New Reduction Algorithm Based on Decision Power of Decision Table

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29 Οκτ 2013 (πριν από 3 χρόνια και 7 μήνες)

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New Reduction
Algorithm Based on
Decision Power of
Decision Table

Jiucheng Xu, Lin Sun

College of Computer &
Information Technology,
Henan Normal University,
Xinxiang Henan, China

Introduction


Rough set theory is a valid mathematical tool that deals with
imprecise, uncertain, vague or incomplete knowledge of a
decision system (see [1]). Reduction of knowledge is always one
of the most important topics. Pawlak (see [1]) first proposed
attribute reduction from the algebraic point of view. Wang (see [2,
3]) proposed some reduction theories based on the information
point of view, and introduced two novel heuristic algorithms of
knowledge reduction with the time complexity
O
(|
C
||
U
|
2
) +
O
(|
U
|
3
)
and

O
(|
C
|
2
|
U
|) +
O
(|
C
||
U
|
3
) respectively, where
|C|
denotes the
number of conditional attributes and
|U|
is the number of objects
in
U
, and the heuristic algorithm based on the mutual information
(see [4]) with the time complexity
O
(|
C
||
U
|
2
) +
O
(|
U
|
3
). These
presented reduction algorithms have still their own limitations,
such as sensitivity to noises, relatively high complexities,
nonequivalence in the representation of knowledge reduction and
some drawbacks in dealing with inconsistent decision tables.


It is known that reliability and coverage of a decision rule are all
the most important standards for estimating the decision quality
(see [5, 6]), but these algorithms (see
[

1, 2, 3, 7, 8, 9]) can’t
reflect the change of decision quality objectively. To compensate
for their limitations, we construct a new method for separating
consistent objects from inconsistent objects, and the
corresponding judgment criterion with an inequality used in
searching for the minimal or optimal reducts. Then we design a
new heuristic reduction algorithm with relatively lower time
complexity. For the large decision tables, since usually |
U
| >> |
C
|,
the reduction algorithm is more efficient than the algorithms
discussed above. Finally, six data sets from UCI repository are
used to illustrate the performance of the proposed algorithm and a
comparison with the existing methods is reported.

The Proposed Approach


Limitations of Current Reduction Algorithms



Hence, one can analyze algorithms based on the positive region and the
conditional entropy deeply. Firstly, if for any
P C
, the
P
-
quality of
approximation relative to
D
is equal to the
C
-
quality of approximation relative
to
D
, i.e.,
γ
P
(
D
) =
γ
C
(
D
), and there is no
P* P

such that
γ
P*
(
D
) =
γ
C
(
D
), then
P
is called the reduct of
C
relative to
D
(see [1, 7, 8, 9]). In these algorithms,
whether or not any conditional attributes is redundant depends on whether the
lower approximation corresponding to decision set is changed or not after the
attribute is deleted. Accordingly if new inconsistent objects are added to the
decision table, it is not taken into account whether the conditional probability
distributionof the primary inconsistent objects are changed in every
corresponding decision class (see [10]). Hence, if the generated deterministic
decision rules are the same, they will support the same important standards for
estimating decision quality. Suppose the generated deterministic decision rules
are the same, that is, the prediction of these rules is not changing. Thus it is
seen that these presented algorithms only take into account whether or not the
prediction of deterministic decision rules is changing after reduction.



Secondly, if for any
P C
,
H
(
D|P
) =
H
(
D|C
) and
P
is independent relative to
D
, then
P
is called the reduct of
C
relative to
D
(see [2, 3, 10, 11]). Hence, whether any
conditional attributes is redundant or not depends on whether the conditional entropy
of decision table is changed or not, after the attribute is deleted. It is known that the
conditional entropy generated by
POS
C
(
D
) is 0, thus
U
-
POS
C
(
D
) can lead to a change
of conditional entropy. Due to the new added and primary inconsistent objects in every
corresponding decision class, if their conditional probability distribution changes, it
will cause the change of conditional entropy of the whole decision table. Therefore, as
it goes, the main criterions of these algorithms for estimating decision quality include
two aspects, the invariability of the deterministic decision rules, the invariability of the
reliability of nondeterministic decision rules.


So, some researchers above only think about the change of reliability for all decision
rules after reduction. However, in decision application, besides the reliability of
decision rules, the object coverage of decision rules is also one of the most important
standards of estimating decision quality. So these current reduction algorithms above
can’t reflect the change of decision quality objectively. Meanwhile, the significance of
attribute is regarded as the quantitative computation of radix for the positive region,
which merely describes the subsets of certain classes in
U
, while from the information
point of view, the significance of attribute only indicates the detaching objects of
different decision classes in the equivalence relation of conditional attribute subset.
However, for the inconsistent objects, these current measures for attribute reduction
lack of dividing
U
into consistent object sets and inconsistent object sets for the
inconsistent decision table. Therefore, these algorithms will not be equivalent in the
representation of knowledge reduction for inconsistent decision tables (see [12]). It is
necessary to seek for a new kind of measure to search for the precise reducts
effectively.


Representation of Decision Power on Decision Table


Now, in a decision table

S
= (
U
,
C
,
D
,
V
,
f
), suppose
D
0 =
U


POS
C
(
D
),

from

the definition of positive region, we have
C
D
0 =
D
0.
Suppose that any set of {
A
D
0,
A
D
1,
A
D
2,…,
A
Dm
} isn’t empty, then
the sets must be also a decision partition of
U
, if there is an empty
decision class

A
Di
, then the
A
Di

is called a redundant set of the new
decision partition.

After the redundant sets are taken out, it makes no
difference to the decision partition.


Suppose that condition attributes subset
A

is a reduction of
C
, thus the
partition {
A
D
0,
A
D
1,
A
D
2,…,
A
Dm
} is divided into consistent and
inconsistent objects set respective1y, and all inconsistent objects
detached form the unattached set. On the basis of the idea mentioned
above, the

new

partition of condition attributes

set

C

is {
C
D
0,
C
D
1,
C
D
2,…,
C
Dm
}, then we have a new equivalent relation generated by
the

new

partition, which is denoted by
RD
,

U
/
RD
= {
C
D
0,
C
D
1,
C
D
2,…,
C
Dm
}. Accordingly it shows that the presented decision
partition

U
/
RD

has not only detached consistent objects from different
decision classes in
U
, but also separated consistent objects from
inconsistent objects, while
U
/
D

is gained through detaching objects
from different decision classes corresponding to equivalent classes.


Definition 1.
Given a decision table

S
= (
U
,
C
,
D
,
V
,
f
), let
P C
(
U
/
P
= {
X
1,
X
2,…,
Xt
}),
D
= {
d
} (
U
/
D
= {
Y
1,
Y
2,…,
Ym
}), and
U
/
RD
= {
C
Y
0,
C
Y
1,
C
Y
2,…,
C
Ym
}, then the decision power of equivalent relation
RD

with respect
to
P

is denoted by

S
(
RD
;
P
), defined thus


.



Theorem 1.

Let
r


P C
, then we have
S
(
RD
;
P
)

S
(
RD
;
P


{
r
}).


Theorem 2.

If
S
is a consistent one, then
U/RD
=
U/D
. Assume that




,
then
S
(
RD
;
P
) =
S
(
RD
;
P


{
r
})
H
(
D
|
P
) =
H
(
D
|
P



{
r
})
γ
P
(
D
) =
γ
p
-

{
r
}
(
D
). If
S

is an inconsistent
decision table, due to
C
Y
0 =
Y
0

.
Assume
that , then




S
(
RD
;
P
) =
S
(
RD
;
P


{
r
})
γ
P
(
D
) =
γ
p
-
{
r
}

(
D
).


Theorem 3.
Let
P

be a subset of condition attributes

set

C
on
U
, and any
r

P

is said to be dispensable in
P
with
respect to
D

if and only if
S
(
RD
;
P
) =
S
(
RD
;
P


{
r
}).


Definition 2.

If
P C
, then the significance of any attribute
r

C


P

with respect to
D

is defined in algebra view, denoted
by


SGF
(
r
,
P
,
D
) =
S
(
RD
;
P


{
r
})



S
(
RD
;
P
). (2)


Definition 3.

Let
P C

be equivalent relations

on
U
, then
P
is an attribute reduction of
C

with respect to
D
, which
satisfies
S
(
RD
;
P
) =
S
(
RD
;
C
) and
S
(
RD
;
P
*) <
S
(
RD
;
P
), for
any
P
*
P
.


Design of Reduction Algorithm Based on Decision
Power


Input:

Decision table

S
= (
U
,
C
,
D
,
V
,
f
).


Output:
A relative reduction
P
.


(1) Calculating
POSC
(
D
) and
U


POSC
(
D
) for the new partition
U
/
RD
.


(2)

Calculating

S
(
RD
;
C
),
CORED
(
C
), and let
P
=

CORED
(
C
).


(3)

If

P
=
Ø
,

then

turn to

(4), and i
f

S
(
RD
;
P
) =
S
(
RD
;
C
),
then

turn to (6).


(4) Calculating
S
(
RD
;
P
{
r
}), for any attribute
r

C


P
, select an attribute
r
with
the maximum

of
S
(
RD
;
P
{
r
}), and i
f

this
r

is not only,
then

select that with the
maximum

of
|U
/

(
P


{
r
})
|
.


(5)

P
=
P


{
r
}, and i
f

S
(
RD
;
P
) ≠
S
(
RD
;
C
),
then

turn to (4), else {
P*
=
P


CORED
(
C
)

t
= |
P*
|




for
(
i

= 1;
i


t
;
i

++)


{

ri

P*

P*
=
P*


{
ri
}



if

S
(
RD
;
P*CORED
(
C
)) <
S
(
RD
;
P
) then
P*
=
P*


{
ri
};}


P
=
P*

CORED
(
C
)

}


(6) The output
P
is a minimum relative reduction.


(7)
End
.



Experimental Results


Example 1.

S
= (
U
,
C
,
D
,
V
,
f
) can be seen in Table 1 below, where
U
= {
x
1,
x
2,…,
x
10},
C
= {
a
1,
a
2,…,
a
5}, and
D
= {
d
}.





In Table 2 below, there is the significance of attribute relative to the
core {
a
2}

and the relative reducts, the Algorithm in
[7],CEBARKCC in [3], Algorithm 2 in [12], and the proposed
Algorithm are denoted by A1, A2, A3, and A4 respectively, and let
m, n
be the number of attributes and universe respectively.


From Table 2, the significance of attribute in [3, 7]
a
4 is relatively
minimum, and their reducts are {
a
1
, a
2
, a
3
, a
5}, rather than the
minimum relative reduct {
a
2
, a
4
, a
5}. However, the
SGF
(
a
4
,
{
a
2}
,D
) is relatively maximum. Thus we get the minimum relative
reduction
{a
2
, a
4
, a
5
}
generated by A3 and A4. Compared with A1
and A2, the new proposed algorithm does not need much
mathematical computation, logarithm computation in particular.
Meanwhile, we know that the general schema of adding attributes is
typical for old approaches to forward selection of attributes although
they are using different evaluation measures, but it is clear that on
the basis of
U/RD
, the proposed decision power is feasible to
discuss the roughness of rough sets. Hence, the new heuristic
information will compensate for the proposed limitations of those
current algorithms. Therefore, this algorithm’s effects on reduction
of knowledge are well remarkable.



Here we choose six discrete data sets from UCI repository and
five algorithms to do more experiments on PC (P4 2.6G, 256M
RAM, WINXP) under DK1.4.2 in Table 3 below, where T or F
indicates that the data sets are consistent or not,
m, n
are the
number of primal attributes and after reduction respectively,
t
is
the time of operation, and A5 denotes the algorithm in [6].


Conclusion


In this paper, to reflect the change of decision quality
objectively, a measure for reduction of knowledge and
its judgment theorem with an inequality are established
by introducing the decision power from the algebraic
point of view. To compensate for these current
disadvantages of classical algorithms, we design an
efficient complete algorithm for reduction of knowledge
with the time complexity reduced to
O
(
|C|
2
|U|
) (In
preprocessing, the complexity for computing
U/C
based
on radix sorting is cut down to
O
(
|C||U|
), and the
complexity for measuring attribute importance based on
the positive region is descended to
O
(
|C


P||U
´

-

U
´

P
|
)
(see [9]).), and the result of this method is objective.


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MUCH!