - 2*y/(x+1)) : Normal Kernel
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## There is a small typo in Darrens code.
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## The full conditional for the normal has the wrong variance
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## It should be 1/sqrt(2*(x+1)) not 1/sqrt(1+x)
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## This we can verify ...
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## The actual conditional (say for x=3) can be computed as follows
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## First - Construct the Unnormalized Conditional
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fy.unnorm <- function(y) fun(3,y)
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## Then - Find the appropriate Normalizing Constant
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K <- integrate(fy.unnorm,-Inf,Inf)
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## Finally - Construct Actual Conditional
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fy <- function(y) fy.unnorm(y)/K$val
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## Now - The corresponding Normal should be
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fy.dnorm <- function(y) {
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x <- 3
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dnorm(y,1/(1+x),sqrt(1/(2*(1+x))))
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}
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## and not ...
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fy.dnorm.wrong <- function(y) {
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x <- 3
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dnorm(y,1/(1+x),sqrt(1/((1+x))))
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}
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if (interactive()) {
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## Graphical check
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## Actual (gray thick line)
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curve(fy,-2,2,col='grey',lwd=5)
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## Correct Normal conditional (blue dotted line)
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curve(fy.dnorm,-2,2,col='blue',add=T,lty=3)
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## Wrong Normal (Red line)
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curve(fy.dnorm.wrong,-2,2,col='red',add=T)
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}
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## Here is the actual Gibbs Sampler
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## This is Darren Wilkinsons R code (with the corrected variance)
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## But we are returning only his columns 2 and 3 as the 1:N sequence
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## is never used below
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Rgibbs <- function(N,thin) {
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mat <- matrix(0,ncol=2,nrow=N)
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x <- 0
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y <- 0
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for (i in 1:N) {
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for (j in 1:thin) {
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x <- rgamma(1,3,y*y+4)
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y <- rnorm(1,1/(x+1),1/sqrt(2*(x+1)))
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}
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mat[i,] <- c(x,y)
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}
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mat
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}
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## Now for the Rcpp version -- Notice how easy it is to code up!
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cppFunction("NumericMatrix RcppGibbs(int N, int thn){
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NumericMatrix mat(N, 2); // Setup storage
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double x = 0, y = 0; // The rest follows the R version
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for (int i = 0; i < N; i++) {
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for (int j = 0; j < thn; j++) {
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x = R::rgamma(3.0,1.0/(y*y+4));
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y = R::rnorm(1.0/(x+1),1.0/sqrt(2*x+2));
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}
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mat(i,0) = x;
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mat(i,1) = y;
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}
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return mat; // Return to R
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}")
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## Use of the sourceCpp() is preferred for users who wish to source external
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## files or specify their headers and Rcpp attributes within their code.
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## Code here is able to easily be extracted and placed into its own C++ file.
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## Compile and Load
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sourceCpp(code="
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#include <RcppGSL.h>
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#include <gsl/gsl_rng.h>
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#include <gsl/gsl_randist.h>
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using namespace Rcpp; // just to be explicit
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// [[Rcpp::depends(RcppGSL)]]
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// [[Rcpp::export]]
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NumericMatrix GSLGibbs(int N, int thin){
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gsl_rng *r = gsl_rng_alloc(gsl_rng_mt19937);
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double x = 0, y = 0;
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NumericMatrix mat(N, 2);
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for (int i = 0; i < N; i++) {
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for (int j = 0; j < thin; j++) {
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x = gsl_ran_gamma(r,3.0,1.0/(y*y+4));
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y = 1.0/(x+1)+gsl_ran_gaussian(r,1.0/sqrt(2*x+2));
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}
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mat(i,0) = x;
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mat(i,1) = y;
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}
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gsl_rng_free(r);
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return mat; // Return to R
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}")
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## Now for some tests
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## You can try other values if you like
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## Note that the total number of interations are N*thin!
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Ns <- c(1000,5000,10000,20000)
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thins <- c(10,50,100,200)
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tim_R <- rep(0,4)
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tim_Rgsl <- rep(0,4)
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tim_Rcpp <- rep(0,4)
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for (i in seq_along(Ns)) {
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tim_R[i] <- system.time(mat <- Rgibbs(Ns[i],thins[i]))[3]
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tim_Rgsl[i] <- system.time(gslmat <- GSLGibbs(Ns[i],thins[i]))[3]
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tim_Rcpp[i] <- system.time(rcppmat <- RcppGibbs(Ns[i],thins[i]))[3]
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cat("Replication #", i, "complete \n")
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}
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## Comparison
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speedup <- round(tim_R/tim_Rcpp,2);
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speedup2 <- round(tim_R/tim_Rgsl,2);
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summtab <- round(rbind(tim_R, tim_Rcpp,tim_Rgsl,speedup,speedup2),3)
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colnames(summtab) <- c("N=1000","N=5000","N=10000","N=20000")
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rownames(summtab) <- c("Elasped Time (R)","Elapsed Time (Rcpp)", "Elapsed Time (Rgsl)",
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"SpeedUp Rcpp", "SpeedUp GSL")
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print(summtab)
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## Contour Plots -- based on Darren's example
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if (interactive() && require(KernSmooth)) {
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op <- par(mfrow=c(4,1),mar=c(3,3,3,1))
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x <- seq(0,4,0.01)
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y <- seq(-2,4,0.01)
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z <- outer(x,y,fun)
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contour(x,y,z,main="Contours of actual distribution",xlim=c(0,2), ylim=c(-2,4))
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fit <- bkde2D(as.matrix(mat),c(0.1,0.1))
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contour(drawlabels=T, fit$x1, fit$x2, fit$fhat, xlim=c(0,2), ylim=c(-2,4),
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main=paste("Contours of empirical distribution:",round(tim_R[4],2)," seconds"))
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fitc <- bkde2D(as.matrix(rcppmat),c(0.1,0.1))
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contour(fitc$x1,fitc$x2,fitc$fhat,xlim=c(0,2), ylim=c(-2,4),
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main=paste("Contours of Rcpp based empirical distribution:",round(tim_Rcpp[4],2)," seconds"))
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fitg <- bkde2D(as.matrix(gslmat),c(0.1,0.1))
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contour(fitg$x1,fitg$x2,fitg$fhat,xlim=c(0,2), ylim=c(-2,4),
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main=paste("Contours of GSL based empirical distribution:",round(tim_Rgsl[4],2)," seconds"))
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par(op)
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}
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## also use rbenchmark package
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N <- 20000
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thn <- 200
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res <- benchmark(Rgibbs(N, thn),
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RcppGibbs(N, thn),
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GSLGibbs(N, thn),
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columns=c("test", "replications", "elapsed",
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"relative", "user.self", "sys.self"),
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order="relative",
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replications=10)
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print(res)
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## And we are done
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