PART II Concept lattices and related structures in a fuzzy setting Radim BELOHLAVEK Dept. Computer Science Palack´ y University, Olomouc Czech Republic Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 1 / 62
Outline – ordinary FCA—overview – fuzzy attributes, concept-forming operators, and fuzzy concept lattices – basic theorem – closure structures – representation theorems – algorithms for fuzzy concept lattices – non-classical issues – factorization by similarity – measure-theoretic-like result – other approaches – factor analysis with formal concepts as factors Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 2 / 62
Introduction to Formal Concept Analysis (FCA) – FCA = method of analysis of tabular data (R. Wille, TU Darmstadt) – based on order-theoretic concepts (Birkhoff, Ore, Monjardet) – input : objects (rows) × attributes (columns) table y 1 y 2 y 3 y 1 y 2 y 3 x 1 1 1 1 x 1 X X X � 1 1 1 � x 2 1 0 1 or x 2 X X or 1 0 1 0 1 1 x 3 0 1 1 x 3 X X . . . . . . . . . . . . – output : 1. concept lattice 2. (non-redundant) dependencies A ⇒ B , e.g. { y 1 , y 3 } ⇒ { y 2 } Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 3 / 62
FCA: Basic Notions X = { x 1 , . . . } . . . objects, Y = { y 1 , . . . } . . . attributes I y 1 y 2 y 3 induced operators ↑ : 2 X → 2 Y , ↓ : 2 Y → 2 X x 1 X X A ↑ = { y ∈ Y | ∀ x ∈ A : ( x , y ) ∈ I } x 2 X X B ↓ = { x ∈ X | ∀ y ∈ B : ( x , y ) ∈ I } x 3 X → A ↑ . . . attributes common to all objects from A A �− → B ↓ . . . objects sharing all attributes from B B �− example: { x 1 , x 2 } ↑ = { y 3 } , { y 3 } ↓ = { x 1 , x 2 } Definition (formal concept = fixpoint of ↑ , ↓ ) Formal concept in data is a pair � A , B � s.t. A ↑ = B and B ↓ = A . – formal concepts = interesting clusters in data – inspired by Port-Royal logic: concept = extent A + intent B – e.g. DOG: extent = { poodle, . . . } , intent = { barks, has tail, . . . } Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 4 / 62
formal concepts = maximal rectangles I y 1 y 2 y 3 y 4 I y 1 y 2 y 3 y 4 I y 1 y 2 y 3 y 4 x 1 1 1 1 1 x 1 1 1 1 1 x 1 1 1 1 1 x 2 1 0 1 1 x 2 1 0 1 1 x 2 1 0 1 1 x 3 0 1 1 1 x 3 0 1 1 1 x 3 0 1 1 1 x 4 0 1 1 1 x 4 0 1 1 1 x 4 0 1 1 1 x 5 1 0 0 0 x 5 1 0 0 0 x 5 1 0 0 0 ( A 1 , B 1 ) = ( { x 1 , x 2 , x 3 , x 4 } , { y 3 , y 4 } ) ( A 2 , B 2 ) = ( { x 1 , x 3 , x 4 } , { y 2 , y 3 , y 4 } ) ( A 3 , B 3 ) = ( { x 1 , x 2 } , { y 1 , y 3 , y 4 } ) Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 5 / 62
FCA: Basic Notions (cntd.) Definition (concept lattice = formal concepts + conceptual hierarchy) Concept lattice (Galois lattice) of a data table is the set B ( X , Y , I ) = {� A , B � | A ↑ = B , B ↓ = A } of all formal concepts plus conceptual hierarchy ≤ defined by � A 1 , B 1 � ≤ � A 2 , B 2 � iff A 1 ⊆ A 2 (iff B 2 ⊆ B 1 ). – conceptual hierarchy: DOG ≤ MAMMAL ≤ ANIMAL – concept1=(extent1,intent1) ≤ concept2=(extent2,intent2) ⇔ extent1 ⊆ extent2 ( ⇔ intent1 ⊇ intent2) – concept lattice is visualized by its diagram: Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 6 / 62
Theorem (Basic theorem of concept lattices, Wille 1982) B ( X , Y , I ) equipped with ≤ is a complete lattice where infima and suprema are given by j ∈ J B j ) ↓↑ � , � j ∈ J � A j , B j � = � � j ∈ J A j , ( � j ∈ J A j ) ↑↓ , � � j ∈ J � A j , B j � = � ( � j ∈ J B j � . A complete lattice V = � V , ∧ , ∨� is isomorphic to B ( X , Y , I ) iff there are mappings γ : X → V , µ : Y → V such that γ ( X ) is � -dense in V, µ ( Y ) is � -dense in V, and � x , y � ∈ I iff γ ( x ) ≤ µ ( y ) . – generalization of previous results on lattices of fixpoints (Birkhoff, Schmidt, Banaschewski) – for a partially ordered set � V , ≤� , B ( V , V , ≤ ) is the Dedekind-MacNeille completion of � V , ≤� , – every complete lattice is isomorphic to some concept lattice, – every closure operator is the closure operator ↑↓ associated with some � X , Y , I � , etc. Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 7 / 62
Illustrative Example: persons × activities previous criminal activity expired driving license access to explosives multiple accounts finnancial records address changes money transfers fake or no SSN person A × × × × person B × × × × × × × person C × × × × person D × × × × × × × person E × × × × person F × × × × × person G × × × × person H × × × × person I × × × × × × × person J × × × × × × × person K × × × × person L × × × × × × × Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 8 / 62
fake SSN money transfers criminal multiple accounts address changes C 1 C 2 C 3 financial records C 4 C 5 C 6 F expired license explosives C 7 C 8 C 9 G, K A, C, E, H C 10 C 11 D, I, L B, J Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 9 / 62
Mathematics behind FCA Galois connection between X and Y . . . a pair � ↑ , ↓ � of operators satisfying A ⊆ A ↑↓ , B ⊆ B ↓↑ A 1 ⊆ A 2 implies A ↑ 2 ⊆ A ↑ B 1 ⊆ B 2 implies B ↓ 2 ⊆ B ↓ 1 , 1 Closure operator in X . . . a mapping c : 2 X → 2 X satisfying A ⊆ c ( A ) , A 1 ⊆ A 2 implies c ( A 1 ) ⊆ c ( A 2 ) , c ( A ) = c ( c ( A )) . Complete lattice � V , ≤� . . . arbitrary infs and sups exist (visualization by Hasse diagrams, drawing algorithms) Algorithms Kuznetsov & Obiedkov: Comparing performance of algorithms for generating concept lattices. J. Experimental and Theoretical Artificial Intelligence 14 (2–3)(2002), 189–216. Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 10 / 62
Concept lattices from data with fuzzy attributes – Basic approach and related structures – RB: Fuzzy Galois connections. Math. Logic Quarterly 45 ,4(1999). – RB: Fuzzy closure operators. J. Math. Anal. Appl. 262 (2001). – RB: Concept lattices and order in fuzzy logic. Ann. Pure and Appl. Logic 128 (2004), 277–298. – Relationship to classical (bivalent) case – RB: Reduction and a simple proof . . . fuzzy concept lattices. Fund. Informaticae 46 (4)(2001). – Non-classical and further issues – RB: Similarity relations in concept lattices. J. Log. Computation Vol. 10 No. 6(2000). – RB, Dvorak J., Outrata J.: Fast factorization by similarity . . . J. Computer and System Sciences 73 (6)(2007). – RB+V. Vychodil: Fuzzy concept lattices constrained by hedges. J. Adv. Comp. Int. & Intel. Informatics 11 (6)(2007). – RB+V. Vychodil: Reducing the size of fuzzy concept lattices by fuzzy closure operators. SCIS & ISIS 2006, Tokyo, pp. 309–314. Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 11 / 62
FCA of data with graded (fuzzy) attributes input data I y 1 y 2 y 3 y 4 I y 1 y 2 y 3 y 4 x 1 1 1 0.7 0.8 x 1 X X X X x 2 0.8 0.1 0.6 0.9 x 2 X X X instead of x 3 0 0.9 0.9 0.8 x 3 X X X x 4 1 0.5 0.6 0.5 x 4 X X X x 5 1 0 0 0.4 x 5 X – what are: ↑ , ↓ , formal concept, concept lattice? – basic properties, related structures, computationally feasibility, – several approaches: Burusco and Fuentes-Gonz´ ales ’94, Pollandt ’97, Bˇ elohl´ avek ’98, Ben Yahia et al. ’99, Sn´ aˇ sel, Vojt´ aˇ s et al. ’01, Krajˇ ci ’01, etc. . . . – in the following: using residuated (fuzzy) implication Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 12 / 62
Preliminaries: Residuated Lattices classical logic : two-element Boolean algebra fuzzy logic : several possibilities, a general one: complete residuated lattice: L = � L , ∧ , ∨ , ⊗ , → , 0 , 1 � , where – � L , ∧ , ∨ , 0 , 1 � . . . complete lattice, – � L , ⊗ , 1 � . . . commutative monoid, – �⊗ , →� . . . adjoint pair ( a ⊗ b ≤ c iff a ≤ b → c ). – introduced by Dilworth and Ward (Trans. AMS, 1939), – proposed as structure of truth degrees by Goguen (Synthese 1969), – from logical point of view: natural requirements for modus ponens Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 13 / 62
Examples on on [0 , 1] L = � [0 , 1] , min , max , ⊗ , → , 0 , 1 � given by left-continuous (continuous) ⊗ . – � Lukasiewicz : a ⊗ b = max( a + b − 1 , 0), a → b = min(1 − a + b , 1). � 1 if a ≤ b , – G¨ odel (minimum) : a ⊗ b = min( a , b ), a → b = b otherwise. � 1 if a ≤ b , – Goguen (product) : a ⊗ b = a · b , a → b = b otherwise. a Finite structures of truth degrees – finite chains L = { a 0 = 0 , a 1 , . . . , a n = 1 } , – L = { 0 , 1 } . . . two-element Boolean algebra of classical logic Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 14 / 62
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