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(|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.

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]) 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.

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., γP(D) = γC(D), and there is no P* P such that γP*(D) = γC(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(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 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 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.

6. Representation of Decision Power on Decision Table • Now, in a decision table S = (U, C, D, V, f), suppose D0 = U – POSC(D),fromthe definition of positive region, we have CD0 = D0. Suppose that any set of {AD0, AD1, AD2,…, ADm} isn’t empty, then the sets must be also a decision partition of U, if there is an empty decision classADi, 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, thenewpartition of condition attributesset C is {CD0, CD1, CD2,…, CDm}, then we have a new equivalent relation generated by thenewpartition, 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(D|P) = H(D|P – {r}) γP(D) = γp- {r}(D). If S is an inconsistent decision table, due to CY0 = Y0 .Assume that , then S(RD; P) = S(RD; P – {r}) γP(D) = γp-{r} (D).

8. Theorem 3. Let P be a subset of condition attributesset 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 relationson 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.

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) IfP = Ø,thenturn to(4), and ifS(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 maximumof S(RD; P{r}), and if this r is not only, then select that with the maximumof |U/(P∪{r})|. • (5) P = P∪{r}, and ifS(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}； • ifS(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.

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 algorithm’s 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(|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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