Variables Commands

Variables Commands Table of Contents

Variables Description

The variables section in a DAKOTA input file specifies the parameter set to be iterated by a particular method. This parameter set is made up of design, uncertain, and state variables. Design variables can be continuous or discrete and consist of those variables which an optimizer adjusts in order to locate an optimal design. Each of the design parameters can have an initial point, a lower bound, an upper bound, and a descriptive tag.

Uncertain variables are continuous variables which are characterized by probability distributions. The distribution type can be normal, lognormal, uniform, loguniform, triangular, exponential, beta, gamma, gumbel, frechet, weibull, or histogram. In addition to the uncertain variables defined by probability distributions, we have an uncertain variable type called interval, where the uncertainty in a variable is described by one or more interval values in which the variable may lie. The interval uncertain variable type is used in epistemic uncertainty calculations, specifically Dempster-Shafer theory of evidence.

Each uncertain variable specification contains descriptive tags and most contain, either explicitly or implicitly, distribution lower and upper bounds. Distribution lower and upper bounds are explicit portions of the normal, lognormal, uniform, loguniform, triangular, and beta specifications, whereas they are implicitly defined for histogram and interval variables from the extreme values within the bin/point/interval specifications. When used with design of experiments and multidimensional parameter studies, distribution bounds are also inferred for normal and lognormal (if optional bounds are unspecified) as well as for exponential, gamma, gumbel, frechet, and weibull (which have no bounds specification); these bounds are [0, $\mu + 3 \sigma$ ] for exponential, gamma, frechet, weibull and unspecified lognormal, and [ $\mu - 3 \sigma$ , $\mu + 3 \sigma$ ] for gumbel and unspecified normal. In addition to tags and bounds specifications, normal variables include mean and standard deviation specifications, lognormal variables include mean and either standard deviation or error factor specifications, triangular variables include mode specifications, exponential variables include beta specifications, beta, gamma, gumbel, frechet, and weibull variables include alpha and beta specifications, histogram variables include bin pairs and point pairs specifications, and interval variables include basic probability assignments per interval.

State variables can be continuous or discrete and consist of "other" variables which are to be mapped through the simulation interface. Each state variable specification can have an initial state, lower and upper bounds, and descriptors. State variables provide a convenient mechanism for parameterizing additional model inputs, such as mesh density, simulation convergence tolerances and time step controls, and can be used to enact model adaptivity in future strategy developments.

Several examples follow. In the first example, two continuous design variables are specified:

variables,
	continuous_design = 2
	  initial_point    0.9    1.1
	  upper_bounds     5.8    2.9
	  lower_bounds     0.5   -2.9
	  descriptors   'radius' 'location'

In the next example, defaults are employed. In this case, initial_point will default to a vector of 0. values, upper_bounds will default to vector values of DBL_MAX (the maximum number representable in double precision for a particular platform, as defined in the platform's float.h C header file), lower_bounds will default to a vector of -DBL_MAX values, and descriptors will default to a vector of 'cdv_i' strings, where i ranges from one to two:

variables,
	continuous_design = 2

In the following example, the syntax for a normal-lognormal distribution is shown. One normal and one lognormal uncertain variable are completely specified by their means and standard deviations. In addition, the dependence structure between the two variables is specified using the uncertain_correlation_matrix.

variables,
        normal_uncertain    =  1
          means             =  1.0
          std_deviations    =  1.0
          descriptors       =  'TF1n'
        lognormal_uncertain =  1
          means             =  2.0
          std_deviations    =  0.5
          descriptors       =  'TF2ln'
        uncertain_correlation_matrix =  1.0 0.2
                                        0.2 1.0

An example of the syntax for a state variables specification follows:

variables,
        continuous_state = 1
          initial_state       4.0
          lower_bounds        0.0
          upper_bounds        8.0
          descriptors        'CS1'
        discrete_state = 1
          initial_state       104
          lower_bounds        100
          upper_bounds        110
          descriptors        'DS1'

And in a more advanced example, a variables specification containing a set identifier, continuous and discrete design variables, normal and uniform uncertain variables, and continuous and discrete state variables is shown:

variables,
	id_variables = 'V1'
	continuous_design = 2
	  initial_point    0.9    1.1
	  upper_bounds     5.8    2.9
	  lower_bounds     0.5   -2.9
	  descriptors   'radius' 'location'
	discrete_design = 1
	  initial_point    2
	  upper_bounds     1
	  lower_bounds     3
	  descriptors   'material'
	normal_uncertain = 2
	  means             =  248.89, 593.33
	  std_deviations    =   12.4,   29.7
	  descriptors       =  'TF1n'   'TF2n'
	uniform_uncertain = 2
	  lower_bounds =  199.3,  474.63
	  upper_bounds =  298.5,  712.
	  descriptors       =  'TF1u'   'TF2u'
	continuous_state = 2
	  initial_state = 1.e-4  1.e-6
	  descriptors = 'EPSIT1' 'EPSIT2'
	discrete_state = 1
	  initial_state = 100
	  descriptors = 'load_case'

Refer to the DAKOTA Users Manual [Eldred et al., 2007] for discussion on how different iterators view these mixed variable sets.

Variables Specification

The variables specification has the following structure:

variables,
	<set identifier>
	<continuous design variables specification>
	<discrete design variables specification>
	<normal uncertain variables specification>
	<lognormal uncertain variables specification>
	<uniform uncertain variables specification>
	<loguniform uncertain variables specification>
	<triangular uncertain variables specification>
	<exponential uncertain variables specification>
	<beta uncertain variables specification>
	<gamma uncertain variables specification>
	<gumbel uncertain variables specification>
	<frechet uncertain variables specification>
	<weibull uncertain variables specification>
	<histogram uncertain variables specification>
	<interval uncertain variables specification>
	<uncertain correlation specification>
	<continuous state variables specification>
	<discrete state variables specification>

Referring to dakota.input.txt, it is evident from the enclosing brackets that the set identifier specification, the uncertain correlation specification, and each of the variables specifications are all optional. The set identifier and uncertain correlation are stand-alone optional specifications, whereas the variables specifications are optional group specifications, meaning that the group can either appear or not as a unit. If any part of an optional group is specified, then all required parts of the group must appear.

The optional status of the different variable type specifications allows the user to specify only those variables which are present (rather than explicitly specifying that the number of a particular type of variables = 0). However, at least one type of variables must have nonzero size or an input error message will result. The following sections describe each of these specification components in additional detail.

Variables Set Identifier

The optional set identifier specification uses the keyword id_variables to input a unique string for use in identifying a particular variables set. A model can then identify the use of this variables set by specifying the same string in its variables_pointer specification (see Model Independent Controls). For example, a model whose specification contains variables_pointer = 'V1' will use a variables specification containing the set identifier id_variables = 'V1'.

If the id_variables specification is omitted, a particular variables set will be used by a model only if that model omits specifying a variables_pointer and if the variables set was the last set parsed (or is the only set parsed). In common practice, if only one variables set exists, then id_variables can be safely omitted from the variables specification and variables_pointer can be omitted from the model specification(s), since there is no potential for ambiguity in this case. Table 7.1 summarizes the set identifier inputs.

Table 7.1 Specification detail for set identifier
Description Keyword Associated Data Status Default

Variables set identifier id_variables string Optional use of last variables parsed

Table 7.1 Specification detail for set identifier
Description	Keyword	Associated Data	Status	Default
Variables set identifier	`id_variables`	string	Optional	use of last variables parsed

Design Variables

Within the optional continuous design variables specification group, the number of continuous design variables is a required specification and the initial point, lower bounds, upper bounds, scaling factors, and variable names are optional specifications. Likewise, within the optional discrete design variables specification group, the number of discrete design variables is a required specification and the initial guess, lower bounds, upper bounds, and variable names are optional specifications. Table 7.2 summarizes the details of the continuous design variable specification and Table 7.3 summarizes the details of the discrete design variable specification.

Table 7.2 Specification detail for continuous design variables
Description Keyword Associated Data Status Default

Continuous design variables continuous_design integer Optional group no continuous design variables

Initial point initial_point list of reals Optional vector values = 0.

Lower bounds lower_bounds list of reals Optional vector values = -DBL_MAX

Upper bounds upper_bounds list of reals Optional vector values = +DBL_MAX

Scaling types scale_types list of strings Optional vector values = 'none'

Scales scales list of reals Optional vector values = 1. (no scaling)

Descriptors descriptors list of strings Optional vector of 'cdv_i' where i = 1,2,3...

Table 7.2 Specification detail for continuous design variables
Description	Keyword	Associated Data	Status	Default
Continuous design variables	`continuous_design`	integer	Optional group	no continuous design variables
Initial point	`initial_point`	list of reals	Optional	vector values = `0`.
Lower bounds	`lower_bounds`	list of reals	Optional	vector values = `-DBL_MAX`
Upper bounds	`upper_bounds`	list of reals	Optional	vector values = `+DBL_MAX`
Scaling types	`scale_types`	list of strings	Optional	vector values = `'none'`
Scales	`scales`	list of reals	Optional	vector values = `1`. (no scaling)
Descriptors	`descriptors`	list of strings	Optional	vector of `'cdv_i'` where `i = 1,2,3...`

Table 7.3 Specification detail for discrete design variables
Description Keyword Associated Data Status Default

Discrete design variables discrete_design integer Optional group no discrete design variables

Initial point initial_point list of integers Optional vector values = 0

Lower bounds lower_bounds list of integers Optional vector values = INT_MIN

Upper bounds upper_bounds list of integers Optional vector values = INT_MAX

Descriptors descriptors list of strings Optional vector of 'ddv_i' where i = 1,2,3,...

Table 7.3 Specification detail for discrete design variables
Description	Keyword	Associated Data	Status	Default
Discrete design variables	`discrete_design`	integer	Optional group	no discrete design variables
Initial point	`initial_point`	list of integers	Optional	vector values = `0`
Lower bounds	`lower_bounds`	list of integers	Optional	vector values = `INT_MIN`
Upper bounds	`upper_bounds`	list of integers	Optional	vector values = `INT_MAX`
Descriptors	`descriptors`	list of strings	Optional	vector of `'ddv_i'` where `i = 1,2,3,...`

The initial_point specifications provide the point in design space from which an iterator is started for the continuous and discrete design variables, respectively. The lower_bounds and upper_bounds restrict the size of the feasible design space and are frequently used to prevent nonphysical designs. The scale_types specification includes strings specifying the scaling type for each component of the continuous design variables vector in methods that support scaling, when scaling is enabled (see Method Independent Controls for details). Each entry in scale_types may be selected from 'none', 'value', 'auto', or 'log', to select no, characteristic value, automatic, or logarithmic scaling, respectively. If a single string is specified it will apply to all components of the continuous design variables vector. Each entry in scales may be a user-specified nonzero real characteristic value to be used in scaling each variable component. These values are ignored for scaling type 'none', required for 'value', and optional for 'auto' and 'log'. If a single real value is specified it will apply to all components of the continuous design variables vector. The descriptors specifications supply strings which will be replicated through the DAKOTA output to help identify the numerical values for these parameters. Default values for optional specifications are zeros for initial values, positive and negative machine limits for upper and lower bounds (+/- DBL_MAX, INT_MAX, INT_MIN from the float.h and limits.h system header files), and numbered strings for descriptors. As for linear and nonlinear inequality constraint bounds (see Method Independent Controls and Objective and constraint functions (optimization data set)), a nonexistent upper bound can be specified by using a value greater than the "big bound size" constant (1.e+30 for continuous design variables, 1.e+9 for discrete design variables) and a nonexistent lower bound can be specified by using a value less than the negation of these constants (-1.e+30 for continuous, -1.e+9 for discrete), although not all optimizers currently support this feature (e.g., DOT and CONMIN will treat these large bound values as actual variable bounds, but this should not be problematic in practice).

Uncertain Variables

Uncertain variables involve one of several supported probability distribution specifications, including normal, lognormal, uniform, loguniform, triangular, exponential, beta, gamma, gumbel, frechet, weibull, or histogram distributions. Each of these specifications is an optional group specification. There also is an uncertain variable type called an interval variable. This is not a probability distribution, but is used in specifying the inputs necessary for an epistemic uncertainty analysis using Dempster-Shafer theory of evidence.

The inclusion of lower and upper distribution bounds for all uncertain variable types (either explicitly defined, implicitly defined, or inferred; see Variables Description) allows the use of these variables with methods that rely on a bounded region to define a set of function evaluations (i.e., design of experiments and some parameter study methods). In addition, distribution bounds can be used to truncate the tails of distributions for normal and lognormal uncertain variables (see "bounded normal", "bounded lognormal", and "bounded lognormal-n" distribution types in [Wyss and Jorgensen, 1998]). Default upper and lower bounds are positive and negative machine limits (+/- DBL_MAX from the float.h system header file), respectively, for non-logarithmic distributions and positive machine limits and zeros, respectively, for logarithmic distributions. The uncertain variable descriptors provide strings which will be replicated through the DAKOTA output to help identify the numerical values for these parameters. Default values for descriptors are numbered strings. Tables 7.4 through 7.17 summarize the details of the uncertain variable specifications.

Normal Distribution

Within the normal uncertain optional group specification, the number of normal uncertain variables, the means, and standard deviations are required specifications, and 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 mean becomes approximately normal, regardless of the distribution of the original variables.

The density function for the normal distribution is:

$f(x) = \frac{1}{\sqrt{2\pi}\sigma_N} e^{-\frac{1}{2}\left(\frac{x-\mu_N}{\sigma_N}\right)^2}$

where $\mu_N$ and $\sigma_N$ 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. 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.

Table 7.4 Specification detail for normal uncertain variables
Description Keyword Associated Data Status Default

normal uncertain variables normal_uncertain integer Optional group no normal uncertain variables

normal uncertain means means list of reals Required N/A

normal uncertain standard deviations std_deviations list of reals Required N/A

Distribution lower bounds lower_bounds list of reals Optional vector values = -DBL_MAX

Distribution upper bounds upper_bounds list of reals Optional vector values = +DBL_MAX

Descriptors descriptors list of strings Optional vector of 'nuv_i' where i = 1,2,3,...

Table 7.4 Specification detail for normal uncertain variables
Description	Keyword	Associated Data	Status	Default
normal uncertain variables	`normal_uncertain`	integer	Optional group	no normal uncertain variables
normal uncertain means	`means`	list of reals	Required	N/A
normal uncertain standard deviations	`std_deviations`	list of reals	Required	N/A
Distribution lower bounds	`lower_bounds`	list of reals	Optional	vector values = `-DBL_MAX`
Distribution upper bounds	`upper_bounds`	list of reals	Optional	vector values = `+DBL_MAX`
Descriptors	`descriptors`	list of strings	Optional	vector of `'nuv_i'` where `i = 1,2,3,...`

Lognormal Distribution

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 time to perform some task. It can also be used to model variables which are the product of a large number of other quantities, by the Central Limit Theorem. Finally, the lognormal is used to model 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.

For the lognormal variables, DAKOTA's uncertainty quantification methods standardize on the use of statistics of the actual lognormal distribution, as opposed to statistics of the underlying normal distribution. This approach diverges from that of [Wyss and Jorgensen, 1998], which assumes that a specification of means and standard deviations provides parameters of the underlying normal distribution, whereas a specification of means and error factors provides statistics of the actual lognormal distribution. By binding the mean, standard deviation, and error factor parameters consistently to the actual lognormal distribution, inputs are more intuitive and require fewer conversions in most user applications. The conversion equations from lognormal mean $\mu_{LN}$ and either lognormal error factor $\epsilon_{LN}$ or lognormal standard deviation $\sigma_{LN}$ to the mean $\mu_N$ and standard deviation $\sigma_N$ of the underlying normal distribution are as follows:

$\sigma_N = \frac{ln(\epsilon_{LN})}{1.645}$

$\sigma_N^2 = ln(\frac{\sigma_{LN}^2}{\mu_{LN}^2} + 1.)$

$\mu_N = ln(\mu_{LN}) - \frac{\sigma_N^2}{2}$

Conversions from $\mu_N$ and $\sigma_N$ back to $\mu_{LN}$ and $\epsilon_{LN}$ or $\sigma_{LN}$ are as follows:

$\mu_{LN} = e^{\mu_N + \frac{\sigma_N^2}{2}}$

$\sigma_{LN}^2 = e^{2\mu_N + \sigma_N^2}(e^{\sigma_N^2} - 1.)$

$\epsilon_{LN} = e^{1.645\sigma_N}$

The density function for the lognormal distribution is:

$f(x) = \frac{1}{\sqrt{2\pi}\sigma_N x} e^{-\frac{1}{2}\left(\frac{ln x-\mu_N}{\sigma_N}\right)^2}$

Table 7.5 Specification detail for lognormal uncertain variables
Description Keyword Associated Data Status Default

lognormal uncertain variables lognormal_uncertain integer Optional group no lognormal uncertain variables

lognormal uncertain means means list of reals Required N/A

lognormal uncertain standard deviations std_deviations list of reals Required (1 of 2 selections) N/A

lognormal uncertain error factors error_factors list of reals Required (1 of 2 selections) N/A

Distribution lower bounds lower_bounds list of reals Optional vector values = 0.

Distribution upper bounds upper_bounds list of reals Optional vector values = +DBL_MAX

Descriptors descriptors list of strings Optional vector of 'lnuv_i' where i = 1,2,3,...

Table 7.5 Specification detail for lognormal uncertain variables
Description	Keyword	Associated Data	Status	Default
lognormal uncertain variables	`lognormal_uncertain`	integer	Optional group	no lognormal uncertain variables
lognormal uncertain means	`means`	list of reals	Required	N/A
lognormal uncertain standard deviations	`std_deviations`	list of reals	Required (1 of 2 selections)	N/A
lognormal uncertain error factors	`error_factors`	list of reals	Required (1 of 2 selections)	N/A
Distribution lower bounds	`lower_bounds`	list of reals	Optional	vector values = `0`.
Distribution upper bounds	`upper_bounds`	list of reals	Optional	vector values = `+DBL_MAX`
Descriptors	`descriptors`	list of strings	Optional	vector of `'lnuv_i'` where `i = 1,2,3,...`

Uniform Distribution

Within the uniform uncertain optional group specification, the number of uniform uncertain variables and the distribution lower and upper bounds are required specifications, and variable descriptors is an optional specification. The uniform distribution has the density function:

$f(x) = \frac{1}{U_U-L_U}$

where $U_U$ and $L_U$ are the upper and lower bounds of the uniform distribution, respectively. The mean of the uniform distribution is $\frac{U_U+L_U}{2}$ and the variance is $\frac{(U_U-L_U)^2}{12}$ . Note that this distribution is a special case of the more general beta distribution.

Table 7.6 Specification detail for uniform uncertain variables
Description Keyword Associated Data Status Default

uniform uncertain variables uniform_uncertain integer Optional group no uniform uncertain variables

Distribution lower bounds lower_bounds list of reals Required N/A

Distribution upper bounds upper_bounds list of reals Required N/A

Descriptors descriptors list of strings Optional vector of 'uuv_i' where i = 1,2,3,...

Table 7.6 Specification detail for uniform uncertain variables
Description	Keyword	Associated Data	Status	Default
uniform uncertain variables	`uniform_uncertain`	integer	Optional group	no uniform uncertain variables
Distribution lower bounds	`lower_bounds`	list of reals	Required	N/A
Distribution upper bounds	`upper_bounds`	list of reals	Required	N/A
Descriptors	`descriptors`	list of strings	Optional	vector of `'uuv_i'` where `i = 1,2,3,...`

Loguniform Distribution

If the logarithm of an uncertain variable X has a uniform distribution, that is $\log X \sim U(L_{LU},U_{LU})$ , then X is distributed with a loguniform distribution. Within the loguniform uncertain optional group specification, the number of loguniform uncertain variables and the distribution lower and upper bounds are required specifications, and variable descriptors is an optional specification. The loguniform distribution has the density function:

$f(x) = \frac{1}{x(ln U_{LU} - ln {L_{LU}})}$

Table 7.7 Specification detail for loguniform uncertain variables
Description Keyword Associated Data Status Default

loguniform uncertain variables loguniform_uncertain integer Optional group no loguniform uncertain variables

Distribution lower bounds lower_bounds list of reals Required N/A

Distribution upper bounds upper_bounds list of reals Required N/A

Descriptors descriptors list of strings Optional vector of 'luuv_i' where i = 1,2,3,...

Table 7.7 Specification detail for loguniform uncertain variables
Description	Keyword	Associated Data	Status	Default
loguniform uncertain variables	`loguniform_uncertain`	integer	Optional group	no loguniform uncertain variables
Distribution lower bounds	`lower_bounds`	list of reals	Required	N/A
Distribution upper bounds	`upper_bounds`	list of reals	Required	N/A
Descriptors	`descriptors`	list of strings	Optional	vector of `'luuv_i'` where `i = 1,2,3,...`

Triangular Distribution

The triangular distribution is often used when one does not have much data or information, but does have an estimate of the most likely value and the lower and upper bounds. Within the triangular uncertain optional group specification, the number of triangular uncertain variables, the modes, and the distribution lower and upper bounds are required specifications, and variable descriptors is an optional specification.

The density function for the triangular distribution is:

$f(x) = \frac{2(x-L_T)}{(U_T-L_T)(M_T-L_T)}$

if $L_T\leq x \leq M_T$ , and

$f(x) = \frac{2(U_T-x)}{(U_T-L_T)(U_T-M_T)}$

if $M_T\leq x \leq U_T$ , and 0 elsewhere. In these equations, $L_T$ is the lower bound, $U_T$ is the upper bound, and $M_T$ is the mode of the triangular distribution.

Table 7.8 Specification detail for triangular uncertain variables
Description Keyword Associated Data Status Default

triangular uncertain variables triangular_uncertain integer Optional group no triangular uncertain variables

triangular uncertain modes modes list of reals Required N/A

Distribution lower bounds lower_bounds list of reals Required N/A

Distribution upper bounds upper_bounds list of reals Required N/A

Descriptors descriptors list of strings Optional vector of 'tuv_i' where i = 1,2,3,...

Table 7.8 Specification detail for triangular uncertain variables
Description	Keyword	Associated Data	Status	Default
triangular uncertain variables	`triangular_uncertain`	integer	Optional group	no triangular uncertain variables
triangular uncertain modes	`modes`	list of reals	Required	N/A
Distribution lower bounds	`lower_bounds`	list of reals	Required	N/A
Distribution upper bounds	`upper_bounds`	list of reals	Required	N/A
Descriptors	`descriptors`	list of strings	Optional	vector of `'tuv_i'` where `i = 1,2,3,...`

Exponential Distribution

The exponential distribution is often used for modeling failure rates. Within the exponential uncertain optional group specification, the number of exponential uncertain variables and the beta parameters are required specifications, and variable descriptors is an optional specification.

The density function for the exponential distribution is given by:

$f(x) = \frac{1}{\beta} e^{\frac{-x}{\beta}}$

where $\mu_{E} = \beta$ and $\sigma^2_{E} = \beta^2$ . Note that this distribution is a special case of the more general gamma distribution.

Table 7.9 Specification detail for exponential uncertain variables
Description Keyword Associated Data Status Default

exponential uncertain variables exponential_uncertain integer Optional group no exponential uncertain variables

exponential uncertain betas betas list of reals Required N/A

Descriptors descriptors list of strings Optional vector of 'euv_i' where i = 1,2,3,...

Table 7.9 Specification detail for exponential uncertain variables
Description	Keyword	Associated Data	Status	Default
exponential uncertain variables	`exponential_uncertain`	integer	Optional group	no exponential uncertain variables
exponential uncertain betas	`betas`	list of reals	Required	N/A
Descriptors	`descriptors`	list of strings	Optional	vector of `'euv_i'` where `i = 1,2,3,...`

Beta Distribution

Within the beta uncertain optional group specification, the number of beta uncertain variables, the alpha and beta parameters, and the distribution upper and lower bounds are required specifications, and 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

$f(x)= \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\frac{(x-L_B)^{\alpha-1}(U_B-x)^{\beta-1}}{(U_B-L_B)^{\alpha+\beta-1}}$

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 mean and standard deviation from the alpha, beta, upper bound, and lower bound parameters of the beta distribution, the following expressions may be used.

$\mu_B = L_B+\frac{\alpha}{\alpha+\beta}(U_B-L_B)$

$\sigma_B^2 =\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}(U_B-L_B)^2$

Solving these for $\alpha$ and $\beta$ gives:

$\alpha = (\mu_B-L_B)\frac{(\mu_B-L_B)(U_B-\mu_B)-\sigma_B^2}{\sigma_B^2(U_B-L_B)}$

$\beta = (U_B-\mu_B)\frac{(\mu_B-L_B)(U_B-\mu_B)-\sigma_B^2}{\sigma_B^2(U_B-L_B)}$

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

Table 7.10 Specification detail for beta uncertain variables
Description Keyword Associated Data Status Default

beta uncertain variables beta_uncertain integer Optional group no beta uncertain variables

beta uncertain alphas alphas list of reals Required N/A

beta uncertain betas betas list of reals Required N/A

Distribution lower bounds lower_bounds list of reals Required N/A

Distribution upper bounds upper_bounds list of reals Required N/A

Descriptors descriptors list of strings Optional vector of 'buv_i' where i = 1,2,3,...

Table 7.10 Specification detail for beta uncertain variables
Description	Keyword	Associated Data	Status	Default
beta uncertain variables	`beta_uncertain`	integer	Optional group	no beta uncertain variables
beta uncertain alphas	`alphas`	list of reals	Required	N/A
beta uncertain betas	`betas`	list of reals	Required	N/A
Distribution lower bounds	`lower_bounds`	list of reals	Required	N/A
Distribution upper bounds	`upper_bounds`	list of reals	Required	N/A
Descriptors	`descriptors`	list of strings	Optional	vector of `'buv_i'` where `i = 1,2,3,...`

Gamma Distribution

The gamma distribution is sometimes used to model time to complete a task, such as a repair or service task. It is a very flexible distribution. Within the gamma uncertain optional group specification, the number of gamma uncertain variables and the alpha and beta parameters are required specifications, and variable descriptors is an optional specification.

The density function for the gamma distribution is given by:

$f(x) = \frac{{x}^{\alpha-1}{e}^{\frac{-x}{\beta}}}{\beta^{\alpha}\Gamma(\alpha)}$

where $\mu_{GA} = \alpha\beta$ and $\sigma^2_{GA} = \alpha\beta^2$ . Note that the exponential distribution is a special case of this distribution for parameter $\alpha = 1$ .

Table 7.11 Specification detail for gamma uncertain variables
Description Keyword Associated Data Status Default

gamma uncertain variables gamma_uncertain integer Optional group no gamma uncertain variables

gamma uncertain alphas alphas list of reals Required N/A

gamma uncertain betas betas list of reals Required N/A

Descriptors escriptors list of strings Optional vector of 'gauv_i' where i = 1,2,3,...

Table 7.11 Specification detail for gamma uncertain variables
Description	Keyword	Associated Data	Status	Default
gamma uncertain variables	`gamma_uncertain`	integer	Optional group	no gamma uncertain variables
gamma uncertain alphas	`alphas`	list of reals	Required	N/A
gamma uncertain betas	`betas`	list of reals	Required	N/A
Descriptors	`escriptors`	list of strings	Optional	vector of `'gauv_i'` where `i = 1,2,3,...`

Gumbel Distribution

Within the gumbel optional uncertain group specification, the number of gumbel uncertain variables, and the alpha and beta parameters are required specifications. The Gumbel distribution is also referred to as the Type I Largest Extreme Value distribution. The distribution of maxima in sample sets from a population with a normal distribution will asymptotically converge to this distribution. It is commonly used to model demand variables such as wind loads and flood levels.

The density function for the Gumbel distribution is given by:

$f(x) = \alpha e^{-\alpha(x-\beta)} exp(-e^{-\alpha(x-\beta)})$

where $\mu_{GU} = \beta + \frac{0.5772}{\alpha}$ and $\sigma_{GU} = \frac{\pi}{\sqrt{6}\alpha}$ .

Table 7.12 Specification detail for gumbel uncertain variables
Description Keyword Associated Data Status Default

gumbel uncertain variables gumbel_uncertain integer Optional group no gumbel uncertain variables

gumbel uncertain alphas alphas list of reals Required N/A

gumbel uncertain betas betas list of reals Required N/A

Descriptors descriptors list of strings Optional vector of 'guuv_i' where i = 1,2,3,...

Table 7.12 Specification detail for gumbel uncertain variables
Description	Keyword	Associated Data	Status	Default
gumbel uncertain variables	`gumbel_uncertain`	integer	Optional group	no gumbel uncertain variables
gumbel uncertain alphas	`alphas`	list of reals	Required	N/A
gumbel uncertain betas	`betas`	list of reals	Required	N/A
Descriptors	`descriptors`	list of strings	Optional	vector of `'guuv_i'` where `i = 1,2,3,...`

Frechet Distribution

With the frechet uncertain optional group specification, the number of frechet uncertain variables and the alpha and beta parameters are required specifications. The Frechet distribution is also referred to as the Type II Largest Extreme Value distribution. The distribution of maxima in sample sets from a population with a lognormal distribution will asymptotically converge to this distribution. It is commonly used to model non-negative demand variables.

The density function for the frechet distribution is:

$f(x) = \frac{\alpha}{\beta}(\frac{\beta}{x})^{\alpha+1}e^{-(\frac{\beta}{x})^\alpha}$

where $\mu_F = \beta\Gamma(1-\frac{1}{\alpha})$ and $\sigma_F^2 = \beta^2[\Gamma(1-\frac{2}{\alpha})-\Gamma^2(1-\frac{1}{\alpha})]$

Table 7.13 Specification detail for frechet uncertain variables
Description Keyword Associated Data Status Default

frechet uncertain variables frechet_uncertain integer Optional group no frechet uncertain variables

frechet uncertain alphas alphas list of reals Required N/A

frechet uncertain betas betas list of reals Required N/A

Descriptors descriptors list of strings Optional vector of 'fuv_i' where i = 1,2,3,...

Table 7.13 Specification detail for frechet uncertain variables
Description	Keyword	Associated Data	Status	Default
frechet uncertain variables	`frechet_uncertain`	integer	Optional group	no frechet uncertain variables
frechet uncertain alphas	`alphas`	list of reals	Required	N/A
frechet uncertain betas	`betas`	list of reals	Required	N/A
Descriptors	`descriptors`	list of strings	Optional	vector of `'fuv_i'` where `i = 1,2,3,...`

Weibull Distribution

The Weibull distribution is commonly used in reliability studies to predict the lifetime of a device. Within the weibull uncertain optional group specification, the number of weibull uncertain variables and the alpha and beta parameters are required specifications. The Weibull distribution is also referred to as the Type III Smallest Extreme Value distribution. It is also used to model capacity variables such as material strength.

The density function for the weibull distribution is given by:

$f(x) = \frac{\alpha}{\beta} \left(\frac{x}{\beta}\right)^{\alpha-1} e^{-\left(\frac{x}{\beta}\right)^{\alpha}}$

where $\mu_W = \beta \Gamma(1+\frac{1}{\alpha})$ and $\sigma_W = \sqrt{\frac{\Gamma(1+\frac{2}{\alpha})}{\Gamma^2(1+\frac{1}{\alpha})} - 1} \mu_W$

Table 7.14 Specification detail for weibull uncertain variables
Description Keyword Associated Data Status Default

weibull uncertain variables weibull_uncertain integer Optional group no weibull uncertain variables

weibull uncertain alphas alphas list of reals Required N/A

weibull uncertain betas betas list of reals Required N/A

Descriptors descriptors list of strings Optional vector of 'wuv_i' where i = 1,2,3,...

Table 7.14 Specification detail for weibull uncertain variables
Description	Keyword	Associated Data	Status	Default
weibull uncertain variables	`weibull_uncertain`	integer	Optional group	no weibull uncertain variables
weibull uncertain alphas	`alphas`	list of reals	Required	N/A
weibull uncertain betas	`betas`	list of reals	Required	N/A
Descriptors	`descriptors`	list of strings	Optional	vector of `'wuv_i'` where `i = 1,2,3,...`

Histogram

Within the histogram uncertain optional group specification, the number of histogram uncertain variables is a required specification, the bin pairs and point pairs are optional group specifications, and the variable descriptors is an optional specification. When using a histogram variable, one must define at least one set of bin pairs or point pairs. Also note that the total number of histogram variables must be equal to the number of variables defined by bin pairs and point pairs.

For the histogram uncertain variable specification, the bin pairs and point pairs specifications provide sets of (x,y) pairs for each histogram variable. The distinction between the two types is that the former specifies counts for bins of non-zero width, whereas the latter specifies counts for individual point values, which can be thought of as bins with zero width. In the terminology of LHS [Wyss and Jorgensen, 1998], the former is a "continuous linear histogram" and the latter is a "discrete histogram" (although the points are real-valued, the number of possible values is finite). To fully specify a bin-based histogram with n bins where the bins can be of unequal width, n+1 (x,y) pairs must be specified with the following features:

x is the parameter value for the left boundary of a histogram bin and y is the corresponding count for that bin.
the final pair specifies the right end of the last bin and must have a y value of zero.
the x values must be strictly increasing.
all y values must be positive, except for the last which must be zero.
a minimum of two (x,y) pairs must be specified for each bin-based histogram.

Similarly, to specify a point-based histogram with n points, n (x,y) pairs must be specified with the following features:

x is the point value and y is the corresponding count for that value.
the x values must be strictly increasing.
all y values must be positive.
a minimum of one (x,y) pair must be specified for each point-based histogram.

For both cases, the number of pairs specifications provide for the proper association of multiple sets of (x,y) pairs with individual histogram variables. For example, in the following specification

histogram_uncertain = 3
  num_bin_pairs     = 3 4
  bin_pairs         = 5 17 8 21 10 0 .1 12 .2 24 .3 12 .4 0
  num_point_pairs   = 2
  point_pairs       = 3 1 4 1

num_bin_pairs associates the first 3 pairs from bin_pairs ((5,17),(8,21),(10,0)) with one bin-based histogram variable and the following set of 4 pairs ((.1,12),(.2,24),(.3,12),(.4,0)) with a second bin-based histogram variable. Likewise, num_point_pairs associates both of the (x,y) pairs from point_pairs ((3,1),(4,1)) with a single point-based histogram variable. Finally, the total number of bin-based variables and point-based variables must add to the total number of histogram variables specified (3 in this example).

Table 7.15 Specification detail for histogram uncertain variables
Description Keyword Associated Data Status Default

histogram uncertain variables histogram_uncertain integer Optional group no histogram uncertain variables

number of (x,y) pairs for each bin-based histogram variable num_bin_pairs list of integers Optional group no bin-based histogram uncertain variables

(x,y) pairs for all bin-based histogram variables bin_pairs list of reals Optional group no bin-based histogram uncertain variables

number of (x,y) pairs for each point-based histogram variable num_point_pairs list of integers Optional group no point-based histogram uncertain variables

(x,y) pairs for all point-based histogram variables point_pairs list of reals Optional group no point-based histogram uncertain variables

Descriptors descriptors list of strings Optional vector of 'huv_i' where i = 1,2,3,...

Table 7.15 Specification detail for histogram uncertain variables
Description	Keyword	Associated Data	Status	Default
histogram uncertain variables	`histogram_uncertain`	integer	Optional group	no histogram uncertain variables
number of (x,y) pairs for each bin-based histogram variable	`num_bin_pairs`	list of integers	Optional group	no bin-based histogram uncertain variables
(x,y) pairs for all bin-based histogram variables	`bin_pairs`	list of reals	Optional group	no bin-based histogram uncertain variables
number of (x,y) pairs for each point-based histogram variable	`num_point_pairs`	list of integers	Optional group	no point-based histogram uncertain variables
(x,y) pairs for all point-based histogram variables	`point_pairs`	list of reals	Optional group	no point-based histogram uncertain variables
Descriptors	`descriptors`	list of strings	Optional	vector of `'huv_i'` where `i = 1,2,3,...`

Interval Uncertain Variable

The interval uncertain variable is NOT a probability distribution. Although it may seem similar to a histogram, the interpretation of this uncertain variable is different. It is used in epistemic uncertainty analysis, where one is trying to model uncertainty due to lack of knowledge. In DAKOTA, epistemic uncertainty analysis is performed using Dempster-Shafer theory of evidence. In this approach, one does not assign a probability distribution to each uncertain input variable. Rather, one divides each uncertain input variable into one or more intervals. The input parameters are only known to occur within intervals: nothing more is assumed. Each interval is defined by its upper and lower bounds, and a Basic Probability Assignment (BPA) associated with that interval. The BPA represents a probability of that uncertain variable being located within that interval. The intervals and BPAs are used to construct uncertainty measures on the outputs called "belief" and "plausibility." Belief represents the smallest possible probability that is consistent with the evidence, while plausibility represents the largest possible probability that is consistent with the evidence. For more information about the Dempster-Shafer approach, see the nondeterministic evidence method, nond_evidence, in the Methods section of this Reference manual. As an example, in the following specification:

interval_uncertain = 2
  num_intervals    = 3 2
  interval_probs   = 0.2 0.5 0.3 0.4 0.6
  interval_bounds  = 2 2.5 4 5 4.5 6 1.0 5.0 3.0 5.0

there are 2 interval uncertain variables. The first one is defined by three intervals, and the second by two intervals. The three intervals for the first variable have basic probability assignments of 0.2, 0.5, and 0.3, respectively, while the basic probability assignments for the two intervals for the second variable are 0.4 and 0.6. The basic probability assignments for each interval variable must sum to one. The interval bounds for the first variable are [2, 2.5], [4, 5], and [4.5, 6]. Note that the lower bound must always come first in the bound pair. Also note that the intervals can be overlapping. The interval bounds for the second variable are [1.0, 5.0] and [3.0, 5.0]. Table 7.16 summarizes the specification details for the interval_uncertain variable.

Table 7.16 Specification detail for interval uncertain variables
Description Keyword Associated Data Status Default

interval uncertain variables interval_uncertain integer Optional group no interval uncertain variables

number of intervals defined for each interval variable num_intervals list of integers Required group None

basic probability assignments per interval interval_probs list of reals Required group. Note that the probabilities per variable must sum to one. None

bounds per interval interval_bounds list of reals Required group. Specify bounds as (lower, upper) per interval, per variable None

Descriptors descriptors list of strings Optional vector of 'iuv_i' where i = 1,2,3,...

Table 7.16 Specification detail for interval uncertain variables
Description	Keyword	Associated Data	Status	Default
interval uncertain variables	`interval_uncertain`	integer	Optional group	no interval uncertain variables
number of intervals defined for each interval variable	`num_intervals`	list of integers	Required group	None
basic probability assignments per interval	`interval_probs`	list of reals	Required group. Note that the probabilities per variable must sum to one.	None
bounds per interval	`interval_bounds`	list of reals	Required group. Specify bounds as (lower, upper) per interval, per variable	None
Descriptors	`descriptors`	list of strings	Optional	vector of `'iuv_i'` where `i = 1,2,3,...`

Correlations

Uncertain variables may have correlations specified through use of an uncertain_correlation_matrix specification. This specification is generalized in the sense that its specific meaning depends on the nondeterministic method in use. When the method is a nondeterministic sampling method (i.e., nond_sampling), then the correlation matrix specifies rank correlations [Iman and Conover, 1982]. When the method is instead a reliability (i.e., nond_local_reliability or nond_global_reliability) or polynomial chaos (i.e., nond_polynomial_chaos) method, then the correlation matrix specifies correlation coefficients (normalized covariance) [Haldar and Mahadevan, 2000]. In either of these cases, specifying the identity matrix results in uncorrelated uncertain variables (the default). The matrix input should be symmetric and have all $n^2$

entries where n is the total number of uncertain variables (all normal, lognormal, uniform, loguniform, weibull, and histogram specifications, in that order). Table 7.17 summarizes the specification details:

Table 7.17 Specification detail for uncertain correlations
Description Keyword Associated Data Status Default

correlations in uncertain variables uncertain_correlation_matrix list of reals Optional identity matrix (uncorrelated)

Table 7.17 Specification detail for uncertain correlations
Description	Keyword	Associated Data	Status	Default
correlations in uncertain variables	`uncertain_correlation_matrix`	list of reals	Optional	identity matrix (uncorrelated)

State Variables

Within the optional continuous state variables specification group, the number of continuous state variables is a required specification and the initial states, lower bounds, upper bounds, and variable descriptors are optional specifications. Likewise, within the optional discrete state variables specification group, the number of discrete state variables is a required specification and the initial states, lower bounds, upper bounds, and variable descriptors are optional specifications. These variables provide a convenient mechanism for managing additional model parameterizations such as mesh density, simulation convergence tolerances, and time step controls. Table 7.18 summarizes the details of the continuous state variable specification and Table 7.19 summarizes the details of the discrete state variable specification.

Table 7.18 Specification detail for continuous state variables
Description Keyword Associated Data Status Default

Continuous state variables continuous_state integer Optional group No continuous state variables

Initial states initial_state list of reals Optional vector values = 0.

Lower bounds lower_bounds list of reals Optional vector values = -DBL_MAX

Upper bounds upper_bounds list of reals Optional vector values = +DBL_MAX

Descriptors descriptors list of strings Optional vector of 'csv_i' where i = 1,2,3,...

Table 7.18 Specification detail for continuous state variables
Description	Keyword	Associated Data	Status	Default
Continuous state variables	`continuous_state`	integer	Optional group	No continuous state variables
Initial states	`initial_state`	list of reals	Optional	vector values = `0`.
Lower bounds	`lower_bounds`	list of reals	Optional	vector values = `-DBL_MAX`
Upper bounds	`upper_bounds`	list of reals	Optional	vector values = `+DBL_MAX`
Descriptors	`descriptors`	list of strings	Optional	vector of `'csv_i'` where `i = 1,2,3,...`

Table 7.19 Specification detail for discrete state variables
Description Keyword Associated Data Status Default

Discrete state variables discrete_state integer Optional group No discrete state variables

Initial states initial_state list of integers Optional vector values = 0

Lower bounds lower_bounds list of integers Optional vector values = INT_MIN

Upper bounds upper_bounds list of integers Optional vector values = INT_MAX

Descriptors descriptors list of strings Optional vector of 'dsv_i' where i = 1,2,3,...

Table 7.19 Specification detail for discrete state variables
Description	Keyword	Associated Data	Status	Default
Discrete state variables	`discrete_state`	integer	Optional group	No discrete state variables
Initial states	`initial_state`	list of integers	Optional	vector values = `0`
Lower bounds	`lower_bounds`	list of integers	Optional	vector values = `INT_MIN`
Upper bounds	`upper_bounds`	list of integers	Optional	vector values = `INT_MAX`
Descriptors	`descriptors`	list of strings	Optional	vector of `'dsv_i'` where `i = 1,2,3,...`

The initial_state specifications define the initial values for the continuous and discrete state variables which will be passed through to the simulator (e.g., in order to define parameterized modeling controls). The lower_bounds and upper_bounds restrict the size of the state parameter space and are frequently used to define a region for design of experiments or parameter study investigations. The descriptors specifications provide strings which will be replicated through the DAKOTA output to help identify the numerical values for these parameters. Default values for optional specifications are zeros for initial states, positive and negative machine limits for upper and lower bounds (+/- DBL_MAX, INT_MAX, INT_MIN from the float.h and limits.h system header files), and numbered strings for descriptors.

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