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

Description::
The \c 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 \ref topic-variables page to browse the
available variables by type and domain.


<b> Variable Types </b>
- 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 \c initial_value is used as the only value
    for the state variable for all other methods, unless \c active
    \c state is invoked.
  - See more details on the \ref topic-state_variables page.

<b> Variable Domains </b>

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

<b> Ordering of Variables </b>

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:
-# design
 -# continuous 
 -# discrete integer 
 -# discrete string
 -# discrete real
-# aleatory uncertain
 -# continuous 
 -# discrete integer 
 -# discrete string
 -# discrete real
-# epistemic uncertain
 -# continuous 
 -# discrete integer 
 -# discrete string
 -# discrete real
-# state
 -# continuous 
 -# discrete integer 
 -# discrete string
 -# 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.

<b> Active Variables </b>

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 \ref variables-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.

<b>Inferred Default Values and Bounds</b>

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 \c
num_trials) and hypergeometric (0 to min(\c num_drawn, \c
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, \f$\mu + 3 \sigma\f$] for exponential, gamma, frechet,
weibull, poisson, negative binomial, geometric, and unspecified
lognormal, and [\f$\mu - 3 \sigma\f$, \f$\mu + 3 \sigma\f$] for gumbel
and unspecified normal.

When an intial point is needed and not explcitly specified in user
input, it is assigned as described in the \c initial_point or \c
initial_state specification, e.g., \ref
variables-normal_uncertain-initial_point. For example, 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.

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

\verbatim
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'
\endverbatim

In the next example, defaults are employed. In this case, \c
initial_point will default to a vector of \c 0. values, \c
upper_bounds will default to vector values of \c DBL_MAX (the maximum
number representable in double precision for a particular platform),
\c lower_bounds will default to a vector of \c -DBL_MAX values, and \c
descriptors will default to a vector of <tt>'cdv_i'</tt> strings,
where \c i ranges from one to two:

\verbatim
variables,
	continuous_design = 2
\endverbatim

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 \c uncertain_correlation_matrix.

\verbatim
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
\endverbatim

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

\verbatim
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'
\endverbatim

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:

\verbatim
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'
\endverbatim

Theory::
Faq::
See_Also::	
