Many-Valued Concept Lattices for Conceptual Clustering and Information Retrieval Nizar Messai, Marie-Dominique Devignes, Amedeo Napoli, Malika Sma¨ıl-Tabbone ORPAILLEUR Team LORIA: CNRS, INRIA Nancy - Grand Est, Nancy-Universite´ Nancy, France
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}
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B′ = {g ∈ G | ∀m ∈ B, gIm}
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{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
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G a set of objets M a set of attributes I ⊆ G × M a binary relation
Far from Sun
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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
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Big Mass
Formal Concept Large
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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
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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