beta_uncertain

Aleatory uncertain variable - beta

Topics

continuous_variables, aleatory_uncertain_variables

Specification

• Alias: None

• Arguments: INTEGER

• Default: no beta uncertain variables

Child Keywords:

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 for variables
Optional		descriptors	Labels for the variables

Description

The number of beta uncertain variables, the alpha and beta parameters, and the distribution upper and lower bounds are required specifications, while 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

\[f(x)= \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\frac{(x-L)^{\alpha-1}(U-x)^{\beta-1}}{(U-L)^{\alpha+\beta-1}},\]

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.

\[\mu = L+\frac{\alpha}{\alpha+\beta}(U-L)\]

\[\sigma^2 =\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}(U-L)^2\]

Solving these for \(\alpha\) and \(\beta\) gives:

\[\alpha = (\mu-L)\frac{(\mu-L)(U-\mu)-\sigma^2}{\sigma^2(U-L)}\]

\[\beta = (U-\mu)\frac{(\mu-L)(U-\mu)-\sigma^2}{\sigma^2(U-L)}\]

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

Theory

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.