normal_uncertain

Aleatory uncertain variable - normal (Gaussian)

Topics

continuous_variables, aleatory_uncertain_variables

Specification

• Alias: None

• Arguments: INTEGER

• Default: no normal uncertain variables

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Required		means	First parameter of the distribution
Required		std_deviations	Second parameter of the distribution
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

The number of normal uncertain variables, their means, and standard deviations are required specifications, while the distribution lower and upper bounds and variable descriptors are optional specifications. The normal distribution is widely used to model uncertain variables such as population characteristics. It is also used to model the mean of a sample: as the sample size becomes very large, the Central Limit Theorem states that the distribution of the mean becomes approximately normal, regardless of the distribution of the original variables.

The density function for the normal distribution is:

\[f(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp \left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 \right),\]

where \(\mu\) and \(\sigma\) are the mean and standard deviation of the normal distribution, respectively.

Note that if you specify bounds for a normal distribution, the sampling occurs from the underlying distribution with the given mean and standard deviation, but samples are not taken outside the bounds (see “bounded normal” distribution type in [WJ98]). This 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. For example, if you are sampling from a normal distribution with a mean of 5 and a standard deviation of 3, but you specify bounds of 1 and 7, the resulting mean of the samples will be around 4.3 and the resulting standard deviation will be around 1.6. This is because you have bounded the original distribution significantly, and asymetrically, since 7 is closer to the original mean than 1.

Theory

When used with some methods such as design of experiments and multidimensional parameter studies, distribution bounds are inferred to be [ \(\mu - 3 \sigma\) , \(\mu + 3 \sigma\) ]

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.