Information Technology Reference
In-Depth Information
7.3.3 Structure-Based Reducts in Variable Precision
Rough Set Models
Before we define structure-based reducts in VPRSM, we firstly introduce Q-reducts.
They preserve the quality of classification with the parameter
ʲ
.
Definition 8
([
4
,
5
]) Let
ʲ
∈[
0
,
0
.
5
)
be an admissible error rate. A Q-reduct with
ʲ
in VPRSM is a minimal condition attribute subset
A
ↆ
C
satisfying the following
conditions:
ʳ
A
(
)
=
ʳ
C
(
d
d
),
(VPQ1)
B
satisfies (VPQ1)
for all
B
ↇ
A
.
(VPQ2)
Remark 6
In VPRSM, approximations are no longer monotonic with respect to the
set inclusion of condition attributes. Hence, condition (
VPQ1
) is not monotonic with
respect to condition attributes, namely,
A
satisfies (
VPQ1
)but
B
A
does not. In
that case, we modify the preserving condition of reducts by adding a condition like
(
VPQ2
). We notice that Beynon [
4
,
5
] originally proposed Q-reducts (the author
called them
Ↄ
ʲ
-reducts) using only (
VPQ1
).
We define 4 kinds of structure-based reducts in VPRSM [
20
], which are already
discussed in the classical RSM.
Definition 9
([
20
,
33
]) Let
ʲ
∈[
0
,
0
.
5
)
be an admissible error rate.
A P-reduct
2
with
•
ʲ
in VPRSM is a minimal condition attribute subset
A
ↆ
C
satisfying the following conditions:
POS
A
(
POS
C
(
d
)
=
d
),
(VPP1)
B
satisfies (VPP1)
for all
B
ↇ
A
.
(VPP2)
•
An L-reduct with
ʲ
in VPRSM is a minimal condition attribute subset
A
ↆ
C
satisfying the following condition:
LA
A
(
LA
C
(
X
i
)
=
X
i
)
for all
i
∈
V
d
.
(VPL)
•
A B-reduct with
ʲ
in VPRSM is a minimal condition attribute subset
A
ↆ
C
satisfying the following conditions:
BN
A
(
BN
C
(
X
i
)
=
X
i
)
for all
i
∈
V
d
,
(VPB1)
B
satisfies (VPB1)
for all
B
ↇ
A
.
(VPB2)
2
Strictly speaking, P-reducts do not appear in [
20
].
Search WWH ::
Custom Search