ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
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" |
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:
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.
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}
Final Remarks ⛓
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character(*, SK), parameter pm_distUnifPar::MODULE_NAME = "@pm_distUnifPar" |
Definition at line 98 of file pm_distUnifPar.F90.