TD_GENERALIZED_EXTREME_VALUE

	This is part of the ves module
	It is only available if you configure PLUMED with ./configure –enable-modules=ves . Furthermore, this feature is still being developed so take care when using it and report any problems on the mailing list.

Generalized extreme value distribution (static).

Employ a target distribution given by a generalized extreme value distribution that is defined as

\[ p(s) = \frac{1}{\sigma} \, t(s)^{\xi+1} \, e^{-t(s)}, \]

where

\[ t(s) = \begin{cases} \left( 1 + \xi \left( \frac{s-\mu}{\sigma} \right) \right)^{-1/\xi} & \mathrm{if\ }\xi \neq 0 \\ \exp\left(- \frac{s-\mu}{\sigma} \right) & \mathrm{if\ } \xi = 0 \end{cases}, \]

and \(\mu\) is the location parameter which approximately determines the location of the maximum of the distribution, \(\sigma>0\) is the scale parameter that determines the broadness of the distribution, and \(\xi\) is the shape parameter that determines the tail behavior of the distribution. For \(\xi=0\), \(\xi>0\), and \(\xi<0\) the Gumbel, Frechet, and Weibull families of distributions are obtained, respectively.

The location parameter \(\mu\) is given using the LOCATION keyword, the scale parameter \(\sigma\) using the SCALE keyword, and the shape parameter \(\xi\) using the SHAPE keyword.

This target distribution action is only defined for one dimension, for multiple dimensions it should be used in combination with TD_PRODUCT_DISTRIBUTION action.

Examples

Generalized extreme value distribution with \(\mu=0.0\), \(\sigma=2.0\), and \(\xi=0.0\) (Gumbel distribution)

Click on the labels of the actions for more information on what each action computes

td: TD_GENERALIZED_EXTREME_VALUE 
LOCATION
compulsory keyword 
The \f$\mu\f$ parameter of the generalized extreme value distribution. 
=0.0 
SCALE
compulsory keyword 
The \f$\sigma\f$ parameter for the generalized extreme value distribution given as
a positive number. 
=2.0 
SHAPE
compulsory keyword 
The \f$\xi\f$ parameter for the generalized extreme value distribution. 
=0.0 The TD_GENERALIZED_EXTREME_VALUE action with label td

Generalized extreme value distribution with \(\mu=-5.0\), \(\sigma=1.0\), and \(\xi=0.5\) (Frechet distribution)

Click on the labels of the actions for more information on what each action computes

td: TD_GENERALIZED_EXTREME_VALUE 
LOCATION
compulsory keyword 
The \f$\mu\f$ parameter of the generalized extreme value distribution. 
=-5.0 
SCALE
compulsory keyword 
The \f$\sigma\f$ parameter for the generalized extreme value distribution given as
a positive number. 
=1.0 
SHAPE
compulsory keyword 
The \f$\xi\f$ parameter for the generalized extreme value distribution. 
=0.5 The TD_GENERALIZED_EXTREME_VALUE action with label td

Generalized extreme value distribution with \(\mu=5.0\), \(\sigma=2.0\), and \(\xi=-0.5\) (Weibull distribution)

Click on the labels of the actions for more information on what each action computes

td: TD_GENERALIZED_EXTREME_VALUE 
LOCATION
compulsory keyword 
The \f$\mu\f$ parameter of the generalized extreme value distribution. 
=5.0 
SCALE
compulsory keyword 
The \f$\sigma\f$ parameter for the generalized extreme value distribution given as
a positive number. 
=1.0 
SHAPE
compulsory keyword 
The \f$\xi\f$ parameter for the generalized extreme value distribution. 
=-0.5 The TD_GENERALIZED_EXTREME_VALUE action with label td

The generalized extreme value distribution is only defined for one dimension so for multiple dimensions we have to use it in combination with the TD_PRODUCT_DISTRIBUTION action as shown in the following example where we have a Generalized extreme value distribution for argument 1 and uniform distribution for argument 2

Click on the labels of the actions for more information on what each action computes

td_gev: TD_GENERALIZED_EXTREME_VALUE 
LOCATION
compulsory keyword 
The \f$\mu\f$ parameter of the generalized extreme value distribution. 
=-5.0 
SCALE
compulsory keyword 
The \f$\sigma\f$ parameter for the generalized extreme value distribution given as
a positive number. 
=1.0 
SHAPE
compulsory keyword 
The \f$\xi\f$ parameter for the generalized extreme value distribution. 
=0.5 The TD_GENERALIZED_EXTREME_VALUE action with label td_gev
td_uni: TD_UNIFORM The TD_UNIFORM action with label td_uni
td_pd: TD_PRODUCT_DISTRIBUTION 
DISTRIBUTIONS
compulsory keyword 
Labels of the one-dimensional target distribution actions for each argument to be
used in the product distribution. 
=td_gev,td_uni The TD_PRODUCT_DISTRIBUTION action with label td_pd

Glossary of keywords and components

Compulsory keywords

LOCATION	The \(\mu\) parameter of the generalized extreme value distribution.
SCALE	The \(\sigma\) parameter for the generalized extreme value distribution given as a positive number.
SHAPE	The \(\xi\) parameter for the generalized extreme value distribution.

Options

SHIFT_TO_ZERO	( default=off ) Shift the minimum value of the target distribution to zero. This can for example be used to avoid negative values in the target distribution. If this option is active the distribution will be automatically normalized.
NORMALIZE	( default=off ) Renormalized the target distribution over the intervals on which it is defined to make sure that it is properly normalized to 1. In most cases this should not be needed as the target distributions should be normalized. The code will issue a warning (but still run) if this is needed for some reason.
WELLTEMPERED_FACTOR	Broaden the target distribution such that it is taken as [p(s)]^(1/ \(\gamma\)) where \(\gamma\) is the well tempered factor given here. If this option is active the distribution will be automatically normalized.