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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
This module contains classes and procedures for computing various statistical quantities related to the MultiVariate Uniform Ellipsoid (MVUE) distribution. More...
Data Types | |
| type | distUnifEll_type |
| This is the derived type for signifying distributions that are of type MultiVariate Uniform Ellipsoid (MVUE) as defined in the description of pm_distUnifEll. More... | |
| interface | getUnifEllLogPDF |
| Generate and return the natural logarithm of the Probability Density Function (PDF) of the MultiVariate MVUE (MVUE) Distribution. More... | |
| interface | getUnifEllRand |
Generate and return a (collection) of random vector(s) of size ndim from the ndim-dimensional MultiVariate Uniform Ellipsoidal (MVUE) distribution, optionally with the specified input mean(1:ndim) and the specified subset of the Cholesky Factorization of the Gramian matrix of the MVUE distribution. More... | |
| interface | setUnifEllRand |
Return a (collection) of random vector(s) of size ndim from the ndim-dimensional MultiVariate Uniform Ellipsoidal (MVUE) distribution, optionally with the specified input mean(1:ndim) and the specified subset of the Cholesky Factorization of the Gramian matrix of the MVUE distribution. More... | |
Variables | |
| character(*, SK), parameter | MODULE_NAME = "@pm_distUnifEll" |
This module contains classes and procedures for computing various statistical quantities related to the MultiVariate Uniform Ellipsoid (MVUE) distribution.
Specifically, this module contains routines for computing the following quantities of the MultiVariate Uniform Ellipsoid (MVUE) distribution:
An \(\ndim\)-dimensional MVUE distribution is represented by an \(\ndim\)-dimensional hyper-ellipsoid support.
The MVUE distribution is fully determined by the ellipsoid \(\ell\) containing its support.
The ellipsoid \(\ell\) is in turn fully determined by its representative Gramian matrix \(\gramian_\ell\), containing all points in \(\mathbb{R}^\ndim\) that satisfy,
\begin{equation} \large (X - \mu_\ell)^T ~ \gramian_\ell^{-1} ~ (X - \mu_\ell) \leq 1 ~, \end{equation}
where \(\mu_\ell\) represents the center of the ellipsoid, \((X - \mu_\ell)^T\) is the transpose of the vector \((X - \mu_\ell)\), and \(\gramian_\ell^{-1}\) is the inverse of the matrix \(\gramian_\ell\).
The volume of this ellipsoid is given by,
\begin{equation} \large V(\ell) = V_\ndim \sqrt{\left| \gramian_\ell \right|} ~, \end{equation}
where \(\left|\gramian_\ell\right|\) is the determinant of \(\gramian_\ell\) and,
\begin{equation} \large V_\ndim = \frac{\pi^{\ndim / 2}}{\up\Gamma(1 + \ndim / 2)} = \begin{cases} \frac{1}{(\ndim/2)!} \pi^{\ndim/2} & \text{if $\ndim$ is even} \\\\ 2^\ndim \frac{1}{\ndim!} \big( \frac{\ndim-1}{2} \big)! ~ \pi^{(\ndim-1)/2} & \text{if $\ndim$ is odd} \end{cases} \end{equation}
is the volume of an \(\ndim\)-ball (that is, a unit-logChoDia \(\ndim\)-dimensional hyper-sphere).
It is readily seen that the corresponding unit-volume ellipsoid \(\widehat\ell\) has the representative Gramian matrix,
\begin{equation} \large \gramian_{\widehat\ell} = V_\ell^{-2/\ndim} ~ \gramian_\ell ~. \end{equation}
More generally, to scale the ellipsoidal support of an MVUE distribution \(\ell\) by some factor \(\alpha\) along each coordinate axis, it suffices to be used the new scaled ellipsoid \(\ell^*\) with the representative Gramian matrix,
\begin{equation} \large \gramian_{\ell^*} = \alpha^2 ~ \gramian_\ell ~, \end{equation}
in which case, the volume of \(\ell^*\) becomes,
\begin{equation} \large V_{\ell^*} = \alpha^\ndim ~ V_\ell ~. \end{equation}
The Probability Density Function (PDF) of the MVUE distribution with ellipsoidal support \(\ell\) is given by,
\begin{equation} \large \pi(X | \ell) = \frac{1}{V(\ell)} = \frac{1}{V_\ndim \sqrt{\left|\gramian_\ell\right|}} ~. \end{equation}
Random Number Generation
The RNG generic interfaces within this module generate uniformly-distributed random vectors from within an \(\ndim\)-dimensional hyper-ellipsoid generalize the proposed approach of Marsaglia (1972) for choosing a point from the surface of a sphere.
\begin{equation} \unit{r} = \frac{1}{\sqrt{\sum_{i=1}^\ndim g_i^2}} \sum_{j=1}^{\ndim} g_j \unit{r}_j ~, \end{equation}
where \(\unit{r}_j\) are the unit vectors representing the orthogonal basis the of \(\ndim\)-space.\begin{equation} \bs{r}_{\sphere} = u^{1/\ndim} \unit{r} ~, \end{equation}
represents a vector pointing to a uniformly-distributed location inside of the \(\ndim\)-sphere.\begin{equation} \gramian = \mat{L}\mat{L}^H ~, \end{equation}
where \(\mat{L}\) is the left triangular matrix resulting from the Cholesky factorization, and \(\mat{L}^T\) is its Hermitian transpose.\begin{equation} \bs{r}_\ell = \mat{L} ~ \bs{r}_\sphere + \mu_\ell ~, \end{equation}
is uniformly distributed inside ellipsoid \(\ell\) centered at \(\mu_\ell\).The covariance matrix of the MVUE distribution
The covariance matrix of the multivariate uniform ellipsoidal distribution is given by its Gramian matrix as,
\begin{eqnarray*} \Sigma = \frac{\gramian}{\ndim + 2} ~, \end{eqnarray*}
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.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
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.
| character(*, SK), parameter pm_distUnifEll::MODULE_NAME = "@pm_distUnifEll" |
Definition at line 154 of file pm_distUnifEll.F90.
1.9.3