lognormal_uncertain

Aleatory uncertain variable - lognormal

Topics

continuous_variables, aleatory_uncertain_variables

Specification

• Alias: None

• Arguments: INTEGER

• Default: no lognormal uncertain variables

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Required (Choose One)	Lognormal Characterization	lambdas	First parameter of the lognormal distribution (option 3)
		means	First parameter of the lognormal distribution (options 1 & 2)
Optional		lower_bounds	Specify minimum values
Optional		upper_bounds	Specify maximium values
Optional		initial_point	Initial values for variables
Optional		descriptors	Labels for the variables

Description

If the logarithm of an uncertain variable X has a normal distribution, that is \(\log X \sim \mathcal{N}(\mu,\sigma^2)\) , then \(X\) is distributed with a lognormal distribution. The lognormal is often used to model:

• time to perform some task

• variables which are the product of a large number of other quantities, by the Central Limit Theorem

• quantities which cannot have negative values.

The number of lognormal uncertain variables, their means, and either standard deviations or error factors must be specified, while the distribution lower and upper bounds and variable descriptors are optional specifications. These distribution bounds can be used to truncate the tails of lognormal distributions, which as for bounded normal, can result in the mean and the standard deviation of the sample data being different from the mean and standard deviation of the underlying distribution (see “bounded lognormal” and “bounded lognormal-n” distribution types in [WJ98]).

For the lognormal variables, one may specify either the mean \(\mu\) and standard deviation \(\sigma\) of the actual lognormal distribution (option 1), the mean \(\mu\) and error factor \(\epsilon\) of the actual lognormal distribution (option 2), or the mean \(\lambda\) (“lambda”) and standard deviation \(\zeta\) (“zeta”) of the underlying normal distribution (option 3).

The conversion equations from lognormal mean \(\mu\) and either lognormal error factor \(\epsilon\) or lognormal standard deviation \(\sigma\) to the mean \(\lambda\) and standard deviation \(\zeta\) of the underlying normal distribution are as follows:

\[\zeta = \frac{\ln(\epsilon)}{1.645}\]

\[\zeta^2 = \ln\left(\frac{\sigma^2}{\mu^2} + 1\right)\]

\[\lambda = \ln(\mu) - \frac{\zeta^2}{2}\]

Conversions from \(\lambda\) and \(\zeta\) back to \(\mu\) and \(\epsilon\) or \(\sigma\) are as follows:

\[\mu = \exp \left( \lambda + \frac{\zeta^2}{2} \right)\]

\[\sigma^2 = \exp \left( 2\lambda + \zeta^2 \right) \left( \exp \left(\zeta^2\right) - 1\right)\]

\[\epsilon = \exp \left( 1.645\zeta \right)\]

The density function for the lognormal distribution is:

\[f(x) = \frac{1}{\sqrt{2\pi}\zeta x} \exp \left( -\frac{1}{2}\left(\frac{\ln(x)-\lambda}{\zeta}\right)^2 \right)\]

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, correcting to bounds if needed.