formal concept analysis

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Formal Concept Analysis

Intelligent Systems – Lecture 12

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1) Motivation (what is the problem solved)

2) Technical solution• Definitions• Explanations• Procedures• Illustrations by short examples

3) Illustration by a larger example

4) Extensions (Work only sketched)


Motivation

• Formal Concept Analysis is a method used for investigating and processing explicitely given information, in order to allow for meaningful and comprehensive interpretation.

– An analysis of data.

– Structures of formal abstractions of concepts of human thought.

– Formal emphasizes that the concepts are mathematical objects, rather than concepts of mind.


Motivation

• Simple motivating example by means of numbers:

– Numbers are either positive, whole numbers, prime, rational, algebraic, transcendent...

– These number schemes are characterising all known numbers (keeping the complex numbers out for now)

• Formal Concept Analysis help to draw inferences, to group objects, and hence to create concepts.

– All prime numbers are also whole numbers

– The pairs of numbers and characteristics form objects that could for instance represent IN°, IN+, IN-, IR,


Definition: Formal Context

• Context: A triple (G, M, I) is a (formal) context if

– G is a set of objects

– M is a set of attributes

– I is a binary relation between G and M called incidence

• Incidence := I ⊆ G x M– if gG, mM in (g,m)I, then we know that „object

g has attribute m“ and we write gIm.


Definition: Derivation Operators

• A‘ := {mM | A⊆G, (g,m)I for all gA – A‘ is the derivative of A– A‘ is the set of attributes shared by all objects in A

• B‘ := {gG | B⊆M, (g,m)I for all mB – B‘ is the derivative of B– B‘ is the set of objects that have all attributes in B


Derivation Rules

• There is a set of simple rules that follow and are satisfied by the derivation operators (be A, A1G)1) A1⊆A ⇒ A‘⊆A1‘

the larger the number of objects in a set, the smaller the number of shared attributes.

2) A ⊆A‘‘ and A‘ = A‘‘‘

• The dual relationships are valid for B, B1M, and it follows that: A ⊆ B‘ ⇔ B ⊆ A‘.

• This statement implies a Galois connection: if one makes the set of one type larger, they correspond to smaller sets of the other type, and vice versa.


Definition: Formal Concept

• A pair (A,B) is a formal concept of (G,M,I) if and only if

– A ⊆ G– B ⊆ M– A‘ = B, and A = B‘

• Note that at this point the incidence relationship is closed!

• A is called the extent (Umfang) of the concept (A,B), while

• B is called the intent (Inhalt) of the concept (A,B)


Definition: Concept Lattice

• The concepts of a given context are naturally ordered by a subconcept-superconcept relation:

– (A1,B1) ≤ (A2,B2) :⇔ A1⊆A2 (⇔ B2⊆B1).

• The ordered set of all formal concepts in (G,M,I) is denoted by B(G,M,I) and is called concept lattice (Begriffsverband).

• A concept lattice consists of the set of concepts of a formal context and the subconcept-superconcept relation between the concepts.


Theorem 1: Concept Lattices

• The basic theorem on concept lattices:

The concept lattice B(G,M,I) is a complete lattice in which infinum and supremum are given by– asfd– asfd

A complete lattice L is isomorphic to B(G,M,I) if and only if there are mappings γ: G→L and μ: M→L such that γ(G) is supremum-dense in L, μ(M) is infinum-dense in L and gIm is equivalent to γg ≤ μm for all gG and all mM. In particular, L ≅ B(L,L,≤).


Proof


Proof (2)


Example

• G = {Garfield, Snoopy, Flipper, HUND, Nemo}• M = {cartoon, real, dog, mammal}• B(G,M,I) = {

(Ø,{cartoon,real,dog,mammal}),(Snoopy, {cartoon, dog, mammal}),(HUND, {real, dog, mammal}),({Snoopy,HUND},{dog,mammal}),({Garfield,Snoopy,Nemo}, cartoon),({Flipper,HUND},{real,mammal}),

...({Garfield,Snoopy,Flipper, HUND, Nemo},Ø)}


Example (2)

• FIGURE of concept lattice for example


Extent and Intent in a Lattice

• The extent of a formal concept is given by all formal objects on the paths which lead down from the given concept node.

– The extent of an arbitrary concept is then found in the principle ideal generated by that concept.

• The intent of a formal concept is given by all the formal attributes on the paths which lead up from the given concept node.

– The intent of an arbitrary concept is then found in the principle filter generated by that concept.


Subconcepts in the Concept Lattice

• In the figure above, the Concept B is a subconcept of Concept A because the extent of Concept B is a subset of the extent of Concept A and the intent of Concept B is a superset of the intent of Concept A.

• All edges in the line diagram of a concept lattice represent this subconcept-superconcept relationship.


Reduction of Context

• A context (G,M,I) is called clarified if for g, hG and g’=h’ it always follows that g=h and correspondingly, m’=n’ implies m=n for all m,nM; i.e. a context is reduced, if both derivatives are injective.

• A clarified context (G,M,I) is called row-reduced, if every object concept is ∨-irreducible and column-reduced, if every attribute concept is ∧-irreducible. A context, which is both row-reduced and column-reduced is reduced.

• Reducing a context does not change the concept lattice!

• Always reducable are complete rows (objects g with g‘=M) and complete columns (attributes m with m‘=G).


Algorithm to Reduce a Final Context

1. Clarify the context (G,M,I): merge all gG, resp. mM with the same intent g’ resp. extent m’.

2. Remove all complete rows, and complete columns

3. Remove all objects g, for which g‘ can be represented as average of the derivatives h1‘,...,hk‘ of other objects h1,...,hkG

4. Remove all attributes m, for which m‘ is the average of other derivates m1‘,...,mk‘.

• The last three steps can be formalised by means of the so-called Arrow Relations (next slide).


Arrow Relations

• Arrow relations of a context (G, M, I) are defined as follows, with hG, mM:

• It follows: for gG with g‘≠M and mM with m‘≠G

– An object lattice γ(g)=(g‘‘,g‘) is ∨-irreducible ⇔ ∃m*: g ↕m*– An attribute lattice μ(M)=(m‘,m‘‘) is ∧-irreducible ⇔ ∃g*:

g*↕m


Implication

• An implication A → B (between sets A,BM of attributes) holds in a formal context if and only if B⊆A‘‘– i.e. if every object that has all attributes in A also

has all attributes in B.

• The implication determines the concept lattice up to isomorphism and therefore offers an additional interpretation of the lattice structure.


Many Valued Context


Conceptual Scaling


Attribute Exploration

• Attribute exploration is a knowledge acquisition method of FCA that is used to acquire knowledge from a domain expert by asking successive questions.

• Attribute exploration has proven to be a successful method for efficiently capturing knowledge, if it is only “known" to a domain expert.

• Attribute exploration asks the expert questions of the form “is it true that objects having attributes mi1,…,mik also have the attributes mj1,…,mjl?".

• The expert either confirms the question, in which case a new implication of the application domain is found, or rejects it. If the expert rejects the question, a counterexample is given, i.e., an object that has all the attributes mi1,…,mik but lacks at least one of mj1,…,mjl.

• The counterexample is then added to the application domain as a new object, and the next question is asked.

• What makes attribute exploration an attractive method for capturing expert knowledge is that it guarantees to make best use of the expert's answers, and to ask the minimum possible number of questions that suffices to acquire complete knowledge about the application domain.


Formal Concept Analysis Methods for DL

formal concept analysis

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