PART II Concept lattices and related structures in a fuzzy setting - - PowerPoint PPT Presentation

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 y1 y2 y3 x1 1 1 1 x2 1 1 x3 1 1 . . . . . .

• r

y1 y2 y3 x1 X X X x2 X X x3 X X . . . . . .

• r

1 1 1

1 0 1 0 1 1

• – output: 1. concept lattice
• 2. (non-redundant) dependencies

A ⇒ B, e.g. {y1, y3} ⇒ {y2}

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 3 / 62

FCA: Basic Notions

X = {x1, . . . } . . . objects, Y = {y1, . . . } . . . attributes I y1 y2 y3 x1 X X x2 X X x3 X induced operators ↑ : 2X → 2Y , ↓ : 2Y → 2X A↑ = {y ∈ Y | ∀x ∈ A : (x, y) ∈ I} B↓ = {x ∈ X | ∀y ∈ B : (x, y) ∈ I} A − → A↑ . . . attributes common to all objects from A B − → B↓ . . . objects sharing all attributes from B example: {x1, x2}↑ = {y3}, {y3}↓ = {x1, x2}

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 y1 y2 y3 y4 x1 1 1 1 1 x2 1 1 1 x3 1 1 1 x4 1 1 1 x5 1 I y1 y2 y3 y4 x1 1 1 1 1 x2 1 1 1 x3 1 1 1 x4 1 1 1 x5 1 I y1 y2 y3 y4 x1 1 1 1 1 x2 1 1 1 x3 1 1 1 x4 1 1 1 x5 1 (A1, B1) = ({x1, x2, x3, x4}, {y3, y4}) (A2, B2) = ({x1, x3, x4}, {y2, y3, y4}) (A3, B3) = ({x1, x2}, {y1, y3, y4})

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}

• f all formal concepts plus conceptual hierarchy ≤ defined by

A1, B1 ≤ A2, B2 iff A1 ⊆ A2 (iff B2 ⊆ B1). – 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∈JAj, Bj =

j∈J Aj, ( j∈J Bj)↓↑,

• j∈JAj, Bj = (

j∈J Aj)↑↓, j∈J Bj.

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

access to explosives fake or no SSN money transfers address changes multiple accounts expired driving license previous criminal activity finnancial records 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

C3 C5 C4 C7 C8 C10 C11 C1 C2 C6 C9 G, K F D, I, L A, C, E, H B, J address changes financial records money transfers multiple accounts fake SSN criminal expired license explosives

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↓↑ A1 ⊆ A2 implies A↑

2 ⊆ A↑ 1,

B1 ⊆ B2 implies B↓

2 ⊆ B↓ 1

Closure operator in X . . . a mapping c : 2X → 2X satisfying A ⊆ c(A), A1 ⊆ A2 implies c(A1) ⊆ c(A2), 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 y1 y2 y3 y4 x1 1 1 0.7 0.8 x2 0.8 0.1 0.6 0.9 x3 0.9 0.9 0.8 x4 1 0.5 0.6 0.5 x5 1 0.4 instead of I y1 y2 y3 y4 x1 X X X X x2 X X X x3 X X X x4 X X X x5 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). – G¨

• del (minimum): a ⊗ b = min(a, b), a → b =
• 1

if a ≤ b, b

• therwise.

– Goguen (product): a ⊗ b = a · b, a → b =

• 1

if a ≤ b,

b a

• therwise.

Finite structures of truth degrees – finite chains L = {a0 = 0, a1, . . . , an = 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

Truth-stressing hedges (Takeuti+Titani, Baaz, H´ ajek, . . . ) (idempotent) truth-stressing hedge . . . mapping ∗ : L → L satisfying 1∗ = 1, a∗ ≤ a, (a → b)∗ ≤ a∗ → b∗, a∗∗ = a∗, meaning of ∗: truth function of logical connective “very true” Two boundary hedges – identity, i.e. a∗ = a (a ∈ L); – globalization: a∗ = 1 if a = 1,

• therwise.

. . . truth function of “fully true” Example

0.25 0.5 0.75 1 identity ∗1 ∗2 ∗3 globalization Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 15 / 62

Fuzzy sets and relations

L = L, ∨, ∧, →, ⊗, ∗, 0, 1 . . . residuated structure of truth degrees fuzzy set (L-set) A in universe U . . . mapping A: U → L, A(u): “degree to which u belongs to A”, binary fuzzy relation (L-relation) I between X and Y . . . mapping I : X × Y → L, R(u, v): “degree to which u ∈ U and v ∈ V are related” . . . More can be found e.g. in: – H´ ajek P.: Metamathematics of Fuzzy Logic. Kluwer, 1998. – Gottwald S.: A Treatise on Many-Valued Logic. RSP, 2001. – Gerla G.: Fuzzy Logic. Mathematical Tools for Approximate

• Reasoning. Kluwer, 2001.

– Belohlavek R.: Fuzzy Relational Systems. Foundations and Principles. Kluwer, 2002. back to data tables with fuzzy attributes . . .

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 16 / 62

Operators, formal concepts, concept lattices

Given a formal fuzzy context X, Y , I, i.e. I : X × Y → L: – operators ↑ , ↓: for A ∈ LX, B ∈ LY , define fuzzy sets A↑ ∈ LY and B↓ ∈ LX: A↑(y) =

x∈X(A(x) → I(x, y))

B↓(x) =

y∈Y (B(y) → I(x, y))

A → A↑ . . . (fuzzy set of) common attributes, B → B↓ . . . (fuzzy set of) common objects. Why this definition? Using basic principles of predicate FL, A↑(y) = truth degree of “for every x ∈ X: if x is in A then x has y”. Important consequence: Same verbal description and meaning as in the ordinary case. For L = 2 (two-element Boolean algebra), we get the ordinary definitions.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 17 / 62

– formal concept in X, Y , I: A, B s.t. A↑ = B and B↓ = A, – concept lattice of X, Y , I B (X, Y , I) = {A, B ∈ LX × LY | A↑ = B, B↓ = A} – subconcept-superconcept hierarchy ≤ in B (X, Y , I) A1, B1 ≤ A2, B2 iff A1 ⊆ A2 (iff B1 ⊇ B2)

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 18 / 62

Formal concepts are maximal rectangles

input table:

• ne of its formal concepts:

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 19 / 62

Theorem (Wille’s basic theorem in fuzzy setting)

B (X, Y , I) is under ≤ a complete lattice where infima and suprema are given by

• j∈J Aj, Bj =

j∈J Aj, ( j∈J Bj)↓↑,

• j∈J Aj, Bj = (

j∈J Aj)↑↓, j∈J Bj.

A complete lattice V = V , ∧, ∨ is isomorphic to B (X, Y , I) iff there are γ : X × L → V , µ : Y × L → V such that γ(X, L) is -dense in V, µ(Y , L) is -dense in V, and a ⊗ b ≤ I(x, y) iff γ(x, a) ≤ µ(y, b). – for L = 2 (bivalent case), conditions for γ, µ are satisfied iff there are γ′ : X → V , µ′ : Y → V such that γ′(X) is -dense in V, µ′(Y ) is -dense in V, and x, y ∈ I (i.e. I(x, y) = 1) iff γ′(x) ≤ µ′(y). – ⇒ generalization of Wille’s basic theorem (1982)

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 20 / 62

What about Dedekind-McNeille completion?

– In classical case, for a partially ordered set V , ≤, B (V , V , ≤) is just the Dedekind-McNeille completion of V , ≤. – Q: Can we consider an L-valued order (fuzzy order) on B (X, Y , I) and have some analogy in fuzzy setting? – A: Yes: R.B. Concept lattices and order in fuzzy logic. Ann. Pure

• Appl. Logic 128(2004), 277–298:

– partial order, infimum, supremum, . . . in such away that V , → B (X, Y , I), behaves like Dedekind-MacNeille completion – Some recent results on this line by M. Krupka.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 21 / 62

Fuzzy Galois connections: basic structures behind

An (ordinary) Galois connection between X and Y is a pair ↑ : 2X → 2Y and ↓ : 2Y → 2X such that for A’s∈ 2X, B’s∈ 2Y : A ⊆ A↑↓, B ⊆ B↓↑, if A1 ⊆ A2 then A↑

2 ⊆ A↑ 1,

if B1 ⊆ B2 then B↑

2 ⊆ B↑ 1.

Galois connections between X and Y are in bijective correspondence with binary relations between X and Y . In a fuzzy setting, one “replaces 2 by L”. See next slide. Note: “if A1 ⊆ A2 then A↑

2 ⊆ A↑ 1” is replaced by “S(A1, A2) ≤ S(A↑ 2, A↑ 1)”.

To keep “if A1 ⊆ A2 then A↑

2 ⊆ A↑ 1” is possible but one looses the 1-1

correspondence with binary relations.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 22 / 62

Def. A fuzzy Galois connection between X and Y is a pair

↑ : LX → LY and ↓ : LY → LX such that for A’s∈ LX, B’s∈ LY :

A ⊆ A↑↓, B ⊆ B↓↑, S(A1, A2) ≤ S(A↑

2, A↑ 1),

S(B1, B2) ≤ S(B↓

2, B↓ 1).

Other axiomatizations possible, e.g. S(A, B↓) = S(B, A↑).

Theorem (fuzzy Galois connections vs. binary fuzzy relations)

Let I ∈ LX×Y be a fuzzy relation, let ↑I and ↓I be induced by I. Let ↑, ↓ be a fuzzy Galois connection. Then (1) ↑I , ↓I is a fuzzy Galois connection. (2) I↑,↓ defined by I↑,↓(x, y) = { 1 x}↑(y) is a fuzzy relation and we have (3) ↑, ↓ =

↑I↑,↓, ↓I↑,↓ and I = I↑I ,↓I .

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 23 / 62

Cut-like representation of fuzzy Galois connections

Recall: a fuzzy set A ∈ LX is represented by a system SA = {aA | a ∈ L}

• f its a-cuts

aA = {x ∈ X | a ≤ A(x)}.

Q: is there a cut-like representation of fuzzy Galois connections by ordinary Galois connections?

Definition

A system {↑a, ↓a | a ∈ L} of 2-Galois connections between X and Y is called L-nested if (1) for each a, b ∈ L, a ≤ b, A ∈ 2X, B ∈ 2Y , it holds A↑a ⊇ A↑b, B↓a ⊇ B↓b, (2) the set {a ∈ L | y ∈ {x}↑a} contains a greatest element.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 24 / 62

Theorem

For an L-Galois connection ↑, ↓ between X and Y denote C↑,↓ = {↑a, ↓a | a ∈ L} where ↑a : 2X → 2Y and ↓a : 2Y → 2X are defined by A↑a = a(A↑) and B↓a = a(B↓) for A ∈ 2X, B ∈ 2Y . For an L-nested system C = {↑a, ↓a | a ∈ L} of 2-Galois connections between X and Y denote ↑C, ↓C the pair of mappings ↑C : LX → LY and ↓C : LY → LX defined by A↑C(y) = {a | y ∈

b∈L(bA)↑a ⊗ b},

B↓C(x) = {a | x ∈

b∈L(bB)↓a ⊗ b}

for A ∈ LX, B ∈ LY . Then (1) C↑,↓ is a nested system of L-Galois connections between X and Y , (2) ↑C, ↓C is a L-Galois connection between X and Y , (3) C = C↑C ,↓C and ↑, ↓ =

↑C↑,↓, ↓C↑,↓.

Remark Further representations of fuzzy Gal. connections available.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 25 / 62

Another representation of fuzzy Galois connections

fuzzy Galois connections ↑, ↓ between X and Y (bijective correspondence) particular Galois connections ∧, ∨ between X × L and Y × L Main feature: B(X, Y , ↑, ↓) is isomorphic to B(X × L, Y × L, ∧, ∨). ⇒ fuzzy concept lattice B(X, Y , I) is isomorphic to ordinary concept lattice B(X × L, Y × L, I ×) with x, a, y, b ∈ I × iff a ⊗ b ≤ I. ⇒ alternative proof of the above basic theorem (version with ≤). details: R.B.: Reduction and a simple proof . . . Fund. Inf. 46(2001), 277–285. Also: S. Pollandt: Fuzzy Begriffe. Springer, 1997.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 26 / 62

Algorithms for computing B (X, Y , I)

Due to the representation on the previous slide: To compute B(X, Y , I), – transform X, Y , I to ordinary formal context X × L, Y × L, I × – compute B(X × L, Y × L, I ×) using available algorithms for ordinary concept lattices – transform B(X × L, Y × L, I ×) to B(X, Y , I). Possible in principle, but inefficient. – RB: Algorithms for fuzzy concept lattices. RASC 2002. – RB, De Baets, Outrata, Vychodil: Computing the lattice of all fixpoints of a fuzzy closure operator. IEEE Transactions on Fuzzy Systems (2010).

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 27 / 62

Factorization of fuzzy concept lattices by similarity

problem: concept lattice too large idea of solution: simplification by factorizing – put sufficiently similar formal concepts together – instead of individual concepts (large number), consider groups of similar concepts (smaller number) – meaning of “sufficiently similar”: control by a parameter a

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 28 / 62

• utline of technical solution

R.B.: Similarity relations in concept lattices. J. Logic Comput. (2000): degree of similarity ≈ of A1, B1 and A2, B2 on B (X, Y , I): A1, B1 ≈ A2, B2 =

x∈X A1(x) ↔ A2(x) (= y∈Y B1(y) ↔ B2(y))

For a ∈ L (a threshold specified by a user), define relation a≈ (“sufficiently similar”) : (A1, B1, A2, B2) ∈ a≈ iff (A1, B1 ≈ A2, B2) ≥ a

a≈ is a compatible tolerance on B(X, Y , I), thus results by Cz´

edli, Wille ⇒ B (X, Y , I)/a≈ . . . the collection of all a≈-blocks (covering). B (X, Y , I)/a≈ can be partially ordered by , B (X, Y, I)/a≈+ is a complete lattice, so-called factor lattice of B (X, Y, I) by similarity ≈ and a threshold a.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 29 / 62

factorization: example example: similarity threshold a = 0.5

• riginal concept lattice

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 30 / 62

factorization: example example: similarity threshold a = 0.5

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1 4 8 13 20 28 3 7 12 10 26 33 34 27 18 2 6 11 17 25 32 37 36 38 31 24 23 16 10 5 9 15 22 29 35 30 14 21

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12

• riginal concept lattice

blocks of a-similar concepts

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 31 / 62

factorization: example example: similarity threshold a = 0.5

• riginal concept lattice

factorized concept lattice

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 32 / 62

computation of B (X, Y , I)/a≈ first way: by definition data ⇒1 ⇒2 Remark: for both ⇒1 and ⇒2 there are polynomial time delay algorithms second way: directly? data blue? ⇒ ?

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 33 / 62

computation of B (X, Y , I)/a≈ Solution (R.B. et al, J. Computer and System Sciences (2007)): – fuzzy closure operator Ca, the fixpoints of which uniqiely determine the blocks of factor lattice – the operator Ca: A → a → (a ⊗ A)↑↓ – computing fix(Ca) — by the above algorithm Illustration of speed-up: |B (X, Y , I)| = 774 concepts, computation of B (X, Y , I) = 2292ms similarity threshold a 0.2 0.4 0.6 0.8 size

• riginal lattice, |B (X, Y , I)|

774 774 774 774 factorized lattice, |B (X, Y , I)/a≈| 8 57 193 423 time for computing B (X, Y , I)/a≈ “naive” algorithm (ms) 8995 9463 8573 9646 new algorithm (ms) 23 214 383 1517

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 34 / 62

Example of measure-theoretic-like result

To what extent is it important that a particular entry I(x, y) in a formal context is exactly 0.6. What happens if it is 0.7? Some results available. One is (RB, 2006):

Theorem

Let L be equipped with G¨

• del structure (min). If for every x1, y1 and

x2, y2 we have I1(x1, y1) < I1(x2, y2) iff I2(x1, y1) < I2(x2, y2), and if I1(x, y) = 1 iff I2(x, y) = 1, then B(X, Y , I1) and B(X, Y , I2) are isomorphic. ⇒ What matters id the relative ordering of truth degrees (ordinal information), not their absolute values.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 35 / 62

Related approaches and generalizations

The approach presented is a basic one in that it uses a residuated structure of truth degrees. This seems to be crucial. Most advanced. Several other approaches and generalizations of the presented one appeared in the literature (by date): Burusco & Fuentes Gonzales (first paper), Pollandt (first one using residuated implication), Ben Yahia et al., Krajˇ ci (one-sided), Georgescu & Popescu (non-commutative, general approach), RB, Vychodil (concept lattices with hedges), Krajˇ ci (generalized concept lattice), Zhang et al. (domains, category-theoretic viewpoint), Ojeda-Aciego & Medina (multi-adjoint concept lattices), . . . my students Koneˇ cn´ y, Osiˇ cka, Bartl, . . . and possibly others. Also, large number of not very good papers.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 36 / 62

Concept lattices with hedges

– generalization of

– fuzzy concepts and fuzzy concept lattices (presented above) – crisply generated fuzzy concepts and the corresponding concept lattices

– crisply generated formal concepts: particular fuzzy concepts, suggested to us as important by experts when working on evaluation

• f IPAQ questionnaires (RB et al., ICFCA 2005)

(crisply generated: D↓, D↓↑ for some D ∈ {0, 1}Y ) – different, but equivalent definitions concepts independently proposed by Ben Yahia (2001) and Krajˇ ci (2002) (one-sided fuzzy concepts). Definitions and results: – modify arrow operators by hedges ∗1 and ∗2 A↑(y) =

x∈X A(x)∗1 → I(x, y),

B↓(x) =

y∈Y B(y)∗2 → I(x, y).

– recall: hedge ∗ : L → L is a (truth function of) connective “very true” – B (X ∗1, Y ∗2, I) = {A, B | A↑ = B, B↓ = A} . . . c. l. with hedges – basic theorem available – ∗1 and ∗2 are parameters controlling the size of B (X ∗1, Y ∗2, I).

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 37 / 62

All hedges ∗X and ∗Y on a five-element Lukasiewicz chain L

gl.

• L1
• L2
• L3

0.25 0.5 0.75 1 id.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 38 / 62

– hedges ∗X and ∗Y on a five-element Lukasiewicz chain – corresponding B (X ∗X , Y ∗Y , I)’s

gl.

• L1
• L2
• L3

id. gl.

• L1
• L2
• L3

id.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 39 / 62

Remark on c. lattices of isotone Galois connections

Recall: A↑(y) =

x∈X(A(x) → I(x, y))

B↓(x) =

y∈Y (B(y) → I(x, y))

Several papers recently appeared on concept lattice based on the following

• perators:

A∩(y) =

x∈X(A(x) ⊗ I(x, y))

B∪(x) =

y∈Y (I(x, y) → B(y))

These papers study B(X ∩, Y ∪, I); notions, proofs, etc. are very similar to the case of B(X ↑, Y ↓, I). Is this needed (from mathematical viewpoint)?

• No. There exists a general framework in which both ↑, ↓ and ∩, ∪ are two

particular cases of a general case.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 40 / 62

In binary case (or if negation satisfies law of double negation): A∩I = A↑I and B∩I = B

↓I , ∩, ∪ is definable in terms of ↑, ↓ (and vice versa), immediate observation:

D¨ untsch, Gediga: Modal-style operators in qualitative data analysis. ICDM 2002 (plus other reducible operators). In general residuated lattices, the previous “duality” does not work. Yet, a different way showing duality between ↑, ↓ and ∩, ∪ exists: RB: Optimal decompositions of matrices with entries from residuated lattices . . . . – Starts with : L × L → L, distributive with suprema (order on L is ≤

• r its dual).

– Defines residua ◦, ◦. – ↑, ↓ is a particular case for = ⊗;

∩, ∪ is a particular case for =→.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 41 / 62

Factor analysis with formal concepts as factors

– Problem setting and formal concepts as factors – Formal concepts are optimal factors – Example and experiments

– RB: Representation of concept lattices by BAM. Neural Computation 12,10(2000), 2279–2290. – RB+V. Vychodil: On Boolean factor analysis with formal concepts as

• factors. SCIS & ISIS 2006, Tokyo, Japan, pp. 1054-1059.

– RB+V. Vychodil: Factor analysis of incidence data via novel decomposition of matrices. ICFCA 2009, LNCS 5548(2009), 83–97. – RB: Optimal triangular decompositions of matrices with entries from residuated lattices. Int. J. Approximate Reasoning 50(8)(2009), 1250–1258. – RB: Optimal decompositions of matrices with entries from residuated

• lattices. J. Logic and Computation (in revision).

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 42 / 62

Problem Description

given an objects × attributes Boolean or graded matrix I like I = 1 1 0 0 0

1 1 0 0 1 1 1 1 1 0 1 0 0 0 1

• r

I = 1 .9 0 .2 .1

1 .9 .2 .2 .8 .8 .8 .9 .9 .1 1 .2 .1 .1 1

• ,

decompose I ≈ A ◦ B where – A . . . objects × factors matrix, B . . . factors × attributes matrix – aim: no. factors (as small as possible) << no. attributes 1 1 0 0 0

1 1 0 0 1 1 1 1 1 0 1 0 0 0 1

• =

1 0 0

1 0 1 1 1 0 0 0 1

• 1 1 0 0 0

0 0 1 1 0 1 0 0 0 1

• r

1 .9 0 .2 .1

1 .9 .2 .2 .8 .8 .8 .9 .9 .1 1 .2 .1 .1 1

• =

1 0 0

1 .2 .8 .8 .9 .1 0 .1 1

• · · ·

results

• show new way to find factors
• factors = formal concepts
• proposed factorization is optimal (theorem) + algorithms
• examples and experiments

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 43 / 62

idea of factor analysis: – Spearman: General intelligence, objectively determined and measured.

• Amer. J. Psychology (1904)

– according to Harman: “The principal concern of factor analysis is the resolution of a set of variables linearly in terms of (usually) a small number of categories

• r ‘factors’. . . . A satisfactory solution will yield factors which convey all

the essential information of the original set of variables. Thus, the chief aim is to attain scientific parsimony or economy of description.”

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 44 / 62

Matrices for Decomposition

– input: n × m (object×attribute) matrix I – matrix entries: Iij . . . degree to which object i has attribute j – aim: decompose I into I = A ◦ B – example 1: binary matrices, i.e. Iij = 0 or Iij = 1 – example 2: matrices with entries from [0, 1] (unit interval) – general case: Iij ∈ L where L, ∧, ∨, ⊗, →, 0, 1 is a complete residuated lattice

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 45 / 62

Decomposition Operator ◦

Aim: Find decomposition I = A ◦ B n × k matrix A, k × m matrix B, k smallest possible, where (A ◦ B)ij = k

l=1 Ajl ⊗ Blj.

– ◦ with L = [0, 1] used in fuzzy sets (composition of fuzzy relations). – Matrix multiplication scheme with + and · replaced by and ⊗. – Factor model given by I = A ◦ B: I . . . objects×attributes, A . . . objects×factors, B . . . factors×attributes for L = {0, 1}: Iij = (A ◦ B)ij = 1 IFF there is factor k such that object i has k and attribute j is a particular manifestation of k for general L: (A ◦ B)ij = truth degree of “there is factor k . . . manifestation of k”

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 46 / 62

Formal concepts can be used as factors

How? Given a set F = {C1, D1, . . . , Ck, Dk} ⊆ B(X, Y , I)

• f fixpoints, define n × k and k × m matrices AF and BF by

(AF)il = (Cl)(i) and (BF)lj = (Dl)(j). Main result:

Theorem (formal concepts of I are optimal factors for I)

Let I = A ◦ B for n × k and k × m matrices A and B. Then there exists a set F ⊆ B(X, Y , I) of formal concepts with |F| ≤ k such that for the n × |F| and |F| × m matrices AF and BF we have I = AF ◦ BF.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 47 / 62

Illustrative example—factors in graded data

2004 Olympic Games Decathlon—Top 5 Athletes 10 lj sp hj 40 11 di pv ja 15 Sebrle 894 1020 873 915 892 968 844 910 897 680 Clay 989 1050 804 859 852 958 873 880 885 668 Karpov 975 1012 847 887 968 978 905 790 671 692 Macey 885 927 835 944 863 903 836 731 715 775 Warners 947 995 758 776 911 973 741 880 669 693 Legend: 10—100 meters sprint race; lj—long jump; sp—shot put; hj—high jump; 40—400 meters sprint race; 11—110 meters hurdles; di—discus throw; pv—pole vault; ja—javelin throw; 15—1500 meters run. visual representation:

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 48 / 62

7 factors (formal concepts) found which “explain” the data corresponding decomposition: =

• Belohlavek (Palack´

y University) Concept lattices in a fuzzy setting CLA 2010 49 / 62

factors and their interpretation: F1 F2 F3 F4 F5 F6 F7 F1 ≈ speed 100 m, long jump, 110 m hurdles in degree 1 F2 ≈ explosiveness long jump, shot put, high jump, 110 m hurdles, javelin in degree 1 F3 ≈ not very muscular athlete high jump and 1500 m in degree 1 . . .

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 50 / 62

• riginal matrix:

superposition of factors, 1st, 2nd, 3rd, . . . , 7th: F1 (46%) F2 added (72%) F3 added (84%) F4 added (92%) F5 added (96%) F6 added (98%) F7 added (100%)

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 51 / 62

Complexity and algorithms for ◦-decomposition

Theorem

The problem of optimal ◦-decomposition is NP-complete (decision version)/NP-hard (optimization version). – Proof by reduction using set basis problem (which is NP-complete, Stockmeyer, 1975). – Need for an approximation algorithm. – First attempt: compute B(X, Y , I) (fixed points/optimal factors) and use greedy approach to select a small F ⊆ B(X, Y , I) ⇒ Algorithm 1 – Obstacle: Computing B(X, Y , I) slow for large data. – Second attempt: Greedy approach and computing factors from B(X, Y , I) “on demand” ⇒ Algorithm 2

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 52 / 62

Algorithm 1: factor concepts

INPUT: I (Boolean matrix), OUTPUT: F (set of factor concepts) set S to B(X, Y , I) set U to {xi, yj | Iij = 1} set F to ∅ for each C, D ∈ S: if (C, D ∈ O(X, Y , I) ∩ A(X, Y , I)): add C, D to F remove C, D from S for each x, y ∈ C × D: remove x, y from U while (U = ∅): do select C, D ∈ S that maximizes (C × D) ∩ U: add C, D to F remove C, D from S for each x, y ∈ C × D: remove x, y from U return F

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 53 / 62

Disadvantage: compute the whole concept lattice first. Algorithm 2 (next slide), overcomes this by: – construct factor concepts by adding sequentially “promising columns”: – start with D = ∅, – if there is y ∈ D s.t. (D ∪ {y})↓, (D ∪ {y})↓↑ is better factor (covers more uncovered 1s), then add the best such y – and ask again: can still further attribute be added? ⇒ greedy approach: select y ∈ Y which maximizes |D ⊕ y| = |((D ∪ {y})↓×(D ∪ {y})↓↑) ∩ U|.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 54 / 62

Algorithm 2: factor concepts

INPUT: I (Boolean matrix), OUTPUT: F (set of factor concepts) set U to {i, j | Iij = 1} set F to ∅ while (U = ∅): set D to ∅ set V to 0 while there is j ∈ D such that |D ⊕ j| > V : do select j ∈ D that maximizes D ⊕ j: set D to (D ∪ {j})↓↑ set V to (D↓×D) ∩ U set C to D↓ add C, D to F for each i, j ∈ C × D: remove i, j from U return F

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 55 / 62

Experiments: Algorithm 1 vs. Algorithm 2

UCI MLR Mushroom data set 8124 objects × 119 attributes, 238, 710 formal concepts ANSI C, Intel Xeon 4 CPU 3.20 GHz, 1 GB RAM Algorithm 1 Algorithm 2 Time 18 min., 5.66 sec. 3.47 sec. Memory 97 MB RAM 2 MB RAM

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 56 / 62

Experiments (approximate factorization)

The number of factors delivered by Algorithm 1 and 2 needed to cover a prescribed percentage of 1s in I drops rapidly when the required percentage drops. The behavior is similar for Algorithm 1 and 2. For the Mushroom dataset, the results are:

number of factor concepts Algorithm 1 20 40 60 80 100 120 20% 40% 60% 80% 100% number of factor concepts Algorithm 2 20 40 60 80 100 120 20% 40% 60% 80% 100%

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 57 / 62

Experiments (comparing size of computed factors)

Both algorithms tend to deliver formal concepts which cover areas of the input matrix of approximately of the same size. Plot a sequence r1, s1, . . . , rk, rk where ri and si are the numbers of 1’s covered by i-th factor delivered by Algorithm 1 and Algorithm 2: close to the diagonal.

5 15 50 500 10000 5 15 50 500 10000

Figure: Corresponding factors cover approximately the same number of 1s.

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 58 / 62

Finding the least number of factors (best decompositions). We used 80,000 artificially created datasets with known least number of factors: Optimal Algorithm 1 Algorithm 2 Factors number of factors number of factors 5 5.313 ± 0.520 5.255 ± 0.473 6 6.545 ± 0.683 6.443 ± 0.666 7 7.785 ± 0.832 7.672 ± 0.781 8 9.182 ± 1.032 9.139 ± 1.097 9 10.542 ± 1.150 10.416 ± 1.107 10 11.718 ± 1.210 11.641 ± 1.213 11 12.775 ± 1.259 12.740 ± 1.290 12 13.800 ± 1.226 13.751 ± 1.229 13 14.652 ± 1.235 14.623 ± 1.306 14 15.284 ± 1.124 15.401 ± 1.162 15 16.068 ± 1.054 16.056 ± 1.011

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 59 / 62

Number of factors for data with varying density. 80,000 artificially created datasets. Density Algorithm 1 Algorithm 2 % of 1s number of factors number of factors 94 % 6.740 ± 0.821 6.710 ± 0.836 92 % 7.026 ± 0.894 7.008 ± 0.861 90 % 7.538 ± 0.889 7.532 ± 0.903 88 % 8.278 ± 0.935 8.338 ± 1.005 85 % 9.478 ± 1.012 9.592 ± 1.106 75 % 11.620 ± 1.091 11.844 ± 1.141 50 % 14.651 ± 0.529 14.791 ± 0.412 25 % 14.590 ± 0.645 14.556 ± 0.649 15 % 13.452 ± 1.095 13.374 ± 1.127 12 % 11.893 ± 1.471 11.753 ± 1.462 10 % 10.914 ± 1.685 10.782 ± 1.632 8 % 9.876 ± 1.672 9.744 ± 1.633 7 % 8.912 ± 1.683 8.810 ± 1.651

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 60 / 62

Extension to other types of decomposition

– Extension to other products such as triangular products of binary/graded matrices possible. – (A ⊳ B)ij = k

l=1 Ail → Blj

– Instead of maximal rectangles, I-beams and H-beams are used here   

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

   ,   

1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1

   ,

and

  

1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1

   . – Examples in submitted paper (RB, Vychodil).

Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 61 / 62

Factorization of Large Data with Triangular Product

MUSHROOM Database: 119 × 8192 binary matrix – UCI Machine Learning Repository (benchmark data) – 8192 mushrooms, 119 binary attributes – algorithm: decomposition into 119 × 85 and 85 × 8192 matrices. – ⇒ MUSHROOM can be represented in 85D space, without any loss. Approximate factorization of MUSHROOM – number of factors vs. % data “explained”

number of factors 20 40 60 85 20% 40% 60% 80% 100% Belohlavek (Palack´ y University) Concept lattices in a fuzzy setting CLA 2010 62 / 62