On the Family of Concept Forming Operators in Polyadic FCA

On the Family of Concept Forming Operators in

Polyadic FCA

Dmitry I. Ignatov

National Research University Higher School of Economics, Moscow, Russia

Department of Data Analysis and Machine Intelligence, Faculty of Computer

Science &

Laboratory of Intelligent Systems and Structural Analysis

FCA4KDMoscow2017

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Outline

1 Formal Concept Analysis (FCA)

2 Triadic FCA

3 Towards a closure operator for 3-adic case

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Motivation

A large amount of structured and unstructured data generatestriadic relations.E.g. folksonomy is a set of triples (user, object, tag)

Concrete examples:

Bibsonomy.org (user,bookmark, tag)

Social networking sites(user, group, interest)

Delicious (user, link,tag)

Figure: Folksonomy as a graph.

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Main research question

Are there concept-forming operators for n-ry case?(answered more than two decades ago)

What about associated closure operators?

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Formal Concept Analysis[R.Wille, 1982], [B.Ganter and R.Wille, 1999]

G, a set of objects

M , a set of attributes

relation I ⊆ G×M such that (g,m) ∈ I if and only if the object g hasthe attribute m.

K := (G,M, I) is a formal context.

Derivation operators: A ⊆ G, B ⊆M

A′ def= {m ∈M | gIm for all g ∈ A}, B′ def= {g ∈ G | gIm for all m ∈ B}

A formal concept is a pair (A,B): A ⊆ G, B ⊆M, A′ = B, and B′ = A.

A is the extent and B is the intent of the concept (A,B).

The concepts, ordered by (A1, B1) ≥ (A2, B2) ⇐⇒ A1 ⊇ A2

(B2 ⊇ B1)

form a complete lattice, called the concept lattice B(G,M, I).

(·)′′ is a closure operator (idempotent, extensive, monotone)

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Road Map

1 Formal Concept Analysis (FCA)

2 Triadic FCA

3 Towards a closure operator for 3-adic case

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Triadic Formal Concept Analysis[F.Lehman & R.Wille, 1995]

De�nition 1

Triadic context K = (G,M,B, I) consists of set G (objects), M(attributes), B (conditions) and ternary relationsY ⊆ G×M ×B. Triple (g,m, b) ∈ Y means that the object ghas the attribute m under the condition b.

De�nition 2

(Formal) triconcept of K is a triple (X,Y, Z) which is maximalw.r.t. its components inclusion, i.e. X ⊆ G, Y ⊆M , Z ⊆ B èX × Y × Z ⊆ Y

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For convenience, a triadic context is denoted byK = (X1, X2, X3, Y ). A triadic context K = (X1, X2, X3, Y )gives rise to the following dyadic contextsK(1) = (X1, X2 ×X3, Y

(1)), K(2) = (X2, X1 ×X3, Y(2)), and

K(3) = (X3, X1 ×X2, Y(3)) where

gY (1)(m, b) :⇔ mY (2)(g, b) :⇔ bY (3)(g,m) :⇔ (g,m, b) ∈ Y .The derivation operators (primes or concept-forming operators)induced by K(i) are denoted by (.)(i). For each induced dyadiccontext we have two kinds of such derivation operators.

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That is, for {i, j, k} = {1, 2, 3} with j < k and for Z ⊆ Xi andW ⊆ Xj ×Xk, the (i)-derivation operators are de�ned by:

Z 7→ Z(i) = {(xj , xk) ∈ Xj ×Xk|xi, xj , xk are related by Y for all xi ∈ Z},

W 7→W (i) = {xi ∈ Xi|xi, xj , xk are related by Y for all (xj , xk) ∈W}.

Formally, a triadic concept of a triadic contextK = (X1, X2, X3, Y ) is a triple (A1, A2, A3) ofA1 ⊆ X1, A2 ⊆ X2, A3 ⊆ X3 such that for every{i, j, k} = {1, 2, 3} with j < k we have (Aj ×Ak)

(i) = Ai.

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The set of all triadic concepts of K = (X1, X2, X3, Y ) is denotedby T(K).There is a quasiorder .i for each i ∈ {1, 2, 3} and itscorresponding equivalence relation ∼i is de�ned by

(A1, A2, A3) .i (B1, B2, B3) :⇐⇒ Ai ⊆ Bi and

(A1, A2, A3) ∼i (B1, B2, B3) :⇐⇒ Ai = Bi.

These quasiorders satisfy the antiordinal dependencies: For{i, j, k} = {1, 2, 3} and all triconcepts (A1, A2, A3) and(B1, B2, B3) from T(K) it holds that(A1, A2, A3) .i (B1, B2, B3) and (A1, A2, A3) .j (B1, B2, B3)imply (A1, A2, A3) &k (B1, B2, B3).

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Folksonomy[T. van der Wal, 2004]

De�nition

A quadruple (U, T,R, Y ) is called folksonomy, where U is a set of users, T isa set of tags, R is a set of resources, and Y ⊆ U × T ×R.

A triple (u, t, r) ∈ Y denotes that the user u assigns the tag t to the

resource r.

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Folksonomy exampleIt is inspired by Bibsonomy (http://bibsonomy.org).

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http://bibsonomy.org)

TRIAS[J�aschke et al., 2006]

Trias is a method for �nding triadic formal concepts, that areclosed 3-sets. Triadic formal concepts can be interpreted asabsolutely dense triclusters.

NextClosure algorithm enumerates all formal concepts ofthe dyadic context in their lexicographical order

Trias is a NextClosure extension to the triadic case

Minimal support constraints are added (triclusters with toosmall extent, intent or modus are skipped)

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TRIASIgnatov et al.: Triadic Formal Concept Analysis and triclustering: searching foroptimal patterns. Machine Learning (2015)

High elapsed time

Too large number of small well-interpreted triclusters (triconcepts)

Examples of the triconcepts for the IMDB context:

1 {The Princess Bride (1987), Pirates of the Caribbean: The Curse ofthe Black Pearl (2003)}, {Pirate}, {Fantasy, Adventure}

2 {Platoon (1986), Letters from Iwo Jima (2006)}, {Battle},{Drama,War}

3 {V for Vendetta (2005)}, {Fascist, Terrorist, Government, SecretPolice ,Fight}, {Action, Sci-Fi, Thriller}

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Trisets

De�nition

(Trabelsi et al., 2012) Let K = (G,M,B, I) be a formaltricontext. A triple (X,Y, Z) is called a triset of K i�X × Y × Z ⊆ I.

Note that Cerf et al.(2009) de�ne a triset of K di�erently:X × Y × Z ∈ 2G × 2M × 2B.For trisets t1 = (A1, B1, C1) and t2 = (A2, B2, C2), t1 v t2means that A1 ×B1 × C1 ⊆ A2 ×B2 × C2, i.e. every triple(a, b, c) ∈ (A1, B1, C1) is in (A2, B2, C2).

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Triset-based concept forming operator

De�nition

(Trabelsi et al., 2012) Let S = (X,Y, Z) be a tri-set ofK = (G,M,B, I ⊆ G×M ×B). The mappingh : 2G × 2M × 2B ∩ 2I → 2G × 2M × 2B is de�ned as follows:h(S) = {(U, V,W ) | U = {g ∈ G | ∀m ∈ Y,∀b ∈ Z : (g,m, b) ∈I}, V = {m ∈M | ∀g ∈ U,∀b ∈ Z : (g,m, b) ∈ Y },W = {b ∈ B | ∀g ∈ U,∀m ∈ V : (g,m, b) ∈ Y }

Note that every triconcept is a maximal or closed triset, i.e. atriset that cannot be extended by triples from I being a triset.

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Proposition

h(·) is extensive and idempotent by v onT = {t | t is a triset of K} = {(X,Y, Z) ∈ 2G × 2M × 2B | (X,Y, Z) ⊆ I}and every �xpoint f of h (i.e. h(f) = f) is a triconcept of K.

Proof. One can �nd the proof of extensivity and idempotency in(Trabelsi et al, 2012). It is easy to see that every formaltriconcept is a �xpoint of h(·) and every triset (X,Y, Z) istransformed by h(·) to the triconcept((Y ×Z)(1), ((Y ×Z)(1)×Z)(2), ((Y ×Z)(1)×((Y ×Z)(1)×Z)(2))(3).Indeed, all formal triconcepts should be listed since a triset isallowed to be a triple with at least one component being ∅.

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Theorem

For a given tricontext K = (G,M,B, I ⊆ G×M ×B) and itsassociated triset systemT = {(X,Y, Z) ∈ 2G × 2M × 2B | (X,Y, Z) ⊆ I} operator h isnot monotone w.r.t. v.

Proof. To construct a violating example, one needs two di�erenttriconcepts with the same extent, c1 = (X,Y1, Z1) andc2 = (X,Y2, Z2) of K such that Y1 ⊂ Y2 and Z1 ⊃ Z2.Consider the tri-set s = (X,Y1, Z2):

s v c1 ⇒ h(s) = c2 6v h(c1) = c1.

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Example

For the tricontext in Figure 2, the violating example formonotonicity of h(·) is as follows:

x = ({u1, u2}, {t1}, {r1}) v y = ({u1, u2}, {t1}, {r1, r2})⇒

h(x)) = ({u1, u2}, {t1, t2}, {r1}) 6v h(y) = ({u1, u2}, {t1}, {r1, r2}).

t1 t2 t3u1 × ×u2 × ×u3

r1

t1 t2 t3u1 ×u2 ×u3

r2

Figure: A small example with Bibsonomy data19 / 24

Switching generators

De�nition

Let K = (K1,K2,K3, I) be a triadic formal context. A triset Sis called a (maximal) switching generator of the context K i�this is a non-empty component-wise intersection of c1 and c2 ,where c1 and c2 are concepts of K.

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Triset-based concept forming operators

For tricontext K = (K1,K2,K3, I) we consider a family ofoperators

{σijk|σijk : 2K1 × 2K2 × 2K3 → 2K1 × 2K2 × 2K3 such that

σijk : (X1, X2, X3) 7→ (Y1, Y2, Y3), where

Yi = (Xj×Xk)(i), Yj = (Yi×Xk)

(j), Yk = (Yi×Yj)(k), and {i, j, k} = {1, 2, 3}}.

The cardinality of the family is 3! = 6 and n! for its n-ary casegeneralisation.

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Proposition

Operators σijk(·) are not commutative, i.e.σijk(σlmn(·)) 6= σlmn(σijk(·)), where (i, j, k) 6= (l,m, n) and{i, j, k} = {1, 2, 3}.

Proof. Consider a tricontext given below.t1 t2 t3 t4

u1 × × × ×u2 × × ×u3 × × ×u4 ×

r1

t1 t2 t3 t4u1 × × ×u2 × × ×u3 × ×u4

r2

t1 t2 t3 t4u1 ×u2

u3

u4

r3The system of all switching generators S contains s1 = {u1, t4, r1} ands2 = {u1, u2, t3, t4, r1}.s1 proves that σi__(·) 6= σj__(·) 6= σk__(·) ands2 proves that σijk(·) 6= σikj(·) for {i, j, k} = {1, 2, 3}.The fact that σlmnσijk(·) = σijk(·) proves the proposition.

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Theorem

For K = (K1,K2,K3, I) and the associated triset system Tthere is no an associated closure operator in case there exist atleast two concepts c1 = (X1, Y1, Z1) and c2 = (X1, Y2, Z2) suchthat they have the common non-empty maximal switchinggenerator s, i.e. the intersection of the unions of theircomponets, respectively, is not an �empty� triset.

Proof. Let σ be a closure operator for K. Since s @ c1 ands @ c2 then σ(s) should result in ci which is either c1 or c2 (orone of other concepts ck with s @ ck if any exist). So letσ(s) = ci and consider s v cj ; it implies thatσ(s) = ci 6v σ(cj) = cj for i 6= j, and {i, j} = {1, 2}.Contradiction.

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For more details see https://arxiv.org/abs/1602.07267

Thank you!

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https://arxiv.org/abs/1602.07267