Information Technology Reference
In-Depth Information
For each object
u
i
∈
U
, symbols
∗
in the row of
u
i
indicate the objects in
D
−
(
in the column indicate the objects in
D
+
(
u
i
)
∗
u
i
)
, while symbols
. For example,
D
−
(
and
D
+
(
u
3
)
={
u
3
,
u
4
,
u
5
,
u
6
,
u
7
}
u
3
)
={
u
1
,
u
2
,
u
3
}
.
There are three decision classes
X
b
={
u
5
,
u
7
}
,
X
m
={
u
2
,
u
4
,
u
6
}
and
X
g
=
{
for bad, med and good, respectively. The upward and downward unions of
those decision classes are,
u
1
,
u
3
}
X
b
=
X
m
={
X
g
={
,
u
1
,
u
2
,
u
3
,
u
4
,
u
6
}
,
u
1
,
u
3
}
,
U
X
b
={
X
m
={
X
g
=
u
5
,
u
7
}
,
u
2
,
u
4
,
u
5
,
u
6
,
u
7
}
,
U
.
Given a decision table, the inconsistency with respect to the dominance principle
is captured by the difference between upper and lower approximations of the unions
of decision classes. Given a condition attribute set
A
ↆ
∈
C
, and
i
V
d
,thelower
X
i
X
i
of
X
i
approximation LA
A
(
)
and the upper approximation UA
A
(
)
are defined,
respectively, by:
X
i
D
A
(
X
i
}
,
LA
A
(
)
={
u
∈
U
|
u
)
ↆ
X
i
D
A
(
X
i
UA
A
(
)
={
∈
|
)
∩
=∅}
.
u
U
u
X
i
of
X
i
Similarly, the lower approximation LA
A
(
)
and upper approximation
X
i
UA
A
(
)
are defined, respectively, by:
X
i
D
A
(
X
i
}
,
LA
A
(
)
={
u
∈
U
|
u
)
ↆ
X
i
D
A
(
X
i
UA
A
(
)
={
u
∈
U
|
u
)
∩
=∅}
.
then all objects dominating
u
do not belong to
X
i
−
1
,
i.e., there exists no evidence for
u
X
i
If
u
belongs to LA
A
(
)
X
t
−
1
in view of the monotonicity assumption.
Therefore, we can say that
u
certainly belongs to
X
i
∈
. On the other hand if
u
belongs to
X
i
then
u
is dominating an object belonging to
X
i
UA
A
(
)
, i.e., there exists evidence
X
i
in view of the monotonicity assumption. Therefore, we can say that
u
possibly belongs to
X
i
for
u
∈
X
i
. The similar interpretations can be applied to LA
A
(
)
and
X
i
UA
A
(
.
The difference between the upper and lower approximations is called a boundary.
The boundaries of an upward union
X
i
)
and a downward union
X
i
, denoted by
X
i
X
i
BN
A
(
)
and BN
A
(
)
, are defined by:
X
i
X
i
X
i
BN
A
(
)
=
UA
A
(
)
\
LA
A
(
),
X
i
X
i
X
i
BN
A
(
)
=
UA
A
(
)
\
LA
A
(
).
Objects in the boundary region of an upward or downward union are classified
neither to that union nor to the complement with certainty.
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