Philipp K. Janert
and 1 contributors

NAME

Math::Gauss - Gaussian distribution function and its inverse

SYNOPSIS

  use Math::Gauss ':all';

  $p = pdf( $z );
  $p = pdf( $x, $m, $s );

  $c = cdf( $z );
  $c = cdf( $x, $m, $s );

  $z = inv_cdf( $z );

DESCRIPTION

This module calculates the Gaussian probability density, the cumulative distribution function (Gaussian distribution function), and the inverse Gaussian distribution function.

EXPORT

None by default. The :all tag is recognized, or import individual functions.

FUNCTIONS

pdf( $x, $m, $s )

The Gaussian probability density function (pdf)

  exp( -0.5 ( (x-m)/s )**2 )/sqrt(2*pi*s**2)

Only the first argument is mandatory, the other two are optional. If they are not supplied, they default as follows: pdf( $z ) == pdf( $x, 0, 1 ).

The parameter $s must be strictly positive, $x and $m can be arbitrary.

If you choose to supply the z-score, $z = ($x-$m)/$s as argument to the pdf() instead of supplying the mean and the standard deviation as separate arguments, you must divide the return value by the standard deviation to obtain a properly normalized result. The following is true:

  pdf( $x, $m, $s ) == pdf( ($x-$m)/$s )/$s

cdf( $x, $m, $s )

The Gaussian cumulative distribution function (cdf). This is the integral of pdf(t,m,s) from -infinity to x over t:

  int_{-infty}^x exp( -0.5 ( (t-m)/s )**2 )/sqrt(2*pi*s**2) dt

Only the first argument is mandatory, the other two are optional. If they are not supplied, they default as follows: cdf( $z ) == cdf( $x, 0, 1 ).

The parameter $s must be strictly positive, $x and $m can be arbitrary.

The implementation is guaranteed to have a maximum absolute error of less than 7.5e-8 for all x.

inv_cdf( $z )

The inverse of the Gaussian cumulative distribution function. This function is only defined for arguments strictly greater than 0 and strictly less than 1: 0 < $z < 1.

The implementation is guaranteed to have a maximum absolute error of less than 4.5e-4 for all x.

USAGE NOTES

This module favors simplicity and portability over accuracy. The accuracy should be good enough for most applications; if you need higher accuracy, see the resources below.

The cumulative distribution function and its inverse defined by this module are the so-called "probability function" versions. They are related to the "error function" erf() and its inverse inverf() by:

  erf(x) = 2 cdf( sqrt(2)*x ) - 1
  inverf(x) = inv_cdf( (x+1)/2 )/sqrt(2)

SEE ALSO

The numerical algorithms are taken from:

Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables by Milton Abramowitz and Irene A. Stegun; Dover (1965).

Alternative algorithms (including a high-accuracy algorithm for the inverse distribution functions) can be found here:

  • http://home.online.no/~pjacklam/notes/invnorm/

  • http://homepages.physik.uni-muenchen.de/~Winitzki/erf-approx.pdf

AUTHOR

Philipp K. Janert, <janert at ieee dot org>, http://www.beyondcode.org

COPYRIGHT AND LICENSE

Copyright (C) 2011 by Philipp K. Janert

This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself, either Perl version 5.10.0 or, at your option, any later version of Perl 5 you may have available.