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POS
A
(
d
)
=
POS
A
(
X
i
).
i
∈
V
d
ↆ
A generalized decision function [
1
,
45
] with respect to
A
C
, denoted by
2
V
d
, provides a useful representation of RSM. For
u
∂
A
:
ₒ
∈
∂
A
(
)
is a set
of decision attribute values or decision classes to which
u
is possibly classified:
U
U
,
u
∂
A
(
u
)
={
i
∈
V
d
|
X
i
∩
R
A
(
u
)
=∅}
.
The generalized decision function gives an object-wise view of RSM. The lower
and upper approximations can be expressed by the generalized decision function:
LA
A
(
X
i
)
={
u
∈
U
|
∂
A
(
u
)
={
i
}}
,
UA
A
(
X
i
)
={
u
∈
U
|
∂
A
(
u
)
i
}
.
Because
∂
A
(
u
)
is defined based on
R
A
(
u
)
,wehave
u
)
u
)
∈
∂
A
(
u
)
=
∂
A
(
if
(
u
,
R
A
,
and because each object
u
is included in at least one upper approximation, we have
∂
A
(
u
)
=∅
.
The monotonic property of upper approximations is represented as:
ↆ
⃒
∂
B
(
)
ↇ
∂
A
(
)
∈
.
B
A
u
u
for all
u
U
Example 4
Remember
D =
(
U
,
C
∪{
d
}
,
{
V
a
}
)
in Table
7.1
. The generalized deci-
sion function
∂
C
is obtained as follows.
∂
C
(
u
1
)
={
unacc
}
,
(
u
2
)
=
∂
C
(
u
3
)
={
unacc
,
acc
}
,
C
∂
C
(
u
4
)
={
acc
}
,
(
u
5
)
=
∂
C
(
u
6
)
={
acc
,
good
}
,
C
∂
C
(
u
7
)
={
good
}
.
C
, a quality of classification (or quality of approximation) of the decision
attribute
d
with respect to
A
is defined by:
For
A
ↆ
)
=
|
POS
A
(
d
)
|
ʳ
A
(
d
.
(7.8)
|
U
|
It measures to what degree objects are correctly classified by RSM.
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