Uncertain variables may be categorized as either aleatory or epistemic and either continuous or discrete. Continuous aleatory uncertain variables include normal, lognormal, uniform, loguniform, triangular, exponential, beta, gamma, gumbel, frechet, weibull, and histogram bin distributions. Discrete aleatory uncertain variables include poisson, binomial, negative binomial, geometric, hypergeometric, and histogram point distributions. In addition to aleatory uncertain variables defined by probability distributions, DAKOTA also supports epistemic uncertain variables that are non-probabilistic. The interval type specification is a continuous epistemic type that supports both simple bounded intervals as well as basic probability assignment (BPA) belief structures, where a BPA defines the uncertainty in a variable through providing one or more intervals in which the variable may lie along with varying levels of belief for each interval.
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 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. 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, weibull, poisson, negative binomial, and geometric (which have no bounds specifications); these bounds are [0, ] for exponential, gamma, frechet, weibull, poisson, negative binomial, geometric, and unspecified lognormal, and [
,
] for gumbel and unspecified normal. For other types of parameter studies (vector and centered), an inferred initial starting point is needed for the uncertain variables. All uncertain variables are initialized to their means for these studies, where mean values for bounded normal and bounded lognormal may additionally be repaired to satisfy any 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.
In addition to tags and bounds specifications, normal variables include mean and standard deviation specifications, lognormal variables include lambda and zeta, mean and standard deviation, or mean and 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 bin variables include abscissa and either ordinate or count specifications, poisson variables include lambda specifications, binomial and negative binomial variables include probability per trial and number of trials specifications, geometric variables include probability per trial specifications, hypergeometric variables include the specification of the total population, selected population, and number drawn, histogram point variables include abscissa and count 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 and descriptors. Continuous and discrete range types include lower and upper bounds, and discrete set types include the admissible set values. 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.
The ordering of variables is important, and a consistent ordering is employed throughout the DAKOTA software. It is the same ordering as shown in dakota.input.summary and as presented in the outline of this chapter. That ordering can be summarized as continuous followed by discrete integer followed by discrete real within each of the following types: design, aleatory uncertain, epistemic uncertain, and state. Ordering of variable types below this granularity (e.g., from normal to histogram bin within continuous aleatory uncertain) is defined somewhat arbitrarily, but is enforced consistently throughout the code.
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_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_int = 1 initial_state = 100 set_values = 100 212 375 descriptors = 'load_case'
Refer to the DAKOTA Users Manual [Adams et al., 2010] for discussion on how different iterators view these mixed variable sets.
variables, <set identifier> <continuous design variables specification> <discrete design range variables specification> <discrete design set integer variables specification> <discrete design set real 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 bin uncertain variables specification> <poisson uncertain variables specification> <binomial uncertain variables specification> <negative binomial uncertain variables specification> <geometric uncertain variables specification> <hypergeometric uncertain variables specification> <histogram point uncertain variables specification> <uncertain correlation specification> <interval uncertain variables specification> <continuous state variables specification> <discrete state range variables specification> <discrete state set integer variables specification> <discrete state set real variables specification>
Referring to dakota.input.summary, 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 is zero). However, at least one type of variables that are active for the iterator in use must have nonzero size or an input error message will result. The following sections describe each of these specification components in additional detail.
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.
Description | Keyword | Associated Data | Status | Default |
Variables set identifier | id_variables | string | Optional | use of last variables parsed |
initial_point
specifications provide the point in design space from which an iterator is started and default to either zeros (continuous and discrete range) or middle values (discrete sets). The descriptors
specifications supply strings which will be replicated through the DAKOTA output to identify the numerical values for these parameters; these default to numbered strings.
For continuous and discrete range variables, the lower_bounds
and upper_bounds
restrict the size of the feasible design space and are frequently used to prevent nonphysical designs. Default values are 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). 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 variables, 1.e+9 for discrete integer 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 integer), 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).
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 . (repaired to bounds, if required) |
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... |
For continuous variables, 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.
Description | Keyword | Associated Data | Status | Default |
Discrete design range variables | discrete_design_range | integer | Optional group | no discrete design variables |
Initial point | initial_point | list of integers | Optional | vector values = 0 (repaired to bounds, if required) |
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 'ddriv_i' where i = 1,2,3,... |
discrete_design_set_integer
(Table 7.4) and discrete_design_set_real
(Table 7.5), respectively.Description | Keyword | Associated Data | Status | Default |
Discrete design set of integer variables | discrete_design_set_integer | integer | Optional group | no discrete design set of integer variables |
Initial point | initial_point | list of integers | Optional | middle set values (mean indices, rounded down) |
Number of values for each variable | num_set_values | list of integers | Optional | equal distribution |
Set values | set_values | list of integers | Required | N/A |
Descriptors | descriptors | list of strings | Optional | vector of 'ddsiv_i' where i = 1,2,3,... |
Description | Keyword | Associated Data | Status | Default |
Discrete design set of real variables | discrete_design_set_real | integer | Optional group | no discrete design set of real variables |
Initial point | initial_point | list of reals | Optional | middle set values (mean indices, rounded down) |
Number of values for each variable | num_set_values | list of integers | Optional | equal distribution |
Set values | set_values | list of reals | Required | N/A |
Descriptors | descriptors | list of strings | Optional | vector of 'ddsrv_i' where i = 1,2,3,... |
These specifications of probability distributions directly support the use of probabilistic uncertainty quantification methods such as sampling, reliability, and stochastic expansion methods. However, the inclusion of lower and upper distribution bounds for all uncertain variable types (either explicitly defined, implicitly defined, or inferred; see Variables Description) also allows the use of these variables within methods that rely on a bounded region to define a set of function evaluations (i.e., design of experiments and some parameter study methods). Each distribution also provides optional uncertain variable descriptors (default values are numbered strings) that supply identifiers that help associate the numerical values with the uncertain parameters as they appear within the DAKOTA output. Tables 7.6 through 7.23 summarize the details of the aleatory uncertain variable specifications.
The density function for the normal distribution is:
where and
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 [Wyss and Jorgensen, 1998]). 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.
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,... |
For the lognormal variables, one may specify either the mean and standard deviation
of the actual lognormal distribution, the mean
and error factor
of the actual lognormal distribution, or the mean
("lambda") and standard deviation
("zeta") of the underlying normal distribution. 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:
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 (1 of 3 selections) | N/A |
lognormal uncertain standard deviations | std_deviations | list of reals | Required (1 of 3 selections) | N/A |
lognormal uncertain error factors | error_factors | list of reals | Required (1 of 3 selections) | N/A |
lognormal uncertain lambdas | lambdas | list of reals | Required (1 of 3 selections) | N/A |
lognormal uncertain zetas | zetas | list of reals | Required (1 of 3 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,... |
where and
are the upper and lower bounds of the uniform distribution, respectively. The mean of the uniform distribution is
and the variance is
. Note that this distribution is a special case of the more general beta distribution.
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,... |
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,... |
The density function for the triangular distribution is:
if , and
if , and 0 elsewhere. In these equations,
is the lower bound,
is the upper bound, and
is the mode of the triangular distribution.
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,... |
The density function for the exponential distribution is given by:
where and
. Note that this distribution is a special case of the more general gamma distribution.
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,... |
where is the gamma function and
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.
Solving these for and
gives:
Note that the uniform distribution is a special case of this distribution for parameters .
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,... |
The density function for the gamma distribution is given by:
where and
. Note that the exponential distribution is a special case of this distribution for parameter
.
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 | descriptors | list of strings | Optional | vector of 'gauv_i' where i = 1,2,3,... |
The density function for the Gumbel distribution is given by:
where and
.
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,... |
The density function for the frechet distribution is:
where and
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,... |
The density function for the weibull distribution is given by:
where and
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,... |
Within the histogram bin uncertain optional group specification, the number of histogram bin uncertain variables is a required specification, the number of pairs is an optional key for apportionment of abscissas/ordinates/counts, specification of abscissas and either ordinates or counts is required, and the variable descriptors is an optional specification. When using a histogram bin variable, one must define at least one bin (with two bounding value pairs).
The abscissas
specification define abscissa values ("x" coordinates) for the PDF of each histogram variable. When paired with counts
, the specifications provide sets of (x,c) pairs for each histogram variable where
c
defines a count (i.e., a frequency or relative probability) associated with a bin. If using bins of unequal width and specification of probability densities is more natural, then the counts
specification can be replaced with a ordinates
specification ("y" coordinates) in order to support interpretation of the input as (x,y) pairs defining the profile of a "skyline" PDF. Conversion between the two specifications is straightforward: a count/frequency is a cumulative probability quantity defined from the product of the ordinate density value and the
x
bin width. Thus, in the cases of bins of equal width, ordinate and count specifications are equivalent. In addition, ordinates and counts may be relative values; it is not necessary to scale them as all user inputs will be normalized.
To fully specify a bin-based histogram with n
bins (potentially of unequal width), n+1
(x,c) or
(x,y) pairs must be specified with the following features:
x
is the parameter value for the left boundary of a histogram bin and c
is the corresponding count for that bin. Alternatively, y
defines the ordinate density value for this bin within a skyline PDF. The right boundary of the bin is defined by the left boundary of the next pair. c
or y
value of zero. x
values must be strictly increasing. c
or y
values must be positive, except for the last which must be zero.
(x,c) or
(x,y) pairs with individual histogram variables. For example, in the following specification
histogram_bin_uncertain = 2 num_pairs = 3 4 abscissas = 5 8 10 .1 .2 .3 .4 counts = 17 21 0 12 24 12 0
num_pairs
associates the first 3 (x,c) pairs from
abscissas
and counts
((5,17),(8,21),(10,0)) with one bin-based histogram variable, where one bin is defined between 5 and 8 with a count of 17 and another bin is defined between 8 and 10 with a count of 21. The following set of 4
(x,c) pairs
((.1,12),(.2,24),(.3,12),(.4,0)) defines a second bin-based histogram variable containing three equal-width bins with counts 12, 24, and 12 (middle bin is twice as probable as the other two).
Description | Keyword | Associated Data | Status | Default |
histogram bin uncertain variables | histogram_bin_uncertain | integer | Optional group | no histogram bin uncertain variables |
key to apportionment among bin-based histogram variables | num_pairs | list of integers | Optional | equal distribution |
sets of abscissas for bin-based histogram variables | abscissas | list of reals | Required | N/A |
sets of ordinates for bin-based histogram variables | ordinates | list of reals | Required (1 of 2 selections) | N/A |
sets of counts for bin-based histogram variables | counts | list of reals | Required (1 of 2 selections) | N/A |
Descriptors | descriptors | list of strings | Optional | vector of 'hubv_i' where i = 1,2,3,... |
The density function for the poisson distribution is given by:
where is the frequency of events happening, and
x
is the number of events that occur. The poisson distribution returns samples representing number of occurrences in the time period of interest.
Description | Keyword | Associated Data | Status | Default |
poisson uncertain variables | poisson_uncertain | integer | Optional group | no poisson uncertain variables |
poisson uncertain lambdas | lambdas | list of reals | Required | N/A |
Descriptors | descriptors | list of strings | Optional | vector of 'puv_i' where i = 1,2,3,... |
n
independent tests or trials, where each trial has the probability p
of failing or being defective. Each particular test can be considered as a Bernoulli trial.The density function for the binomial distribution is given by:
where p
is the probability of failure per trial and n
is the number of trials.
Description | Keyword | Associated Data | Status | Default |
binomial uncertain variables | binomial_uncertain | integer | Optional group | no binomial uncertain variables |
binomial uncertain prob_per_trial | prob_per_trial | list of reals | Required | N/A |
binomial uncertain num_trials | num_trials | list of integers | Required | N/A |
Descriptors | descriptors | list of strings | Optional | vector of 'biuv_i' where i = 1,2,3,... |
n
successes, where each test has a probability p
of success.The density function for the negative binomial distribution is given by:
where p
is the probability of success per trial and n
is the number of successful trials.
Description | Keyword | Associated Data | Status | Default |
negative binomial uncertain variables | negative_binomial_uncertain | integer | Optional group | no negative binomial uncertain variables |
negative binomial uncertain success prob_per_trial | prob_per_trial | list of reals | Required | N/A |
negative binomial uncertain success num_trials | num_trials | list of integers | Required | N/A |
Descriptors | descriptors | list of strings | Optional | vector of 'nbuv_i' where i = 1,2,3,... |
The density function for the geometric distribution is given by:
where p
is the probability of failure per trial.
Description | Keyword | Associated Data | Status | Default |
geometric uncertain variables | geometric_uncertain | integer | Optional group | no geometric uncertain variables |
geometric uncertain prob_per_trial | prob_per_trial | list of reals | Required | N/A |
Descriptors | descriptors | list of strings | Optional | vector of 'geuv_i' where i = 1,2,3,... |
The density function for the hypergeometric distribution is given by:
where N is the total population, m is the number of items in the selected population (e.g. the number of white balls in the full urn of N items), and n is the number of balls drawn.
Description | Keyword | Associated Data | Status | Default |
hypergeometric uncertain variables | hypergeometric_uncertain | integer | Optional group | no hypergeometric uncertain variables |
hypergeometric uncertain total_population | total_population | list of integers | Required | N/A |
hypergeometric uncertain selected_population | selected_population | list of integers | Required | N/A |
hypergeometric uncertain num_drawn | num_drawn | list of integers | Required | N/A |
Descriptors | descriptors | list of strings | Optional | vector of 'hguv_i' where i = 1,2,3,... |
Point histograms are similar to Discrete Design Real Set Variables and Discrete State Real Set Variables, but differ in the inclusion of information on the relative probabilities of observing the different values within the set.
Within the histogram point uncertain optional group specification, the number of histogram point uncertain variables is a required specification, the number of pairs is an optional key for apportionment of abscissas and counts, the sets of abscissas and counts are required, and the variable descriptors are optional. When using a histogram point variable, one must define at least one set of abscissa/count pairs. As for Histogram Bin Distribution, the abscissas
specifications define abscissa values ("x" coordinates) for the PDF of each histogram variable. When paired with counts
, the specifications provide sets of (x,c) pairs for each histogram variable where
c
defines a count (i.e., a frequency or relative probability) associated with a point.
To fully specify a point-based histogram with n
points, n
(x,c) pairs (note that
(x,c) and
(x,y) are equivalent in this case) must be specified with the following features:
x
is the point value and c
is the corresponding count for that value. x
values must be strictly increasing. c
values must be positive. num_pairs
specification provides for the proper association of multiple sets of
(x,c) or
(x,y) pairs with individual histogram variables. For example, in the following specification,
histogram_point_uncertain = 2 num_pairs = 2 3 abscissas = 3 4 100 200 300 counts = 1 1 1 2 1
num_pairs
associates the (x,c) pairs ((3,1),(4,1)) with one point-based histogram variable (where the values 3 and 4 are equally probable) and associates the
(x,c) pairs ((100,1),(200,2),(300,1)) with a second point-based histogram variable (where the value 200 is twice as probable as either 100 or 300).
Description | Keyword | Associated Data | Status | Default |
histogram point uncertain variables | histogram_point_uncertain | integer | Optional group | no histogram point uncertain variables |
key to apportionment among point-based histogram variables | num_pairs | list of integers | Optional | equal distribution |
sets of abscissas for point-based histogram variables | abscissas | list of reals | Required | N/A |
sets of counts for point-based histogram variables | counts | list of reals | Required | N/A |
Descriptors | descriptors | list of strings | Optional | vector of 'hupv_i' where i = 1,2,3,... |
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 stochastic expansion (i.e., nond_polynomial_chaos
or nond_stoch_collocation
) 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 Description | Keyword | Associated Data | Status | Default |
correlations in aleatory uncertain variables | uncertain_correlation_matrix | list of reals | Optional | identity matrix (uncorrelated) |
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 = 2.0 2.5 4.0 5.0 4.5 6.0 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], and the interval bounds for the second variable are [1.0, 5.0] and [3.0, 5.0].
Note that the intervals can be overlapping or disjoint. Table 7.25 summarizes the specification details for the interval_uncertain
variable.
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,... |
initial_state
specifications provide the initial values for the state variables which will be passed through to the simulator (e.g., in order to define parameterized modeling controls). The remaining specifications are analagous to those for Design 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 . (repaired to bounds, if required) |
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,... |
Description | Keyword | Associated Data | Status | Default |
Discrete state range variables | discrete_state_range | integer | Optional group | No discrete state variables |
Initial states | initial_state | list of integers | Optional | vector values = 0 (repaired to bounds, if required) |
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 'dsriv_i' where i = 1,2,3,... |
Description | Keyword | Associated Data | Status | Default |
Discrete state set of integer variables | discrete_state_set_integer | integer | Optional group | no discrete state set of integer variables |
Initial state | initial_state | list of integers | Optional | middle set values (mean indices, rounded down) |
Number of values for each variable | num_set_values | list of integers | Optional | equal distribution |
Set values | set_values | list of integers | Required | N/A |
Descriptors | descriptors | list of strings | Optional | vector of 'dssiv_i' where i = 1,2,3,... |
Description | Keyword | Associated Data | Status | Default |
Discrete state set of real variables | discrete_state_set_real | integer | Optional group | no discrete state set of real variables |
Initial state | initial_state | list of reals | Optional | middle set values (mean indices, rounded down) |
Number of values for each variable | num_set_values | list of integers | Optional | equal distribution |
Set values | set_values | list of reals | Required | N/A |
Descriptors | descriptors | list of strings | Optional | vector of 'dssrv_i' where i = 1,2,3,... |