An algebraic structure for Dominance-based Rough Set Approach to ordinal classification Salvatore Greco University of Catania, Italy Benedetto Matarazzo University of Catania, Italy Roman Słowiński Poznań University of Technology, Poland
Bipolar Algebraic Structures for Dominance-based Rough Set Approach: Theoretical aspects of reasoning about ordinal data Salvatore Greco University of Catania, Italy Benedetto Matarazzo University of Catania, Italy Roman Słowiński Poznań University of Technology, Poland
Why Classical Rough Set Approach has to be adapted to take into account order in the data? Classical rough set approach does not detect inconsistency w.r.t. order in the data (Pareto principle) Student Mathematics ( M ) Physics ( Ph ) Literature ( L ) Overall class S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good S6 good good good good S7 bad bad bad bad S8 bad bad medium bad 3
What type of Rough Set Approach to deal with ordinal data? In the last decade, an extension of Classical Rough Set Theory (Indisceribility Rough Set Approach – IRSA) permitting to deal with ordinal data has been proposed, applied, and thoroughly investigated Greco, S., Matarazzo, B., Słowiński, R.: Rough sets theory for multicriteria decision analysis. European J. of Operational Research , 129 (2001) no.1, 1-47 It is called Dominance-based Rough Set Approach ( DRSA ) DRSA proved to be useful in many real world applications ranging from investment analysis to credit risk evaluation, from customer satisfaction analysis to technical and medical diagnoses 4
Classical Rough Set Theory vs. Dominance-based Rough Set Theory from indiscernibility principle to dominance principle Classical Rough Set Theory Indiscernibility principle If x and y are indiscernible with respect to all relevant attributes , then x should be classified to the same class as y Dominace-based Rough Set Theory Dominance principle If x is at least as good as y with respect to all relevant criteria , then x should be classified at least as good as y S.Greco, B.Matarazzo, R.Słowiński: Rough sets theory for multicriteria decision analysis. European J. of Operational Research , 129 (2001) no.1, 1-47 5
Dominance principle as monotonicity principle Interpretation of the dominance principle The better the evaluation of x with respect to considered criteria, the better its comprehensive evaluation Many other relationships of this type, e.g.: The faster the car, the more expensive it is The higher the inflation, the higher the interest rate The larger the mass and the smaller the distance, the larger the gravity The colder the weather, the greater the energy consumption The Dominance-based Rough Set Approach does not only permit representation and analysis of decision problems but, more generally, representation and analysis of all phenomena involving monotonicity 6
Dominance-based Rough Set Approach and fuzzy-rough hybridization Dominance-based Rough Set Approach = Ordinal data + Rough approximation = Graduation + Granulation = Fuzzy sets + Rough Sets different from other fuzzy-rough hybridiztion because a) DRSA does not use any fuzzy operators such as t-norms, t-conorms, fuzzy implication, b) DRSA takes into account only ordinal properties. 7
Fuzzy operators are not purely ordinal Consider the t- conorm of Łukasiewicz : T* ( , ) = min{ + , 1}, the following values of arguments: =0.5, =0.3, =0.2, =0.1 and their order preserving transformation: ’ =0.4, ’ =0.3, ’ =0.2, ’ =0.05. The values of the t-conorm are: T* ( , ) = 0.6 > T* ( , ) = 0.5, T* ( ’ , ’ ) = 0.45 < T* ( ’ , ’ ) = 0.5. The order of the results has changed after the order preserving transformation of the arguments. This means that the Łukasiewicz t -conorm takes into account not only the ordinal properties of the membership degrees, but also their cardinal properties. 8
Fuzzy operators are ...subjective and arbitrary For instance, what conorm to choose? Max: T* ( , ) = max{ , } ? t- conorm of Łukasiewicz : T* ( , ) = min{ + , 1} ? Probabilistic sum: T* ( , ) = + ? 0 if α 0 or β 0 Drastic t-conorm: T* ( , ) = ? 1 otherwise max α , β if α β 1 Nilpotent maximum: T* ( , ) = ? 1 otherwise 1 x 1 y λ 1 λ 1 Frank T-conorm: T* ( , ) = 1 log 1 ? λ λ 1 … 9
DRSA decision rules: no fuzzy operators Set of decision rules in terms of {M,Ph,L} representing preferences: If L good, then student good {S5,S6} If M medium & L medium, then student medium {S3,S4,S5,S6} If M medium & L bad, then student is bad or medium {S1,S2} If M bad, then student bad {S7,S8} If L bad, then student medium {S1,S2,S7} Greco, S., Matarazzo, B., Słowiński, R.: Axiomatic characterization of a general utility function and its particular cases in terms of conjoint measurement and rough-set decision rules. European J. of Operational Research , 158 (2004) no. 2, 271-292 10
Abstract algebra and rough set theory Many algebraic models have been proposed for rough set theory: Nelson algebra (Pagliani 1998), Heyting algebra (Pagliani 1998), Wajsberg algebra (Polkowski 2002), Stone algebra (Pomyka ł a & Pomyka ł a 1988) , Ł ukasiewicz algebra (Pagliani 1998), Brouwer-Zadeh algebra (Cattaneo & Nisticò 1989), ... These algebra models give elegant representations of basic properties of rough set theory. 11
In search of algebraic models for DRSA: bipolarity 12
In search of algebraic models for DRSA: bipolarity In Classical Rough Set Theory we approximate subsets of the universe, e.g. bad students, medium students, good students. In Dominance based Rough Set Theory we approximate downward cumulations of sets, e.g.: at most bad students (i.e., bad or worse students) at most medium students (i.e., medium or worse students) or upward cumulations of sets, e.g.: at least medium students (i.e. medium or better students) at least good students (i.e. good or better students) 13
In search of algebraic models for DRSA: bipolarity Critical remarks Lower and upper approximations for downward cumulations are different operators from lower and upper approximations for upward cumulations. The complement of a downward cumulation is an upward cumulation: • e.g., the complement of the set of at most medium students is the set of at least good students The complement of an upward cumulation is a downward cumulation: • e.g., the complement of the set of at least medium students is the set of at most bad students 14
In search of algebraic models for DRSA: bipolarity A typical algebra for Classical Rough Set Theory: A system < , , ,’, ,0,1> is a Brouwer-Zadeh distributive lattice if the following properties hold … A typical algebra for Dominance-based Rough Set Theory: A system < , + , - , , ,’ + , ’ - , + , - ,0,1> is a bipolar Brouwer- Zadeh distributive lattice if the following properties hold… 15
In search of algebraic models for DRSA:bipolarity Negations in a typical algebra for Classical Rough Set Theory: ’ : is a Kleene complementation,… : is a Brouwer complementation, … Negations in a typical algebra for Dominance-based Rough Set Theory: ’ + : + - and ’ - : - + are Kleene complementations,… + : + - and - : - + are Brouwer complementations, … 16
Algebra and Dominance-based Rough Set Theory:bipolarity We can generalize to Dominance-based Rough Set Approach all algebra models proposed for Classical Rough Set Theory: Nelson algebra Bipolar Nelson algebra , Heyting algebra Bipolar Heyting algebra, Wajsberg algebra Bipolar Wajsberg algebra, Stone algebra Bipolar Stone algebra, Ł ukasiewicz algebra Bipolar Łukasiewicz algebra , Brouwer-Zadeh algebra Bipolar Brouwer-Zadeh algebra, ... These algebra models give elegant representations of basic properties of Dominance – based Rough Set Theory. (Greco, S., Matarazzo, B., S ł owi ń ski, R.: Algebra and Topology for Dominance-based Rough Set Approach, in Z.W. Ra ś , W. Ribarsky (editors), Studies in Computational Intelligence, Springer, 2009, to appear) 17
In search of algebraic models for DRSA: graduality 18
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