EXISTING MATHEMATICAL METHODS FOR UNCERTAINTY ASSESSMENT - Modelling Uncertainty in Flood Forecasting Systems - page 42

3.5.2 Transformation by principle of uncertainty invariance

The principle of uncertainty invariance, which is based on measures of uncertainty in
various theories, provides some basis for establishing meaningful relationships between
uncertainty theories. In particular, the transformation from one theory, say T 1 to another,
say T 2 , with the principle of uncertainty invariance requires that (Klir and Yuan, 1995;
Klir and Wierman, 1998):

1. The amount of uncertainty associated with a given piece of information or a
given situation be preserved when we move from T 1 to T 2 .
2. The degree of belief in T 1 be converted to its counterparts in T 2 by an appropriate
scale.

The requirement 1 guarantees that no uncertainty is added or eliminated solely by
changing the mathematical theory by which a particular phenomenon is formalised.
Therefore the amount of uncertainty measured before and after the conversion based on
the respective theories must remain equal. The requirement 2 guarantees that certain
properties, which are considered essential in a given context (such as ordering or
proportionality of relevant values) are preserved by the transformation.
The transformation under discussion here is from probability (defined by a PDF) to
possibility (defined by a MF as a possibility distribution) and vice versa. The expressions
for the uncertainty measures in the probability and possibility theories are given below. A
detailed coverage on these measures is beyond the scope of this thesis. The readers are
referred to, e.g. Klir and Yuan (1995) and Klir and Wierman (1998), Ayyub (2001) for
more coverage on this topic.

Uncertainty measures in probability theory
When uncertainty is formalised in terms of probability theory, the only measure of
uncertainty applicable is the Shannon entropy measure (Klir, 1992). The Shannon entropy
measure given by Shannon (1948) measures uncertainty due to conflict that arises from
probability mass function of a body of information (Ayyub, 2001). This uncertainty is
also referred to as strife or discord . More specifically, it measures the average uncertainty
(in bits) associated with the prediction of outcomes in a random experiment. The
Shannon entropy, represented by H(p), is given by

(3 .40)

As can be observed in the H(p) function, its range is form 0 (for p X (x) =1 , x X ) to log 2 | X |

(for uniform probability distribution with p X (x) =1/| X | for all x X ).
For the continuous random variable, the Shannon entropy can be extended to

(3.41)