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Dakota Reference Manual
Version 6.15
Explore and Predict with Confidence
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Aleatory uncertain variable - lognormal
This keyword is related to the topics:
Alias: none
Argument(s): INTEGER
Default: no lognormal uncertain variables
Child Keywords:
Required/Optional | Description of Group | Dakota Keyword | Dakota Keyword Description | |
---|---|---|---|---|
Required (Choose One) | Lognormal Characterization (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 for variables | ||
Optional | descriptors | Labels for the variables |
If the logarithm of an uncertain variable X has a normal distribution, that is , then X is distributed with a lognormal distribution. The lognormal is often used to model:
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[93]).
For the lognormal variables, one may specify either the mean and standard deviation
of the actual lognormal distribution (option 1), the mean
and error factor
of the actual lognormal distribution (option 2), or the mean
("lambda") and standard deviation
("zeta") of the underlying normal distribution (option 3).
The conversion equations from lognormal mean and either lognormal error factor
or lognormal standard deviation
to the mean
and standard deviation
of the underlying normal distribution are as follows:
Conversions from and
back to
and
or
are as follows:
The density function for the lognormal distribution is:
When used with some methods such as design of experiments and multidimensional parameter studies, distribution bounds are inferred to be [0, ].
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.