C Library Functions - gamran (3)
NAME
gamran(3f) - [M_datapac:RANDOM] generate gamma random numbers
CONTENTS
Synopsis
Description
Algorithm
Input Arguments
Output Arguments
Examples
Author
Maintainer
License
References
SYNOPSIS
SUBROUTINE GAMRAN(N,Gamma,Iseed,X)
INTEGER,intent(in) :: N INTEGER,intent(inout) :: Iseed REAL(kind=wp),intent(in) :: Gamma REAL(kind=wp),intent(out) :: X(:)
DESCRIPTION
GAMRAN(3f) generates a random sample of size N from the gamma distribution with tail length parameter value = GAMMA.The prototype gamma distribution used herein has mean = GAMMA and standard deviation = sqrt(GAMMA). This distribution is defined for all positive X, and has the probability density function
f(X) = (1/constant) * (X**(GAMMA-1)) * exp(-X)where the constant is equal to the Gamma function evaluated at the value GAMMA.
ALGORITHM
Generate N Gamma Distribution random numbers using Greenwood’s Rejection Algorithm--
1. Generate a normal random number; 2. Transform the normal variate to an approximate gamma variate using the Wilson-Hilferty approximation (see the Johnson and Kotz reference, page 176); 3. Form the rejection function value, based on the probability density function value of the actual distribution of the pseudo-gamma variate, and the probability density function value of a true gamma variate. 4. Generate a uniform random number; 5. If the uniform random number is less than the rejection function value, then accept the pseudo-random number as a gamma variate; if the uniform random number is larger than the rejection function value, then reject the pseudo-random number as a gamma variate.
INPUT ARGUMENTS
N The desired integer number of random numbers to be generated. GAMMA The value of the tail length parameter. GAMMA should be positive. GAMMA should be larger than 1/3 (algorithmic restriction).
| ISEED | An integer seed value. Should be set to a non-negative value to start a new sequence of values. Will be set to -1 on return to indicate the next call should continue the current random sequence walk. |
OUTPUT ARGUMENTS
X A vector (of dimension at least N) into which the generated random sample from the gamma distribution will be placed.
EXAMPLES
Sample program:
program demo_gamran use m_datapac, only : gamran, plott, label, plotxt, sort implicit none integer,parameter :: n=4000 real :: x(n) integer :: iseed real :: gamma call label(’gamran’) gamma=3.4 iseed=12345 call gamran(n,gamma,iseed,x) call plotxt(x,n) call sort(x,n,x) ! sort to show distribution call plotxt(x,n) end program demo_gamranResults:
THE FOLLOWING IS A PLOT OF X(I) (VERTICALLY) VERSUS I (HORIZONTALLY) I-----------I-----------I-----------I-----------I 0.1547529E+02 - X X 0.1483860E+02 I 0.1420192E+02 I 0.1356523E+02 I X 0.1292854E+02 I X 0.1229185E+02 I X X 0.1165516E+02 - X 0.1101848E+02 I X X X 0.1038179E+02 I XX X X X X X X X 0.9745100E+01 I X X X XX X X XX X XX X X 0.9108413E+01 I X X X XX X XXX XX 0.8471725E+01 I X X XX XX X XXXXX XXX X XX X X X X XX X 0.7835037E+01 - X XXX XX X XXX X XX XXXXXXX XX XXXX XX X XX 0.7198349E+01 I X XXXXX XXXXX XXXX X X XXX XXXXX XXX XXX X X 0.6561661E+01 I XXXXXXXXXX XXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXX 0.5924973E+01 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.5288285E+01 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.4651597E+01 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.4014910E+01 - XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.3378222E+01 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.2741534E+01 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.2104846E+01 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.1468158E+01 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.8314705E+00 I XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.1947823E+00 - X X X X XX X X X X XX XXXX X X X XX I-----------I-----------I-----------I-----------I 0.1000E+01 0.1001E+04 0.2000E+04 0.3000E+04 0.4000E+04================================================================================ ‘‘‘THE FOLLOWING IS A PLOT OF X(I) (VERTICALLY) VERSUS I (HORIZONTALLY) I-----------I-----------I-----------I-----------I 0.1547529E+02 - X 0.1483860E+02 I 0.1420192E+02 I 0.1356523E+02 I X 0.1292854E+02 I X 0.1229185E+02 I X 0.1165516E+02 - X 0.1101848E+02 I X 0.1038179E+02 I X 0.9745100E+01 I X 0.9108413E+01 I XX 0.8471725E+01 I X 0.7835037E+01 - XX 0.7198349E+01 I X 0.6561661E+01 I XXX 0.5924973E+01 I XXX 0.5288285E+01 I XXXX 0.4651597E+01 I XXXXX 0.4014910E+01 - XXXXXX 0.3378222E+01 I XXXXXXXX 0.2741534E+01 I XXXXXXXX 0.2104846E+01 I XXXXXXXX 0.1468158E+01 I XXXXXXX 0.8314705E+00 I XXXX 0.1947823E+00 - X I-----------I-----------I-----------I-----------I 0.1000E+01 0.1001E+04 0.2000E+04 0.3000E+04 0.4000E+04
AUTHOR
The original DATAPAC library was written by James Filliben of the Statistical Engineering Division, National Institute of Standards and Technology.
MAINTAINER
John Urban, 2022.05.31
LICENSE
CC0-1.0
REFERENCES
o Greenwood, ’A Fast generator for Gamma-Distributed Random Variables’, Compstat 1974, Proceedings in Computational Statistics, Vienna, September, 1974, pages 19-27. o Tocher, The Art of Simulation, 1963, pages 24-27. o Hammersley and Handscomb, Monte Carlo Methods, 1964, pages 36-37. o Wilk, Gnanadesikan, and Huyett, ’Probability Plots for the Gamma Distribution’, Technometrics, 1962, pages 1-15. o Johnson and Kotz, Continuous Univariate Distributions--1, 1970, pages 166-206. o Hastings and Peacock, Statistical Distributions--A Handbook for Students and Practitioners, 1975, pages 68-73. o National Bureau of Standards Applied Mathematics Series 55, 1964, page 952.
| Nemo Release 3.1 | gamran (3) | July 22, 2023 |
