Formal concept analysis
The term was introduced by Rudolf Wille in 1981, and builds on the mathematical theory of lattices and ordered sets that was developed by Garrett Birkhoff and others in the 1930s.
The original motivation of formal concept analysis was the search for real-world meaning of mathematical order theory.
Then the mathematical theory of formal concept analysis may be helpful, e.g., for decomposing the lattice into smaller pieces without information loss, or for embedding it into another structure which is easier to interpret.
The theory in its present form goes back to the early 1980s and a research group led by Rudolf Wille, Bernhard Ganter and Peter Burmeister at the Technische Universität Darmstadt.
Its basic mathematical definitions, however, were already introduced in the 1930s by Garrett Birkhoff as part of general lattice theory.
Restructuring lattice theory is an attempt to reinvigorate connections with our general culture by interpreting the theory as concretely as possible, and in this way to promote better communication between lattice theorists and potential users of lattice theoryThis aim traces back to the educationalist Hartmut von Hentig, who in 1972 pleaded for restructuring sciences in view of better teaching and in order to make sciences mutually available and more generally (i.e. also without specialized knowledge) critiqueable.
Hence, formal concept analysis is oriented towards the categories extension and intension of linguistics and classical conceptual logic.
[3] In his late philosophy, Peirce assumed that logical thinking aims at perceiving reality, by the triade concept, judgement and conclusion.
Mathematics is an abstraction of logic, develops patterns of possible realities and therefore may support rational communication.
The data table represents a formal context, the line diagram next to it shows its concept lattice.
In this diagram the concept immediately to the left of the label reservoir has the intent stagnant and natural and the extent puddle, maar, lake, pond, tarn, pool, lagoon, and sea.
In this matrix representation, each formal concept corresponds to a maximal submatrix (not necessarily contiguous) all of whose elements equal 1.
It is however misleading to consider a formal context as boolean, because the negated incidence ("object g does not have attribute m") is not concept forming in the same way as defined above.
For this reason, the values 1 and 0 or TRUE and FALSE are usually avoided when representing formal contexts, and a symbol like × is used to express incidence.
But pairs of attributes which are negations of each other often naturally occur, for example in contexts derived from conceptual scaling.
It offers a general way of understanding change of concrete or abstract objects in continuous, discrete or hybrid space and time.
[15] TCA generalizes the above mentioned case by considering temporal data bases with an arbitrary key.
That leads to the notion of distributed objects which are at any given time at possibly many places, as for example, a high pressure zone on a weather map.
Most of these tools are academic open-source applications, such as: A formal context can naturally be interpreted as a bipartite graph.
The same remark applies for recommender systems where one is interested in local patterns characterizing groups of users that strongly share almost the same tastes for a subset of items.
[28] Relaxed FCA-based versions of biclustering and triclustering include OA-biclustering[29] and OAC-triclustering[30] (here O stands for object, A for attribute, C for condition); to generate patterns these methods use prime operators only once being applied to a single entity (e.g. object) or a pair of entities (e.g. attribute-condition), respectively.
Such a pair can be represented as an inclusion-maximal rectangle in the numerical table, modulo rows and columns permutations.
The complements of knowledge states therefore form a closure system and may be represented as the extents of some formal context.
