Diff´ erentes extensions floues de l’analyse formelle de concepts Yassine Djouadi 1 , Didier Dubois 2 , Henri Prade 2 1 Universit´ e de Tizi-Ouzou - Alg´ erie 2 IRIT - Universit´ e Paul Sabatier - France 1
CONTENT • Reminder on Formal Concept Analysis (FCA) • Different semantics for degrees • Fuzzy context with gradual summaries • Typicality in concept analysis • Uncertainty in concept analysis • Possibility theory and Formal Concept Analysis: A new Galois connexion 2
FORMAL CONCEPT ANALYSIS • formal context : a relation R between a set of objects Obj and a set of properties Prop . • ( x, y ) ∈ R means x has property y . • Formal concept – Pair ( X, Y ) s.t. X = { x ∈ Obj | R ( x ) ⊇ Y } and Y = { y ∈ Prop | R − 1 ( y ) ⊇ X } The X’s share the properties in Y and are the only ones – Equivalently defined as a maximal pair ( X, Y ) s.t. X × Y ⊆ R . • The 2 definitions of formal concepts will not always coincide when R becomes fuzzy • starting point of the theoretical basis for data mining 3
Example objects 1 2 3 4 5 6 7 8 × × × × a × × b × × × × c × × × × × d × e × × × f × × × × g × × × h × × i 4
Example : R ∆ ( { 5 , 6 } ) = { a, b, c, d, f } Properties satisfied by at least both 5 and 6 objects 5 6 1 2 3 4 7 8 ⊗ ⊗ × × a ⊗ ⊗ b × × × × c × ⊗ ⊗ × × d × e ⊗ ⊗ × f × × × × g × × × h × × i 5
DIFFERENT SEMANTICS FOR DEGREES (1) English Married Age × Peter 0.9 Young Sophie (0,1] ? Age-domain Mike 0 25 Nahla [0.2,0.4] (0.7; 1) [20, 22] • Gradual property: e.g. to what extent one masters English. 1. Turn some × into a degree R − 1 ( y ) : R is the support of the fuzzy set of objects having property y . 2. Turn some blank into a degree R − 1 ( y ) : R is the core of the fuzzy set of objects having property y . 3. Do both : R is the 1 / 2 -cut of the fuzzy set of objects having property y . 6
Example : Gradual Data Table The core of R is the previous Boolean table objects 1 2 3 4 5 6 7 8 0 . 3 0 . 4 × × × × a 0 . 7 0 . 8 × × 0 . 5 b × × × × 0 . 4 c × × × × × 0 . 9 d × 0 . 3 0 . 7 e × × × 0 . 8 f × × × × 0 . 2 0 . 6 g 0 . 3 × × × 0 . 3 0 . 4 0 . 9 h 0 . 5 × 0 . 3 × 0 . 7 i 7
DIFFERENT SEMANTICS FOR DEGREES (2) English Married Age × Peter 0.9 Young (0,1] ? Age-domain Sophie Mike 0 25 Nahla [0.2,0.4] (0.7; 1) [20, 22] • Ignorance, uncertainty. • Ill-known gradual property ≡ interval of degrees. • Moving from binary properties to many-valued attribute values. 8
GRADUAL SUMMARIES OF A CONTEXT R 2 Peter Sophie Mike Joe age ≥ 20 + + + + age ≥ 25 + + age ≥ 30 + salary ≥ 1000 + + + + salary ≥ 1200 + + + salary ≥ 1400 + + R 2 is be re-encoded in a more compact way, using fuzzy sets. 9
GRADUAL SUMMARIES OF A CONTEXT (continued) R 3 Peter Sophie Mike Joe age ‘young’ 1 0.7 0.6 1 salary ‘small’ 1 0.8 0.6 0.6 A fuzzy concept can be built from its α -cuts R α = { ( x, y ) : R ( x, y ) ≥ α } , yielding formal concepts ( X α , Y α ) Fuzzy concept complemented by the gradual relation R ( x, young ) ≤ R ( x, small ) for x ∈ { Peter, Sophie, Mike } . 10
Fuzzy concepts: choice of connectives • If ⊤ is a t-norm and → ⊤ is the residuated implication: A formal fuzzy concept (Belohlavek) is a pair ( X, Y ) of fuzzy sets such that x ∈ Obj µ X ( x ) → µ R ( x, y ) = µ Y ( y ); min y ∈ P rop µ Y ( y ) → µ R ( x, y ) = µ X ( x ) min • If ⊤ = min and → ⊤ is a G¨ odel implication : A formal fuzzy concept is a nested family of crisp formal concepts ( X α , Y α ) that are maximal sets such that X α × Y α ⊆ R α . • α → ⊤ β = (1 − α ) ⊥ β = 1 − α ⊤ (1 − β ) has been also proposed (Burusco and Fuentes-Gonzalez), but lack of closure properties • Needs more investigation to understand which choice of operators is natural/possible in the context of applications. 11
Introducing typicality in formal concept analysis concepts as ( extension , intension )-pairs ( X, Y ) s. t. X = { x ∈ Obj | R ( x ) ⊇ Y } and Y = { y ∈ Prop | R − 1 ( y ) ⊇ X } • Gradualness in properties can be taken into account by allowing R to be fuzzy (Belohlavek) • Typicality can be introduced in FCA by keeping R crisp , and introducing degrees among objects and among properties. 12
two principles • (A) An object x is all the more normal (or typical ) w.r.t. a set of properties Y as it has all the properties y ∈ Y that are sufficiently important ; • (B) A property y is all the more important w.r.t. a set of objects X as all the objects x ∈ X that are sufficiently normal have it. 13
Bird example Table 1: R eggs 2 legs feather fly + + + + albatross parrot + + + + penguin + + + kiwi + + 14
Example: What is a bird? birds: X = { albatross, parrot, penguin, kiwi } bird properties: Y = { ‘ laying eggs ’ , ‘ having two legs ’ , ‘ flying ’ , ‘ having feathers ’ } ) X t ( albatross ) = X t ( parrot ) = 1 , X t ( penguin ) = α , X t ( kiwi ) = β typicality X t with 1 > α > β (kiwis do not fly and have no feathers). • fuzzy set of important properties , according to (B) Y i ( y ) = min x X t ( x ) → R ( x, y ) , with a → 1 = 1 and a → 0 = 1 - a It expresses that an object not having property y makes a property all the less important for the concept bird as this bird is considered as more typical • Let Y i ( y ) define the degree of importance of property y , in the definition of bird , ∀ y . fuzzy set of typical objects , according to (A) µ ( x ) = min y Y i ( y ) → R ( x, y ) , using (1 − a ) → 0 = a We get µ ( albatross ) = Y i ( parrot ) = 1 , µ ( penguin ) = α ; µ ( kiwi ) = β • We have ∀ x µ ( x ) = X t ( x ) a (fuzzy) Galois connection 15
Representing ‘Tweety is a bird’ ‘Tweety is a bird’ Y i bird the fuzzy set of important properties for birds ∀ y ∈ Prop π Tweety ( y c ) = 1 − Y i bird ( y ) where y c is the negation of y • the possibility that Tweety has not property y is all the greater as y is less important for birds • the certainty that Tweety has property y is all the greater as y is more important for birds π y ( Tweety ) ( no ) = 1 − Y i bird ( y ) • to be paralleled with ‘Tweety is young’ π age ( Tweety ) ( u ) = µ young ( u ) 16
Uncertainty in FCA (1) • Incomplete information in data tables : changing the representation convention from – × = an object has a property – blank space = object does not have the property, to the case when it is unknown whether n object has a property • Introducing a new symbol in the table, for ignorance : ? 17
Example : incomplete formal context objects 1 2 3 4 5 6 7 8 × × × × ? ? a × ? b × × ? c × × × × d ? × e ? × × × f × ? × × ? g × × ? h × ? i 18
Uncertainty in FCA (2) • An incomplete formal context stands for a set of relations R ∗ ⊆ R ⊆ R ∗ where – R ∗ is obtained by changing ? into blank. – R ∗ is obtained by changing ? into × . • An ill-known formal concept ( X , Y ) = (( X ∗ , X ∗ ) , ( Y ∗ , X ∗ )) such that – X ∗ ⊆ X ∗ , Y ∗ ⊆ X ∗ – ( X ∗ , Y ∗ ) is a formal concept from R ∗ ; ( X ∗ , Y ∗ ) is a formal concept from R ∗ . 19
Uncertainty in Boolean data tables • Introducing gradual uncertainty valued on an ordinal scale L : ( α, β ) ∈ L 2 with min( α, β ) = 0 in place ( x, y ) . – α = N ( xRy ) is the certainty degree that object x satisfies property y : ( α, 0) – β = N ( ¬ ( xRy )) is the certainty degree that object x does not satisfy property y With conventions – × : (1 , 0) generalized by ( α, 0) , α > 0 – Blank : (0 , 1) generalized by (0 , β ) , β > 0 – The symbol ? is encoded by (0 , 0) : ignorance – The λ -cut of an uncertain data table is an incomplete one. • One can compute, in the spirit of possibilistic logic: – the degree of possibility that a data table fits with an uncertain one. – a possibility distribution over possible formal concepts For further research.... 20
Example : Uncertain formal context L = { 0 , 0 . 1 , 0 . 2 , . . . , 0 . 9 , 1 } objects 1 2 3 4 5 6 7 8 (0 , 0 . 2) (0 , 1) (0 , 0) (0 , 0) (1 , 0) (0 . 7 , 0) (0 . 5 , 0) (0 . 8 , 0) a (0 , 0 . 7) (0 , 0 . 6) (0 , 0 . 2) (0 , 0 . 3) (0 , 0) (1 , 0) (0 , 9) (0 , 1) b (0 , 1) (0 , 0 . 6) (0 , 0 . 4) (0 , 0 . 2) (0 , 1) (0 . 8 , 0) (0 , 0) (1 , 0) c (0 , 0 . 8) (0 , 0 . 4) (0 , 0 . 7) (0 , 1) (0 . 9 , 0) (0 . 6 , 0) (0 . 5 , 0) (1 , 0) d (0 , 0 . 4) (0 , 1) (0 , 0) (0 , 1) (0 , 1) (0 , 0 . 7) (1 , 0) (0 , 0 . 5) e (0 , 1) (0 , 0) (0 , 0 . 6) (0 , 0 . 7) (0 . 3 , 0) (0 . 4 , 0) (0 , 1) (1 , 0) f (1 , 0) (0 , 0) (0 . 8 , 0) (0 . 6 , 0) (0 , 0 . 2) (0 , 1) (0 , 0) (0 , 0 . 8) g (0 , 0 . 5) (0 . 4 , 0) (1 , 0) (0 , 0) (0 , 1) (0 , 0 . 8) (0 , 0 . 5) (0 , 0 . 6) h (0 , 0 . 4) (0 , 0) (0 , 1) (1 , 0) (0 , 0 . 7) (0 , 0 . 3) (0 , 0 . 1) (0 , 0 . 9) i 21
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