A formal context is a triple (G,M,I) consisting of a set of formal objects G, a set of formal attributes M, and a binary relation I inclus G x M (expressing for each object which attributes it has) (Wille R., 1982, 1997a, 1997a, 2001 and Ganter B., Wille R., (1997, 1999).). Generally, the relation is an equivalence, but it may also be a similarity (see Slowinski and Vanderpooten 2000).

A formal concept of (G,M,I) is a pair (A,B) of sets satisfying :
       A inclus G, B inclus M, A'= B, A = B'
where A' et B' are the operators :
       A' := {m inclus M | (g,m) inclus I for all g inclus A} (A' is the set of attributes which are possessed by all the objects of A)
       B' := {g inclus G | (g,m) inclus I for all m inclus B}  (B' is the set of objects which possess all the attributes of B)
The set A is called the extent of the formal concept (A, B), and the set B is called its intent.

Concepts are ordered by:
       (A1,B1) LE (A2,B2) :   implique A1 inclus A2   ( implique B2 inclus B1 )
The set B (G,M,I) of all concepts of (G,M,I) with this order is a complete lattice, called the concept lattice of (G,M,I).

NB: Multi-valued-contexts must be transformed into one-valued contexts by using nominal scaling or plain scaling (each value of a multi-valued attribute becomes nominally a one-valued attribute).

     FCA example

List of available software: click here

Concept Approximation by a similarity index (see Saquer and Deogun)

A method giving the formal concept which is the closest approximation to a set of objects A, a set of attributes B or a pair {A,B} is implemented in Semana. It is based on a similarity index proposed by Saquer and Deogun.
Another method based on Rough Set Theory has been proposed by the same authors (see RST).

Alpha Galois Lattices
(see Pernelle et al.)
A method to reduce the size of the lattice when the number of nodes is very large. The method is based on a preliminary partition of the full BD in "types" (basic concepts)

Association Rules
A list of the Association Rules (A => B) (cf. Agrawal and Srikant 1994) may be obtained with their support up to a given threshold.
A is a set of attributes called premise, B is a set of attributes called conclusion. Example:
      {D,F} => {H}    (12/12)     100
means that, when an object has features D and F, it has also feature H in 100% of the 12 occurrences in the formal context. Such a rule, observed without exception within the formal context, is called Duquenne-Guigues implication.


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