1Blurb::
2Epistemic uncertain variable - values from one or more continuous intervals
3
4Description::
5Continuous interval uncertain variables are epistemic types.
6They can specify a single interval per variable which may
7be used in interval analysis, where the goal is to determine the
8interval bounds on the output corresponding to the interval bounds on
9the input. All values between the bounds are permissible.
10More detailed continuous interval representations can
11specify a set of belief structures based on intervals that may be
12contiguous, overlapping, or disjoint. This is used in specifying the
13inputs necessary for an epistemic uncertainty analysis using
14Dempster-Shafer theory of evidence.
15
16Other epistemic types include:
17\li \ref variables-discrete_interval_uncertain
18\li discrete_uncertain_set \ref variables-discrete_uncertain_set-integer
19\li discrete_uncertain_set \ref variables-discrete_uncertain_set-string
20\li discrete_uncertain_set \ref variables-discrete_uncertain_set-real
21
22<!--
23\li \ref variables-continuous_interval_uncertain
24
25In addition to continuous and discrete aleatory probability
26distributions, %Dakota provides support for continuous and discrete
27epistemic uncertainties through the keywords:
28
29Interval-based and set variables do not represent probability distributions.
30-->
31
32Topics::	continuous_variables, epistemic_uncertain_variables
33Examples::
34
35The following specification is for an interval analysis:
36\verbatim
37continuous_interval_uncertain = 2
38 lower_bounds = 2.0 4.0
39 upper_bounds = 2.5 5.0
40\endverbatim
41
42
43The following specification is for a Dempster-Shafer analysis:
44\verbatim
45continuous_interval_uncertain = 2
46 num_intervals = 3 2
47 interval_probs = 0.25 0.5 0.25 0.4 0.6
48 lower_bounds = 2.0 4.0 4.5 1.0 3.0
49 upper_bounds = 2.5 5.0 6.0 5.0 5.0
50\endverbatim
51
52Here 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.
53The 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.  The BPA for the first variable indicates that it is twice as likely that the value occurs
54on the interval [4,5] than either [2,2.5] or [4.5,6].
55
56Theory::
57The continuous 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.
58The continuous interval uncertain variable is used in both interval analysis and in Dempster-Shafer theory of evidence.
59
60\li interval analysis
61-only one interval is allowed for each \c continuous_interval_uncertain variable
62-the interval is defined by lower and upper bounds
63-the value of the random variable lies somewhere in this interval
64-output is the minimum and maximum function value conditional
65on the specified interval
66
67\li Dempster-Shafer theory of evidence
68-multiple intervals can be assigned to each \c continuous_interval_uncertain variable
69-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.
70-each interval is defined by lower and upper bounds
71-outputs are called "belief" and "plausibility."
72Belief represents the smallest possible probability that is consistent with the evidence,
73while plausibility represents the largest possible probability that is consistent with the evidence.
74Evidence is the intervals together with their BPA.
75
76Faq::
77See_Also::
78