pm_distMultiNorm Module Reference

This module contains classes and procedures for computing various statistical quantities related to the MultiVariate Normal (MVN) distribution. More...

Data Types
type	distMultiNorm_type
	This is the derived type for signifying distributions that are of type MultiVariate Normal (MVN) as defined in the description of pm_distMultiNorm. More...
interface	getMultiNormLogPDF
	Generate and return the natural logarithm of the Probability Density Function (PDF) of the MultiVariate Normal distribution as defined in the description of pm_distMultiNorm. More...
interface	getMultiNormLogPDFNF
	Generate and return the natural logarithm of the normalization coefficient of the Probability Density Function (PDF) of the MultiVariate Normal distribution as defined in the description of pm_distMultiNorm. More...
interface	getMultiNormRand
	Generate and return a (collection) of random vector(s) of size ndim from the ndim-dimensional MultiVariate Normal (MVN) distribution, optionally with the specified input mean(1:ndim) and the specified subset of the Cholesky Factorization of the Covariance matrix of the MVN distribution. More...
interface	setMultiNormRand
	Return a (collection) of random vector(s) of size ndim from the ndim-dimensional MultiVariate Normal (MVN) distribution, optionally with the specified input mean(1:ndim) and the specified subset of the Cholesky Factorization of the Covariance matrix of the MVN distribution. More...

Variables
character(*, SK), parameter	MODULE_NAME = "@pm_distMultiNorm"

Detailed Description

This module contains classes and procedures for computing various statistical quantities related to the MultiVariate Normal (MVN) distribution.

Specifically, this module contains routines for computing the following quantities of the MultiVariate Normal distribution:

1. the Probability Density Function (PDF)
2. the Cumulative Distribution Function (CDF)
3. the Random Number Generation from the distribution (RNG)
4. the Inverse Cumulative Distribution Function (ICDF) or the Quantile Function

The PDF of the MVN distribution with the mean vector \(\bu{\mu}\) and covariance matrix \(\bu{\Sigma}\) at a given input point \(X\) is defined as,

\begin{equation} \large \pi\big(\bu{X} ~|~\bu{\mu}, \bu{\Sigma}\big) = \frac{1}{\sqrt{\big| 2\pi\bu{\Sigma} \big|}} ~ \exp\bigg( -\frac{1}{2}(\bu{X}-\bu{\mu})^T ~ \bu{\Sigma}^{-1} ~ (\bu{X}-\bu{\mu}) \bigg) ~, \end{equation}

which is defined if and only if the \(\bu{\Sigma}\) is positive-definite. The term \(\large \frac{1}{\sqrt{\big| 2\pi\bu{\Sigma} \big|}}\) is the Normalization Factor or the Normalization Constant of the MVN PDF whose natural logarithm can be computed via getMultiNormLogPDFNF in this module.

See also

pm_distUnif
pm_distNorm
pm_distMultiNorm
pm_distUnifEll
pm_distUnifPar
Applied Multivariate Statistical Analysis, Johnson, Wichern, 1998, 4th ed.

Test:

test_pm_distMultiNorm

Final Remarks ⛓

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Copyright

Computational Data Science Lab

Author:

Amir Shahmoradi, Oct 20, 2009, 9:12 PM, Michigan

Variable Documentation

MODULE_NAME

character(*, SK), parameter pm_distMultiNorm::MODULE_NAME = "@pm_distMultiNorm"

Definition at line 70 of file pm_distMultiNorm.F90.