tdr.new: UNU.RAN generator based on Transformed Density Rejection (TDR)

Description

UNU.RAN random variate generator for continuous distributions with given probability density function (PDF). It is based on the Transformed Density Rejection method (‘TDR’).

[Universal] -- Rejection Method.

Usage

tdr.new(pdf, dpdf=NULL, lb, ub, islog=FALSE, ...)
tdrd.new(distr)

Value

An object of class "unuran".

Arguments

pdf: probability density function. (R function)
dpdf: derivative of pdf. (R function)
lb: lower bound of domain; use -Inf if unbounded from left. (numeric)
ub: upper bound of domain; use Inf if unbounded from right. (numeric)
islog: whether pdf is given as log-density (the dpdf must then be the derivative of the log-density). (boolean)
...: (optional) arguments for pdf.
distr: distribution object. (S4 object of class "unuran.cont")

Author

Josef Leydold and Wolfgang H\"ormann unuran@statmath.wu.ac.at.

Details

This function creates an unuran object based on ‘TDR’ (Transformed Density Rejection). It can be used to draw samples of a continuous random variate with given probability density function using ur.

The density pdf must be positive but need not be normalized (i.e., it can be any multiple of a density function). The derivative dpdf of the (log-) density is optional. If omitted, numerical differentiation is used. Notice, however, that this might cause some round-off errors such that the algorithm fails. This is in particular the case when the density function is provided instead of the log-density.

The given pdf must be $T_{- 0.5}$ -concave (with implies unimodal densities with tails not higher than $(1 / x^{2})$ ; this includes all log-concave distributions).

It is recommended to use the log-density (instead of the density function) as this is numerically more stable.

Alternatively, one can use function tdrd.new where the object distr of class "unuran.cont" must contain all required information about the distribution.

The setup time of this method depends on the given PDF, whereas its marginal generation times are almost independent of the target distribution.

There exists a variant of ‘TDR’ which is numerically more stable (albeit a bit slower and less flexible) which is avaible via the ars.new function.

References

W. H\"ormann, J. Leydold, and G. Derflinger (2004): Automatic Nonuniform Random Variate Generation. Springer-Verlag, Berlin Heidelberg. See Chapter 4 (Tranformed Density Rejection).

Examples

Run this code

## Create a sample of size 100 for a Gaussian distribution
pdf <- function (x) { exp(-0.5*x^2) }
gen <- tdr.new(pdf=pdf, lb=-Inf, ub=Inf)
x <- ur(gen,100)

## Create a sample of size 100 for a 
## Gaussian distribution (use logPDF)
logpdf <- function (x) { -0.5*x^2 }
gen <- tdr.new(pdf=logpdf, islog=TRUE, lb=-Inf, ub=Inf)
x <- ur(gen,100)

## Same example but additionally provide derivative of log-density
## to prevent possible round-off errors
logpdf <- function (x) { -0.5*x^2 }
dlogpdf <- function (x) { -x }
gen <- tdr.new(pdf=logpdf, dpdf=dlogpdf, islog=TRUE, lb=-Inf, ub=Inf)
x <- ur(gen,100)

## Draw sample from Gaussian distribution with mean 1 and
## standard deviation 2. Use 'dnorm'.
gen <- tdr.new(pdf=dnorm, lb=-Inf, ub=Inf, mean=1, sd=2)
x <- ur(gen,100)

## Draw a sample from a truncated Gaussian distribution
## on domain [5,Inf)
logpdf <- function (x) { -0.5*x^2 }
gen <- tdr.new(pdf=logpdf, lb=5, ub=Inf, islog=TRUE)
x <- ur(gen,100)

## Alternative approach
distr <- udnorm()
gen <- tdrd.new(distr)
x <- ur(gen,100)

Run the code above in your browser using DataLab

Data engineering and BI courses are free!