Information Technology Reference
In-Depth Information
Table 9.1.
Decision System
ab
c
d
x
1
2
1
2
L
x
2
2
1
2
L
x
3
1
1
0
H
x
4
1
1
0
H
x
5
2
3
2
H
x
6
2
3
2
H
x
7
2
1
1
L
x
8
2
1
1
L
x
9
2
2
1
L
x
10
2
3
0
L
x
11
1
1
2
H
x
12
1
1
1
H
As an example of a decision table we take
S
=(
{x
1
,x
2
,x
3
,x
4
,x
5
,x
6
,x
7
,x
8
,
x
9
,x
10
,x
11
,x
12
}
lists
stable attributes,
b
is a flexible attribute and
d
is a decision attribute. Also, we
assume that
H
denotes a
high
profit and
L
denotes a
low
one.
In order to induce rules in which the THEN part consists of the decision
attribute
d
and the IF part consists of attributes belonging to
A
St
∪
,
{
a, c
}∪{
b
}∪{
d
}
) represented by Table 9.1. The set
{
a, c
}
A
Fl
,for
instance system
LERS
[4] can be used for rules extraction.
Alternatively, we can extract rules from sub-tables (
U, B
)of
S
,where
B
is a
d
-reduct (see [11]) of
S
, to improve eciency of the algorithm when
the number of attributes is large. The set B is called
d
-reduct in S if there is
no proper subset
C
of
B
such that
d
depends on
C
. The concept of
d
-reduct
in
S
was introduced to induce eciently rules from
S
describing values of the
attribute
d
depending on minimal subsets of
A
St
∪
∪{
d
}
A
Fl
.
By
L
(
r
) we mean all attributes listed in the IF part of a rule
r
. For example,
if
r
1
=[(
a
1
,
2)
.
By
d
(
r
1
) we denote the decision value of that rule. In our example
d
(
r
1
)=8.
If
r
1
,
r
2
are rules and
B
∧
(
a
2
,
1)
∧
(
a
3
,
4)
−→
(
d,
8)] is a rule then
L
(
r
1
)=
{
a
1
,a
2
,a
3
}
A
Fl
is a set of attributes, then
r
1
/B
=
r
2
/B
means that the conditional parts of rules
r
1
,
r
2
restricted to attributes
B
are
the same. For example if
r
2
=[(
a
2
,
1)
⊆
A
St
∪
∗
(
a
3
,
4)
−→
(
d,
1)], then
r
1
/
{
a
2
,a
3
}
=
r
2
/
{
a
2
,a
3
}
.
In our example, we get the following certain rules with support greater or
equal to 2:
(
b,
3)
∗
(
c,
2)
−→
(
d, H
), (
a,
1)
∗
(
b,
1)
−→
(
d, L
),
(
a,
1)
∗
(
c,
1)
−→
(
d, L
), (
b,
1)
∗
(
c,
0)
−→
(
d, H
),
(
a,
1)
−→
(
d, H
)
w
)denotesthefactthatthevalueofat-
tribute
a
has been changed from
v
to
w
. Similarly, the term (
a, v
Now, let us assume that (
a, v
−→
−→
w
)(
x
)
