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
FCA4KD
Moscow
2017
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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 generates
triadic 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
‘‚F‡illeD IWVP“D ‘fFq—nter —nd ‚F‡illeD IWWW“
GD — set of o˜je™ts
MD — set of —ttri˜utes
rel—tion I ⊆ G × M su™h th—t (g, m) ∈ I if —nd only if the o˜je™t g h—s
the —ttri˜ute mF
K := (G, M, I) is — form—l ™ontextF
heriv—tion oper—torsX A ⊆ GD B ⊆ M
A
def
= {m ∈ M | gIm for all g ∈ A}, B
def
= {g ∈ G | gIm for all m ∈ B}
e form—l ™on™ept is — p—ir (A, B)X A ⊆ G, B ⊆ M, A = B, and B = A.
A is the extent —nd B is the intent of the ™on™ept (A, B)F
„he ™on™eptsD ordered ˜y (A1, B1) ≥ (A2, B2) ⇐⇒ A1 ⊇ A2
@B2 ⊇ B1A
form — ™omplete l—tti™eD ™—lled the ™on™ept l—tti™e B(G, M, I)F
(·) is — ™losure oper—tor @idempotentD extensiveD monotoneA
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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
‘pFvehm—n 8 ‚F‡illeD IWWS“
Denition 1
Triadic context K = (G, M, B, I) consists of set G (objects), M
(attributes), B (conditions) and ternary relations
Y ⊆ G × M × B. Triple (g, m, b) ∈ Y means that the object g
has the attribute m under the condition b.
Denition 2
(Formal) triconcept of K is a triple (X, Y, Z) which is maximal
w.r.t. its components inclusion, i.e. X ⊆ G, Y ⊆ M, Z ⊆ B è
X × Y × Z ⊆ Y
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For convenience, a triadic context is denoted by
K = (X1, X2, X3, Y ). A triadic context K = (X1, X2, X3, Y )
gives rise to the following dyadic contexts
K(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 dyadic
context 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 and
W ⊆ Xj × Xk, the (i)-derivation operators are dened by:
Z → Z(i)
= {(xj, xk) ∈ Xj × Xk|xi, xj, xk —re rel—ted ˜y ‰ for —ll xi ∈ Z},
W → W(i)
= {xi ∈ Xi|xi, xj, xk —re rel—ted ˜y ‰ for —ll (xj, xk) ∈ W}.
Formally, a triadic concept of a triadic context
K = (X1, X2, X3, Y ) is a triple (A1, A2, A3) of
A1 ⊆ 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 denoted
by T(K).
There is a quasiorder i for each i ∈ {1, 2, 3} and its
corresponding equivalence relation ∼i is dened 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
‘„F v—n der ‡—lD PHHR“
he(nition
e qu—druple (U, T, R, Y ) is ™—lled folksonomyD where U is — set of usersD T is
— set of t—gsD R is — set of resour™esD —nd Y ⊆ U × T × R.
e triple (u, t, r) ∈ Y denotes th—t the user u —ssigns the t—g t to the
resour™e rF
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Folksonomy example
st is inspired ˜y fi˜sonomy @httpXGG˜i˜sonomyForgAF
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TRIAS
‘t¤—s™hke et —lFD PHHT“
Trias is a method for nding triadic formal concepts, that are
closed 3-sets. Triadic formal concepts can be interpreted as
absolutely dense triclusters.
NextClosure algorithm enumerates all formal concepts of
the dyadic context in their lexicographical order
Trias is a NextClosure extension to the triadic case
Minimal support constraints are added (triclusters with too
small extent, intent or modus are skipped)
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TRIAS
sgn—tov et —lFX „ri—di™ porm—l gon™ept en—lysis —nd tri™lusteringX se—r™hing for
optim—l p—tternsF w—™hine ve—rning @PHISA
righ el—psed time
„oo l—rge num˜er of sm—ll wellEinterpreted tri™lusters @tri™on™eptsA
ix—mples of the tri™on™epts for the swhf ™ontextX
1 {„he €rin™ess fride @IWVUAD €ir—tes of the g—ri˜˜e—nX „he gurse of
the fl—™k €e—rl @PHHQA}D {€ir—te}D {p—nt—syD edventure}
2 {€l—toon @IWVTAD vetters from swo tim— @PHHTA}D {f—ttle}D
{hr—m—D‡—r}
3 {† for †endett— @PHHSA}D {p—s™istD „erroristD qovernmentD ƒe™ret
€oli™e D
pight}D {e™tionD ƒ™iEpiD „hriller}
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Trisets
Denition
(Trabelsi et al., 2012) Let K = (G, M, B, I) be a formal
tricontext. A triple (X, Y, Z) is called a triset of K i
X × Y × Z ⊆ I.
Note that Cerf et al.(2009) dene a triset of K dierently:
X × Y × Z ∈ 2G × 2M × 2B.
For trisets t1 = (A1, B1, C1) and t2 = (A2, B2, C2), t1 t2
means 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
Denition
(Trabelsi et al., 2012) Let S = (X, Y, Z) be a tri-set of
K = (G, M, B, I ⊆ G × M × B). The mapping
h : 2G × 2M × 2B ∩ 2I → 2G × 2M × 2B is dened 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. a
triset that cannot be extended by triples from I being a triset.
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Proposition
h(·) is extensive and idempotent by on
T = {t | t is — 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 formal
triconcept is a xpoint of h(·) and every triset (X, Y, Z) is
transformed 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 is
allowed 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 its
associated triset system
T = {(X, Y, Z) ∈ 2G × 2M × 2B | (X, Y, Z) ⊆ I} operator h is
not monotone w.r.t. .
Proof. To construct a violating example, one needs two dierent
triconcepts with the same extent, c1 = (X, Y1, Z1) and
c2 = (X, Y2, Z2) of K such that Y1 ⊂ Y2 and Z1 ⊃ Z2.
Consider the tri-set s = (X, Y1, Z2):
s c1 ⇒ h(s) = c2 h(c1) = c1.
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Example
For the tricontext in Figure 2, the violating example for
monotonicity of h(·) is as follows:
x = ({u1, u2}, {t1}, {r1}) y = ({u1, u2}, {t1}, {r1, r2}) ⇒
h(x)) = ({u1, u2}, {t1, t2}, {r1}) h(y) = ({u1, u2}, {t1}, {r1, r2}).
t1 t2 t3
u1 × ×
u2 × ×
u3
r1
t1 t2 t3
u1 ×
u2 ×
u3
r2
Figure: A small example with Bibsonomy data
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Switching generators
Denition
Let K = (K1, K2, K3, I) be a triadic formal context. A triset S
is 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 of
operators
{σijk|σijk : 2K1
× 2K2
× 2K3
→ 2K1
× 2K2
× 2K3
su™h th—t
σijk : (X1, X2, X3) → (Y1, Y2, Y3), where
Yi = (Xj×Xk)(i)
, Yj = (Yi×Xk)(j)
, Yk = (Yi×Yj)(k)
D —nd {i, j, k} = {1, 2, 3}}.
The cardinality of the family is 3! = 6 and n! for its n-ary case
generalisation.
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Proposition
Operators σijk(·) are not commutative, i.e.
σijk(σlmn(·)) = σlmn(σijk(·)), where (i, j, k) = (l, m, n) and
{i, j, k} = {1, 2, 3}.
€roofF gonsider — tri™ontext given ˜elowF
t1 t2 t3 t4
u1 × × × ×
u2 × × ×
u3 × × ×
u4 ×
r1
t1 t2 t3 t4
u1 × × ×
u2 × × ×
u3 × ×
u4
r2
t1 t2 t3 t4
u1 ×
u2
u3
u4
r3
„he system of —ll swit™hing gener—tors S ™ont—ins s1 = {u1, t4, r1} —nd
s2 = {u1, u2, t3, t4, r1}F
s1 proves th—t σi__(·) = σj__(·) = σk__(·) —nd
s2 proves th—t σijk(·) = σikj(·) for {i, j, k} = {1, 2, 3}F
„he f—™t th—t σlmnσijk(·) = σijk(·) proves the propositionF
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Theorem
For K = (K1, K2, K3, I) and the associated triset system T
there is no an associated closure operator in case there exist at
least two concepts c1 = (X1, Y1, Z1) and c2 = (X1, Y2, Z2) such
that they have the common non-empty maximal switching
generator s, i.e. the intersection of the unions of their
componets, respectively, is not an empty triset.
Proof. Let σ be a closure operator for K. Since s c1 and
s c2 then σ(s) should result in ci which is either c1 or c2 (or
one of other concepts ck with s ck if any exist). So let
σ(s) = ci and consider s cj; it implies that
σ(s) = ci σ(cj) = cj for i = 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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On the Family of Concept Forming Operators in Polyadic FCA

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