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equivalently,
UNP
A
(
UA
A
(
d
)
=
U
−
X
i
).
i
∈
V
d
The unpredictable region is the set of all objects which cannot be classified to any
decision class.
We can define the positive region of
d
with respect to
ʲ
and
A
in the same manner
of RSM,
POS
A
(
POS
A
(
d
)
=
X
i
).
i
∈
V
d
The quality of classification of
d
can be also defined in the same manner,
POS
A
(
)
=
|
d
)
|
ʳ
A
(
d
.
|
U
|
The generalized decision function in RSM can be extended in VPRSM. However,
differently from RSM, we define two functions. They are called lower and upper
generalized decision functions, denoted by
ʻ
and
˅
, respectively. For each
u
∈
U
,
ʻ
A
(
A
u
)
={
i
∈
V
d
|
μ
X
i
(
u
)
≥
1
−
ʲ
}
,
˅
A
(
A
u
)
={
i
∈
V
d
|
μ
X
i
(
u
)>ʲ
}
.
The lower generalized decision of
u
is the set of the decision values to which the
membership degree of
u
is more than or equal to 1
. The upper generalized
decision of
u
is the set of the decision values to which the membership degree of
u
is
more than
−
ʲ
. The upper generalized decision corresponds to the generalized decision
in RSM. By the definitions, the lower and upper generalized decision functions are
closely related to the lower and upper approximations,
ʲ
∈
ʻ
A
(
LA
A
(
i
u
)
⃔
u
∈
X
i
),
∈
˅
A
(
UA
A
(
)
⃔
∈
X
i
).
i
u
u
So, they have the inclusion relation:
ʻ
A
(
)
ↆ
˅
A
(
u
u
).
(7.13)
Any two objects in the same equivalence class take the same values of generalized
decision functions.
R
A
,ʻ
A
(
)
=
ʻ
A
(
˅
A
(
)
=
˅
A
(
u
)
∈
u
)
u
).
For each
(
u
,
u
and
u
(7.14)
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