pm_distNormShell Module Reference

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"

Detailed Description

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.

See also

pm_distMultiNorm
pm_distEggBox

Test:

test_pm_distNormShell

Final Remarks ⛓

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Copyright

Computational Data Science Lab

Author:

Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Variable Documentation

MODULE_NAME

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

Definition at line 52 of file pm_distNormShell.F90.