Dakota Reference Manual  Version 6.15
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gumbel_uncertain


Aleatory uncertain variable - gumbel

Topics

This keyword is related to the topics:

Specification

Alias: none

Argument(s): INTEGER

Default: no gumbel uncertain variables

Child Keywords:

Required/Optional Description of Group Dakota Keyword Dakota Keyword Description
Required alphas First parameter of the gumbel distribution
Required betas Second parameter of the gumbel distribution
Optional initial_point

Initial values for variables

Optional descriptors

Labels for the variables

Description

The Gumbel distribution is also referred to as the Type I Largest Extreme Value distribution. The distribution of maxima in sample sets from a population with a normal distribution will asymptotically converge to this distribution. It is commonly used to model demand variables such as wind loads and flood levels.

The density function for the Gumbel distribution is given by:

\[f(x) = \alpha \exp \left( -\alpha(x-\beta) \right) \exp \left( -e^{-\alpha(x-\beta)} \right), \]

$\mu = \beta + \frac{0.5772}{\alpha},$ and $\sigma = \frac{\pi}{\sqrt{6}\alpha}$.

Theory

When used with some methods such as design of experiments and multidimensional parameter studies, distribution bounds are inferred to be [ $\mu - 3 \sigma$, $\mu + 3 \sigma$]

For some methods, including vector and centered parameter studies, an initial point is needed for the uncertain variables. When not given explicitly, these variables are initialized to their means.