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7.2.3 Reducts in Rough Set Models
7.2.3.1 Preserving Positive Region, Quality, and Generalized Decisions
Attribute reduction is to find important subsets of condition attributes by dropping
as many as possible other condition attributes while preserving some specific infor-
mation of RSM for a decision table
. A minimal subset of condition attributes
preserving the information is called a relative (or decision) reduct. In this chapter,
we call it “reduct” for short. Reducts are originally defined to preserve the positive
region POS
C
(
D
d
)
[
36
,
43
].
Definition 1
([
36
,
43
]) A reduct is a minimal condition attribute subset
A
ↆ
C
satisfying the following condition:
POS
A
(
d
)
=
POS
C
(
d
).
(P)
Here, the minimality is defined in terms of the set inclusion, i.e., there is no proper
subset
A
ↂ
A
satisfying (
P
).
A condition attribute subset
A
satisfying (
P
) preserves the information of the
certain classification in the decision table. Generally, there exist more than one reduct
in a decision table. The intersection of all reducts is called the core. Every element in
the core is an essential condition attribute to preserve the information of POS
C
(
)
.
The core can be empty. On the other hand, the condition attributes which do not
belong to any reducts can be dropped without deterioration of the information. We
call the original reduct a P-reduct.
d
Remark 2
Condition (
P
) is monotonic with respect to the set inclusion of condition
attributes, i.e., for
A
ↆ
A
ↆ
C
we have POS
A
(
d
)
=
POS
C
(
d
)
implies POS
A
(
d
)
=
POS
C
(
. Hence, the above minimality condition for
A
is equivalent to that there is
no condition attribute
a
d
)
∈
A
such that
A
\{
a
}
satisfies (
P
).
Example 5
Remember
in Table
7.1
. The set of all condition
attributes
C
obviously satisfies condition (
P
), but it is not a P-reduct because a proper
subset
A
D =
(
U
,
C
∪{
d
}
,
{
V
a
}
)
satisfies (
P
). On the other hand,
A
is a P-reduct because all of
proper subsets of
A
do not preserve the positive region: POS
={
Pr
,
Ma
}
}
(
d
)
={
u
1
,
u
7
}
and
{
Pr
POS
}
(
d
)
=
POS
∅
(
d
)
=∅
.
{
Ma
We can define another kind of reducts preserving the quality of classification
[
23
,
39
,
40
].
Definition 2
([
40
]) A Q-reduct is a minimal condition attribute subset
A
ↆ
C
satisfying the following condition:
ʳ
A
(
d
)
=
ʳ
C
(
d
).
(Q)
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