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P
(
E
)
|
P
(
X
|
E
)
−
P
(
X
)
|
E
∈
U
/
C
ʻ(
X
|
C
)
=
.
(6.22)
2
P
(
X
)(
1
−
P
(
X
))
The non-parametric
ʻ
—
dependency
ʻ(
X
|
C
)
is a normalized expected degree of
deviation of the conditional probability
P
(
X
|
E
)
from the prior probability
P
(
X
)
.
The main practical advantage of the non-parametric
—
dependency
is the absence of
any external parameters, whichmay be difficult to obtain, to compute the dependency.
Another useful advantage is its
monotonicity
with respect to condition attributes, as
explained in the next section.
ʻ
6.6
λ
—
Dependency
-Based Reduct
The application of idea of
reduct
, introduced by Pawlak [
10
,
11
], allows for opti-
mization of representation of classification knowledge by providing a technique
for removal of redundant attributes. The concept of reduct generated considerable
amount of research interest, primarily as a method for feature selection [
1
,
2
,
6
,
8
,
12
-
14
,
16
,
19
-
21
,
23
-
25
]. The general notion of reduct is applicable to the optimiza-
tion of classification tables and probabilistic decision tables. The following theorem
[
13
] demonstrates that the
—
dependency
measure is
monotonic
, which means that
expanding the set of condition attributes
B
ʻ
ↆ
C
will not result in the decrease of the
dependency level
ʻ(
X
|
B
)
.
Theorem 1
C be a subset of condition attributes on U and let “a” be any
condition attribute. Then the following relation holds:
Let B
ↆ
ʻ(
X
|
B
)
≤
ʻ(
X
|
B
∪{
a
}
).
(6.23)
As a consequence of the Theorem, the notion of the
probabilistic reduct
of
attributes
RED
ↆ
C
can be defined as a minimal subset of attributes preserving
ʻ
(
,
¬
)
the
.
The reduct satisfies the following two important properties:
—
dependency
with the target classification
X
X
ʻ(
X
|
RED
)
=
ʻ(
X
|
C
)
(6.24)
and for any attribute
a
∈
RED
:
ʻ(
X
|
RED
−{
a
}
)<ʻ(
X
|
RED
).
(6.25)
—
reducts
, can be computed using any methods
available for reduct computation in the framework of the Pawlak's original rough
set approach, and in particular, a single
The probabilistic reducts, called
ʻ
ʻ
—
reduct
can be easily computed from a
classification table using the following
ʻ
—
Reduction
algorithm:
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