Rough Sets Modeling - Probabilistic Composition of Preferences, Theory and Applications - page 105

Table 10.1 Data aggregation

Alternatives D P 1 P 2 D 0

A

2 1 1 3

B

3 1 2 3

C

5 2 3 5

D

6 3 3 6

E

7 4 4 7

F

8 5 5 8

attribute. Indeed, if a set P 1 of condition attributes contained in the set of attributes
P is a reduct, then no indiscernibility according to attributes in P/P 1 is able to
withdraw alternatives from any boundary. With the expansion of the classes
determined by the decision attributes there is no way to raise the possibility of
occurrence of any such withdrawal. Thus the size of the reducts cannot increase.
On the other hand, joining classes opens the possibility of appearing new reducts
of smaller size. Indeed, if, before aggregation, P was a reduct and P 1 ⊆ P was not,

this is necessarily due to the existence of at least one indiscernible pair of alter-
natives (x, y) according to some element of P/P 1 that would place such alternatives
in a boundary situation. With the aggregation of classes, such alternatives may
become members of the same class.
The following example demonstrates concretely how the quality of the
approximation may increase and the size of the reducts decrease with the union of
classes.
In Table 10.1 ,D 0 results from joining two classes at the lower end of the range
of values of D. With D as the decision attribute, P 1 and P 2 are needed to achieve the
maximum quality of approximation of 1. But replacing D by D 0 , it is easy to see
that P 1 alone offers this quality of approximation of 1.

10.5 A Probabilistic Aggregation Algorithm

In this strategy, the values of the decision attribute are treated as the result of
distortion of unknown values by random disturbances. This allows, as in preceding
chapters, for replacing them by the probabilities of presenting an extreme value.
The probability of presenting the highest value for two alternatives with the same
evaluation will be the same and will be lower than that of an alternative belonging
to a higher class. This argument is reversed when the transformation is based on the
probability of presenting the lowest value.
What is explored to increase the index of quality of approximation is the fact that
the probability of maximizing (or minimizing) becomes so close to zero for alter-
natives in extreme classes that, with a suitable decimal approximation, values in
neighbor classes may be considered equal.