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


Aleatory uncertain variable - beta

Topics

This keyword is related to the topics:

Specification

Alias: none

Argument(s): INTEGER

Default: no beta uncertain variables

Required/Optional Description of Group Dakota Keyword Dakota Keyword Description
Required alphas First parameter of the beta distribution
Required betas Second parameter of the beta distribution
Required lower_bounds Specify minimum values
Required upper_bounds Specify maximium values
Optional initial_point

Initial values

Optional descriptors

Labels for the variables

Description

Within the beta uncertain optional group specification, the number of beta uncertain variables, the alpha and beta parameters, and the distribution upper and lower bounds are required specifications, and the variable descriptors is an optional specification. The beta distribution can be helpful when the actual distribution of an uncertain variable is unknown, but the user has a good idea of the bounds, the mean, and the standard deviation of the uncertain variable. The density function for the beta distribution is

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

where $\Gamma(\alpha)$ is the gamma function and $B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}$ is the beta function. To calculate the mean and standard deviation from the alpha, beta, upper bound, and lower bound parameters of the beta distribution, the following expressions may be used.

\[\mu_B = L_B+\frac{\alpha}{\alpha+\beta}(U_B-L_B)\]

\[\sigma_B^2 =\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}(U_B-L_B)^2\]

Solving these for $\alpha$ and $\beta$ gives:

\[\alpha = (\mu_B-L_B)\frac{(\mu_B-L_B)(U_B-\mu_B)-\sigma_B^2}{\sigma_B^2(U_B-L_B)}\]

\[\beta = (U_B-\mu_B)\frac{(\mu_B-L_B)(U_B-\mu_B)-\sigma_B^2}{\sigma_B^2(U_B-L_B)}\]

Note that the uniform distribution is a special case of this distribution for parameters $\alpha = \beta = 1$.

Theory

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.