urhitro: UNU.RAN transformend density rejection generator

Description

UNU.RAN random variate generator for continuous multivariate distributions with given probability density function (PDF). It is based on the Hit-and-Run algorithm in combinaton with the Ratio-of-Uniforms method (HITRO).

Usage

urhitro(n, dim=1, pdf, mode=NULL, center=NULL, ll=NULL, ur=NULL, thinning=1, burnin = 0, ...)

Arguments

size of required sample.

dim

number of dimensions of the distribution. (integer)

pdf

probability density function. (Rfunction)

mode

location of the mode -- optional. (numeric vector)

center

point in typical region of distribution, e.g. the approximate location of the mode. If omitted the mode is implicitly used. If the mode is not given either, the origin is used. (numeric vector -- opti

ll,ur

lower left and upper right vertex of a rectangular domain of the pdf. The domain is only set if both vertices are not NULL. Otherwise, the domain is unbounded by default. (numeric vectors)

thinning

thinning factor. (positive integer)

burnin

lenght of burin-in phase. (positive integer)

...

(optional) arguments for pdf

Details

This is a Markov chain sampler that generates a sequence of continuous random vectors with given probability density function. This function must be provided by pdf, a function which must return non-negative numbers and which need not be normalized (i.e., it can be any multiple of a density function).

The center is used as starting point of the Hit-and-Run algorithm. It is thus important, that the center is contained in the (closure of the) domain. Alternatively, one could the mode. However, this requires its exact position whereas center allows any point in the typical region of the distribution. The mode is used a starting point of the Hit-and-Run algorithm. It is thus important, that the mode is contained in the (closure of the) domain. Notice that MCMC samplers do not produce a sequence of independent variates. The drawn sample follows the target distribution only appropriately. Beware: MCMC sampling can be dangerous! The algorithm works best with log-concave distributions. Other distributions work as well but convergence can be slower.

References

R. Karawatzki, J. Leydold, and K. P"otzelberger (2005): Automatic Markov Chain Monte Carlo Procedures for Sampling from Multivariate Distributions. Research Report Series / Department of Statistics and Mathematics, Nr. 27, December 2005 Department of Statistics and Mathematics, Wien, Wirtschaftsuniv., 2005. http://epub.wu-wien.ac.at/dyn/virlib/wp/showentry?ID=epub-wu-01_8cb

Examples

Run this code

## Create a sample of size 100 for a 
## Gaussian distribution
mvpdf <- function (x) { exp(-sum(x^2)) }
x <- urhitro(20, dim=2, pdf=mvpdf)
x

## use:
##  mode at (0,0)
##  thinning factor 3
##    (only every 3rd vector in the sequence is returned)
##  burn-in of length 100
##    (the first 1000 vectors in the sequence are discarded)
x <- urhitro(20, dim=2, pdf=mvpdf, mode=c(0,0), thinning=3, burnin=100)
x

## Create a sample of size 100 for a 
## Gaussian distribution restricted to first quadrant
mvpdf <- function (x) { exp(-sum(x^2)) }
x <- urhitro(20, dim=2, pdf=mvpdf, ll=c(0,0), ur=c(Inf,Inf))
x

## Create a sample of size 100 for a 
## Gaussian distribution restricted to the rectangle [1,2]x[1,2]
## (don't forget to provide a starting point using 'center')

mvpdf <- function (x) { exp(-sum(x^2)) }
x <- urhitro(20, dim=2, pdf=mvpdf, center=c(1.1,1.1), ll=c(1,1), ur=c(2,2))
x