Dominance-Bases Rough Set Approach: Features, Extensions and Application

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

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

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 sortingconcerns an assignment of the objectsto pre-defined classes (concepts) that are preference-ordered.

The Philosophy of Dominance-Based Rough Set Approach • Analyzed objects are described using criteria • Criteriaareattributes 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

H I G H L O W 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.

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

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

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) xdominates ywhen 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

Preliminaries of DRSA • An Example

BEST c1 + + + + 40 + + + + + + o + o + o + 20 o + o o o o o - - o - - - - - - c2 0 20 40 WORST Preliminaries of DRSA • An Example o o

BEST c1 + + + + 40 + + + + + + + o o + o + 20 o + o o o o o - - o - - - - - - c2 0 20 40 WORST Preliminaries of DRSA • Granules of Knowledge: Dominating and Dominated Sets o o

BEST c1 + + + + 40 + + + + + + o + o + + o 20 o + o o o o o - - o - - - - - - c2 0 20 40 WORST Preliminaries of DRSA • Granules of Knowledge: Dominating and Dominated Sets o o

Preliminaries of DRSA • Lower and Upper Approximation of the class unions BEST c1 + + + + 40 + + + o + + + o + o + o + 20 o + o o o o o o - - o - - - - - - c2 0 20 40 WORST

Preliminaries of DRSA • Inducing of Decision Rules BEST c1 + + + + 40 + + + o + + + o + o + o + 20 o + o o o o o o - - o - - - - - - c2 0 20 40 WORST

Preliminaries of DRSA • Form of Decision Rules if f(x, c1)  25 and f(x, c2)  19, then x is at least High if f(x, c1)  20 and f(x, c2)  17, then x could be at least High if f(x, c1)  20 and f(x, c2)  17 and f(x, c1)  22 and f(x, c2)  19.5, then x belongs to High or Medium

Preliminaries of DRSA • Inducing of Decision Rules with Hyperplanes BEST c1 + + + + 40 + + + o + + + o + o + o + 20 o + o o o o o o - - o - - - - - - c2 0 20 40 WORST

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

BEST c1 + + + + 40 + + + + + + o + o + + o 20 o + o o o o o - - o - - - - - - c2 0 20 40 WORST Variable-Consistency DRSA • Lower Approximation consists of limited counterexamples controlled by pre-defined level of certainty o o

Multi-Valued DRSA • 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

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

Continuous Decision Criterion • 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

Continuous Decision Criterion • Decision Rules if f(x, c1)  34.4, then x is at least 34.5 if f(x, c2)  25, then x is at least 25.4 if f(x, c1)  20, then x is at least 21.5 if f(x, c1)  8.9, then x is at least 4.3 if f(x, c1)  17.1, then x is at most 20.1

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

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

Dominance-Bases Rough Set Approach: Features, Extensions and Application