Dependency Analysis and Attribute Reduction in the Probabilistic Approach to Rough Sets - Feature Selection for Data and Pattern Recognition - page 100

Once the approximation region of each elementary set E was determined, the
classification table can be converted into a probabilistic decision table. The creation
of the probabilistic decision table involves adding an extra column, technically of a
newdecision attribute called Region , tomark the approximation region designation of
each elementary set. The decision table created in that way is fully deterministic with
respect to the new Region decision attribute which is representing the corresponding
three approximation regions: POS , NEG and BND . This is illustrated in the example
probabilistic decision Table 6.1 , derived from the classification Table 6.2 , with l

=

0 . 3 and u = 0 . 8.

6.5 Dependencies in Decision Tables

In this section, dependencies between attributes occurring in classification tables and
probabilistic decision tables are discussed. Specifically, our interest is in the depen-
dencies occurring between condition attributes C , or their subset, and the two-class
classification

X .Thisclas-
sification is numerically represented in both classification and probabilistic decision
tables, by values of the conditional probability P

( X , ¬ X ) formed by the target set X and its complement ¬

( X | E ) . Technically, the columns

P(E) and P can be treated as extra “attributes” associating some real values

with elementary sets of the classification generated by condition attributes. In par-
ticular, the attribute P

( X | E )

describes the distribution of the degrees of association
across different elementary sets E with the target set X . Consequently, it can be used,
in conjunction with the attribute P

( X | E )

( )

, for computing the overall degree of associa-
tion of the set of condition attributes, or of its subset, with the binary classification
of the universe U , as defined by the target set X and its complement

E

X .
In our research, we identified two dependencies, called γ — dependency and

¬

—
dependency , which provide useful measures for evaluating probabilistic decision
tables. They also provide criteria for decision table optimization through reduction
of redundant condition attributes.

ʻ

6.5.1 Functional and Partial Functional Dependencies

Functional dependencies and partially functional dependencies between attributes
of decision tables were originally explored in [ 11 ]. We will refer to them as γ —
dependencies . They capture the quality of approximation of the target set X U/D

in terms of the elementary sets of the approximation space induced by condition
attributes. We generalize them within the framework of the VPRS model by defining
the γ — dependencies [ 33 ] as a relative size of the positive region of the two class
partition

∈

( X , ¬ X ) , subject to prior setting of the values of the control parameters l

and u :