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γ
l
,
u
(
X
|
C
)
=
P
(
POS
u
(
X
|
C
)
∪
NEG
l
(
X
|
C
)),
(6.19)
where
POS
u
(
, respectively are positive and negative regions
of
X
in the approximation space induced on
U
by the set of condition attributes
C
.
This dependency measure reflects the proportion of objects in the universe
U
that
can be classified as members of the target set
X
, or a complement of the target set
X
,
with sufficient certainty, as given by the parameters
l
and
u
.
The
γ
l
,
u
(
X
|
C
)
and
NEG
l
(
X
|
C
)
X
|
C
)
measure was inspired by the partial functional dependency mea-
sure
γ
(
introduced by Pawlak [
11
], which is given as a fraction of objects of
the universe
U
that can be uniquely classified, based on their condition attributes
value combinations, as members of some classes of the decision attribute
D
.More
precisely, in the VPRS model terms:
D
|
C
)
γ
(
D
|
C
)
=
P
(
POS
1
(
F
|
C
)).
(6.20)
F
∈
U
/
D
The above measures play useful role in decision table analysis and reduction of
condition attributes.
6.5.2
λ
—Dependency Measure
Another kind of dependency, unrelated to the the
γ
—
dependencies
measure and
conveying different kind of information, is a parametric
ʻ
—
dependency
, denoted as
ʻ
l
,
u
(
[
33
]. It captures the average, or expected degree of the probabilistic con-
nection between elementary sets
E
(
E
X
|
C
)
∈
U/C
) and the binary classification
(
X
,
¬
X
)
corresponding to the target set
X
and its complement
X
. The dependency is defined
as a normalized expected degree of deviation of the conditional probability
P
¬
(
X
|
E
)
(
)
from the prior probability
P
X
:
P
(
E
)
|
P
(
X
|
E
)
−
P
(
X
)
|
E
ↆ
POS
u
(
X
|
C
)
∪
NEG
l
(
X
|
C
)
ʻ
l
,
u
(
X
|
C
)
=
,
(6.21)
2
P
(
X
)(
1
−
P
(
X
))
where 2
P
is a normalization factor equal to the theoreticallymaximum
value of the numerator summation, achievable only when
X
is definable in Pawlak's
rough set's sense, independent of settings of the parameters
l
and
u
. The higher the
deviation, the stronger the probabilistic connection between conditional attributes
C
and the decision partition
(
X
)(
1
−
P
(
X
))
(
X
,
¬
X
)
, and vice versa, with the total probabilistic
independence occurring at
0.
In the framework of the Bayesian rough set model, the parametric
ʻ
l
,
u
(
X
|
C
)
=
ʻ
—
dependency
ʻ
reduces to non-parametric
—
dependency
defined as:
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