Miao Duoqian, Qian Jin, Li Wen, Zhang Zehua

1. Mining Hierarchical Decision Rules from Hybrid Data with Categorical and Continuous Valued Attributes Miao Duoqian, Qian Jin, Li Wen, Zhang Zehua

2. Outline Introduction Similarity-based Rough Set Model Attribute reduction Mining Hierarchical decision rules Conclusion

3. Introduction • Rough set theory, proposed by Pawlak, is a useful mathematical framework to deal with imprecise, uncertain information. • Classical attribute reduction methods mainly deal with categorical data. • In practice, there exist continuous-valued (numerical) attributes in real application systems.

4. Existing Methods • Discretization methods These methods are too categorical and may bring information loss in some cases because the degrees of membership of numerical values to discretized values are not considered. • Extended rough set model • Fuzzy rough set model • Tolerance rough set model • Neighborhood rough set model • Similarity rough set model • ……

5. Decision rule Attribute reduction Similarity class Similarity relation Similarity rough set model

6. Similarity • The similarity class of x, denoted by R(x), is the set of objects which are similar to x. • Notice that the statements yRx, which means “y is similar to x”, is directional. It has a subject y and a referent x.

7. Symmetry and Transitivity? • Symmetry? • The most controversial property is symmetry. • Although yRx is directional, most authors dealing with similarity relation do impose this property. • Transitivity? • Imposing transitivity to R is even more questionable. • The reason for this is that, sometimes, a series of negligible differences cannot be propagated.

8. Similarity Measure • For numerical attributes • For categorical attributes

9. Similarity • Local similarity • Global similarity

10. If a global similarity measure threshold equals 1, the similarity-based rough set model degenerates into classical rough set model. • Researchers pointed out empirically that in some contexts, similarity does not necessarily have features like symmetry or subadditivity implied by distance measures.

11. New Similarity Distance Measure

12. Similarity distance measure? • This inherent weakness of the distance-based similarity measure comes from a lack of consideration of the contribution of the similarity direction when comparing the similarity of two objects.

13. Similarity direction measure =

14. Similarity direction measure Definition 9. Given two objects x and y, the similarity direction measure of both objects is defined as = If D (y, x) >=0, the object y is similar to x; otherwise y is dissimilar to x.

15. However, if we employ such similarity direction measure, similarity relation is not symmetric in most cases, even if the similarity direction differences between two objects are very small. Furthermore, each similarity direction measure may not possess subadditivity.

16. Definition 10. Given two objects x and y, the similarity direction measure of both objects is defined as . = If D (y, x)>= , the object y is similar to x; otherwise y is dissimilar to x. In general, the same similarity direction is good. Here we give a constraint parameter to extend similarity.

17. Similarity relation • Construction of a rational, reliable and practical similarity measure is a fundamental and substantial research topic in the field of decision making, otherwise the accuracy and validity of a similarity measure could be challenged.

18. Attribute reduction Definition 11. Let DT be a decision table, and , we will say that x is a consistent object under similarity measure parameters and if for all y; otherwise x is an inconsistent object. All consistent objects set and inconsistent objects set are denoted by and

19. Attribute reduction Definition 12. Let DT be a decision table, and , we will say that x and y are dissimilar under similarity measure parameters and if . Definition 13 Let DT be a decision table, and , the discernibility matrix = is defined as

20. Mining Hierarchical decision rules

21. Example

22. Decision rules Fig 3. A similarity relation graph with =0.75 and =-0.01

23. Fig 4. A similarity relation graph with =0.75 Without considering similarity direction parameter, we can not discern object 4 and object 9 under =0.75. In such case, we will generate some inconsistent decision rules. Choosing a level in concept hierarchy, we can mine hierarchical decision rules.

24. Conclusion • This paper mainly discusses similarity distance measure and similarity direction measure, and proposes an algorithm for mining hierarchical decision rules . • Future work Both theoretical and experimental comparison of mining hierarchical decision rules.

25. Thanks !