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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
    any set of AD0, AD1, AD2,, ADm isnt empty,
    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
    reduction of C, thus the partition AD0, AD1,
    AD2,, ADm 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 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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15
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16
  • THANK YOU VERY MUCH!
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