1. Dominance-Bases Rough Set Approach: Features, Extensions and Application Krzysztof Dembczyński Institute of Computing Science, Poznań University of Technology, Poland Salvatore Greco Faculty of Economy, University of Catania, Italy Roman Słowiński Institute of Computing Science, Poznań University of Technology, Poland

2. 2 Topics Philosophy of Dominance-Based Rough Set Approach (DRSA) Preliminaries of DRSA Extensions of DRSA Variable Consistency DRSA Multi-Valued DRSA Continuous Decision Criterion and DRSA Conclusion

3. 3 The Philosophy of Dominance-Based Rough Set Approach The aim of the decision analysis is to answer two questions: To explain decisions in terms of the circumstances in which they were made. To give a recommendation how to make a good decision under specific circumstances. One of decision problems is the multicriteria sorting Multicriteria sorting concerns an assignment of the objects to pre-defined classes (concepts) that are preference-ordered.

4. 4 The Philosophy of Dominance-Based Rough Set Approach Analyzed objects are described using criteria Criteria are attributes with preference-ordered domain Decision criterion shows the class of any object Multicriteria decision problem has no solution unless a preference model is defined Functional Relational Decision rules

5. 5 The Philosophy of Dominance-Based Rough Set Approach Data are very often inconsistent with dominance principle that requires that an object having a better (not worse) evaluation on considered criteria cannot be assigned to a worse class. H I G H L O W

6. 6 The Philosophy of Dominance-Based Rough Set Approach Greco, Matarazzo and Słowiński have proposed Dominance- Based Rough Set Approach The Classical Rough Set Approach, proposed by Pawlak, has been proved as excellent tool for data analysis, however, it was falling for multicriteria sorting problem The analyzed objects may be considered only in the perspective of available information

7. 7 The Philosophy of Dominance-Based Rough Set Approach The rough set approaches features: Information has granular structure Approximation of one knowledge by another knowledge Analysis of uncertain and inconsistent data Inducing of “if…, then” decision rules In DRSA the set of decision rules plays a role of comprehensive preference model The rules syntax is concordant with Dominance Principle

8. 8 Topics Philosophy of Dominance-Based Rough Set Approach (DRSA) Preliminaries of DRSA Extensions of DRSA Variable Consistency DRSA Multi-Valued DRSA Continuous Decision Criterion and DRSA Conclusion

9. 9 Preliminaries of DRSA Basic notions Outranking relation x is at least so good as y with respect to criterion q Dominance relation (reflexive and transitive) x dominates y when on all criteria x outranks y (x is at least so good then y) Data are often presented as a table Because of preference order of classes it is possible to consider upward and downward unions of classes

10. 10 Preliminaries of DRSA An Example First CriterionSecond CriterionDecision Criterion 34.423.4High 30.322.1High 2519High 2017High 2219.5Medium 1225Medium 15.516.8Medium 17.117.6Medium 8.910.1Low 1113.5Low 97 12.54Low

11. 11 Preliminaries of DRSA An Example c1c1 c2c2 0 40 20 + + + + o o o o -- - - + + + + + + + + + + BEST WORST o o o o o o - - - - o o

12. 12 Preliminaries of DRSA Granules of Knowledge: Dominating and Dominated Sets c1c1 c2c2 0 40 20 + + + + o o o o -- - - + + + + + + + + + + BEST WORST o o o o o o - - - - o o

13. 13 Preliminaries of DRSA Granules of Knowledge: Dominating and Dominated Sets c1c1 c2c2 0 40 20 + + + + o o o o -- - - + + + + + + + + + + BEST WORST o o o o o o - - - - o o

14. 14 Preliminaries of DRSA Lower and Upper Approximation of the class unions c1c1 c2c2 0 40 20 + + + + o o o o -- - - + + + + + + + + + + BEST WORST o o o o o o - - - - o o

15. 15 Preliminaries of DRSA c1c1 c2c2 0 40 20 + + + + o o o o -- - - + + + + + + + + + + WORST o o o o o o - - - - o o BEST Inducing of Decision Rules

16. 16 Preliminaries of DRSA Form of Decision Rules if f(x, c 1 )  25 and f(x, c 2 )  19, then x is at least High if f(x, c 1 )  20 and f(x, c 2 )  17, then x could be at least High if f(x, c 1 )  20 and f(x, c 2 )  17 and f(x, c 1 )  22 and f(x, c 2 )  19.5, then x belongs to High or Medium

17. 17 Preliminaries of DRSA c1c1 c2c2 0 40 20 + + + + o o o o -- - - + + + + + + + + + + WORST o o o o o o - - - - o o BEST Inducing of Decision Rules with Hyperplanes

18. 18 Preliminaries of DRSA Inducing of Decision Rules with Hyperplanes c1c1 c2c2 0 40 20 + + + + o o o o -- - - + + + + + + + + + + WORST o o o o o o - - - - o o BEST

19. 19 Preliminaries of DRSA Features Analysis of multicriteria sorting problems with inconsistent information It is possible to analyze objects described by criteria and regular attributes Continuous domain of criteria (discretization is not needed) Sorting of new objects

20. 20 Topics Philosophy of Dominance-Based Rough Set Approach (DRSA) Preliminaries of DRSA Extensions of DRSA Variable Consistency DRSA Multi-Valued DRSA Continuous Decision Criterion and DRSA Conclusion

21. 21 Variable-Consistency DRSA c1c1 c2c2 0 40 20 + + + + o o o o -- - - + + + + + + + + + + BEST WORST o o o o o o - - - - o o Lower Approximation consists of limited counterexamples controlled by pre-defined level of certainty

22. 22 Multi-Valued DRSA C1C1 C2C2 Decision 34.423.4High 30.322.1High 25-2119High 20-1717High 18-1519.5Medium 1225Medium 15.516.8Medium 17.117.6Medium 8.910.1Low 1113.5Low 97 12.54Low Interval order object x is not worse than y with respect to a single criterion, if there exist a value describing x that is not worse than at least one value describing y Form of the rules: if u(x)  21 then, x is at least High

23. 23 Extensions of DRSA VC-DRSA and MV-DRSA are only examples of extensions of DRSA. Another example is the methodology that allows deal with missing values There exist different strategies of induction of decision rules It is also possible to induces decision trees using rough approximations

24. 24 Topics Philosophy of Dominance-Based Rough Set Approach (DRSA) Preliminaries of DRSA Extensions of DRSA Variable Consistency DRSA Multi-Valued DRSA Continuous Decision Criterion and DRSA Conclusion

25. 25 Continuous Decision Criterion C1C1 C2C2 D1D1 DCDC 34.423.4High34.5 30.322.1High31.5 2519High25.4 2017High22.1 2219.5Medium21.5 1225Medium20.1 15.516.8Medium17.4 17.117.6Medium16 8.910.1Low10.7 1113.5Low9.5 97Low4.3 12.54Low3.5 What we can do? Pre-discretization of decision criterion Or Analyzing data with continues decision Large number of classes and unions of classes? This is more inconsistencies Looking for good association on the conditional part of the decision table

26. 26 Continuous Decision Criterion Decision Rules if f(x, c 1 )  34.4, then x is at least 34.5 if f(x, c 2 )  25, then x is at least 25.4 if f(x, c 1 )  20, then x is at least 21.5 if f(x, c 1 )  8.9, then x is at least 4.3 if f(x, c 1 )  17.1, then x is at most 20.1

27. 27 Topics Philosophy of Dominance-Based Rough Set Approach (DRSA) Preliminaries of DRSA Extensions of DRSA Variable Consistency DRSA Multi-Valued DRSA Discussion about Continuous Decision Criterion and DRSA Conclusion

28. 28 Conclusion It is proven that: The preference model in the form of rules derived from examples is more general then the classic functional or relational model and it is more understandable for the users because of its natural syntax. It fulfils both explanation and recommendation tasks that are principal aims of decision analysis. DRSA is still developing DRSA in the Malaria Vulnerability Case Study in IIASA during YSSP