pm_distUnifPar Module Reference

This module contains classes and procedures for setting up and computing the properties of the MultiVariate Uniform Parallelepiped (MVUP) Distribution. More...

Data Types
type	distUnifPar_type
	This is the derived type for signifying distributions that are of type MultiVariate Uniform Parallelepiped (MVUP) as defined in the description of pm_distUnifPar. More...
interface	getUnifParLogPDF
	Generate and return the natural logarithm of the Probability Density Function (PDF) of the MultiVariate Uniform Parallelepiped (MVUP) Distribution. More...
interface	getUnifParRand
	Generate and return a random vector from the \(\ndim\)-dimensional MultiVariate Uniform Parallelepiped (MVUP) Distribution. More...
interface	setUnifParRand
	Return a random vector from the \(\ndim\)-dimensional MultiVariate Uniform Parallelepiped (MVUP) Distribution. More...

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

Detailed Description

This module contains classes and procedures for setting up and computing the properties of the MultiVariate Uniform Parallelepiped (MVUP) Distribution.

Specifically, this module contains routines for computing the following quantities of the MultiVariate Uniform Parallelepiped (MVUP) distribution:

1. the Probability Density Function (PDF)
2. the Random Number Generation from the distribution (RNG)

An \(\ndim\)-dimensional parallelepiped \(P\) in vector space \(\mathbb{R}^{\ndim}\) can be defined by a set of arbitrary but independent column vectors \(v_1, \ldots, v_\ndim\) as the set,

\begin{equation} P = {\sum_{i = 1}^{\ndim} t_i v_i ~~~,~~~ 0 \leq t_i < 1} ~, \end{equation}

where \(t_i\) are a set of coefficients whose defined range allows full coverage of the parallelepiped.

A 3-dimensional parallelepiped.

The above parallelepiped can be expressed in the form of a square representative matrix of edges of rank \(\ndim\),

\begin{equation} M_R = \begin{pmatrix} v_1 ~,~ \vdots ~,~ v_i ~,~ \vdots ~,~ v_{\ndim} \end{pmatrix} \end{equation}

where \(v_i\) is a column vector representing the \(i\)th edge of the parallelepiped.
The corresponding (positive definite) Gramian matrix of the parallelepiped is,

\begin{equation} M_G = M_R^T M_R ~, \end{equation}

where \(M_R^T\) is the transpose of \(M_R\).
The hyper-volume occupied by the parallelepiped is the given by,

\begin{equation} \ms{Vol}(P) = |M_R| = \sqrt{|M_G|} ~, \end{equation}

where \(|M_G|\) represents the determinant of \(M_G\).

The Probability Density Function (PDF) of the Uniform Parallelepiped distribution with support \(P\) is given by,

\begin{equation} \pi(X | P) = \frac{1}{\ms{Vol}(P)} = \frac{1}{|M_R|} = \frac{1}{\sqrt{|M_G|}} ~. \end{equation}

Note

1. A parallelepiped with a diagonal representative matrix \(M_R\) is a hyper-rectangle.
   
   1. A hyper-rectangle is uniquely determined by lengths of its \(\ndim\) edges.
      
   2. The corresponding Gramian matrix \(M_G\) of a hyper-rectangle is also diagonal.
      
2. A hyper-rectangle with equal diagonal elements in the representative matrix \(M_R\) is a hyper-cube.
   
   1. A hyper-cube is uniquely determined by a single scalar number representing the length of its edge that is the same along all axes.
      
   2. The representative matrix \(M_R\) of a hyper-cube is a multiple of the Identity matrix.
      
   3. The corresponding Gramian matrix \(M_G\) a hyper-cube is also a multiple of the Identity matrix.
      

See also

pm_distUnifEll
pm_distUnifPar
pm_distUnifEll
pm_distUnifPar

Test:

test_pm_distUnifPar

Final Remarks ⛓

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For details on the naming conventions, see this page.
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Copyright

Computational Data Science Lab

Author:

Amir Shahmoradi, April 23, 2017, 1:36 AM, Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin

Variable Documentation

MODULE_NAME

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

Definition at line 98 of file pm_distUnifPar.F90.