Information Technology Reference
In-Depth Information
6.3.2 Absolute Dependency Gain
Another dependency measure is an
absolute dependency gain
, which is a bi-
directional dependency measure used to evaluate the degree of the two-way con-
nection between any two sets. Given two arbitrary subsets
X
and
Y
of the universe
U
, the absolute dependency gain, denoted as
dabs(X,Y)
, is defined by:
dabs
(
X
,
Y
)
=|
P
(
X
∩
Y
)
−
P
(
X
)
P
(
Y
)
|
.
(6.13)
The absolute dependency gain reflects the degree of probabilistic dependency
between sets
X
and
Y
by quantifying the amount of deviation from the probabilistic
independence between sets
X
and
Y
, as represented by the product
P
.
Similar to the absolute certainty gain, in an approximation space (
U, IND
), if a
subset
Y
is definable, then the absolute dependency gain between the subsets
X
and
Y
can be computed directly from the available probabilistic knowledge based on the
following formula:
(
X
)
P
(
Y
)
dabs
(
X
,
Y
)
=|
ʣ
E
ↆ
Y
P
(
E
)
P
(
X
|
E
)
−
P
(
X
)ʣ
E
ↆ
Y
P
(
E
)
|
.
(6.14)
The absolute boundary region of the target set
X
can alternatively be expressed
by the absolute dependency gain as:
BND
∗
(
)
=∪{
:
(
,
)
=
}
.
X
E
dabs
X
E
0
(6.15)
In other words, the absolute boundary region is an area with no dependency gain.
6.3.3 Average Dependency Gain
The average, or expected gain function, denoted as
egabs
, is a measure
of the degree of probabilistic dependency between classification represented by the
indiscernibility relation
IND
and the classification
(
X
|
IND
)
(
X
,
¬
X
)
of the universe
U
induced
by the target set
X
, and its complement
¬
X
. It is a measure of dependency between
two partitions of the universe
U
:
IND
∗
|
(
∩
)
−
(
)
(
)
|=
(
,
).
egabs
(
X
|
IND
)
=
P
X
E
P
X
P
E
dabs
X
E
(6.16)
IND
∗
E
∈
E
∈
When the dependency is functional, i.e. when set
X
is definable in Pawlak's sense
[
11
], we have:
egabs
(
X
|
IND
)
=
IND
∗
|
P
(
X
∩
E
)
−
P
(
X
)
P
(
E
)
|
(6.17)
E
∈
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