.. _variables-discrete_interval_uncertain:

"""""""""""""""""""""""""""
discrete_interval_uncertain
"""""""""""""""""""""""""""


Epistemic uncertain variable - values from one or more discrete intervals




**Topics**


discrete_variables, epistemic_uncertain_variables



.. toctree::
   :hidden:
   :maxdepth: 1

   variables-discrete_interval_uncertain-num_intervals
   variables-discrete_interval_uncertain-interval_probabilities
   variables-discrete_interval_uncertain-lower_bounds
   variables-discrete_interval_uncertain-upper_bounds
   variables-discrete_interval_uncertain-initial_point
   variables-discrete_interval_uncertain-descriptors


**Specification**

- *Alias:* None

- *Arguments:* INTEGER

- *Default:* No discrete interval uncertain variables


**Child Keywords:**

+-------------------------+--------------------+----------------------------+---------------------------------------------+
| Required/Optional       | Description of     | Dakota Keyword             | Dakota Keyword Description                  |
|                         | Group              |                            |                                             |
+=========================+====================+============================+=============================================+
| Optional                                     | `num_intervals`__          | Specify the number of intervals for each    |
|                                              |                            | variable                                    |
+----------------------------------------------+----------------------------+---------------------------------------------+
| Optional                                     | `interval_probabilities`__ | Assign probability mass to each interval    |
+----------------------------------------------+----------------------------+---------------------------------------------+
| Required                                     | `lower_bounds`__           | Specify minimum values                      |
+----------------------------------------------+----------------------------+---------------------------------------------+
| Required                                     | `upper_bounds`__           | Specify maximium values                     |
+----------------------------------------------+----------------------------+---------------------------------------------+
| Optional                                     | `initial_point`__          | Initial values for variables                |
+----------------------------------------------+----------------------------+---------------------------------------------+
| Optional                                     | `descriptors`__            | Labels for the variables                    |
+----------------------------------------------+----------------------------+---------------------------------------------+

.. __: variables-discrete_interval_uncertain-num_intervals.html
__ variables-discrete_interval_uncertain-interval_probabilities.html
__ variables-discrete_interval_uncertain-lower_bounds.html
__ variables-discrete_interval_uncertain-upper_bounds.html
__ variables-discrete_interval_uncertain-initial_point.html
__ variables-discrete_interval_uncertain-descriptors.html



**Description**

 
Discrete interval uncertain variables are epistemic
types.  They can specify a single interval per variable which may be
used in interval analysis, where the goal is to determine the interval
bounds on the output corresponding to the interval bounds on the
input. Permissible values are any integer within the bound, e.g., [1,
4], allowing discrete values of 1, 2, 3, or 4)

Discrete variables may be used to represent things like epistemic
model form uncertainty. For example, if one wants to analyze the
effect of model 1 vs. model 2 vs. model 3 in an epistemic analysis
(either an interval analysis or a Dempster-Shafer evidence theory
analysis), one can use a discrete epistemic variable to represent the
uncertainty in the model form.

More detailed continuous interval representations can specify a set of
belief structures based on intervals that may be contiguous,
overlapping, or disjoint. This is used in specifying the inputs
necessary for an epistemic uncertainty analysis using Dempster-Shafer
theory of evidence.

Other epistemic types include:

- :dakkw:`variables-continuous_interval_uncertain`
- :dakkw:`variables-discrete_uncertain_set-integer`
- :dakkw:`variables-discrete_uncertain_set-string`
- :dakkw:`variables-discrete_uncertain_set-real`




**Examples**


Let d1 be 2, 3 or 4 with probability 0.2, 4 or 5 with probability 0.5 and 6 with probability 0.3.
Let d2 be 4, 5 or 6 with probability 0.4 and 6, 7 or 8 with probability 0.6.
The following specification is for a Dempster-Shafer analysis:

.. code-block::

    discrete_interval_uncertain = 2
     num_intervals = 3 2
     interval_probs = 0.2 0.5 0.3 0.4 0.6
     lower_bounds = 2 4 6 4 6
     upper_bounds = 4 5 6 6 8





**Theory**



**Dempster-Shafer Theory of Evidence**

- Multiple intervals can be assigned to each ``discrete_interval_uncertain`` variable
- A Basic Probability Assignment (BPA) is associated with each interval. The BPA represents a probability that the value of the uncertain variable is located within that interval.
- Each interval is defined by lower and upper bounds
- Outputs are 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. Evidence is the intervals together
  with their BPA.