### Formal Concept Analysis

• 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

### Formal Concept Analysis

• Formal Concept Analysis takes as input a matrix specifying a set of objects and the properties thereof, called attributes, and finds both all the “natural” clusters of attributes and all the “natural” clusters of objects in the input data, where
• a “natural” object cluster is the set of all objects that share a common subset of attributes, and
• a “natural” property cluster is the set of all attributes shared by one of the natural object clusters

• Natural property clusters correspond one-for-one with natural object clusters, and a concept is a pair containing both a natural property cluster and its corresponding natural object cluster
• The family of these concepts obeys the mathematical axioms defining a lattice, and is called a concept lattice

### Definition: Formal Context

• Context: A triple (G, M, I) is a (formal) context if
• G is a set of objects (Gegenstand)
• M is a set of attributes (Merkmal)
• I is a binary relation between G and M called incidence
• Incidence relation: I ⊆ G x M
• if g∈G, m∈M in (g,m)∈I, then we know that “ object g has attribute m „ and we write gIm
• Derivation operators:
• For A ⊆ G, A‘={m∈M | (g,m)∈I for all g∈A}
• For B ⊆ M, B‘={g∈G | (g,m)∈I for all m∈B}

### 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 ; i.e. all objects of the concept carry all its attributes and that there is no other object in G carrying all attributes of the concept
• A is called the extent (Umfang) of the concept (A,B)
• B is called the intent (Inhalt) of the concept (A,B)

### Generating Formal Concepts

• Using the derivation operators we can derive formal concepts from our formal context with the following routine:

1. Pick a set of objects A

2. Derive the attributes A'

3. Derive (A')'

4. (A'',A') is a formal concept

• A dual approach can be taken starting with an attribute

### Example

1. Pick any set of objects A, e.g. A={duck}.
2. Derive the attributes A'={small, two legs, feathers, fly, swim}
3. Derive (A')'={small, two legs, feathers, fly, swim}'={duck, goose}
4. (A'',A')=({duck, goose},{small, two legs, feathers, fly, swim}) is a formal concept.

Creator: OlliG

Contributors:
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