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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 Sowiski Pozna University of Technology, Poland

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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

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SLIDE 2

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

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3

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)

bad medium bad bad S8 bad bad bad bad S7 good good good good S6 good good medium good S5 good medium medium medium S4 medium medium medium medium S3 medium bad medium medium S2 bad bad medium good S1 Overall class Literature (L) Physics (Ph) Mathematics (M) Student

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4

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

  • rdinal data has been proposed, applied, and thoroughly investigated

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

Greco, S., Matarazzo, B., Słowiński, R.: Rough sets theory for multicriteria decision analysis. European J. of Operational Research, 129 (2001) no.1, 1-47

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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

Classical Rough Set Theory vs. Dominance-based Rough Set Theory from indiscernibility principle to dominance principle

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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

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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.

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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.

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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*(,) = + ?

Drastic t-conorm: T*(,) =

Nilpotent maximum: T*(,) =

Frank T-conorm: T*(,) =

… ?

  • therwise

β

  • r

α if 1     

 

     ?

  • therwise

β α if β , α max 1 1

  

? λ λ λ log

y x λ

1 1 1 1 1

1 1

            

 

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10

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

DRSA decision rules: no fuzzy operators

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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 (Pomykała & Pomykała 1988) ,

Łukasiewicz algebra (Pagliani 1998),

Brouwer-Zadeh algebra (Cattaneo & Nisticò 1989),

...

These algebra models give elegant representations of basic properties

  • f rough set theory.
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In search of algebraic models for DRSA: bipolarity

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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

  • f sets, e.g.:

at most bad students (i.e., bad or worse students)

at most medium students (i.e., medium or worse students)

  • r 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)

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In search of algebraic models for DRSA: bipolarity

Critical remarks

Lower and upper approximations for downward cumulations are different

  • perators 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
  • f 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
  • f at most bad students
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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…

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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, …

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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,

Łukasiewicz 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., Słowiński, R.: Algebra and Topology for Dominance-based Rough Set Approach, in Z.W. Raś, W. Ribarsky (editors), Studies in Computational Intelligence, Springer, 2009, to appear)

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In search of algebraic models for DRSA: graduality

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In search of algebraic models for DRSA: graduality

In Classical Rough Set Theory we approximate subsets of the universe which can be considered as disjoint concepts, e.g.:

the subset of Bad Students, corresponding to the of Bad Student,

the subset of Medium Students, corresponding to the of Medium Student,

the subset of Good Students , corresponding to the of Good Student.

In Dominance based Rough Set Theory we approximate gradual concepts, e.g., taking into account the concept of Good Student

Bad Students is the set of elements belonging with the lowest level to the the concept of Good Student,

Medium Students is the set of elements belonging with an intermediate level to the the concept of Good Student,

Good Students is the set of elements belonging with the highest level to the the concept of Good Student.

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In search of algebraic models for DRSA: graduality

Critical remarks

At least Good Students  At least Medium Students  At least Bad Students Lower Approximation (and Upper Approximation) of at least Good Students  Lower Approximation (and Upper Approximation) of at least Medium Students  Lower Approximation (and Upper Approximation) of at most Bad Students

At most Good Students  At least Medium Students  At most Bad Students Lower Approximation (and Upper Approximation) of at most Good Students  Lower Approximation (and Upper Approximation) of at most Medium Students  Lower (and Upper Approximation) Approximation of at most Bad Students

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In search of algebraic models for DRSA: graduality

A typical algebra for Classical Rough Set Theory:

A system <,,,’,,0,1> is a Brouwer-Zadeh distributive lattice if the following properties hold …  = A(U)=(I(X),E(X)), XU, I(X)=R(X), E(X)=U-

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…

nU= (X1,…,Xn): X1,…,Xn U, , XiXj =, i,j=1,…,n

= (I1,…,In-1;E1,…,En-1): Ii,Ei U, Ii Ei =, i=1,…,n-1

+=

, (X1,…,Xn) nU 

-=

, (X1,…,Xn) nU 

 

X R

U X

n 1 i i  

       

      

        n 2 n 2

X R U ,..., X R U ; X R ,..., X R

       

      

          1 n 1 1 n 1

X R U ,..., X R U ; X R ,..., X R

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In search of algebraic models for DRSA: graduality

Ordinal classifications and their DRSA approximations:

 nU= (X1,…,Xn): X1,…,Xn U, , XiXj =, i,j=1,…,n  nU= (Bad Students,Medium Students, Good Students)  +=

, (X1,…,Xn) nU 

 += R+(at least Medium Students), R+(at least Good Students);

U- (at least Medium Students), U- (at least Good Students)

 -= R-(at most Bad Students), R-(at most Medium Students);

U- (at most Bad Students), U- (at most Medium Students)

U X

n 1 i i  

       

      

        n 2 n 2

X R U ,..., X R U ; X R ,..., X R

R

R

R

R

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23

Algebra and Dominance-based Rough Set Theory: graduality

We can generalize to Dominance-based Rough Set Approach all algebra models proposed for Classical Rough Set Theory:

Nelson algebra of rough sets  Bipolar Nelson algebra of rough ord. class.,

Heyting algebra of rough sets  Bipolar Heyting algebra of rough ord. class.,

Wajsberg algebra of rough sets  Bipolar Wajsberg algebra of rough ord. class.,

Stone algebra of rough sets  Bipolar Stone algebra of rough ord. class.,

Łukasiewicz algebra of rough sets  Bipolar Łukasiewicz algebra of rough ord. class.,

Brouwer-Zadeh algebra of rough sets  Bipolar Brouwer-Zadeh algebra

  • f rough ord. class.,

...

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Plan

The reasons of Dominance-based Rough Set Approach (DRSA)

De Morgan Brouwer-Zadeh lattice

De Morgan Brouwer-Zadeh lattice as a model for Indiscernibility-based Rough Set Approach (IRSA)

Bipolar De Morgan Brouwer-Zadeh lattice

Bipolar de Morgan Brouwer-Zadeh lattice as a model for DRSA

Ordinal classification

Bipolar de Morgan Brouwer-Zadeh lattice as a model for DRSA approximation of ordinal classification

Another algebraic model: Nelson algebra and bipolar Nelson algebra

Conclusions

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The reasons of Dominance-based Rough Set Approach (DRSA)

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What type of Rough Set Approach?

In the last decade, an extension of Classical Rough Set Theory (Indisceribility Rough Set Approach – IRSA) permitting to deal with decision problems has been proposed, applied, and thoroughly investigated

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

Greco, S., Matarazzo, B., Słowiński, R.: Rough sets theory for multicriteria decision analysis. European J. of Operational Research, 129 (2001) no.1, 1-47

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27

Classical Rough Set Approach (Z.Pawlak)

The granules of indiscernible objects are used to approximate classes

bad medium bad bad S8 bad medium bad bad S7 good good good good S6 good good medium good S5 good medium medium medium S4 medium medium medium medium S3 medium bad medium medium S2 bad bad medium good S1 Overall class Literature (L) Physics (Ph) Mathematics (M) Student

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28

Classical Rough Set Approach (Z.Pawlak)

Lower approximation of class „good”

bad medium bad bad S8 bad medium bad bad S7 good good good good S6 good good medium good S5 good medium medium medium S4 medium medium medium medium S3 medium bad medium medium S2 bad bad medium good S1 Overall class Literature (L) Physics (Ph) Mathematics (M) Student Lower Approximation

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Classical Rough Set Approach (Z.Pawlak)

Lower and upper approximation of class „good”

bad medium bad bad S8 bad medium bad bad S7 good good good good S6 good good medium good S5 good medium medium medium S4 medium medium medium medium S3 medium bad medium medium S2 bad bad medium good S1 Overall class Literature (L) Physics (Ph) Mathematics (M) Student Lower Approximation Upper Approximation

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Classical Rough Set Approach (Z.Pawlak)

The information in the decision table can be represented by the following “if ..., then ...” decision rules

1) If Literature=“good”, then the student is good (s5,s6)

2) If Mathematics=“bad”, then the student is bad (s7,s8)

3) If Mathematics=“good” and Literature=“bad”, then the student is bad (s1)

4) If Mathematics=“medium” and Literature=“bad”, then the student is medium (s2)

5) If Physics=“medium” and Literature=“medium”, then the student is medium or good (s3,s4)

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Classification in strict sense and

  • rdinal classification (sorting)
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Classification to preferentially non-ordered classes (classification in the strict sense)

Class 1

... x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

A

Class 2 Class p

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Ordinal classification to preference-ordered classes (sorting)

Class 1

... x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

A

Class 2 Class p

Class 1  Class 2  ...  Class p

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Classification in the strict sense – example of traffic signs

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Ordinal classification – example of multiple criteria sorting

  • f students

bad medium bad bad S8 bad bad bad bad S7 good good good good S6 good good medium good S5 good medium medium medium S4 medium medium medium medium S3 medium bad medium medium S2 bad bad medium good S1 Overall class Literature (L) Physics (Ph) Mathematics (M) Student

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Rough set approach and decision problems

The only class of decision problems which can be dealt with Classical Rough Set Theory is classification in the strict sense

This class is rather limited (even if it is practically the only class

  • f decision problems considered within data mining and knowledge

discovery!)

Why classical rough set approach is not useful to deal with choice problems, sorting problems and ranking problems?

Because Classical Rough Set Approach does not consider preferences

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Why Classical Rough Set Approach has to be adapted to MCDM?

Classical rough set approach does not detect inconsistency w.r.t. dominance (Pareto principle)

bad medium bad bad S8 bad bad bad bad S7 good good good good S6 good good medium good S5 good medium medium medium S4 medium medium medium medium S3 medium bad medium medium S2 bad bad medium good S1 Overall class Literature (L) Physics (Ph) Mathematics (M) Student

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Dominance-based Rough Set Approach (DRSA)

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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

Classical Rough Set Theory vs. Dominance-based Rough Set Theory from indiscernibility principle to dominance principle

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40

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

Rough Set approach to multiple-criteria sorting

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41

CRSA – illustration of formal definitions

Example Objects = firms

Investments Sales Value 40 17,8 High 35 30 High 32.5 39 High 31 35 High 27.5 17.5 High 24 17.5 High 22.5 20 High 30.8 19 Medium 27 25 Medium 21 9.5 Medium 18 12.5 Medium 10.5 25.5 Medium 9.75 17 Medium 17.5 5 Low 11 2 Low 10 9 Low 5 13 Low

slide-42
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42

CRSA – illustration of formal definitions

Objects in condition attribute space

attribute 1 (Investment) attribute 2 (Sales) 40 40 20 20

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43

CRSA – illustration of formal definitions

a1 a2 40 40 20 20

Indiscernibility sets Quantitative attributes are discretized according to perception of the user

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44

a1 40 40 20 20

CRSA – illustration of formal definitions

Granules of knowlegde are sets IP(x)

a2

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45

a1 40 40 20 20

CRSA – illustration of formal definitions

Lower approximation of class High

a2

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46

a1 40 40 20 20

Upper approximation of class High

a2

CRSA – illustration of formal definitions

slide-47
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47

CRSA – illustration of formal definitions

a1 a2 40 40 20 20

Lower approximation of class Medium

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SLIDE 48

48

CRSA – illustration of formal definitions

a1 a2 40 40 20 20

Upper approximation of class Medium

slide-49
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49

a1 40 40 20 20

CRSA – illustration of formal definitions

Boundary set of classes High and Medium

a2

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50

a1 40 40 20 20

CRSA – illustration of formal definitions

Lower = Upper approximation of class Low

a2

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51

DRSA – Evaluation of firms

 

Investements  Sales  Value  40 17,8 High 35 30 High 32.5 39 High 31 35 High 27.5 17.5 High 24 17.5 High 22.5 20 High 30.8 19 Medium 27 25 Medium 21 9.5 Medium 18 12.5 Medium 10.5 25.5 Medium 9.75 17 Medium 17.5 5 Low 11 2 Low 10 9 Low 5 13 Low

Value:

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52

Sales Investments 40 40 20 20

DRSA – illustration of formal definitions

Objects in condition criteria space

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53

Sales 40 40 20 20

DRSA – illustration of formal definitions

Granular computing with dominance cones

Investments

} : { ) ( x yD U y x D

P P

 

} : { ) ( y xD U y x D

P P

 

x

slide-54
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54

Sales 40 40 20 20

DRSA – illustration of formal definitions

Granular computing with dominance cones

Investments

} : { ) ( x yD U y x D

P P

 

} : { ) ( y xD U y x D

P P

 

slide-55
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55

Sales 40 40 20 20

DRSA – illustration of formal definitions

Lower approximation of upward union of class High

Investments

 

 

 

  

  

t P t

Cl x D U x Cl P :

slide-56
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56

Sales 40 40 20 20

DRSA – illustration of formal definitions

Upper approximation and the boundary of upward union of class High

Investments

 

 

 

 

   

     

t

Cl x P t

  • P

t

x D Cl x D U x Cl P :

     

Cl P Cl P Cl Bn

t t t P   

 

slide-57
SLIDE 57

57

Sales 40 40 20 20

DRSA – illustration of formal definitions

Lower = Upper approximation of upward union of class Medium

Investments

 

 

 

  

  

t P t

Cl x D U x Cl P :

 

 

 

 

   

     

t

Cl x P t

  • P

t

x D Cl x D U x Cl P :

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58

Sales 40 40 20 20

DRSA – illustration of formal definitions

Lower = upper approximation of downward union of class Low

Investements

 

 

 

 

    

     

t

Cl x P t P t

x D Cl x D U x Cl P :

 

 

 

  

  

t P t

Cl x D U x Cl P :

slide-59
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59

Sales 40 40 20 20

DRSA – illustration of formal definitions

Lower approximation of downward union of class Medium

Investments

 

 

 

  

  

t P t

Cl x D U x Cl P :

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SLIDE 60

60

Sales 40 40 20 20

DRSA – illustration of formal definitions

Upper approximation and the boundary of downward union of class Medium

Investements

 

 

 

 

    

     

t

Cl x P t P t

x D Cl x D U x Cl P :

     

Cl P Cl P Cl Bn

t t t P   

 

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61

Sales Investments 40 40 20 20

DRSA vs. CRSA

Comparison of CRSA and DRSA

a1 a2 40 40 20 20

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DRSA – preference modeling by decision rules

A set of (D D D)-rules induced from rough approximations represents a preference model of a Decision Maker

Traditional preference models:

 utility function (e.g. additive, multiplicative, associative, Choquet

integral, Sugeno integral),

 binary relation (e.g. outranking relation, fuzzy relation)

Decision rule model is the most general model of preferences: a general utility function, Sugeno or Choquet integral, or outranking relation exists if and only if there exists the decision rule model

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

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63

De Morgan Brouwer-Zadeh lattice (Cattaneo & Nisticò 1989)

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64

Quasi Brouwer-Zadeh Distributive Lattices (1)

 A system <,,,’,,0,1> is quasi Brouwer-Zadeh distributive

lattice if the following properties hold

 <,,,0,1> is a distributive lattice  ’ :  is a Kleene complementation, that is for all a,b

 (K1) a’’=a  (K2) (a  b)’=a’  b’  (k3) a  a’  b  b’

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Quasi Brouwer-Zadeh Distributive Lattices (2)

  :  is a Brouwer complementation, that is for all a,b

 (B1) a  a  =a  (B2) (a  b)  =a   b   (B3) a  a =0

 (win) for all a, a  a’  A quasi Brouwer-Zadeh lattice is Brouwer-Zadeh lattice if

stronger interconnection rule is satisfied:

 (in) for all a, a  = a ’

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Brouwer-Zadeh Distributive De Morgan Lattices

 A Brouwer-Zadeh lattice is a de Morgan Brouwer-Zadeh lattice if

it is satisfied the -de Morgan property:

 (B2a) for all a,b, (a b) =a b 

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De Morgan Brouwer-Zadeh lattice as a model for Indiscernibility-based Rough Set Approach (IRSA)

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Indiscernibility Rough Sets

 U, universe  R, equivalence relation on U  For any yU, [y]R is the equivalence class of y  For any XU,

 Lower approximation: R(X)= y U: [y]RX  Upper approximation: = y U: [y]RX

 

X R

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Interior-exterior of Rough Sets

For any XU,

interior of X: I(X)=R(X),

exterior of X: E(X)=U

For any X,Y U,

union: (I(X),E(X))(I(Y),E(Y))=(I(X)I(Y), E(X)E(Y))

intersection: (I(X),E(X))(I(Y),E(Y))=(I(X)I(Y), E(X)E(Y))

  • Kleene negation: (I(X),E(X))-=(E(X),I(X))
  • Brouwer negation: (I(X),E(X))=(E(X),U-E(X))

Minimal element: (,U), Maximal element: (U,)

 

X R

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Indiscernibility based rough sets modeled as de Morgan Brouwer-Zadeh Lattice

 A(U)=(I(X),E(X)), XU  The system

<A(U),,,  , - ,(,U), (U,) > is a de Morgan Brouwer-Zadeh distributive lattice.

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Bipolar De Morgan Brouwer-Zadeh lattice

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Bipolar quasi Brouwer-Zadeh Distributive Lattices (1)

A system <,+,-,,,’+, ’-,+,-,0,1> is a bipolar quasi Brouwer-Zadeh distributive lattice if the following properties hold:

<,,,0,1> is a distributive lattice,

<+,,,0,1>, <-,,,0,1> are distributive lattices with +,-,

’+: +- and ’-: -+ are bipolar Kleene complementations, that is for all a,b+ and c,d-

 (K1b) a’+’- =a, c’-’+ =c,  (K2b) (a  b)’+ =a’+ b’+, (c  d)’- =c’-  d’-,  (k3b) a  a’+  b  b’+, c  c’-  d  d’-.

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Bipolar Quasi Brouwer-Zadeh Distributive Lattices (2)

+:+- and -:-+ are bipolar Brouwer complementations, that is for all a,b+ and c,d-:

 (B1b) a a+- =a, c c-+ =c,  (B2b) (a  b)+=a+ b+ , (c  d)+ =c+ d+ ,  (B3b) a  a+=0, c  c-=0,

(win-b) for all a+ and b-, a+ a’+ and b- b’-.

A bipolar quasi Brouwer-Zadeh lattice is a bipolar Brouwer-Zadeh lattice if stronger interconnection rule is satisfied:

(in-b) for all a+ and b-, a+- = a+’- and b-+ = b-’+.

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Bipolar Brouwer-Zadeh Distributive De Morgan Lattices

 A bipolar Brouwer-Zadeh lattice is a bipolar de Morgan Brouwer-

Zadeh lattice if it is satisfied the -de Morgan property:

 (B2a-b) for all a,b+ and c,d- ,  (ab)+=a+b+, (cd)-=c-d- .

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Bipolar De Morgan Brouwer-Zadeh lattice as a model for DRSA

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76

Dominance-based Rough Sets

U, universe

R, partial preorder relation on U (i.e. R is reflexive and transitive) (very often R is obtained as a dominance relation)

For any yU, R+(y)=xU:xRy and R-(y)=xU:yRx

For any XU,

 Upward lower approximation: R+(X) = yU:R+(y)X,  Upward upper approximation: = y U: R-(y)X,  Downward lower approximation: R-(X) = yU:R-(y)X,  Downward upper approximation: = y U: R+(y)X.

 

X R

 

X R

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Bipolarization of the universe

(W+,W-)  2U2U is a bipolarization of universe U if,

(1) for any X1,X2W+, X1X2 W+ and X1X2 W+,

(2) for any Y1,Y2W-, Y1Y2 W- and Y1Y2 W-,

(3) for any X W+, U-X W-,

(4) for any Y W-, U-YW+,

(5) ,UW+.

(3) and (4) are equivalent to: W-=Y2U:XW+ such that Y=U-X or, equivalently, W+=X2U:YW- such that X=U-Y.

For (3) and (4), (5) implies ,UW- .

(1), (3) and (4) imply (2); (2), (3) and (4) imply (1).

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Interpretation of the bipolarization of the universe

Intuitively, W+ contains positive concepts such as “at least good students”, “at least medium students”, while W- contains negative concepts such as “at most medium students”, “at most bad students”.

Observe that one possible bipolarization is (W+,W-)=(U,U)

(U, W+, W- ) is a bitopology.

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Interior-exterior of Dominance-based Rough Sets

Given a bipolarization (W+,W-) of universe U, for any XW+ and YW-:

 interior of X and Y: I(X)=R+(X), I(Y)=R-(Y),  exterior of X and Y: E(X)=U-

, E(Y)=U-

For any X,Y U,

 union: (I(X),E(X))(I(Y),E(Y))=(I(X)I(Y), E(X)E(Y))  intersection: (I(X),E(X))(I(Y),E(Y))=(I(X)I(Y), E(X)E(Y))

 

X R

 

Y R

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Bipolar negations for interior-exterior represenations

  • f Dominance-based Rough Sets

Given a bipolarization (W+,W-) of universe U, for any XW+ and YW-

  • bipolar Kleene negation:
  • (I(X),E(X))-+=(E(X),I(X)),
  • (I(Y),E(Y))--=(E(Y),I(Y)).
  • bipolar Brouwer negation:
  • (I(X),E(X))+=(E(X),U-E(X)),
  • (I(Y),E(Y)) -=(E(Y),U-E(Y)).
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Dominance based rough sets modeled as bipolar de Morgan Brouwer-Zadeh Lattice

 B=(I,E): I,EU and IE=,  B+=(I,E): XW+ such that I=I(X),E=E(X),  B-=(I,E): YW- such that I=I(Y),E=E(Y),

The system <B,B+,B-,,, - - ,-+, +, -,(,U), (U,) > is bipolar de Morgan Brouwer-Zadeh distributive lattice.

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Bipolar De Morgan Brouwer-Zadeh lattice as a model for DRSA approximation

  • f ordinal classifications
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Dominance based rough sets for ordinal classification modeled as bipolar de Morgan Brouwer-Zadeh Lattice

Consider in an universe U the set of all ordered n-fold partitions: nU= (X1,…,Xn): X1,…,Xn U, , XiXj =, i,j=1,…,n

(X1,…,Xn) can be identified with the set of decision classes (Cl1,…,Cln)

For i=1,…,n

B= (I1,…,In-1;E1,…,En-1): Ii,Ei U, Ii Ei =, i=1,…,n-1

B+= , (X1,…,Xn) nU 

B-= , (X1,…,Xn) nU 

U X

n 1 i i  

 

i 1 j j i n i j j i

X X , X X

   

 

       

      

        n 2 n 2

X R U ,..., X R U ; X R ,..., X R

       

      

          1 n 1 1 n 1

X R U ,..., X R U ; X R ,..., X R

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Union and intersection of ordinal classifications in Dominance-based Rough Set Approach

For any (I1,…,In-1;E1,…,En-1), (I’1,…,I’n-1;E’1,…,E’n-1)B

 union:

(I1,…,In-1;E1,…,En-1)(I’1,…,I’n-1;E’1,…,E’n-1) = (I1 I’1 ,…, In-1 I’n-1 ; E1E’1 ,…,En-1E’n-1 )

 intersection:

(I1,…,In-1;E1,…,En-1)(I’1,…,I’n-1;E’1,…,E’n-1) = (I1I’1 ,…, In-1I’n-1 ; E1E’1 ,…,En-1E’n-1 )

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Bipolar negations for Dominance-based Rough ordinal classifications

Given (I1,…,In-1;E1,…,En-1)B+ and (I’1,…,I’n-1;E’1,…,E’n-1)B-

  • bipolar Kleene negation:
  • (I1,…,In-1;E1,…,En-1)-+=(E1,…,En-1;I1,…,In-1),
  • (I’1,…,I’n-1;E’1,…,E’n-1)--=(E’1,…,E’n-1;I’1,…,I’n-1).
  • bipolar Brouwer negation:
  • (I1,…,In-1;E1,…,En-1)+=(E1,…,En-1,U-E1,…,U-En-1),
  • (I’1,…,I’n-1;E’1,…,E’n-1)-=(E’1,…,E’n-1,U-E’1,…,U-E’n-1).
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Dominance based rough ordinal classifications modeled as bipolar de Morgan Brouwer-Zadeh Lattice The system <B,B+,B-,,, - - ,-+, +, -,(,…, ;U,…,U), (U,…,U; ,…,) > is bipolar de Morgan Brouwer-Zadeh distributive lattice.

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Another algebraic model: Nelson algebra and bipolar Nelson algebra

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Nelson Algebra

A system <,,,,,’, 0,1>, where ’,  and  are unary operations, is a Nelson Algebra if the following properties hold

<,,,0,1> is a distributive lattice

(a  b)’=a’  b’ for all a,b,

a’’=a for all a,

a  a’  b  b’ for all a,b,

a  c  (a’  b) iff c  a  b for all a,b,c,

a (b c)=(a  b) c,

  • a=a  a’ =a  0
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Operations for Nelson algebra of Rough Sets (Pagliani 1998)

For any XU,

interior of X: I(X)=R(X),

exterior of X: E(X)=U

For any X,Y U,

union: (I(X),E(X))(I(Y),E(Y))=(I(X)I(Y), E(X)E(Y))

intersection: (I(X),E(X))(I(Y),E(Y))=(I(X)I(Y), E(X)E(Y))

  • quasi complementation: (I(X),E(X))’ =(E(X),I(X))
  • pseudo-complementation: (I(X),E(X))=(U-I(X),I(X))
  • Weak relative psudo-complementation:

(I(X),E(X))(I(Y),E(Y))=((U-I(X))I(Y), I(X)E(Y))

 

X R

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Bipolar Nelson Algebra (1)

A system <,-,+,,,,,’, 0,1>, where

 +,-  ’-: - +, -: - +, ’+: + -, +: + -,  +-: +- -, -+: -+ +,

is a Nelson Algebra if the following properties hold

<,,,0,1>, <+,,,0,1> and <-,,,0,1> are distributive lattices

(a  b)’+=a’+  b’+ for all a,b+, (c  d)’-=c’-  d’- for all c,d-,

a’+’-=a for all a+, b’-’+=b for all b-,

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Bipolar Nelson Algebra (2)

 a  a’+  b  b’+ for all a,b+, c  c’-  d  d’- for all c,d-,  a  c  (a’+  b) iff c  a +- b for all a+,b-,c,  a  c  (a’-  b) iff c  a -+ b for all a-,b+,c,  a +-(b +- c)=(a  b) +-c for all a,b+, c-,  a -+(b -+c)=(a  b) -+c for all a,b-, c+,  +a=a +-a’+=a +-0,  -a=a -+a’-=a -+0.

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Operations for Bipolar Nelson algebra of Rough Sets: quasi complementation and pseudo complementation

 Given a bipolarization (W+,W-) of universe U, for any XW+ and

YW-

  • quasi complementation:
  • (I(X),E(X))’+=(E(X),I(X)),
  • (I(Y),E(Y))’-=(E(Y),I(Y))
  • pseudo-complementation:
  • +(I(X),E(X)) =(U-I(X),I(X)),
  • -(I(Y),E(Y)) =(U-I(Y),I(Y))
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Operations for Bipolar Nelson algebra of Rough Sets: weak relative pseudo-complementation

 For any (I,E)B+ and (I’,E)B-

  • weak relative pseudo-complementation:
  • (I,E) +- (I’,E) =((U-I) I’, IE’,)
  • (I’,E’) -+ (I,E) = ((U-I’) I, I’E)
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94

Operations for Bipolar Nelson algebra of Rough ordinal classifications: quasi complementation and pseudo complementation

 For any (I1,…,In-1;E1,…,En-1)B+ and (I’1,…,I’n-1;E’1,…,E’n-1)B-

  • quasi complementation:
  • (I1,…,In-1;E1,…,En-1)’+=(E1,…,En-1;I1,…,In-1),
  • (I’1,…,I’n-1;E’1,…,E’n-1)’-=(E’1,…,E’n-1;I’1,…,I’n-1).
  • pseudo-complementation:
  • +(I1,…,In-1;E1,…,En-1)=(U-I1,…,U-In-1;I1,…,In-1),
  • -(I’1,…,I’n-1;E’1,…,E’n-1)’-=(U-I’1,…,U-I’n-1;I’1,…,I’n-1).
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Operations for Bipolar Nelson algebra of Rough ordinal classifications: weak relative pseudo-complementation

 For any (I1,…,In-1;E1,…,En-1)B+ and (I’1,…,I’n-1;E’1,…,E’n-1)B-

  • weak relative pseudo-complementation:
  • (I1,…,In-1;E1,…,En-1) +- (I’1,…,I’n-1;E’1,…,E’n-1)=

=((U-I1) I’1,…, (U-In-1) I’n-1; I1E’1,…,In-1E’n-1)

  • (I’1,…,I’n-1;E’1,…,E’n-1) -+ (I1,…,In-1;E1,…,En-1)

=((U-I’1) I1,…, (U-I’n-1) In-1; I’1E1,…,I’n-1En-1)

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Conclusions

We considered the problem of giving an algebraic model to DRSA.

We observed that these algebraic models should represent two basic features of DRSA: bipolarity and graduality.

We introduced the bipolar de Morgan Brouwer-Zadeh lattice.

We showed how the bipolar de Morgan Brouwer-Zadeh lattice can model DRSA approximation of ordinal classifications

Other algebraic models can be generalized to model DRSA approximation of

  • rdinal classifications: here we presented the bipolar Nelson algebra

Topological approach to DRSA is another interesting subject involving interesting concepts such as bitopological space and Priestely space

(see Greco, S., Matarazzo, B., Słowiński, R.: Algebra and Topology for Dominance-based Rough Set Approach, in Z.W. Raś, W. Ribarsky (editors), Studies in Computational Intelligence, Springer, 2009, to appear)