A FCA perspective on Rough Set Theory Bernhard Ganter & - PowerPoint PPT Presentation

Functional and ordinal dependencies Rough Set Theory Generalisations A FCA perspective on Rough Set Theory Bernhard Ganter & Christian Meschke Institut f¨ ur Algebra Technische Universit¨ at Dresden October 20, 2010

Functional and ordinal dependencies Rough Set Theory Generalisations Survey Functional and ordinal dependencies Indiscernibility Dependencies Rough Set Theory Approximations Lattices of approximations Generalisations Approximations from arbitrary kernel-closure pairs Contextual Representation

Functional and ordinal dependencies Rough Set Theory Generalisations Indiscernibility by attribute values • Let ( G , M , W , I ) a many-valued context. • Let B ⊆ M be a set of attributes. One defines m ( g , h ) ∈ Ind B : ⇐ ⇒ ∀ m ∈ B : m ( g ) = m ( h ) . • Ind B is an indiscernibility equivalence relation. g w • D ⊆ M is functionally dependent on B if Ind B ⊆ Ind D . • In this case one says that the functional dependency B → D is valid.

Functional and ordinal dependencies Rough Set Theory Generalisations Indiscernibility by attribute values • Let ( G , M , W , I ) a many-valued context. • Let B ⊆ M be a set of attributes. One defines m ( g , h ) ∈ Ind B : ⇐ ⇒ ∀ m ∈ B : m ( g ) = m ( h ) . • Ind B is an indiscernibility equivalence relation. g w • D ⊆ M is functionally dependent on B if Ind B ⊆ Ind D . • In this case one says that the functional dependency B → D is valid.

Functional and ordinal dependencies Rough Set Theory Generalisations Indiscernibility by attribute values • Let ( G , M , W , I ) a many-valued context. • Let B ⊆ M be a set of attributes. One defines B ( g , h ) ∈ Ind B : ⇐ ⇒ ∀ m ∈ B : m ( g ) = m ( h ) . • Ind B is an indiscernibility equivalence g 4 sick + relation. • D ⊆ M is functionally dependent on B if 4 sick + h Ind B ⊆ Ind D . • In this case one says that the functional dependency B → D is valid.

Functional and ordinal dependencies Rough Set Theory Generalisations Indiscernibility by attribute values • Let ( G , M , W , I ) a many-valued context. • Let B ⊆ M be a set of attributes. One defines D B ( g , h ) ∈ Ind B : ⇐ ⇒ ∀ m ∈ B : m ( g ) = m ( h ) . • Ind B is an indiscernibility equivalence g hot 4 sick + relation. • D ⊆ M is functionally dependent on B if hot 4 sick + h Ind B ⊆ Ind D . • In this case one says that the functional dependency B → D is valid.

Functional and ordinal dependencies Rough Set Theory Generalisations Reducts • Let C , D ⊆ M be sets of condition and of decision attributes. • A subset R ⊆ C is called a reduct of C if Ind R ⊆ Ind C . A reduct is called minimal if no proper subset is a reduct. • One defines Pos( C , D ) := { g ∈ G | [ g ] Ind C ⊆ [ g ] Ind D } , • and calls R ⊆ C a D -relative reduct of C if Pos( R , D ) = Pos( C , D ) .

Functional and ordinal dependencies Rough Set Theory Generalisations Reducts • Let C , D ⊆ M be sets of condition and of decision attributes. • A subset R ⊆ C is called a reduct of C if Ind R ⊆ Ind C . A reduct is called minimal if no proper subset is a reduct. • One defines Pos( C , D ) := { g ∈ G | [ g ] Ind C ⊆ [ g ] Ind D } , • and calls R ⊆ C a D -relative reduct of C if Pos( R , D ) = Pos( C , D ) .

Functional and ordinal dependencies Rough Set Theory Generalisations Functional dependencies • Consider to the data table ( G , M , W , I ) the formal context ( G × G , M , I fun ) , where ( g , h ) I fun m : ⇐ ⇒ m ( g ) = m ( h ) . • In this formal context, for B ⊆ M it holds that B ′ = Ind B . • Remark: since ( g , h ) I fun = ( h , g ) I fun one is free to “reduce” the context’s object set G × G to the set of all two-elementary subsets of G .

Functional and ordinal dependencies Rough Set Theory Generalisations Functional dependencies • Consider to the data table ( G , M , W , I ) the formal context ( G × G , M , I fun ) , where ( g , h ) I fun m : ⇐ ⇒ m ( g ) = m ( h ) . • In this formal context, for B ⊆ M it holds that B ′ = Ind B . • Remark: since ( g , h ) I fun = ( h , g ) I fun one is free to “reduce” the context’s object set G × G to the set of all two-elementary subsets of G .

Functional and ordinal dependencies Rough Set Theory Generalisations Functional Dependencies • Hence, for C , D ⊆ M the following two statements are equivalent: (a) the attribute implication C → D holds in ( G × G , M , I fun ), (b) the functional depenceny C → D holds in ( G , M , W , I ). • The minimal reducts are the minimal generators in the closure system of the intents of ( G , M , I fun ). • The intents of ( G , M , I fun ) are referred to as coreducts . • The intent C ′′ contains precisely those attributes whose vaulues (on total G ) can be inferred from the values of C .

Functional and ordinal dependencies Rough Set Theory Generalisations Functional Dependencies • Hence, for C , D ⊆ M the following two statements are equivalent: (a) the attribute implication C → D holds in ( G × G , M , I fun ), (b) the functional depenceny C → D holds in ( G , M , W , I ). • The minimal reducts are the minimal generators in the closure system of the intents of ( G , M , I fun ). • The intents of ( G , M , I fun ) are referred to as coreducts . • The intent C ′′ contains precisely those attributes whose vaulues (on total G ) can be inferred from the values of C .

Functional and ordinal dependencies Rough Set Theory Generalisations Ordinal dependencies • The functional dependency C → D holds in ( G , M , W , I ) iff there is a function f : W C → W D such that for every object g ∈ G it holds that f ( c ( g ) | c ∈ C ) �− → ( d ( g ) | d ∈ D ) . • It would be interesting if such a mapping f is “somehow” order-preserving. • Let us assume that for every attribute m ∈ M ( W m , ≤ m ) is a partially ordered set with m [ G ] ⊆ W m ⊆ W .

Functional and ordinal dependencies Rough Set Theory Generalisations Ordinal dependencies • The functional dependency C → D holds in ( G , M , W , I ) iff there is a function f : W C → W D such that for every object g ∈ G it holds that f ( c ( g ) | c ∈ C ) �− → ( d ( g ) | d ∈ D ) . • It would be interesting if such a mapping f is “somehow” order-preserving. • Let us assume that for every attribute m ∈ M ( W m , ≤ m ) is a partially ordered set with m [ G ] ⊆ W m ⊆ W .

Functional and ordinal dependencies Rough Set Theory Generalisations Ordinal dependencies • One says, the ordinal dependency C → D holds if all objects g , h ∈ G satisfy � � � � ∀ c ∈ C : c ( g ) ≤ c c ( h ) ⇒ ∀ d ∈ D : d ( g ) ≤ d d ( h ) = . • Ordinal dependency implies functional dependency (regardless of the choice of the order relations ≤ m ). • If one chooses x ≤ m y : ⇐ ⇒ x = y we receive the functional dependencies again.

Functional and ordinal dependencies Rough Set Theory Generalisations Ordinal dependencies • The ordinal dependency C → D holds in ( G , M , W , I ) iff the attribute implication C → D holds in the formal context ( G × G , M , I ord ) , where ( g , h ) I ord m : ⇐ ⇒ m ( g ) ≤ m m ( h ) . • Finally: let us take a look at an example.

Functional and ordinal dependencies Rough Set Theory Generalisations Ordinal dependencies • The ordinal dependency C → D holds in ( G , M , W , I ) iff the attribute implication C → D holds in the formal context ( G × G , M , I ord ) , where ( g , h ) I ord m : ⇐ ⇒ m ( g ) ≤ m m ( h ) . • Finally: let us take a look at an example.

Functional and ordinal dependencies Rough Set Theory Generalisations Paul the octopus

Functional and ordinal dependencies Rough Set Theory Generalisations An example Germany’s results at the FIFA World Cup 2010 and the predictions of Paul the octopus. game goals result Paul’s outcome opponent stage GER OPP predict. 1 Australia group stage 4 0 win win correct 2 Serbia group stage 0 1 loss loss correct 3 Ghana group stage 1 0 win win correct 4 England round of 16 4 1 win win correct 5 Argentina quarter-finals 4 0 win win correct 6 Spain semi-finals 0 1 loss loss correct 3 rd place play-off 7 Uruguay 3 2 win win correct • 7 objects, G = { 1 , . . . , 7 } ,

Functional and ordinal dependencies Rough Set Theory Generalisations An example Germany’s results at the FIFA World Cup 2010 and the predictions of Paul the octopus. game goals result Paul’s outcome opponent stage GER OPP predict. 1 Australia group stage 4 0 win win correct 2 Serbia group stage 0 1 loss loss correct 3 Ghana group stage 1 0 win win correct 4 England round of 16 4 1 win win correct 5 Argentina quarter-finals 4 0 win win correct 6 Spain semi-finals 0 1 loss loss correct 3 rd place play-off 7 Uruguay 3 2 win win correct • 7 objects, G = { 1 , . . . , 7 } ,