gumbel_uncertain

Aleatory uncertain variable - gumbel

Topics

continuous_variables, aleatory_uncertain_variables

Specification

• Alias: None

• Arguments: 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:

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

where \(\mu = \beta + \frac{0.5772}{\alpha},\) 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.