EXISTING MATHEMATICAL METHODS FOR UNCERTAINTY ASSESSMENT - Modelling Uncertainty in Flood Forecasting Systems - page 43

where a and b are the two extremes (minimum and maximum) of x defined in the PDF,
p X (x) .
Uncertainty measures in possibility theory
The possibility theory here is defined as fuzzy measures considering the membership function as
a possibility distribution. The uncertainty existence in possibility theory can be measured
in terms of nonspecificity and strife . Indeed the possibilistic uncertainty is defined as the sum of
nonspecificity and strife (Eq. (3.44)). The nonspecificity type of uncertainty arises from lack of
specificity in choosing one alternative (the true one) from several alternatives. This type
of uncertainty vanishes, and complete certainty is achieved, when only one alternative
is present. Therefore, the amount of uncertainty associated with a set of alternatives may
thus be measured by the amount of information needed to remove the uncertainty. The
nonspecificity type uncertainty in possibility theory, N(r) can be expressed as (Klir, 1992):
(3.42)
The other type of uncertainty, strife, in possibility theory, which arises from conflicts
among various sets of alternatives can be expressed as
(3.43)
Therefore the total uncertainty U T (r) in possibility theory is given by
(3.44)
Equation (3.44) can be rewritten as
(3.45)