and 1 contributors

# NAME

Math::Random::SkewNormal - Handy, easy-to-use Skew-Normal random number generator

# SYNOPSIS

``````  use Math::Random::SkewNormal;

\$skewness = 1;
\$delta    = [0, 0];
\$omega    = [[1, 0], [0, 1]];

# following functions generate realizations:
generate_sn(\$skewness)            # SkewNormal(0,1,\$skewness)
generate_sn_multi(\$delta, \$omega) # SkewNormal_n(\$delta, \$omega)``````

# DESCRIPTION

This module transforms uniformly spaced random variable realizations into realizations that follow the Skew-Normal (SN) Probability Density Function (PDF).

We accept following definition of the Skew-Normal Distribution:

1-dimensional SN is determined by PDF: f(x,a) = 2 phi(x) Phi(a x), where phi is the PDF and Phi is the CDF of Normal distribution
Let X = (X_0, ..., X_k) follows distribution N_k+1 (0, Omega*), where Omega* is the matrix of the form:
``````       [   1   delta ]
[ delta Omega ]``````

Omega is a matrix and delta is a vector. Then (X_1, ..., X_n)|(X_0 > 0) follows n-dimensional Skew Normal distribution.

# ALGORITHM

We use following algorithms:

• Box-Muller transformation; for generation Normal realizations

• Choleski's decomposition; for inverting matrices

• A lemma mentioned e. g. in article from A. Azalini: The skew-normal distribution and related multivariate families; for generating Skew-Normal realizations

# FUNCTIONS

## generate_sn

Returns realization of SN(0,1, a), where a is the only parameter. The meaning of a is "skewness" (see definition above).

## generate_sn_multi

Returns realization of centralized n-dimensional Skew-Normal distribution.

1st parameter is correlation matrix of the form `[[row_1], [row_2], ..., [row_n]]`, for example `[[1, 0], [0, 1]]` for 2-dimensional unit matrix.

2nd parameter is delta vector of the form `[delta_1, ..., delta_n]`, e. g. `[0, 1]`.

The examples in examples/ subdirectory of this distribution contains example of generated data with histogram.

The scripts in bin/ directory of this distribution contains tools for generating realizations on command line.

# AUTHOR

Jiri Vaclavik, <my name dot my last name at gmail dot com>