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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 Uniform Spherical distribution. More...
Data Types | |
| type | distUnifSphere_type |
| This is the derived type for signifying distributions that are of type MultiVariate Uniform Spherical as defined in the description of pm_distUnifSphere. More... | |
| interface | getUnifSphereLogPDF |
| Generate and return the natural logarithm of the Probability Density Function (PDF) of the uniform distribution on an \(n\)-sphere embedded in an \(\ndim\)-dimensional space. More... | |
| interface | getUnifSphereRand |
Generate and return a (collection) of random vector(s) of size ndim uniformly distributed on the surface of an \(n\)-sphere, optionally with the specified input mean(1:ndim) and optionally affine-transformed to a non-uniform distribution on the surface of an \((n+1)\)-ellipsoid represented by the Cholesky Factorization of its Gramian matrix. More... | |
| interface | setUnifSphereRand |
Return a (collection) of random vector(s) of size ndim uniformly distributed on the surface of an \(n\)-sphere, optionally with the specified input mean(1:ndim) and optionally affine-transformed to a non-uniform distribution on the surface of an \((n+1)\)-ellipsoid represented by the Cholesky Factorization of its Gramian matrix. More... | |
Variables | |
| character(*, SK), parameter | MODULE_NAME = "@pm_distUnifSphere" |
This module contains classes and procedures for computing various statistical quantities related to the Uniform Spherical distribution.
Specifically, this module contains routines for computing the following quantities of the uniform distribution of points on the surface of a an arbitrary n-dimensional sphere:
An \(n\)-sphere or a hypersphere is a topological space that is homeomorphic to a standard \(n\)-sphere, which is the set of points in \((n + 1)\)-dimensional Euclidean space that are situated at a constant distance \(r\) from a fixed point, called the center.
It is the generalization of an ordinary sphere in the ordinary three-dimensional space.
The radius of a sphere is the constant distance of its points to the center.
When the sphere has unit radius, it is usual to call it the unit \(n\)-sphere or simply the \(n\)-sphere for brevity.
In terms of the standard norm, the \(n\)-sphere is defined as,
\begin{equation} S^{n} = \left\{x\in\mathbb{R}^{n+1}:\left\|x\right\|=1\right\} ~, \end{equation}
and an n-sphere of radius \(r\) can be defined as
\begin{equation} S^{n}(r)=\left\{x\in \mathbb {R} ^{n+1}:\left\|x\right\|=r\right\} ~. \end{equation}
The dimension of \(n\)-sphere is \(n\), and must not be confused with the dimension \((n + 1)\) of the Euclidean space in which it is naturally embedded.
An \(n\)-sphere is the surface or boundary of an \((n + 1)\)-dimensional ball.
In particular:
The PDF of an \(n\)-sphere can be readily computed as the inverse of the surface area of the sphere, which has the closed form,
\begin{equation} S_{n-1} = {\frac {n\pi^{\frac{n}{2}}}{\Gamma\left({\frac{n}{2}} + 1\right)}} r^{n-1} = {\frac{2\pi^{\frac{n}{2}}}{\Gamma\left({\frac{n}{2}}\right)}} r^{n-1} = n\frac{V_{n}}{r}~. \end{equation}
for the \((n−1)\)-dimensional surface of the sphere \(S_{n−1}\) of radius \(r\), where \(V_{n + 1}\) is the volume of the corresponding \(n\)-ball.
Unlike the volume and the surface area of an \(n\)-sphere, the computation of the surface area of an arbitrary hyper-ellipsoid is very much involved.
See, for example, the discussion here for more information.
The RNG generic interfaces within this module generate uniformly-distributed random vectors on the surface an \(n\)-sphere using a generalization of 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} \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 non-uniformly distributed on the surface of ellipsoid \(\ell\) centered at the original \(n\)-sphere.
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_distUnifSphere::MODULE_NAME = "@pm_distUnifSphere" |
Definition at line 134 of file pm_distUnifSphere.F90.
1.9.3