Continuous random variate generation by fast numerical inversion

Abstract

The inversion method for generating non-uniform random variates has some advantages compared to other generation methods, since it monotonically transforms uniform random numbers into nonuniform random variates. Hence it is the method of choice in the simulation literature. However, except for some simple cases where the inverse of the cumulative distribution function is a simple function we need numerical methods. Often inversion by "brute force" is used, applying either very slow iterative methods or linear interpolation of the CDF and huge tables. But then the user has to accept unnecessarily large errors or excessive memory requirements, that slow down the algorithm. In this paper we demonstrate that with Hermite interpolation of the inverse CDF we can obtain very small error bounds close to machine precision. Using our adaptive interval splitting method this accuracy is reached with moderately sized tables that allow for a fast and simple generation procedure.

Efficient Hardware Generation of Random Variates with Arbitrary Distributions

“…In [6] a mechanical method is developed for the approximation of arbitrary inverse CDFs by adaptively splitting the domain into intervals and then approximating these intervals with Hermite polynomials. Another automatic method for sampling from arbitrary distributions is based on the ratio-of-uniforms method [10], which relies on sampling from a 2D plane in such a way that the ratio of the co-ordinates has the correct distribution.…”

Random Number Generation

“…When shape parameters are involved (e.g., the gamma and beta distributions), things are more complicated because F −1 then depends on the parameters in a more fundamental manner. Hörmann and Leydold (2003) propose a general adaptive and automatic method that constructs a highly accurate Hermite interpolation method of F −1 . In a one-time setup, their method produces tables for the interpolation Random Number Generation 25 points and coefficients.…”

Implementaciones de Funciones Elementales en Dispositivos FPGA

“…Para algunas distribuciones tenemos multitud de algoritmos que nos permiten generar muestras aleatorias. Para el caso de la generación de variables aleatorias Gaussianas, vamos a clasificar los distintos algoritmos en cuatro categorías básicas [59]: Inversión de la Función de Distribución Acumulada (ICDF) [60], métodos basados en transformaciones [61], métodos aceptación-rechazo [62][63] y los métodos recursivos [64]. Los métodos basados en la ICDF simplemente invierten la CDF de la distribución Gaussiana para generar muestras aleatorias Gaussianas.…”

“…La gran ventaja de este método es que permite generar variables aleatorias con CDF arbitrarias [60]. Además la calidad de las muestras generadas únicamente dependerá de la calidad de las muestras del generador de números uniformemente distribuidos.…”

“…La gran ventaja de este método es que permite generar variables aleatorias con CDF arbitrarias [60]. El mayor problema para la implementación de este método es la gran complejidad en la aproximación de la ICDF de la distribución Gaussiana debido a las altas no-linealidades presentes en los extremos de la misma.…”