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In-Depth Information
POS
A
(
X
)
=
LA
A
(
X
),
BND
A
(
)
=
BN
A
(
),
X
X
NEG
A
(
X
)
=
U
\
UA
A
(
X
).
In the rest of this section, we consider RSM for decision tables, namely, we only
deal with approximations of decision classes
X
={
X
1
,
X
2
,...,
X
p
}
with respect
to subsets of condition attributes
A
ↆ
C
.
Example 3
Remember the decision classes
X
unacc
={
u
1
,
u
2
}
,
X
acc
={
u
3
,
u
4
,
u
5
}
and
X
good
={
u
6
,
u
7
}
of the decision table in Table
7.1
. The lower and upper approx-
imations with respect to
C
of
X
unacc
,
X
cc
and
X
good
are obtained as follows:
LA
C
(
X
unacc
)
={
u
1
}
,
UA
C
(
X
unacc
)
={
u
1
,
u
2
,
u
3
}
,
LA
C
(
X
acc
)
={
u
4
}
,
UA
C
(
X
acc
)
={
u
2
,
u
3
,
u
4
,
u
5
,
u
6
}
,
LA
C
(
X
good
)
={
u
7
}
,
UA
C
(
X
good
)
={
u
5
,
u
6
,
u
7
}
.
good.
Moreover, we can also see that each approximation is the union of equivalence
classes included in the approximation, e.g., UA
C
(
We can see that LA
C
(
X
i
)
ↆ
X
i
ↆ
UA
C
(
X
i
)
for each
i
=
unacc
,
acc
,
X
acc
)
={
u
2
,
u
3
}∪{
u
4
}∪{
u
5
,
u
6
}
.
We reduce condition attributes to
A
={
Pr
}
. The approximations become:
LA
A
(
X
unacc
)
={
u
1
}
,
UA
A
(
X
unacc
)
={
u
1
,
u
2
,
u
3
,
u
4
,
u
5
,
u
6
}
,
LA
A
(
X
acc
)
=∅
,
UA
A
(
X
acc
)
={
u
2
,
u
3
,
u
4
,
u
5
,
u
6
}
,
u
7
}
.
The approximations with respect to
A
are coarser than those with respect to
C
,
namely, LA
A
(
LA
A
(
X
good
)
={
u
7
}
,
UA
A
(
X
good
)
={
u
2
,
u
3
,
u
4
,
u
5
,
u
6
,
X
i
)
ↆ
LA
C
(
X
i
)
and UA
A
(
X
i
)
ↇ
UA
C
(
X
i
)
for each
i
=
unacc
,
acc
,
good.
For every
X
i
, the lower approximation LA
A
(
X
i
)
and the boundary BN
A
(
X
i
)
can be represented using all upper approximations of decision classes UA
A
(
X
1
)
,
UA
A
(
X
2
),...,
UA
A
(
X
p
)
:
LA
A
(
X
i
)
=
UA
A
(
X
i
)
\
UA
A
(
X
j
),
(7.5)
j
∈
V
d
\{
i
}
BN
A
(
X
i
)
=
UA
A
(
X
i
)
∩
UA
A
(
X
j
).
(7.6)
j
∈
V
d
\{
i
}
All upper approximations form a cover of
U
:
U
=
UA
A
(
X
i
).
(7.7)
i
∈
V
d
A positive region with respect to
A
ↆ
C
is also defined for the decision attribute
d
or equivalently for the decision table
. It is the union of all positive regions of
decision classes, i.e., the set of objects which are certainly classified to exactly one
of the decision classes:
D
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