knowledge discovery with fca

1/67

Knowledge Discovery with FCA

Lecture 1: Introduction to Formal Concept Analysis

Babes-Bolyai University, Computer Science Department, Cluj-Napocacsacarea@cs.ubbcluj.ro

February 27, 2018

2/67

WHAT IS FORMAL CONCEPT ANALYSIS?BRANCH OF APPLIED MATHEMATICS AND ARTIFICIAL INTELLIGENCE

� Based on Lattice Theory developed by Garrett Birkhoffand others in the 1930s

� Employs algebra in order to formalize notions of conceptand conceptual hierarchy

� Term Formal Concept Analysis (short: FCA) introduced byRudolf Wille in the 1980s.

3/67

WHY FORMAL CONCEPT ANALYSIS

Because...The methods of Formal Concept Analysis offers an algebraicapproach to data analysis and knowledge processing.

� Strengths of FCA are� a solid mathematical and philosophical foundation,� more than 2000 research publications,� experience of several hundred application projects,� an expressive and intuitive graphical representation,� Due to its elementary yet powerful formal theory, FCA can

express other methods, and therefore has the potential tounify the methodology of data analysis.

4/67

APPLICATIONS

Formal Concept Analysis has recently been applied in� Description Logics, for checking completeness of

knowledge bases,� Linguistics, for the investigation of thesauri and

ontologies,� Software Engineering, for modelling type hierarchies with

role types,� Biomathematics, for analysing gene expression data, item

Machine Learning, for discovering website duplicates,� Data Mining, for pattern matching problems,� Rough Set Theory, for studying granular data,� Web Usage Mining, for discovering usage patterns� Medicine, etc...

5/67

WHAT IS FORMAL CONCEPT ANALYSIS?FORMAL CONCEPT ANALYSIS (FCA) IS A

� mathematization/formalization of the philosophicalunderstanding of concepts

� human-centered method to structure and analyze data� method to visualize data and its inherent structures,

implications and dependencies

6/67

AGENDA

1 Concept Lattices� What is a concept?� Formal Context� Derivation Operators� Formal Concept� Concept Lattice� Computing All Concepts� Drawing Concept Lattices� Clarifying and Reducing a Formal Context� Interlude: ConExp, FCA Tools Bundle

7/67

WHAT IS A CONCEPT?

8/67

WHAT IS A CONCEPT?

Consider the concept bird. What drives us to call something abird?� Every object with certain attributes is called bird:

� A bird has feathers.� A bird has two legs.� A bird has a bill.

� All objects having these attributes are called birds:� Duck, goose, owl and parrot are birds.� Penguins are birds, too.� . . .

9/67

WHAT IS A CONCEPT?

We need� objects� attributes� . . .

� what else?� What makes a concept to be a concept?

9/67

WHAT IS A CONCEPT?

We need� objects� attributes� . . .

� what else?� What makes a concept to be a concept?

10/67

WHAT IS A CONCEPT IN FCA?

Formal Concept Analysis models concepts as units of thoughtthat consist of two parts:� The concept extent comprises all objects that belong to the

concept.� The concept inten contains all attributes that all of the

objects have in common.

FCA is used, amongst others, in data analysis, informationretrieval, data mining and software engineering.

11/67

WHAT IS A CONCEPT?FCA is working on the conceptual layer. The representationallayer plays only a minor role.

12/67

WHAT IS A (FORMAL) CONCEPT?

13/67

THE UNIVERSE OF DISCOURSE

13/67

THE UNIVERSE OF DISCOURSE

14/67

THE (FORMAL) CONTEXT

14/67

THE (FORMAL) CONTEXT

15/67

THE (FORMAL) CONTEXT

DefinitionA formal context is a triple (G,M, I), where G is a set of objects, M isa set of attributes, and I is a relation between G and M.

� What is a relation?� What types of relations do we know?

We read (g,m) ∈ I as object g has attribute m.

15/67

THE (FORMAL) CONTEXT

DefinitionA formal context is a triple (G,M, I), where G is a set of objects, M isa set of attributes, and I is a relation between G and M.

� What is a relation?� What types of relations do we know?

We read (g,m) ∈ I as object g has attribute m.

15/67

THE (FORMAL) CONTEXT

DefinitionA formal context is a triple (G,M, I), where G is a set of objects, M isa set of attributes, and I is a relation between G and M.

� What is a relation?� What types of relations do we know?

We read (g,m) ∈ I as object g has attribute m.

16/67

EXAMPLE 1

17/67

EXAMPLE 2

18/67

DATA REPRESENTATION

� By a formal context we represent data...� Nice... but...� However, why FCA and not SQL, Data Mining, etc?

19/67

DATA BIPLOT OF INTERVIEW DATA

20/67

CONCEPT LATTICE OF INTERVIEW DATA

21/67

UNFOLDING DATA IN A CONCEPT LATTICETHE BASIC PROCEDURE OF FORMAL CONCEPT ANALYSIS:

� Data is represented in a very basic data type, called formalcontext.

� Each formal context is transformed into a mathematicalstructure called concept lattice. The information containedin the formal context is preserved.

� The concept lattice is the basis for further data analysis. Itmay be represented graphically to supportcommunication, or it may be investigated with withalgebraic methods to unravel its structure.

22/67

WHAT IS A CONCEPT LATTICE?

� Graphical diagram� Is it a graph? Yes/No?

� Order diagram?� What is an order?� What is a lattice?

22/67

WHAT IS A CONCEPT LATTICE?

� Graphical diagram� Is it a graph? Yes/No?� Order diagram?� What is an order?

� What is a lattice?

22/67

WHAT IS A CONCEPT LATTICE?

� Graphical diagram� Is it a graph? Yes/No?� Order diagram?� What is an order?� What is a lattice?

23/67

MORE EXAMPLESDIVISOR LATTICE OF 200

24/67

MORE EXAMPLESRECOMMENDED SERVING TEMPERATURE FOR RED WINES

25/67

MORE EXAMPLESRECOMMENDED SERVING TEMPERATURE FOR WHITE WINES

26/67

HOW DO WE COMPUTE CONCEPTS?For the mathematical definition of formal concepts weintroduce the derivation operator ′.

For a set of objects A ⊆ G, A′ is defined as:

A′ = {all attributes in M common to the objects of A}.

For a set of attributes B ⊆M, B′ is defined as:

B′ = {objects in G having all attributes of B}.

� We are looking for pairs (A,B) of objects A and attributes Bthat satisfy the conditions

A′ = B and B′ = A

and we call these pairs formal concepts.

27/67

DERIVATION OPERATORS

28/67

29/67

30/67

31/67

32/67

33/67

34/67

35/67

36/67

37/67

38/67

39/67

40/67

41/67

COMPUTING ALL CONCEPTS

There are several algorithms to compute all concepts:� naive approach� intersection method� Next-Closure (Ganter 1984)� Titanic (Stumme et al. 2001)� Inclose family� and several incremental algorithms

42/67

43/67

44/67

45/67

46/67

47/67

48/67

49/67

50/67

51/67

52/67

53/67

54/67

55/67

56/67

57/67

58/67

59/67

60/67

61/67

62/67

63/67

64/67

65/67

66/67

67/67