Action: TD_GENERALIZED_EXTREME_VALUE

Module	ves
Description	Usage
Generalized extreme value distribution (static).

Details and examples

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_VALUEGeneralized extreme value distribution (static). More details  LOCATIONThe mu parameter of the generalized extreme value distribution=0.0  SCALEThe sigma parameter for the generalized extreme value distribution given as a positive number=2.0 SHAPEThe xi parameter for the generalized extreme value distribution=0.0
The TD_GENERALIZED_EXTREME_VALUE action with label td calculates something

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_VALUEGeneralized extreme value distribution (static). More details  LOCATIONThe mu parameter of the generalized extreme value distribution=-5.0  SCALEThe sigma parameter for the generalized extreme value distribution given as a positive number=1.0 SHAPEThe xi parameter for the generalized extreme value distribution=0.5
The TD_GENERALIZED_EXTREME_VALUE action with label td calculates something

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_VALUEGeneralized extreme value distribution (static). More details  LOCATIONThe mu parameter of the generalized extreme value distribution=5.0  SCALEThe sigma parameter for the generalized extreme value distribution given as a positive number=1.0 SHAPEThe xi parameter for the generalized extreme value distribution=-0.5
The TD_GENERALIZED_EXTREME_VALUE action with label td calculates something

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_VALUEGeneralized extreme value distribution (static). More details  LOCATIONThe mu parameter of the generalized extreme value distribution=-5.0  SCALEThe sigma parameter for the generalized extreme value distribution given as a positive number=1.0 SHAPEThe xi parameter for the generalized extreme value distribution=0.5

The TD_GENERALIZED_EXTREME_VALUE action with label td_gev calculates somethingtd_uni: TD_UNIFORMUniform target distribution (static). More details

The TD_UNIFORM action with label td_uni calculates somethingtd_pd: TD_PRODUCT_DISTRIBUTIONTarget distribution given by a separable product of one-dimensional distributions (static or dynamic). More details DISTRIBUTIONSLabels 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 calculates something

Full list of keywords

The following table describes the keywords and options that can be used with this action

Keyword	Type	Default	Description
LOCATION	compulsory	none	The mu parameter of the generalized extreme value distribution
SCALE	compulsory	none	The sigma parameter for the generalized extreme value distribution given as a positive number
SHAPE	compulsory	none	The xi parameter for the generalized extreme value distribution
WELLTEMPERED_FACTORThis keyword do not have examples	optional	not used	Broaden the target distribution such that it is taken as [p(s)]^(1/gamma) where gamma is the well tempered factor given here
SHIFT_TO_ZEROThis keyword do not have examples	optional	false	Shift the minimum value of the target distribution to zero
NORMALIZEThis keyword do not have examples	optional	false	Renormalized the target distribution over the intervals on which it is defined to make sure that it is properly normalized to 1