Dakota Reference Manual  Version 6.10
Explore and Predict with Confidence
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variables


Specifies the parameter set to be iterated by a particular method.

Topics

This keyword is related to the topics:

Specification

Alias: none

Argument(s): none

Child Keywords:

Required/Optional Description of Group Dakota Keyword Dakota Keyword Description
Optional id_variables

Name the variables block; helpful when there are multiple

Optional active

Set the active variables view a method will see

Optional
(Choose One)
Variable Domain (Group 1) mixed

Maintain continuous/discrete variable distinction

relaxed

Allow treatment of discrete variables as continuous

Optional continuous_design

Design variable - continuous

Optional discrete_design_range

Design variable - discrete range-valued

Optional discrete_design_set

Design variable - discrete set-valued

Optional normal_uncertain Aleatory uncertain variable - normal (Gaussian)
Optional lognormal_uncertain Aleatory uncertain variable - lognormal
Optional uniform_uncertain Aleatory uncertain variable - uniform
Optional loguniform_uncertain Aleatory uncertain variable - loguniform
Optional triangular_uncertain Aleatory uncertain variable - triangular
Optional exponential_uncertain Aleatory uncertain variable - exponential
Optional beta_uncertain Aleatory uncertain variable - beta
Optional gamma_uncertain Aleatory uncertain variable - gamma
Optional gumbel_uncertain Aleatory uncertain variable - gumbel
Optional frechet_uncertain Aleatory uncertain variable - Frechet
Optional weibull_uncertain Aleatory uncertain variable - Weibull
Optional histogram_bin_uncertain

Aleatory uncertain variable - continuous histogram

Optional poisson_uncertain Aleatory uncertain discrete variable - Poisson
Optional binomial_uncertain Aleatory uncertain discrete variable - binomial
Optional negative_binomial_uncertain Aleatory uncertain discrete variable - negative binomial
Optional geometric_uncertain Aleatory uncertain discrete variable - geometric
Optional hypergeometric_uncertain Aleatory uncertain discrete variable - hypergeometric
Optional histogram_point_uncertain

Aleatory uncertain variable - discrete histogram

Optional uncertain_correlation_matrix

Correlation among aleatory uncertain variables

Optional continuous_interval_uncertain

Epistemic uncertain variable - values from one or more continuous intervals

Optional discrete_interval_uncertain

Epistemic uncertain variable - values from one or more discrete intervals

Optional discrete_uncertain_set

Epistemic uncertain variable - discrete set-valued

Optional continuous_state

State variable - continuous

Optional discrete_state_range

State variables - discrete range-valued

Optional discrete_state_set

State variable - discrete set-valued

Optional linear_inequality_constraint_matrix Define coefficients of the linear inequality constraints
Optional linear_inequality_lower_bounds Define lower bounds for the linear inequality constraint
Optional linear_inequality_upper_bounds Define upper bounds for the linear inequality constraint
Optional linear_inequality_scale_types Specify how each linear inequality constraint is scaled
Optional linear_inequality_scales Define the characteristic values to scale linear inequalities
Optional linear_equality_constraint_matrix Define coefficients of the linear equalities
Optional linear_equality_targets Define target values for the linear equality constraints
Optional linear_equality_scale_types Specify how each linear equality constraint is scaled
Optional linear_equality_scales Define the characteristic values to scale linear equalities

Description

The variables specification in a Dakota input file specifies the parameter set to be iterated by a particular method. In the case of

  • An optimization study: These variables are adjusted in order to locate an optimal design.
  • Parameter studies/sensitivity analysis/design of experiments: These parameters are perturbed to explore the parameter space.
  • Uncertainty analysis: The variables are associated with distribution/interval characterizations which are used to compute corresponding distribution/interval characterizations for response functions.

To accommodate these different studies, Dakota supports different:

  • Variable types
    • design
    • aleatory uncertain
    • epistemic uncertain
    • state
  • Variable domains
    • continuous
    • discrete
      • discrete range
      • discrete integer set
      • discrete string set
      • discrete real set

Use the variables page to browse the available variables by type and domain.

Variable Types

  • Design Variables
    • Design variables are those variables which are modified for the purposes of seeking an optimal design.
    • The most common type of design variables encountered in engineering applications are of the continuous type. These variables may assume any real value within their bounds.
    • All but a handful of the optimization algorithms in Dakota support continuous design variables exclusively.
  • Aleatory Uncertain Variables
    • Aleatory uncertainty is also known as inherent variability, irreducible uncertainty, or randomness.
    • Aleatory uncertainty is predominantly characterized using probability theory. This is the only option implemented in Dakota.
  • Epistemic Uncertain Variables
    • Epistemic uncertainty is uncertainty due to lack of knowledge.
    • In Dakota, epistemic uncertainty is assessed by interval analysis or the Dempster-Shafer theory of evidence
    • Continuous or discrete interval or set-valued variables are used to define set-valued probabilities or basic probabiliy assignments (BPA) which define a belief structure.
    • Note that epistemic uncertainty can also be modeled with probability density functions (as done with aleatory uncertainty). Dakota does not support this capability.
  • State Variables
    • State variables consist of "other" variables which are to be mapped through the simulation interface, in that they are not to be used for design and they are not modeled as being uncertain.
    • State variables provide a convenient mechanism for managing additional model parameterizations such as mesh density, simulation convergence tolerances, and time step controls.
    • Only parameter studies and design of experiments methods will iterate on state variables.
    • The initial_value is used as the only value for the state variable for all other methods, unless active state is invoked.
    • See more details on the state_variables page.

Variable Domains

Continuous variables are typically defined by bounds. Discrete variables can be defined in one of three ways, which are discussed on the page discrete_variables.

Ordering of Variables

The ordering of variables is important, and a consistent ordering is employed throughout the Dakota software. The ordering is shown in dakota.input.summary (and in the hierarchical order of this reference manual) and can be summarized as:

  1. design
    1. continuous
    2. discrete integer
    3. discrete string
    4. discrete real
  2. aleatory uncertain
    1. continuous
    2. discrete integer
    3. discrete string
    4. discrete real
  3. epistemic uncertain
    1. continuous
    2. discrete integer
    3. discrete string
    4. discrete real
  4. state
    1. continuous
    2. discrete integer
    3. discrete string
    4. discrete real

Ordering of variable types below this granularity (e.g., from normal to histogram bin within aleatory uncertain - continuous ) is defined somewhat arbitrarily, but is enforced consistently throughout the code.

Active Variables

The reason variable types exist is that methods have the capability to treat variable types differently. All methods have default behavior that determines which variable types are "active" and will be assigned values by the method. For example, optimization methods will only vary the design variables - by default.

The default behavior should be described on each method page, or on topics pages that relate to classes of methods. In addition, the default behavior can be modified using the active keyword.

At least one type of variables that are active for the method in use must have nonzero size (at least 1 active variable) or an input error message will result.

Inferred Default Values and Bounds

The concept of active variables allows any Dakota variable type to be used in any method context. Some methods, e.g., bound-constrained optimization or multi-dimensional or centered parameter studies, require bounds and/or an initial point on the variables, however uncertain variables may not be naturally defined in terms of these characteristics.

Distribution lower and upper bounds are explicit portions of the normal, lognormal, uniform, loguniform, triangular, and beta specifications, whereas they are implicitly defined for others. For example, bounds are naturally defined for histogram bin, histogram point, and interval variables (from the extreme values within the bin/point/interval specifications) as well as for binomial (0 to num_trials) and hypergeometric (0 to min(num_drawn, num_selected)) variables.

If not specified, distribution bounds are also inferred for normal and lognormal (if optional bounds are unspecified) as well as for exponential, gamma, gumbel, frechet, weibull, poisson, negative binomial, and geometric (which have no bounds specifications); these bounds are [0, $\mu + 3 \sigma$] for exponential, gamma, frechet, weibull, poisson, negative binomial, geometric, and unspecified lognormal, and [ $\mu - 3 \sigma$, $\mu + 3 \sigma$] for gumbel and unspecified normal.

When an intial point is needed, uncertain variables are initialized to their means, where mean values for bounded normal and bounded lognormal may be further adjusted to satisfy any user-specified distribution bounds, mean values for discrete integer range distributions are rounded down to the nearest integer, and mean values for discrete set distributions are rounded to the nearest set value.

Examples

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), 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_range = 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_range = 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_set
          integer = 1
        initial_state = 100
        set_values    = 100 212 375
        descriptors   = 'load_case'