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  • a "natural" ''property'' cluster is a set of ''properties'' in the input matrix that are individually necessary and jointly sufficient for picking out some non-empty subset of the ''objects'' in the input data, and

  • a "natural" ''object'' cluster is a set of ''objects'' in the input matrix that can be picked out exactly by one of the natural ''property'' clusters.

  • There's a one-to-one correspondence between natural property clusters and natural object clusters, and a concept is a pair containing both a natural property cluster and its corresponding natural object cluster.


Note the strong parallel between "natural" property clusters and Definition s in terms of individually necessary and jointly sufficient conditions, on one hand, and between "natural" object clusters and the Extension s of such definitions, on the other.

...it also gives you a lattice.


FORMAL PRESENTATION

[[
'''''Given a set of objects O, a set of Attribute s A, and an indication of which objects have which attributes, concept analysis:

# finds all the "concepts" in the input dataset, where a concept is defined as an (Oc ⊆ O, Ac ⊆ A) pair such that A) every object in Oc has every attribute in Ac and B) every object in O (''not'' Oc) that has every attribute in Ac is in Oc.
# produces a Lattice indicating which concepts are strict subconcepts of which other concepts.''''']]


MISC


Provided the input objects and input concepts provide a complete description of the world (never true in practice, but perhaps a reasonable approximation), then:

# the set of attributes in each concept can be interpreted as a set of singly necessary and jointly sufficient conditions for defining the set of objects in the concept.
# if a set of attributes is ''not'' identified as a concept by the algorithm, then those attributes are not singly necessary and jointly sufficient for defining ''any'' non-empty subset of objects in the world.

...


SEE ALSO




FURTHER READING


  • {Link without Title} : a set of lectures describing formal concept analysis for computer scientists

  • [http://www.fcahome.org.uk/ A Formal Concept Analysis Homepage]