Information Technology Reference
In-Depth Information
In the VPRS approach, each equivalence class
E
of the indiscernibility relation
IND
is assigned two measures which are: the relative “size” of the class
E
within
universe
U
, referred to as the probability
P(E)
of
E
, and the relative “size” of the
target set
X
within an elementary set
E
, referred to as the conditional probability
P
. The conditional probability, in this context, is just a measure of the degree
of overlap between the target set
X
and the elementary set
E
. These two measures
can be approximated from data respectively by:
(
X
|
E
)
card
(
E
)
P
(
E
)
=
(6.1)
card
(
U
)
and
card
(
X
∩
E
)
P
(
X
|
E
)
=
(6.2)
card
(
E
)
where
card
denotes set cardinality.
The target set
X
may be undefinable [
11
], which informally means that, in gen-
eral, it cannot be expressed as a set union of some elementary sets forming our
classification knowledge. That is, in general, the set definability criterion:
IND
∗
:
X
=∪{
E
∈
E
ↆ
X
}
(6.3)
is not satisfied.
This lack of definability is more common than not in applications. The original
rough set theory, as introduced by Pawlak [
10
,
11
], deals with this problem via the
notions of lower and upper set approximations. However, in many applications, when
the target set is not definable, this approach is not sufficient due to the absence of
numeric assessments of the degree of association of elementary sets with the target
set
X
.
The VPRS approach extends the rough set model to make it more flexible, by
replacing the full inclusion relation with the overlap relation in the definitions of
set approximations. Two precision control parameters called lower limit
l
and upper
limit
u
are used in the definition of lower approximation of the target set
X
, or its
complement. In this way, one can control the process of computation of approxima-
tions of the target set to identify such approximations which satisfy user-imposed
criteria, such as for example, characterizing classes of patients with an elevated (or
reduced) risk of a disease.
6.2.1 Set Approximations in the VPRS Approach
The approximations of the target set in the VPRS approach are defined in terms of
unions of some elementary sets, as controlled by lower limit
l
and upper limit
u
precision parameters.
Search WWH ::
Custom Search