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


Aleatory uncertain variable - lognormal

Topics

This keyword is related to the topics:

Specification

Alias: none

Argument(s): INTEGER

Default: no lognormal uncertain variables

Required/Optional Description of Group Dakota Keyword Dakota Keyword Description
Required
(Choose One)
Group 1 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

Optional descriptors

Labels for the variables

Description

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

  1. time to perform some task
  2. variables which are the product of a large number of other quantities, by the Central Limit Theorem
  3. quantities which cannot have negative values.

Within the lognormal uncertain optional group specification, the number of lognormal uncertain variables, the means, and either standard deviations or error factors must be specified, and 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[89]).

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:

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

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

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

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

\[\mu = e^{\lambda + \frac{\zeta^2}{2}}\]

\[\sigma^2 = e^{2\lambda + \zeta^2}(e^{\zeta^2} - 1)\]

\[\epsilon = e^{1.645\zeta}\]

The density function for the lognormal distribution is:

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

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.