Rough Sets Modeling - Probabilistic Composition of Preferences, Theory and Applications - page 106

As in VC-DRSA (where the index l for input objects in the lower approximation
is not fixed), an approximation threshold can be determined according to the cir-

cumstances. The limit to the approximation, denoted by l* may be changed by the
analyst.
As in VC-DRSA, a very low value of l* may result in an excessive relaxation for
entering values in the approximation classes. On the other hand, a high value for l*
allows for the union of classes that are not classes of rare alternatives in the tail of
the classi

cation.
As a rule, the approximation to the third decimal enables a satisfactory number
of unions. This corresponds to consider equal those evaluations differing by less
than 0.0005. With the probabilistic transformation, this means equaling to zero all
evaluations of the probability of being the best below 0.0005.
If the important distinctions are between the alternatives presenting large values,
the small probabilities will be assigned to the alternatives with low values for the
decision attribute. Similarly, if the important distinctions are between the alterna-
tives presenting large values, the small probabilities that will be considered equal
will appear in the classes with high evaluations.
Merging classes by performing changes only in the decision attribute, no pre-
mise for application of DRSA is violated. Only classes determined by the decision
attribute are pasted. Thus, the reduction of classes by the probabilistic transfor-
mation does not prevent the subsequent enforcement of any method of extraction of
rules or imposition of consistency and the method for reduction of the number of
classes can be applied in conjunction with any of the techniques developed to
improve the quality of the approximation starting from VC-DRSA.
A combination of RST with the probabilistic transformation was also performed
in the reverse direction in Sant ' Anna ( 2004 ).

10.6 Example of Car Models

Consider the problem of choice among 20 car models. Assume that we intend to
explain the decision attribute D in Table 10.2 by the condition attributes C 1 and C 2 .
The quality of approximation is 0.4, as eight of the 20 alternatives are consis-
tently classi ed: Car1, Car2, Car3, Car4, Car1, Car12, Car16 and Car17.

Applying the transformation of the decision variable into probabilities of pref-
erence (assuming a normal distribution with variance estimated by the observed
variance) and rounding to three decimal places, the decision attribute receives the
values in the last column. Thus, Car18, Car19 and Car20 are joined in the same
decision class, with the value of 0 for the decision attribute. With this reduction in
the number of classes, these three alternatives become consistently classi

ed and

the quality of approximation increases to 0.55.