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that is:
egabs
(
X
|
IND
)
=
P
(
E
)(
1
−
P
(
X
))
=
1
−
P
(
X
)
=
P
(
¬
X
).
(6.18)
IND
∗
E
∈
Similarly,
egabs
(
¬
X
|
IND
)
=
P
(
X
)
in the functional case.
In the casewhen
egabs
=
0
,
P
(
X
∩
E
)
=
P
(
X
)
P
(
E
)
, for every elementary set
E
.
This means that for every elementary set
E
,
P
.
This is equivalent to saying that all elementary sets are probabilistically independent
from the target set
X
. In practical terms, it means that the occurrence of an object
belonging to any of the elementary sets does not affect in any way our ability to guess
whether the object is the member of the set X, or of its complement
(
X
|
E
)
=
P
(
X
)
and
P
(
E
|
X
)
=
P
(
E
)
¬
X
.
6.4 Probabilistic Decision Tables
Probabilistic decision tables
describe classes of approximation space and their prob-
abilistic relations with a target set. They are composed of combinations of attribute
values, probability values and approximation region designations.
6.4.1 Attributes
In many applications, the information about objects is expressed in terms of values
of observations or measurements, often real-valued, referred to as
features
.Forthe
purpose of rough set-based analysis and classifier construction, the feature values are
typically mapped into finite-valued numeric or symbolic domains to form composite
mappings, referred to as
attributes
. A common kind of mapping is dividing the range
of values of a feature into a number of suitably chosen disjoint subranges via a
discretization procedure (see, for example, [
9
]). Formally, an attribute
a
is a function
on the universe
U
,
a
:
U
ₒ
a
(
U
)
ↆ
V
a
, where
V
a
is a finite set of values called the
domain
of the attribute
a
.
Based on combinations of attributes and their values, a structure of approximation
space can be created and analyzed using general notions and results of rough set
theory and of the VPRSM. Each attribute defines a classification of the universe
U
into classes corresponding to different values of the attribute. Each attribute value
v
, corresponds to a set of objects
E
v
U
such that
E
v
a
−
1
∈
a
(
U
)
ↆ
=
(
v
)
={
e
∈
. The classes
E
v
, referred to as
a-elementary sets
, form a partition of
U
. The equivalence relation corresponding to this partition will be denoted as
IND
a
.
Similarly, an equivalence relation
IND
B
, and the corresponding approximation space,
can be defined on the basis of any non-empty set of attributes
B
.
U
:
a
(
e
)
=
v
}
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