Runuran (version 0.40)

pinv.new: UNU.RAN generator based on Polynomial interpolation of INVerse CDF (PINV)

Description

UNU.RAN random variate generator for continuous distributions with given probability density function (PDF) or cumulative distribution function (CDF). It is based on the Polynomial interpolation of INVerse CDF (‘PINV’).

[Universal] -- Inversion Method.

Usage

pinv.new(pdf, cdf, lb, ub, islog=FALSE, center=0,
         uresolution=1.e-10, smooth=FALSE, ...)
pinvd.new(distr, uresolution=1.e-10, smooth=FALSE)

Value

An object of class "unuran".

Arguments

pdf

probability density function. (R function)

cdf

cumulative distribution function. (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 and cdf are given by their corresponding logarithms. (boolean)

center

“typical” point of distribution. (numeric)

...

(optional) arguments for pdf and cdf.

distr

distribution object. (S4 object of class "unuran.cont")

uresolution

maximal acceptable u-error. (numeric)

smooth

whether the inverse CDF is differentiable. (boolean)

Remark

Using function up generator objects that implement method ‘PINV’ may also be used to approximate the cumulative distribution function of the given distribution when only the density is given. The approximation error is about one tenth of the requested uresolution.

Author

Josef Leydold and Wolfgang H\"ormann .

Details

This function creates an unuran object based on ‘PINV’ (Polynomial interpolation of INVerse CDF). It can be used to draw samples of a continuous random variate with given probability density function pdf or cumulative distribution function cdf by means of ur. It also allows to compute quantiles by means of uq.

Function pdf must be positive but need not be normalized (i.e., it can be any multiple of a density function). The set of points where the pdf is strictly positive must be connected. The center is a point where the pdf is not too small, e.g., (a point near) the mode of the distribution.

If the density pdf is given, then the algorithm automatically computes the CDF using Gauss-Lobatto integration. If the cdf is given but not the pdf then the CDF is used instead of the PDF. However, we found in our experiments that using the PDF is numerically more stable.

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

The algorithm approximates the inverse of the CDF of the distribution by means of Newton interpolation between carefully selected nodes. The approxiating functing is thus continuous. Argument smooth controls whether this function is also differentiable(“smooth”) at the nodes. Using smooth=TRUE requires the pdf of the distribution. It results in a higher setup time and memory consumption. Thus using smooth=TRUE is not not recommended, unless differentiability is important.

The approximation error is estimated by means of the the u-error, i.e., \(|CDF(G(U)) - U|\), where \(G\) denotes the approximation of the inverse CDF. The error can be controlled by means of argument uresolution.

When sampling from truncated distributions with extreme truncation points, it is recommended to provide the log-density by setting islog=TRUE. Then the algorithm is numerically more stable.

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

References

G. Derflinger, W. H\"ormann, and J. Leydold (2010): Random variate generation by numerical inversion when only the density is known. ACM Trans. Model. Comput. Simul., 20:4, #18

See Also

ur, uq, up, unuran.cont, unuran.new, unuran.

Examples

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

## Improve the accuracy of the approximation
gen <- pinv.new(pdf=dnorm, lb=-Inf, ub=Inf, uresolution=1e-15)
x <- ur(gen,100)

## We have to provide a 'center' when PDF (almost) vanishes at 0.
gen <- pinv.new(pdf=dgamma, lb=0, ub=Inf, center=4, shape=5)
x <- ur(gen,100)

## We also can force a smoother approximation
gen <- pinv.new(pdf=dnorm, lb=-Inf, ub=Inf, smooth=TRUE)
x <- ur(gen,100)

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