An Introduction to Formal Concept Analysis Mehdi Kaytoue Mehdi Kaytoue mehdi.kaytoue@insa-lyon.fr http://liris.cnrs.fr/mehdi.kaytoue October 29 th 2013
The Knowledge Discovery Process Identified domain(s) Elements of order theory ↓ Data acquisition (crawling, scraping, interviews) Concept lattice Rough data Algorithms ↓ Selection and preparation Conceptual ↓ Transformation : cleaning and formatting Scaling Pattern Prepared data structures ↓ Data mining (Numerical & symbolic methods) Triadic Concepts Extracted units References ↓ Interpretation and evaluation ↓ Knowledge representation formalism Knowledge units ↓ Knowledge based systems An interactive and iterative process guided by an analyst and knowledge of the domain October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 2/59
The Knowledge Discovery Process Elements of order Large volumes of data from which useful, significant and theory reusable units should be extracted Concept lattice Algorithms Involves several tasks of data and knowledge processing Conceptual Mining: ((closed) frequent ...) pattern mining (itemset, Scaling sequences, graphs,...) Pattern structures Modeling: hierarchy of concepts and relations Triadic Concepts Representing: Concepts and relations as knowledge units References Reasoning and solving problems: classification and case based reasonning Many domains of applications Scientific data (agronomy, astronomy, chemistry, cooking, medicine) Sensors data ((interactions) traces of human/system behaviors) October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 3/59
A basic example: What can say a binary table? Elements of order theory c/p p1 p2 p3 p4 p5 Assume a binary table M ij Concept lattice obtained by an interview c1 x x Algorithms c2 x x x x x A set of clients c i Conceptual Scaling c3 x x A set of products p j Pattern c4 x x The relation states that structures c5 x x x x some clients bought some Triadic Concepts c6 x x x x References products c7 x x x x The table may of course be c8 x x x “ big ” (millions of lines, c9 x x thousands of columns) c10 x x x The table may contain errors October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 4/59
Let’s make the table speak! Elements of order { p 2 , p 3 , p 5 } is an itemset theory c/p p1 p2 p3 p4 p5 of frequency 4 / 10 = 0 . 4. Concept lattice c1 x x { p 3 , p 5 } has 6 / 10 = 0 . 6 as Algorithms c2 x x x x x frequency Conceptual Scaling c3 x x p 3 ∧ p 5 − → p 2 is an Pattern c4 x x association rule with a structures c5 x x x x Triadic Concepts confidence of 4 / 6 = 0 . 66: if c6 x x x x References a client buys p 3 and p 5, c7 x x x x 0 . 66 is the probability he c8 x x x buys also p 2. c9 x x c10 x x x conf ( X → Y ) = sup ( X ∪ Y ) / sup ( X ) October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 5/59
FCA and the Concept lattice, a synthetic view Elements of order theory What can we say about Concept lattice { p 2 } ? and { p 2 , p 3 } ? Algorithms Conceptual What about p 2 → p 3? Scaling Pattern structures What about p 3 → p 5? Triadic Concepts References How to classify object described by { p 2 , p 3 } ? What if lines are products and columns their attributes? October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 6/59
FCA and the Concept lattice, a synthetic view Elements of order theory Formal concepts can be Concept lattice represented in a KR Algorithms formalism (eg.. DLs) Conceptual Scaling Concept 1 ≡ ∃ hasAwR . p 3 Pattern structures Concept 2 ≡ ∃ hasAwR . p 2 Triadic Concepts Concept 2 ⊑ Concept 1 References ... October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 7/59
FCA and the Concept lattice, a synthetic view Useful for many tasks of DM, DB, KR; Gives a formalism Elements of order theory (frequent (closed)) itemsets Concept lattice (partial) implications or association rules Algorithms Possible knowledge units to be reused for problem solving Conceptual Scaling Pattern What happens when structures When there are too much patterns ? Triadic Concepts Closure, iceberg, stability, ... References When the table is not binary? Scaling, pattern structures When the table is n -dimensional? Triadic and polyadic concept analysis When relations arise between objects themselves? Relational concept analysis October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 8/59
Outline Elements of order theory Elements of order theory 1 Concept lattice Algorithms 2 Concept lattice Conceptual Scaling 3 Algorithms Pattern structures Triadic Concepts Conceptual Scaling 4 References Pattern structures 5 6 Triadic Concepts 7 References October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 9/59
Binary relations Definition (Binary relation) Elements of order theory A binary relation R between two arbitrary sets M and N is Concept lattice defined on the Cartesian product M × N and consists of Algorithms pairs ( m , n ) with m ∈ M and n ∈ N . When ( m , n ) ∈ R , we Conceptual Scaling usually write mRn . Pattern structures Triadic Concepts Definition (Order relation) References A binary relation R on a set M is called an order relation (or shortly order) if it satisfies the following conditions for all elements x , y , z ∈ M : 1 (reflexivity) xRx 2 (antisymmetry) xRy and x � = y ⇒ not yRx 3 (transitivity) xRy and yRz ⇒ xRz October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 10/59
Total and partial orders Elements of order theory Definition (Ordered set) Concept lattice Algorithms Given an order relation ≤ on a set M , an ordered set is a pair Conceptual Scaling ( M , ≤ ) . When ≤ is a partial order, ( M , ≤ ) is called partially Pattern ordered set, or poset for short. structures Triadic Concepts Example: Given a set E , ( 2 E , ⊆ ) References Definition (Total order) For any a , b ∈ M , either a ≤ b or b ≤ a . Example: real numbers October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 11/59
Infimum, Supremum Elements of order theory Definition (Infimum, supremum) Concept lattice Algorithms Let ( M , ≤ ) be an ordered set and A a subset of M . A lower Conceptual Scaling bound of A is an element s of M with s ≤ a for all a ∈ A . An Pattern upper bound of A is defined dually. If it exists a largest structures element in the set of all lower bounds of A , it is called the Triadic Concepts infimum of A and is denoted by “inf A” or � A ; dually, a least References upper bound is called supremum and denoted by “sup A” or � A . Infimum and supremum are frequently called respectively meet and join, also denoted respectively by the symbols ⊓ and ⊔ . October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 12/59
Lattice Elements of order theory Definition (Lattice, complete lattice) Concept lattice Algorithms A poset V = ( V , ≤ ) is a lattice, if for any two elements Conceptual Scaling x , y ∈ V the supremum x ∨ y and the infimum x ∧ y always Pattern exist. V is called a complete lattice if for any subset X ⊆ V , structures the supremum � X and the infimum � X exist. Every Triadic Concepts complete lattice V has a largest element � called the unit References element denoted by 1 V . Dually, the smallest element 0 V is called the zero element. We will rather use the symbol bottom ⊥ for 0 V and top ⊤ for the unit element in the following. October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 13/59
Remark Elements of order theory Concept lattice We can reconstruct the order relation from the lattice Algorithms operations infimum and supremum by Conceptual Scaling x ≤ y ⇐ ⇒ x = x ∧ y ⇐ ⇒ x ∨ y = y Pattern structures Triadic Concepts { a } ≤ { a , b } ⇐ ⇒ { a } = { a } ∩ { a , b } References { a } ≤ { a , b } ⇐ ⇒ { a } ∪ { a , b } = { a , b } This remark is important for understanding pattern structures October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 14/59
Hasse diagram of the powerset lattice Elements of order theory Concept lattice Algorithms Conceptual Scaling Pattern structures Triadic Concepts References x ≤ y ⇐ ⇒ x = x ∧ y ⇐ ⇒ x ∨ y = y { a } ≤ { a , b } ⇐ ⇒ { a } = { a } ∩ { a , b } { a } ≤ { a , b } ⇐ ⇒ { a } ∪ { a , b } = { a , b } October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 15/59
Hasse diagram of the partition lattice Elements of order theory Concept lattice Algorithms Conceptual Scaling Pattern structures Triadic Concepts References {{ a , b } , { c } , { d }} ≤ {{ a , b , c } , { d }} {{ a , b } , { c } , { d }} ∨ {{ a , c } , { b } , { d }} = {{ a , b , c } , { d }} {{ a , b , c } , { d }} ∧ {{ a , c , d } , { b }} = {{ a , b } , { c } , { d }} October 29 th 2013 Mehdi Kaytoue An Introduction to Formal Concept Analysis 16/59
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