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plays also an essential role in the definitions
of approximations, also called
approximation regions
: it represents the likelihood
that a random object
e
The notion of prior probability
P
(
X
)
U
is a member of the target set
X
in the absence of any
classification knowledge about the object. If the classification knowledge is available,
as represented by the equivalence relation
IND
, the likelihood of membership in the
set
X
of objects belonging to different elementary sets can either increase, or decrease,
or stay approximately the same as the prior probability
P
∈
. These variations in
the set
X
membership likelihood across different elementary sets are reflected in the
definitions of set approximation regions, which characterize areas of the universe
U
with significantly increased, significantly decreased, or approximately unchanged
target set
X
membership probability.
Each elementary set is classified either into one of approximation regions of the
set
X
, i.e. a positive region
POS
u
, a negative region
NEG
l
, or a boundary region
BND
l
,
u
. The upper limit
u
defines the positive region, or lower approximation, of
the target set
X
, with the constraint 0
(
X
)
<
P
(
X
)<
u
≤
1. It represents the least
acceptable degree of the conditional probability
P
, or the set overlap degree,
to include the elementary set
E
in the positive region. The positive region, or the
lower approximation of the target set
X
, denoted as
POS
u
, is a collection of objects
for which the probability of membership in the target set
X
is significantly higher than
the prior probability
P
(
X
|
E
)
, where the term
significantly higher
is precisely specified
by the parameter
u
(as defined by some external criteria):
(
X
)
POS
u
(
)
=∪{
:
(
|
)
≥
}
.
X
E
P
X
E
u
(6.4)
The lower limit
l
defines the negative region of the target set
X
, with the constraint
0
≤
l
<
P
(
X
)<
1. It is the highest acceptable degree of the conditional probability
to include the elementary set
E
in the negative region. The negative region
of the target set
X
, denoted as
NEG
l
is a collection of objects for which the probability
of membership in the target set X is significantly lower than the prior probability
P
P
(
X
|
E
)
, where the term
significantly lower
is precisely specified by the parameter
l
(as defined by some external, application-related, criteria):
(
X
)
NEG
l
(
X
)
=∪{
E
:
P
(
X
|
E
)
≤
l
}
.
(6.5)
The boundary region denoted as
BND
l
,
u
, is a collection of remaining objects
which cannot be classified with sufficient certainty into either positive or negative
regions. For the boundary area objects, the probability of membership in the target
set
X
is not significantly different from the prior probability
P
(
X
)
, that is:
BND
l
,
u
(
X
)
=∪{
E
:
l
<
P
(
X
|
E
)<
u
}
.
(6.6)
Regardless of the choice of lower and upper limit control parameters, the positive
and negative approximation regions are subsets of
absolute approximation regions
,
as described in the next subsection.
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