Hybrid processing of stochastic and subjective uncertainty data Page: 13 of 22
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about ill-defined situations such as abnormal environment responses, a non-Bayesian
hybrid analysis has a useful role. A reasonable approach to this problem that does not
assume nonexistent stochastic information is to provide for smooth transitioning from
subjective (fuzzy) characterization to stochastic characterization as information about
inputs is obtained.
1) Uniform Scale Factors. First, consider the case where the extent of knowledge about a
problem is fractionally partitioned between stochastic and subjective portions. An input
variable to an analysis whose variation characteristics are known partly stochastically and
partly subjectively can be represented by a hybrid number with the relative
stochastic/subjective information apportioned according to a scaling fraction:
h(x) = axp(x) + (1-a)xf(x) (3)
where a is an estimated scale factor representing the fractional stochasticity of the overall
knowledge (Oas1), and where x and + are operators on x values.
The scale factor is a scalar, which Eqn. 3 suggests can be used to fractionally compress the
abscissa (numeric) representation of the probabilistic constituent of variability by a, along
with compression of the fuzzy constituent of uncertainty by (1-a). The total variation is
then additive along the abscissa, i.e., a scaled sum of the two constituents.
A visual description of the uncertainty represented by a scaled hybrid number is shown in
Fig. 8, for an example scale factor of 1/2. The axes represent the numeric variability due
to the constituents of stochastic knowledge and subjective knowledge. The dashed
indication of a fuzzy function has been scaled down by a factor of two along the x axis
from the subjective estimate. The dot-dashed indication of a probability function has also
been scaled down by a factor of two from the stochastic estimate. The x-axis sum of the
scaled variabilities is shown plotted as a three-dimensional hybrid number (solid lines).
This formulation is understood most clearly if the spread and shape of the fuzzy function
and the probability function do not interact with each other, and if separate stochastic and
fuzzy mathematics are used.
In the limited case for which the scale factor applies uniformly to all input variables in a
mathematical analysis (and therefore also to the output), the conventional mathematical
properties (identities, commutative property, and associative property for addition and
multiplication, and multiplication distributive over addition and subtraction) hold for scale-
factor arithmetic with no further requirements'.
1 An important point is that variability for an operand must be entered only one time in equations in
which there are multiple occurrences. For example, A - A = 0. (We may be uncertain about the value of
A , but not about the result of subtracting any value of A from itself).11
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Cooper, J.A.; Ferson, S. & Ginzburg, L. Hybrid processing of stochastic and subjective uncertainty data, report, November 1, 1995; Albuquerque, New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc673112/m1/13/: accessed March 29, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.