ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
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This module contains procedures and generic interfaces for computing the Multivariate Normal Shell density function or mixtures of such densities with varying parameters. More...
Data Types | |
interface | getNormShellLogUDF |
Generate and return the natural logarithm of the NormShell density function value(s) at the specified input point X , for the specified set of parameters of the single or mixture of NormShell distributions.More... | |
Variables | |
character(*, SK), parameter | MODULE_NAME = "@pm_distNormShell" |
This module contains procedures and generic interfaces for computing the Multivariate Normal Shell density function or mixtures of such densities with varying parameters.
The NormShell density function is frequently used in testing the efficiency of optimization and sampling algorithms.
Given a value \(x\in\mathbb{R}^n\), the NormShell density function with the (location, scale, scale, scale) parameters \((\bu{\mu}, \bu{\Sigma}, \omega, \rho)\) is defined as,
\begin{equation} \large f\left( \bu{X} ~|~\bu{\mu}, \bu{\Sigma}, \omega, \rho \right) = \exp \left( -\frac{\left[\left|(\bu{X} - \bu{\mu})^T ~ \bu{\Sigma}^{-1} ~ (\bu{X} - \bu{\mu})\right| - \rho\right]^2}{2\omega^2} \right) ~, \end{equation}
where \((\bu{\mu}, \bu{\Sigma}, \omega, \rho)\) are the shell center, shell covariance matrix, shell width, and shell radius of the distribution respectively.
Final Remarks ⛓
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character(*, SK), parameter pm_distNormShell::MODULE_NAME = "@pm_distNormShell" |
Definition at line 52 of file pm_distNormShell.F90.