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## New Reduction Algorithm Based on Decision Power of Decision Table

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### Rough set theory is a valid mathematical tool that deals with imprecise, ... 4. Miao, D.Q., Hu, G.R. : A Heuristic Algorithm for Reduction of Knowledge. ... – PowerPoint PPT presentation

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Title: New Reduction Algorithm Based on Decision Power of Decision Table

1
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

2
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(CU2) O(U3) and O(C2U)
O(CU3) 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(CU2)
O(U3). 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.

3
• 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) cant 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 gtgt 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.

4
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.,
?P(D) ?C(D), and there is no P P such that
?P(D) ?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.

5
• Secondly, if for any P C, H(DP) H(DC) 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
POSC(D) is 0, thus U -POSC(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 cant 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.

6
• Representation of Decision Power on Decision
Table
• Now, in a decision table S (U, C, D, V, f),
suppose D0 U POSC(D), from the definition of
positive region, we have CD0 D0. Suppose that
then the sets must be also a decision partition
of U, if there is an empty decision class ADi,
then the ADi 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
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 CD0, CD1, CD2,, CDm, then we have a new
equivalent relation generated by the new
partition, which is denoted by RD, U/RD CD0,
CD1, CD2,, CDm. 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.

7
• Definition 1. Given a decision table S (U, C,
D, V, f), let P C (U/P X1, X2,, Xt), D
d (U/D Y1, Y2,, Ym), and U/RD CY0, CY1,
CY2,, CYm, 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(DP) H(DP r) ?P(D) ?p- r(D). If
S is an inconsistent decision table, due to CY0
Y0 .Assume that
, then
• S(RD P) S(RD P r) ?P(D) ?p-r
(D).

8
• 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) lt S(RD P), for any P P.

9
• 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 if S(RD P)
S(RD C), then turn to (6).
• (4) Calculating S(RD Pr), for any attribute
r?C P, select an attribute r with the maximum
of S(RD Pr), and if this r is not only, then
select that with the maximum of U/ (P? r).
• (5) P P? r, and if S(RD P) ? S(RD C), then
turn to (4), else P P CORED(C)t P
• for(i 1 i t i )
• ri?PP P ri
• if S(RD PCORED(C)) lt S(RD P) then P P?
ri
• P P?CORED(C)
• (6) The output P is a minimum relative reduction.
• (7) End.

10
• Experimental Results
• Example 1. S (U, C, D, V, f) can be seen in
Table 1 below, where U x1, x2,, x10, C
a1, a2,, a5, and D d.

11
• In Table 2 below, there is the significance of
attribute relative to the core a2 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 a4 is relatively minimum, and their
reducts are a1, a2, a3, a5, rather than the
minimum relative reduct a2, a4, a5. However,
the SGF(a4, a2,D) is relatively maximum. Thus
we get the minimum relative reduction a2, a4,
a5 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 algorithms
effects on reduction of knowledge are well
remarkable.

12
• 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.

13
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(C2U)
(In preprocessing, the complexity for computing
U/C based on radix sorting is cut down to
O(CU), and the complexity for measuring
attribute importance based on the positive region
is descended to O(C PU - U P) (see
9).), and the result of this method is
objective.

14
References
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Analysis. International Journal of Information
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• 2. Wang, G.Y. Rough Reduction in Algebra View
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• 3. Wang, G.Y., Yu, H., Yang, D.C. Decision
Table Reduction Based on Conditional Information
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(2002)
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for Reduction of Knowledge. Journal of Computer
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15
• 8. Liu, S.H., Sheng, Q.J., Wu, B., et al.
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16
• THANK YOU VERY MUCH!