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In the Pawlak's rough set model [
11
], the notion of upper approximation of a set
is defined as a union of all elementary sets which have non-empty intersection with
the set. The generalized definition of upper approximation
UPP
l
(
in the VPRS
approach, as in the original rough set model, is a set union of the positive region and
of the boundary region giving:
X
)
UPP
l
(
X
)
=∪{
E
:
P
(
X
|
E
)>
l
}
.
(6.7)
Note that the generalized definition coincides with the Pawlak's definition of upper
approximation when
l
1, it can be easily demonstrated
that the VPRSM definitions of positive, negative and boundary regions, become
equivalent to the original rough set model's definitions of lower approximation,
negative and boundary regions [
11
].
One can also note that, in general, as opposed to Pawlak's rough sets, it is not
true that
POS
u
(
=
0. In addition, when
u
=
. Consequently, the
rough set cannot be defined in the VPRSM as a pair consisting of upper and lower
approximation, as it is done in Pawlak's rough sets [
11
].
A frequently asked question is to how to set, or tune, the values of the precision
control parameters
l
and
u
. The author's point of view is that apart from the general
constraint 0
X
)
ↆ
X
and it is not true that
X
ↆ
UPP
l
(
X
)
1, the settings of the parameters are entirely
dependent on the requirements of a practical application, while being likely subjective
or obtained via the cost-benefit analysis [
27
].
≤
l
<
P
(
X
)<
u
≤
6.2.2 Absolute Set Approximation Regions
To describe the areas of the universe characterized by an unconstrained increase, or
decrease of the set
X
membership probability, the following definitions of absolute
approximation regions are applicable. It this case, no parameters to specify “suffi-
ciently” high increase, or decrease of the set membership probability in those areas
are used. We call these areas
absolute approximation regions
.
The
absolute boundary region
of the target set
X
is a definable region of the uni-
verse
U
consisting of those elementary sets which are characterized by the unchanged
probability of membership in the target set
X
ↆ
U
, that is:
BND
∗
(
X
)
=∪{
E
:
P
(
X
|
E
)
=
P
(
X
)
}
.
(6.8)
As it can be easily verified, in the absolute boundary region, each elementary set
E
is probabilistically independent from the target set
X
, i.e.
P
.
Consequently, the whole boundary region is independent from the target set
X
.In
other words, the objects in the absolute boundary regions can be considered entirely
unrelated with the target set.
(
X
∩
Y
)
=
P
(
X
)
P
(
Y
)
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