ParaMonte Fortran 2.0.0 Parallel Monte Carlo and Machine Learning LibrarySee the latest version documentation.
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,

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

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 $$\ndim$$,

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

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,

$$M_G = M_R^T M_R ~,$$

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

$$\ms{Vol}(P) = |M_R| = \sqrt{|M_G|} ~,$$

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,

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

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.
pm_distUnifEll
pm_distUnifPar
pm_distUnifEll
pm_distUnifPar
Test:
test_pm_distUnifPar

Final Remarks

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

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

## ◆ MODULE_NAME

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

Definition at line 98 of file pm_distUnifPar.F90.