An introduction to RCA Analyse Relationnelle de Concepts: Une approche pour fouiller des ensembles de données multi-relationnels Ecole des Mines d’Alès Séminaire LGI2P février 2013 Marianne Huchard February 14, 2013 Marianne Huchard EMA 2013
An introduction to RCA Brief presentation of FCA – Formal Concept Analysis A methodology for: ◮ data analysis, data mining ◮ knowledge representation ◮ unsupervised learning Roots: ◮ lattice theory, Galois correspondences (Birkhoff, 1940; Barbut & Monjardet, 1970) ◮ concept lattices (Wille, 1982) Marianne Huchard EMA 2013
An introduction to RCA Brief presentation of FCA – Formal Concept Analysis Contexts and concepts ◮ Handled data ◮ entities with characteristics ◮ provided with a Formal Context (a binary table) flying nocturnal feathered migratory with_crest with_membrane flying squirrel × × × × × bat × ostrich × × × flamingo × × × chicken ◮ Concept : maximal group of entities sharing characteristics ◮ Concept lattice : concepts with a partial order relation Marianne Huchard EMA 2013
An introduction to RCA Contexte binaire et ses applications caractéristiques Contexte ( O , A , R ) ◮ deux ensembles finis O et A ◮ une relation binaire R ⊆ O × A . Définition (applications caractéristiques d’une relation binaire) Attributs communs à un ensemble d’objets f : P ( O ) → P ( A ) X �− → f ( X ) = { y ∈ A | ∀ x ∈ X , ( x , y ) ∈ R} Objets partageant un ensemble d’attributs g : P ( A ) → P ( O ) Y �− → g ( Y ) = { x ∈ O | ∀ y ∈ Y , ( x , y ) ∈ R} Marianne Huchard EMA 2013
An introduction to RCA Correspondances de Galois Résultats de Birkhoff 1940, Ore 1944, Barbut et Monjardet 1970 ( f , g ) forme une correspondance de Galois ◮ couple d’applications resp. de ( A , ≤ A ) dans ( B , ≤ B ) et inversement, décroissantes et dont les deux composées f ◦ g et g ◦ f sont extensives Conséquences ◮ f ◦ g et g ◦ f sont des opérateurs de fermeture (monotone croissant, extensif, idempotent) ◮ si ( A , ≤ A ) et ( B , ≤ B ) sont des treillis finis, leurs ensembles de fermés pour f ◦ g et g ◦ f forme deux treillis isomorphes (en utilisant l’infimum et la fermeture du supremum) ◮ Le treillis de Galois est le sous-treillis du treillis du produit des fermés restreint aux couples (x, y) tels que y = f (x) Retenir : Le treillis de concepts est un cas particulier de treillis de Galois Marianne Huchard EMA 2013
An introduction to RCA Concept Un concept formel C est un couple ( E , I ) tel que f ( E ) = I ou de manière équivalente E = g ( I ) E = { e ∈ O | ∀ i ∈ I, (e, i) ∈ R} est l’extension (objets couverts), I = { i ∈ A | ∀ e ∈ E, (e, i) ∈ R} est l’intension (caractéristiques partagées). Marianne Huchard EMA 2013
An introduction to RCA Spécialisation entre concepts L’ensemble de tous les concepts C forme un treillis L lorsqu’il est muni de l’ordre suivant : ( E 1 , I 1 ) ≤ L ( E 2 , I 2 ) ⇔ E 1 ⊆ E 2 (or de manière équivalente I 2 ⊆ I 1 ). On peut déduire de la réduction transitive du treillis l’ensemble minimal non redondant des implications du contexte qui ont un support non nul (il existe des objets vérifiant l’implication) Marianne Huchard EMA 2013
An introduction to RCA Brief presentation of FCA – Formal Concept Analysis Marianne Huchard EMA 2013
An introduction to RCA Brief presentation of FCA – Formal Concept Analysis Marianne Huchard EMA 2013
An introduction to RCA Brief presentation of FCA – Formal Concept Analysis Marianne Huchard EMA 2013
An introduction to RCA Brief presentation of FCA – Formal Concept Analysis Marianne Huchard EMA 2013
An introduction to RCA Brief presentation of FCA – Formal Concept Analysis AOC-poset Attribute-Object-Concept poset Lattice without Concepts 0, 3 and 4) max #concepts in lattice 2 min ( | A | , | O | ) max #concepts in AOC- poset | A | + | O | Marianne Huchard EMA 2013
An introduction to RCA FCA and complex data ◮ many-valued contexts (integers, floats, terms, structures, symbolic objects, etc.) (Ganter et Wille, Polaillon, ...) ◮ fuzzy descriptions (Yahia et al., Belohlavek, ...) ◮ hierarchies on values (Godin et al., Carpineto et Romano, ...) ◮ logical description (Chaudron et al., Ferré et al., ...) ◮ graphs (Liquière, Prediger et Wille, ...) ◮ linked objects (Priss, Hacène-Rouane et al., ...) ◮ etc. Marianne Huchard EMA 2013
An introduction to RCA Relational Concept Analysis (RCA) ◮ Extends the purpose of FCA for taking into account object categories and links between objects ◮ Main principles: ◮ a relational model based on the entity-relationship model ◮ integrate relations between objects as relational attributes ◮ iterative process ◮ RCA provides a set of interconnected lattices ◮ Produced structures can be represented as ontology concepts within a knowledge representation formalism such as description logics (DLs). Joint work with: A. Napoli, C. Roume, M. Rouane-Hacène, P. Valtchev Marianne Huchard EMA 2013
An introduction to RCA Relational Context Family (RCF) A simple entity-relationship model to introduce RCA Relational Context Family ◮ object-attribute contexts ◮ Pizza ◮ Ingredient ◮ object-object context ◮ has-topping ⊆ Pizza × Ingredient Marianne Huchard EMA 2013
An introduction to RCA Relational Context Family (RCF) A RCF F is a pair ( K , R ) with: ◮ K is a set of object-attribute contexts K i = ( O i , A i , I i ) ◮ R is a set of object-object contexts R j = ( O k , O l , I j ) , ◮ ( O k , O l ) are the object sets of formal contexts ( K k , K l ) ∈ K 2 ◮ I j ⊆ O k × O l ◮ K k is the source/domain context , K l is the target/range context . ◮ we may have K k = K l . Marianne Huchard EMA 2013
An introduction to RCA Relational Context Family (RCF) / object-attributes contexts cereal-leguminous fruit-vegetable veg-oil meat dairy fish Ingredient tomato-sauce × calzone cream × thick tomato × thin Pizza × basilic okonomi × olive × alberginia × olive oil × × × margherita soy languedoc × mushroom × four-cheeses × eggplant × three-cheeses × onion × frutti-di-mare × pepper × quebec × × ananas regina × mozza × hawai × goat-cheese × × × lorraine emmental kebab × fourme-ambert × squid × shrimp × mussels × ham × Marianne Huchard EMA 2013 bacon ×
An introduction to RCA Relational Context Family (RCF) / object-object context / part 1 tomato-sauce mushroom eggplant olive oil tomato pepper ananas cream basilic onion olive soy has-topping × × × × okonomi alberginia × × × × × margherita × × × × × × × × × × × × languedoc four-cheeses × three-cheeses × frutti-di-mare × × × quebec × × × regina hawai × × lorraine × × × × × × kebab Marianne Huchard EMA 2013
An introduction to RCA Relational Context Family (RCF) / object-object context / part 2 fourme-ambert goat-cheese maple-sirup emmental mussels chicken shrimp mozza bacon squid ham corn has-topping okonomi alberginia margherita × languedoc × × × × × four-cheeses three-cheeses × × × frutti-di-mare × × × × × × × × quebec regina × × hawai × × lorraine × × kebab × × Marianne Huchard EMA 2013
An introduction to RCA Data patterns we would like to extract Using a classification on ingredients by their categories of topping (fruit-vegetable, dairy, etc.) ◮ All pizzas, even different, except four-cheese and three-cheese, contain at least one topping which is a vegetable ◮ Two pizzas (four-cheese and three-cheese) have all their topping in dairy ingredients ◮ For pizzas: have meat ⇒ have dairy ◮ For pizzas: being thin ⇒ have at least dairy ◮ For pizzas: have only dairy ⇒ being thin Marianne Huchard EMA 2013
An introduction to RCA RCA - Initial Lattice building At the beginning, only the object-attribute contexts are used to build the foundation of the concept lattice family Marianne Huchard EMA 2013
An introduction to RCA RCA - Introducing relations as relational attributes Given an object-object context R j = ( O k , O l , I j ) , There are different notable schemas between an object of domain O k and concepts formed on O l . E. g. ◮ Existential : an object is linked (by R j ) to at least one object of the extent of a concept ◮ Universal : an object is linked (by R j ) only to objects of the extent of a concept Marianne Huchard EMA 2013
An introduction to RCA RCA - Existential relational attributes margherita has one topping in Concept_10 extent: mozza . It has other links to other concept extents. ∃ has-topping.Concept_10 is assigned to margherita Marianne Huchard EMA 2013
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