Information Technology Reference
In-Depth Information
7.4 Structure-Based Attribute Reduction
in Dominance-Based Rough Set Models
7.4.1 Decision Tables Under Dominance Principle
and Dominance-Based Rough Set Models
In Dominance-based Rough Set Model (DRSM), known as Dominance-based Rough
Set Approach [
16
,
18
,
49
], decision tables with order relations are analyzed. Let
D =
(
be a decision table. The attribute set
AT
is
partitioned into
AT
N
and
AT
C
, where
AT
N
is the set of nominal attributes and
AT
C
is the set of criteria (ordinal attributes). For a criterion
a
U
,
AT
=
C
∪{
d
}
,
{
V
a
}
a
∈
AT
)
∈
AT
C
, we suppose a total
order
on its value set
V
a
. Moreover, all criteria are of the gain-type, i.e., the greater
the better. We assume that the decision attribute
d
is a criterion.
In DRSM, it is supposed that if an object
u
is better than or equal to another
object
u
with respect to all condition attributes, then the class of
u
should not be
worse than that of
u
. This is called the dominance principle [
16
].
Remark 8
The setting of DRSM is considered as the monotone or ordinal classifi-
cation problem [
2
,
3
,
32
], where classifiers are restricted to be monotonic. Let
f
be
a classifier, which assigns to each object
u
a class label (decision class value)
f
≥
(
)
u
.
The classifier
f
is monotonic if for any object pair
u
and
u
,wehave
u
u
implying
≤
u
)
. In this chapter, however, we do not discuss classifiers nor algorithms
for building classifiers.
Remark 9
We assume the total order, i.e., antisymmetry, transitivity, and, compara-
bility, on the value set
V
a
of each condition criteria
a
f
(
u
)
≤
f
(
C
. However, regardless
of comparability, the result of this section can be applied without modification. Addi-
tionally, we assume that all criteria are of the gain-type. However, in applications, we
may encounter cost-type criteria, i.e., the smaller the better. For a cost-type criterion,
we can deal with it as the gain-type by reversing the order of its values.
Remark 10
Generally, there is more than one decision attribute in a decision table.
In such a case, the set of decision classes (the partition of objects by the decision
attributes) is partially ordered, while it is totally ordered in the case of a single decision
attribute. In this section, we focus on the case of a single decision attribute (more
generally, the case when the decision classes form a totally ordered set), however,
the results of this section could be straightforwardly extended to that of multiple
decision attributes.
∈
AT
C
∩
For
A
ↆ
C
, a dominance relation
D
A
on
U
is defined by:
(
u
)
∈
U
2
u
),
∀
D
A
=
u
,
|
a
(
u
)
≥
a
(
a
∈
AT
C
∩
A
A
.
u
),
∀
and
a
(
u
)
=
a
(
a
∈
AT
N
∩
u
)
∈
D
A
satisfies reflexivity and transitivity. When
(
u
,
D
A
, we say that
u
domi-
nates
u
with respect to
A
. The relation
u
)
∈
(
,
u
D
A
means “
u
is better than or equal
Search WWH ::
Custom Search