frechet_uncertain

Aleatory uncertain variable - Frechet

Topics

continuous_variables, aleatory_uncertain_variables

Specification

• Alias: None

• Arguments: INTEGER

• Default: no frechet uncertain variables

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Required		alphas	First parameter of the Frechet distribution
Required		betas	Second parameter of the Frechet distribution
Optional		initial_point	Initial values for variables
Optional		descriptors	Labels for the variables

Description

The Frechet distribution is also referred to as the Type II Largest Extreme Value distribution. The distribution of maxima in sample sets from a population with a lognormal distribution will asymptotically converge to this distribution. It is commonly used to model non-negative demand variables.

The density function for the frechet distribution is:

\[f(x) = \frac{\alpha}{\beta} \left( \frac{\beta}{x} \right)^{\alpha+1} \exp \left( -\left(\frac{\beta}{x}\right)^\alpha \right),\]

where \(\mu = \beta\Gamma(1-\frac{1}{\alpha}),\) and \(\sigma^2 = \beta^2[\Gamma(1-\frac{2}{\alpha})-\Gamma^2(1-\frac{1}{\alpha})]\)

Theory

When used with some methods such as design of experiments and multidimensional parameter studies, distribution bounds are inferred to be [0, \(\mu + 3 \sigma\) ].

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.