Dakota Reference Manual  Version 6.4
Large-Scale Engineering Optimization and Uncertainty Analysis
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frechet_uncertain


Aleatory uncertain variable - Frechet

Topics

This keyword is related to the topics:

Specification

Alias: none

Argument(s): INTEGER

Default: no frechet uncertain variables

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

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:

\[f(x) = \frac{\alpha}{\beta}(\frac{\beta}{x})^{\alpha+1}e^{-(\frac{\beta}{x})^\alpha}\]

where $\mu_F = \beta\Gamma(1-\frac{1}{\alpha})$ and $\sigma_F^2 = \beta^2[\Gamma(1-\frac{2}{\alpha})-\Gamma^2(1-\frac{1}{\alpha})]$

Theory

When used with design of experiments and multidimensional parameter studies, distribution bounds are inferred. These bounds are [0, $\mu + 3 \sigma$].

For vector and centered parameter studies, an inferred initial starting point is needed for the uncertain variables. These variables are initialized to their means for these studies.