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To overcome such shortcoming of the classical RSM, the variable precision rough
set model (VPRSM) was proposed [
53
,
54
]. Let
V
a
}
a
∈
AT
)
be a decision table. In definitions of lower and upper approximations in VPRSM,
the following rough membership function of an object
u
with respect to an object set
X
D =
(
,
=
∪{
}
,
{
U
AT
C
d
ↆ
U
and an attribute set
A
ↆ
AT
plays an important role:
)
=
|
R
A
(
u
)
∩
X
|
A
μ
X
(
u
.
|
R
A
(
u
)
|
X
(
gives the degree to which the object
u
belongs to the set
X
under
the attribute set
A
. It can be interpreted as the conditional probability of
u
The value
μ
u
)
∈
X
under
u
.
Because the rough membership function of an object is defined based not on
the object but its equivalence class, we define a rough membership function of an
equivalence class
E
∈
R
A
(
u
)
∈
U
/
R
A
for
X
:
)
=
|
∩
|
E
X
μ
X
(
.
E
|
E
|
An important property of the function is that given two equivalence classes
E
∈
E
falls between those
E
,
U
/
R
A
the rough membership of the union
E
∪
of
E
and
E
, namely,
E
)
}≤
μ
X
(
E
)
≤
E
)
}
.
min
{
μ
X
(
E
), μ
X
(
E
∪
max
{
μ
X
(
E
), μ
X
(
(7.9)
7.3.2 Variable Precision Rough Set Models
≤
ʲ<ʱ
≤
Given precision parameters 0
1, lower and upper approximations of
X
with respect to
A
in VPRSM are defined as:
LA
A
(
X
X
)
={
u
∈
U
|
μ
(
u
)
≥
ʱ
}
,
UA
A
(
X
X
)
={
u
∈
U
|
μ
(
u
)>ʲ
}
.
The boundary of
X
is defined by BN
ʱ,ʲ
A
UA
A
(
LA
A
(
(
X
)
=
X
)
\
X
)
. When
ʱ
=
1
and
ʲ
=
0, the approximations of
X
are the same as those of the classical RSM.
LA
A
(
X
)
is the set of objects whose degrees of membership to
X
are not less than
ʱ
.
On the other hand, UA
A
(
X
)
is the set of objects whose degrees of membership to
X
are more than
ʲ
. In this chapter, we restrict our discussion to the situation that
ʱ
=
. Under that situation, we have the dual property LA
A
(
1
−
ʲ
and
ʲ
∈[
0
,
0
.
5
)
X
)
=
UA
A
(
A
A
U
U
\
U
\
X
)
, because
μ
X
(
u
)
=
1
−
μ
X
(
u
)
. We call
ʲ
an admissible error rate.
\
We denote LA
1
−
ʲ
A
and BN
1
−
ʲ,ʲ
A
by LA
A
(
and BN
A
(
(
X
)
(
X
)
X
)
X
)
, respectively.
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