Diff erentes extensions floues de lanalyse formelle de concepts - - PowerPoint PPT Presentation

Diff´ erentes extensions floues de l’analyse formelle de concepts

Yassine Djouadi1, Didier Dubois2, Henri Prade2

1Universit´

e de Tizi-Ouzou - Alg´ erie

2IRIT - 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

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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

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Example

• bjects

1 2 3 4 5 6 7 8 a × × × × b × × c × × × × d × × × × × e × f × × × g × × × × h × × × i × ×

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Example : R∆({5, 6}) = {a, b, c, d, f}

Properties satisfied by at least both 5 and 6

• bjects

1 2 3 4 5 6 7 8 a ⊗ ⊗ × × b ⊗ ⊗ c × × × × d × ⊗ ⊗ × × e × f ⊗ ⊗ × g × × × × h × × × i × ×

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DIFFERENT SEMANTICS FOR DEGREES (1)

English Married Age Peter 0.9 × Young Sophie (0,1] ? Age-domain Mike 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.

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Example : Gradual Data Table

The core of R is the previous Boolean table

• bjects

1 2 3 4 5 6 7 8 a 0.3 0.4 × × × × b 0.7 0.8 × × 0.5 c 0.4 × × × × d × 0.9 × × × × e 0.3 0.7 × f 0.8 × × × g × × × × 0.2 0.6 h 0.3 × × × 0.3 0.4 0.9 i 0.5 × 0.3 × 0.7

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DIFFERENT SEMANTICS FOR DEGREES (2)

English Married Age Peter 0.9 × Young Sophie (0,1] ? Age-domain Mike 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.

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GRADUAL SUMMARIES OF A CONTEXT

R2 Peter Sophie Mike Joe age ≥ 20 + + + + age ≥ 25 + + age ≥ 30 + salary ≥ 1000 + + + + salary ≥ 1200 + + + salary ≥ 1400 + + R2 is be re-encoded in a more compact way, using fuzzy sets.

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GRADUAL SUMMARIES OF A CONTEXT (continued)

R3 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 Rα = {(x, y) : R(x, y) ≥ α}, yielding formal concepts (Xα, Y α) Fuzzy concept complemented by the gradual relation R(x, young) ≤ R(x, small) for x ∈ {Peter, Sophie, Mike}.

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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 min

x∈Obj µX(x) → µR(x, y) = µY (y); min y∈P rop µY (y) → µR(x, y) = µX(x)

• If ⊤ = min and →⊤ is a G¨
• del implication : A formal fuzzy concept is a nested

family of crisp formal concepts (Xα, Yα) that are maximal sets such that Xα × Yα ⊆ Rα.

• 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.

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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.

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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.

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Bird example

Table 1: R eggs 2 legs feather fly albatross + + + + parrot + + + + penguin + + + kiwi + +

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Example: What is a bird?

birds: X = {albatross, parrot, penguin, kiwi} bird properties: Y ={‘laying eggs’, ‘having two legs’, ‘flying’, ‘having feathers’}) typicality Xt Xt(albatross) = Xt(parrot) = 1, Xt(penguin) = α, Xt(kiwi) = β with 1 > α > β (kiwis do not fly and have no feathers).

• fuzzy set of important properties, according to (B)

Y i(y) = minxXt(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) = minyY 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) = Xt(x)

a (fuzzy) Galois connection

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Representing ‘Tweety is a bird’

‘Tweety is a bird’ Y i

bird the fuzzy set of important properties for birds

∀y ∈ Prop

πTweety(yc) = 1 − Y i

bird(y)

where yc 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

• πy(Tweety)(no) = 1 − Y i

bird(y)

to be paralleled with ‘Tweety is young’ πage(Tweety)(u) = µyoung(u)

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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 : ?

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Example : incomplete formal context

• bjects

1 2 3 4 5 6 7 8 a ? ? × × × × b ? × c × ? × d × × × × e ? × f ? × × × g × ? × × ? h × × ? i ? ×

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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∗.

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Uncertainty in Boolean data tables

• Introducing gradual uncertainty valued on an ordinal scale L:

(α, β) ∈ L2 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....

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Example : Uncertain formal context

L = {0, 0.1, 0.2, . . . , 0.9, 1}

• bjects

1 2 3 4 5 6 7 8 a (0, 0.2) (0, 1) (0, 0) (0, 0) (1, 0) (0.7, 0) (0.5, 0) (0.8, 0) b (0, 0.7) (0, 0.6) (0, 0.2) (0, 0.3) (0, 0) (1, 0) (0, 9) (0, 1) c (0, 1) (0, 0.6) (0, 0.4) (0, 0.2) (0, 1) (0.8, 0) (0, 0) (1, 0) d (0, 0.8) (0, 0.4) (0, 0.7) (0, 1) (0.9, 0) (0.6, 0) (0.5, 0) (1, 0) e (0, 0.4) (0, 1) (0, 0) (0, 1) (0, 1) (0, 0.7) (1, 0) (0, 0.5) f (0, 1) (0, 0) (0, 0.6) (0, 0.7) (0.3, 0) (0.4, 0) (0, 1) (1, 0) g (1, 0) (0, 0) (0.8, 0) (0.6, 0) (0, 0.2) (0, 1) (0, 0) (0, 0.8) h (0, 0.5) (0.4, 0) (1, 0) (0, 0) (0, 1) (0, 0.8) (0, 0.5) (0, 0.6) i (0, 0.4) (0, 0) (0, 1) (1, 0) (0, 0.7) (0, 0.3) (0, 0.1) (0, 0.9)

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POSSIBILITY THEORY and FORMAL CONCEPT ANALYSIS

• in FCA formal concepts are obtained with “sufficiency operator”

the set of properties satisfied by all the objects in X : X∆ = {y ∈ P | ∀x ∈ O (x ∈ X ⇒ xRy)} = {y ∈ P | X ⊆ R(y)} =

x∈X R(x)

X∇ = {y ∈ P | X ∪ R(y) = O} =

x∈X R(x)

• XΠ is the set of properties satisfied by at least one object in X :

XΠ = {y ∈ P | X ∩ R(y) = ∅} = {y ∈ P | ∃x ∈ X, xRy} =

x∈X R(x)

• XN is the set of properties that only the objects in X have :

XN = {y ∈ P | R(y) ⊆ X} = {y ∈ P | ∀x ∈ O (xRy ⇒ x ∈ X)} =

x∈X R(x) 22

A NEW GALOIS CONNEXION

• The pairs (X, Y ) such that XN = Y and Y N = X characterize

independent sub-contexts (i.e. which have in common neither objects nor properties) inside the initial context Proposition The pairs (X, Y ) such that XN = Y and Y N = X are minimal pairs in the inclusion sense such that: (X × Y ) ∪ (X × Y ) ⊇ R Proof XN = Y ⇔

x∈X R(x) = Y

⇔

x∈X R(x) = Y ⇔ R(x) ⊆ X ∪ Y

Similarly: R(y) ⊆ Y ∪ X Then R(x) × R(y) ⊆ (X ∪ Y ) × (Y ∪ X) ⇔ R ⊆ (X × Y ) ∪ (X × Y )

• fuzzy extension ?

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Example

• bjects

1 2 3 4 5 6 7 8 a × × × × b × × c × × × d × × × × e × f × × × g × × × × h × × × i ×

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CONCLUSION

• Fuzzy formal concept analysis has been mainly developed under its mathematical

aspect without much discussing the semantics of the degrees.

• New issues are of particular interest:

– Fuzzy concept summaries, – Typicality of objects and importance of properties – Uncertain concepts – Connections between the generalized possibility theory view of FCA with other Galois connections.

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Example : R∆({5, 6}) = {a, b, c, d, f}

Properties satisfied by at least both 5 and 6

• bjects

1 2 3 4 5 6 7 8 a ⊗ ⊗ × × b ⊗ ⊗ c × × × × d ⊗ × × e × f ⊗ ⊗ × g × × × × h × × × i ×

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