continuous_interval_uncertain

Epistemic uncertain variable - values from one or more continuous intervals

Topics

continuous_variables, epistemic_uncertain_variables

Specification

• Alias: interval_uncertain

• Arguments: INTEGER

• Default: no continuous interval uncertain variables

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
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

Description

Continuous 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. All values between the bounds are permissible. 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:

• discrete_interval_uncertain

• integer

• string

• real

Examples

The following specification is for an interval analysis:

continuous_interval_uncertain = 2
 lower_bounds = 2.0 4.0
 upper_bounds = 2.5 5.0

The following specification is for a Dempster-Shafer analysis:

continuous_interval_uncertain = 2
 num_intervals = 3 2
 interval_probs = 0.25 0.5 0.25 0.4 0.6
 lower_bounds = 2.0 4.0 4.5 1.0 3.0
 upper_bounds = 2.5 5.0 6.0 5.0 5.0

Here 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. The BPA for the first variable indicates that it is twice as likely that the value occurs on the interval [4,5] than either [2,2.5] or [4.5,6].

Theory

The 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. The continuous interval uncertain variable is used in both interval analysis and in Dempster-Shafer theory of evidence.

Interval Analysis

• Only one interval is allowed for each continuous_interval_uncertain variable

• The interval is defined by lower and upper bounds

• The value of the random variable lies somewhere in this interval

• Output is the minimum and maximum function value conditional on the specified interval

Dempster-Shafer Theory of Evidence

• Multiple intervals can be assigned to each continuous_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.