Rough Sets Modeling - Probabilistic Composition of Preferences, Theory and Applications - page 101

New proposals in the same direction were presented in Sant ' Anna and Moreira

Filho ( 2013 ) based on joining neighboring classes. While VC DRSA and variants

of it are based on the size of the classes involved, this last approach is based on the
distance between the values of the decision attribute in the two classes. To evaluate
such distance the probabilistic transformation is used.
The idea behind this probabilistic approach is to reduce the numerical precision
in identifying according to the decision attributes by reducing the number of dif-
ferent possible values for them. This is made possible by the transformation of the
vector of initial observations of the attribute into a vector of probabilities of pre-
senting the highest (or lowest) value, because many alternatives have a very low
probability of being the best (or the worst).
In this development is considered the most frequent case, of a unique decision
attribute. The extension of the methods proposed for the case of more than one
decision attribute is simple, once a dominance relationship is established in
accordance with the set of all decision attributes together. To this end one can also
use the composition of probabilistic preferences.

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10.2 Rough Sets Theory

If two alternatives have the same values for all condition attributes, they are con-
sidered indiscernible. It counts as an inconsistency two alternatives classi ed as

indiscernible having different classi cations according to the decision attributes.

For each set of condition attributes P and each alternative x of the universe U of
alternatives to be classi

ed, denote by P(x) the set of alternatives indiscernible of x

according to P:

P ðÞ¼ y 2 U = x and y are indiscernible by P f g:

ned two
approximations: the lower approximation of X by P and the upper approximation of
XbyP,de

For every subset X of U and every set of condition attributes P are de

ned by

P ðÞ¼ x 2 UP ðÞ X f j g

and

P ðÞ¼ x 2 UP ðÞ\ X 6¼ f j 0 g:

It is easy to see that

P ðÞ X P ðÞ: