July 23, 2008 ECAI’08, Patras, Greece

N Messai, M-D Devignes, A Napoli, M Sma¨ıl-Tabbone

Many-Valued Concept Lattices

Motivation FCA Many-Valued Galois Connection Applications Conclusion

Motivation ◮

FCA-based Data Analysis process

N Messai, M-D Devignes, A Napoli, M Sma¨ıl-Tabbone

Many-Valued Concept Lattices

Motivation FCA Many-Valued Galois Connection Applications Conclusion

Motivation ◮

FCA-based Data Analysis process

N Messai, M-D Devignes, A Napoli, M Sma¨ıl-Tabbone

Many-Valued Concept Lattices

Motivation FCA Many-Valued Galois Connection Applications Conclusion

Motivation ◮

FCA-based Data Analysis process

◮

Goal ◮

Avoid data transformation

N Messai, M-D Devignes, A Napoli, M Sma¨ıl-Tabbone

Many-Valued Concept Lattices

Motivation FCA Many-Valued Galois Connection Applications Conclusion

Motivation ◮

FCA-based Data Analysis process

◮

Goal ◮

Avoid data transformation ◮

Define data analysis methods for complex data more precisely, define Galois Connection for Many-Valued Contexts

N Messai, M-D Devignes, A Napoli, M Sma¨ıl-Tabbone

Many-Valued Concept Lattices

Motivation FCA Many-Valued Galois Connection Applications Conclusion

Outline Motivation Formal Concept Analysis Basic settings Many-Valued Contexts Conceptual Scaling Many-Valued Galois Connection Similarity-based Analysis Formalization Applications Multi-level Clustering the SimBA System Information Retrieval Conclusion N Messai, M-D Devignes, A Napoli, M Sma¨ıl-Tabbone

Many-Valued Concept Lattices

Motivation FCA Many-Valued Galois Connection Applications Conclusion

Basic settings Many-Valued Contexts Conceptual Scaling

Formal Context, Derivation operators

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◮

◮

Example

Derivation operators ◮

A′ = {m ∈ M | ∀g ∈ A, gIm}

◮

B′ = {g ∈ G | ∀m ∈ B, gIm}

◮

◮

◮

{Mercury, Venus}’ = {Large} {Large}’ = {Earth, Mercury, Venus}

PP P

Attr. Obj. PPP Mercury × Venus × Earth × Mars

Galois connection

N Messai, M-D Devignes, A Napoli, M Sma¨ıl-Tabbone

Many-Valued Concept Lattices

× ×

Has satellite

◮

G a set of objets M a set of attributes I ⊆ G × M a binary relation

Far from Sun

◮

Big Mass

Formal Context: K = (G, M, I)

Large

◮

× ×

× ×

Motivation FCA Many-Valued Galois Connection Applications Conclusion

Basic settings Many-Valued Contexts Conceptual Scaling

a pair (A,B) ◮

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◮

XXX Attr. XXX Obj.

A′ = B and B ′ = A

({Earth, Mercury, Venus},{Large}) B(K): all the concepts in K

Mercury Venus Earth Mars

Partial Order Relation ◮

(A1,B1) ≤ (A2,B2) ◮

A1⊆A2 (eq. B2⊆B1)

({Earth, Venus}, {Big Mass, Large}) ≤ ({Earth, Mercury, Venus},{Large}) ◮

Concept Lattice ◮

(B(K),≤)

N Messai, M-D Devignes, A Napoli, M Sma¨ıl-Tabbone

Many-Valued Concept Lattices

× × ×

× ×

Has satellite

◮

Big Mass

Formal Concept Large

◮

Far from Sun

Formal Concept, Concept lattice

× ×

× ×

Motivation FCA Many-Valued Galois Connection Applications Conclusion

Basic settings Many-Valued Contexts Conceptual Scaling

Many-Valued Context

× ×

XX X ◮

Formally, (G, M, I, W ) ◮ ◮ ◮ ◮

G a set of objects M a set of attributes W a set of attribute values I ⊆ G × M × W a ternary relation

N Messai, M-D Devignes, A Napoli, M Sma¨ıl-Tabbone

Obj. Mercury Venus Earth Mars

Attr.

XXX

Many-Valued Concept Lattices

Satellite

× ×

Distance to Sun ( 106 km)

× ×

Real-world mesures about planets

Mass (1021 t)

× × ×

Has satellite

Attr.

XXX

Far from Sun

Obj. Mercury Venus Earth Mars

Big Mass

XX X

◮

Diameter (103 km)

Planets context

Large

◮

48.8 12.1 12.7 6.7

0.33 4.87 5.97 0.64

57.9 108 150 228

1 2

Motivation FCA Many-Valued Galois Connection Applications Conclusion

Basic settings Many-Valued Contexts Conceptual Scaling

Standard approach: Conceptual Scaling Real-world mesures about planets XX X Obj. Mercury Venus Earth Mars

Diameter

Mass

Distance to

(103 km)

(1021 t)

Sun ( 106 km)

48.8 12.1 12.7 6.7

0.33 4.87 5.97 0.64

57.9 108 150 228

Satellite

1 2

N Messai, M-D Devignes, A Napoli, M Sma¨ıl-Tabbone

DS>200

Satellite

×

100