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Relations between Proto-fuzzy Concepts, Crisply Generated Fuzzy Concepts, and Interval Pattern Structures

Vera V. Pankratieva and Sergei O. Kuznetsov

State University Higher School of Economics Moscow, Russia

CLA’2010, Sevilla

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 1 / 26

Motivation

Recently several models for the analysis of fuzzy and numerical data within FCA were proposed: Fuzzy formal concepts [R. Bˇ elohl´ avek, 1999] Protofuzzy formal concepts [O. Kr´ ıdlo, S. Krajˇ ci, 2005] Pattern structures [B. Ganter and S.O. Kuznetsov, 2000]

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 2 / 26

Main research goal

Relationships between protofuzzy concepts, fuzzy concepts, pattern structures, both from theoretical and from experimental viewpoints.

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 3 / 26

Outline

Relationships between Protofuzzy [O. Kr´ ıdlo, S. Krajˇ ci] and fuzzy formal concepts [R. Bˇ elohl´ avek] Fuzzy formal concepts [R. Bˇ elohl´ avek] and pattern structures [B. Ganter and S.O. Kuznetsov] Fuzzy formal concepts [R. Bˇ elohl´ avek] and two-way pattern structures

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 4 / 26

Main definitions of FCA

G is a set, called the set of objects, M is a set, called the set of attributes, I ⊆ G × M is a binary relation. The triple K := (G, M, I) is called a (formal) context. Galois connection between (2G, ⊆) and (2M, ⊆) is a pair of maps (·)′ : 2G → 2M A′ def = {m ∈ M | gIm for all g ∈ A}, (·)′ : 2M → 2G B′ def = {g ∈ G | gIm for all m ∈ B}, with the following properties: ∀A1, A2 ⊆ G, B1, B2 ⊆ M

1 A1 ⊆ A2 ⇒ A′

2 ⊆ A′ 1;

B1 ⊆ B2 ⇒ B′

2 ⊆ B′ 1;

2 A1 ⊆ A′′

1,

B1 ⊆ B′′

1 .

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 5 / 26

Main definitions of FCA

Formal concepts are partially ordered by the relation A1, B1 ≥ A2, B2 ⇐ ⇒ A1 ⊇ A2 B2 ⊇ B1. Implication A → B, where A, B ⊆ M, holds in the context if A′ ⊆ B′. The map (·)′′ is a closure operator, since it is idempotent, extensive, and monotone. A formal concept is a pair A, B, A ⊆ G, B ⊆ M, such that A′ = B, B′ = A.

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 6 / 26

Fuzzy formal concepts [R. Bˇ elohl´ avek]

A fuzzy formal context is a triple X, Y , I, where I is a fuzzy relation between X and Y , taking a pair (x, y) to the truth degree I(x, y) ∈ L, with which object x ∈ X has attribute y ∈ Y . A fuzzy formal concept is a pair A, B, A ⊆ X, B ⊆ Y , such that A↑ = B and B↓ = A, where A↑(y) =

• x∈X

(A(x) → I(x, y)) and B↓(x) =

• y∈Y

(B(y) → I(x, y)).

Example

I = 0.1 0.6 0.5 1 0.2 0.4 0.8 0.3 A = (0.3, 0.9, 1) is the fuzzy tuple of extent B = (0.8, 0.3, 0) is the fuzzy tuple of intent

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 7 / 26

Crisply generated fuzzy formal concepts [R. Bˇ elohl´ avek]

A pair A, B such that A = A↑↓

c = B↓ and B = A↑ = A↑↓↑ c

is called a crisply generated formal concept (Ac is a crisp tuple). I = 0.1 0.6 0.5 1 0.2 0.4 0.8 0.3 Ac = (0, 1, 0) is the crisp tuple of extent; A = (0.1, 1, 0.6) = B↓ = A↑↓

c ; B = (1, 0.2, 0.4) = A↑;

A, B is a crisply generated formal concept. Ac = 1A↑↓

c

is a tuple closed by crisp components (crisply closed tuple).

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 8 / 26

Protofuzzy formal concepts [O. Kr´ ıdlo, S. Krajˇ ci]

Consider mappings ↑l : 2X → 2Y and ↓l : 2Y → 2X defined for a fuzzy context X, Y , I over lattice L, and l ∈ L. ∀A ⊆ X let ↑l (A) = {y ∈ Y : (∀x ∈ A)I(x, y) ≥ l}; ∀B ⊆ Y let ↓l (B) = {x ∈ X : (∀y ∈ B)I(x, y) ≥ l}.

Definition

An l-concept is a pair A, B such that ↑l (A) = B and ↓l (B) = A, i.e., a concept in the l-cut, binary context X, Y , Il, where Il = {(x, y) ∈ X × Y : I(x, y) ≥ l}. By Kl we denote the set of all concepts in the l-cut.

Definition

Triples A, B, l ∈ 2X × 2Y × L such that A, B ∈

k∈L

Kk and l = sup{k ∈ L: A, B ∈ Kk} are called protofuzzy concepts. The set of all protofuzzy concepts is denoted by KP.

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 9 / 26

Protofuzzy formal concepts [O. Kr´ ıdlo, S. Krajˇ ci]

A protofuzzy formal concept is a formal concept that occurs for the “first time” in a γ-cut of a fuzzy context.

Example

I = 0.1 0.6 0.5 1 0.2 0.4 0.8 0.3

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 10 / 26

Protofuzzy formal concepts [O. Kr´ ıdlo, S. Krajˇ ci]

A protofuzzy formal concept is a formal concept that occurs for the “first time” in a γ-cut of a fuzzy context.

Example

I = 0.1 0.6 0.5 1 0.2 0.4 0.8 0.3 X X X X 0.5-cut

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 11 / 26

Protofuzzy formal concepts [O. Kr´ ıdlo, S. Krajˇ ci]

A protofuzzy formal concept is a formal concept that occurs for the “first time” in a γ-cut of a fuzzy context.

Example

I = 0.1 0.6 0.5 1 0.2 0.4 0.8 0.3 X X 0.5-cut

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 12 / 26

Contractions [O. Kr´ ıdlo, S. Krajˇ ci]

Let KP denote the set of all proto-fuzzy concepts.

Definition

Let B ⊆ Y be an arbitrary set of attributes. The contraction of the set of proto-fuzzy concepts subsistent to the set B is KP

B = {A, l ∈ 2X × L : (∃B′ ⊇ B)A, B′, l ∈ KP}.

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 13 / 26

Relationship between protofuzzy formal concepts and fuzzy formal concepts

Theorem

Let there be a protofuzzy formal concept in the l-cut. Then there is a tuple Bc that satisfies the following conditions:

1

B↓

c = (a1, . . . , ak), where ai ≥ l if the ith object is contained in the given

protofuzzy concept, and ai < l if it is not;

2

Bc is a crisply closed tuple, i.e., 1B↓↑

c

= Bc.

Theorem

Let A = (a1, . . . , ak) = B↓

c , B = B↓↑ c . Let us define a tuple Z by the formula

zj =

• 1,

aj ≥ ai, 0, aj < ai. Then the context Z × 1Y corresponds to a protofuzzy concept in the ai-cut

• f the initial context, which can be contracted to 1Y .
• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 14 / 26

Relationship between protofuzzy formal concepts and fuzzy concepts

Results: It is shown that for every protofuzzy concept one can construct a crisply closed tuple; It is shown that for every crisply-closed tuple one can construct a contraction of a protofuzzy concept on the crisp subset of attributes

• f its intent (this contraction may coincide with the intent itself);

There is a method for deciding whether the contraction proposed is an intent of a protofuzzy concept.

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 15 / 26

Pattern structures [B. Ganter, S. Kuznetsov]

Let G be a set, called the set of objects, (D, ⊓) be a lower semilattice and δ: G → D be a mapping.

Definition

The triple (G, D, δ), where D = (D, ⊓), is called a pattern structure if the set δ(G) := {δ(g)|g ∈ G} generates a complete subsemilattice (Dδ, ⊓)

• f (D, ⊓).

For a pattern structure (G, D, δ) we define operations A :=

g∈A δ(g)

for sets A ⊆ G and d := {g ∈ G|d ⊑ δ(g)} for elements of semilattice d ∈ D. A pattern concept is a pair A, d such that A = d, d = A.

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 16 / 26

Interval pattern structures

• bject 1

0.1 0.6 0.5

• bject 2

1 0.2 0.4

• bject 3

0.8 0.3 {1, 2} = ([0.1, 1], [0.2, 0.6], [0.4, 0.5]); ([0.1, 1], [0.2, 0.6], [0.4, 0.5]) = {1, 2}. The pair ( {1, 2} , ([0.1, 1], [0.2, 0.6], [0.4, 0.5]) ) is an interval pattern concept

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 17 / 26

Relationship between fuzzy formal concepts and pattern structures

Theorem

If extents of interval pattern concepts coincide with crisply closed tuples, there are no nontrivial implications in the context.

Theorem

For any crisply-closed tuple the corresponding subset of objects is closed.

Theorem

If the set of crisply closed tuples coincides with the set of extents of interval pattern concepts, each interval pattern concept should satisfy the following condition: for no object from its extent the object intent majorizes the minimum of intervals.

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 18 / 26

Two-way pattern structures

Let (D1, ⊓1, ⊔1) and (D2, ⊓2, ⊔2) be two arbitrary lattices with operations of infimum ⊓1, ⊓2 and supremum ⊔1, ⊔2, and a pair of mappings be given: δ1 : D1 → 2D2 and δ2 : D2 → 2D1. We introduce operations : D1 → D2 and : D2 → D1 as follows: for a ∈ D1 we put a = ⊓2δ1(a); for b ∈ D2 we put b = ⊓1δ2(b). The mapping is a closure operator (both over D1 and D2). Two-way pattern concepts are pairs a, b, a ∈ D1, b ∈ D2, such that a = b and b = a. The tuple (D1, D2, δ1, δ2) makes a two-way pattern structure.

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 19 / 26

Relationship between fuzzy formal concepts, pattern structures, and two-way pattern structures

Results It is shown that nonzero components of any crisply closed extent tuple makes an extent of an interval pattern concept, the inverse is not true in general; A sufficient condition for the extent of an interval pattern concept to coincide with nonzero components of a crisply closed extent tuple was obtained; It is shown that fuzzy formal concepts generate a two-way pattern

• structure. (operation ↑ defines a ).
• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 20 / 26

Experiments in clustering

Given: 11947 objects, 5 attributes, fuzzy relation I. Output: A cover of the set of objects by subsets of objects with “similar” attribute values. Two methods were considered: A method based on interval pattern concepts; A method based on protofuzzy formal concepts. Both methods construct a cover of the set of all objects by concept extents.

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 21 / 26

• Experiments. A method based on interval pattern concepts

Define a parameter δ.

• 1. construct all pattern concepts;
• 2. select those with the “width”(maximal over components difference of

attribute values) not larger than δ;

• 3. select those with maximal extents: they make the clusters;
• 4. delete objects forming the cluster from the context;
• 5. repeat the procedure until the whole set of objects is covered.
• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 22 / 26

• Experiments. A method based on protofuzzy formal concepts

Set a parameter k ≤ 5.

• 0. Let γ = 1;
• 1. Decrease γ by 0.1;
• 2. In γ-cut find a formal concept with the largest intent that contains

not less than k attributes: this concept is a cluster;

• 3. Delete objects from the constructed cluster;
• 4. Repeat the procedure while there are concepts with at least

k attributes in their intents). Return to step 1.

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 23 / 26

Experiments in clustering. Results

δ method 1 (IntPS) k method 2 (protoF) Number of clusters 323 46 Dunn index 0.1 0.447 1 0.053 Number of clusters 108 80 Dunn index 0.2 0.224 2 0.064 Dunn index 53 11 Dunn index 0.3 0.149 3 0.05 Dunn index 29 11 Dunn index 0.4 0.117 4 0.053 Number of clusters 19 10 Dunn index 0.5 0.093 5 0.047 Dunn index = minimal distance between clusters maximal cluster diameter

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 24 / 26

Thank you!

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 25 / 26

References

• B. Ganter and R. Wille, Formal Concept Analysis. Mathematical
• Foundations. Springer, Berlin, 1999.
• R. Bˇ

elohl´ avek, V. Sklen´ aˇ r, and J. Zacpal, “Crisply Generated Fuzzy Concepts,” in: B. Ganter and R. Godin (Eds.): ICFCA 2005, LNCS 3403, pp. 268–283, 2005 (Springer-Verlag, Berlin, Heidelberg, 2005).

• O. Kr´

ıdlo, S. Krajˇ ci, “Proto-fuzzy Concepts, their Retrieval and Usage,” in: B. Ganter and R. Godin (Eds.): CLA 2008, pp. 83–95, ISBN 978-80-244-2111-7, Palack´ y University, Olomouc, 2008.

• B. Ganter and S.O. Kuznetsov, “Pattern Structures and Their

Projections,” preprint MATH-AL-14-2000, Technische Universit¨ at Dresden, Herausgeber, Der Rektor, November 2000.

• V. Pankratieva, S. Kuznetsov (SU-HSE)

Relations between fuzzy concepts CLA’2010, Sevilla 26 / 26