Formal concept analysis
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Formal concept analysis is a principled way of automatically deriving an ontology from a collection of objects and their properties. The term was introduced by Rudolf Wille in 1984, and builds on applied lattice and order theory that was developed by Birkhoff and others in the 1930's.
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[edit] Intuitive description
Formal concept analysis is both an unsupervised machine learning technique but also broadly refers to a method of data analysis. The approach 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 or Galois lattice.
Note the strong parallel between "natural" property clusters and definitions in terms of individually necessary and jointly sufficient conditions, on one hand, and between "natural" object clusters and the extensions of such definitions, on the other. 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. Conversely, if a set of attributes is not identified as a concept in this framework, then those attributes are not singly necessary and jointly sufficient for defining any non-empty subset of objects in the world.
[edit] Example
Consider O = {1,2,3,4,5,6,7,8,9,10}, and A = {composite,even,odd,prime,square}. The smallest concept including the number 3 is the one with objects {3,5,7}, and attributes {odd,prime}, for 3 has both of those attributes and {3,5,7} is the set of objects having that set of attributes. The largest concept involving the attribute of being square is the one with objects {1,4,9} and attributes {square}, for 1, 4 and 9 are all the square numbers and all three of them have that set of attributes. It can readily be seen that both of these example concepts satisfy the formal definitions below
The full set of concepts for these objects and attributes is shown in the illustration. It includes a concept for each of the original attributes: the composite numbers, square numbers, even numbers, odd numbers, and prime numbers. Additionally it includes concepts for the even composite numbers, composite square numbers (that is, all square numbers except 1), even composite squares, odd squares, odd composite squares, even primes, and odd primes.
[edit] Definition of concepts
We take as givens a (formal) context consisting of a set of objects O, a set of attributes A, and an indication of which objects have which attributes.
A concept is defined to be a pair (Oi, Ai) such that
- Oi ⊆ O
- Ai ⊆ A
- every object in Oi has every attribute in Ai
- for every object in O that is not in Oi, there is an attribute in Ai that the object does not have
- for every attribute in A that is not in Ai, there is an object in Oi that does not have that attribute
Oi is called the extent of the concept, Ai the intent.
[edit] Tabular interpretation
A context may be described as a table, with the objects corresponding to the rows of the table, the attributes corresponding to the columns of the table, and a Boolean value (represented graphically below as a checkmark) in cell (x, y) whenever object x has value y:
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composite even odd prime square 1 √ √ 2 √ √ 3 √ √ 4 √ √ √ 5 √ √ 6 √ √ 7 √ √ 8 √ √ 9 √ √ √ 10 √ √
A concept, in this representation, forms a maximal subarray (not necessarily contiguous) such that all cells within the subarray are checked. For instance, the concept highlighted with a different background color in the table above is the one describing the odd prime numbers, and forms a 3 × 2 subarray in which all cells are checked.[1]
[edit] The concept lattice
The concepts (Oi, Ai) defined above can be partially ordered by inclusion: if (Oi, Ai) and (Oj, Aj) are concepts, we define a partial order ≤ by saying that (Oi, Ai) ≤ (Oj, Aj) whenever Oi ⊆ Oj. Equivalently, (Oi, Ai) ≤ (Oj, Aj) whenever Aj ⊆ Ai.
Every pair of concepts in this partial order has a unique greatest lower bound (meet). The greatest lower bound of (Oi, Ai) and (Oj, Aj) is the concept with objects Oi ∩ Oj; it has as its attributes the union of Ai, Aj, and any additional attributes held by all objects in Oi ∩ Oj.
Symmetrically, every pair of concepts in this partial order has a unique least upper bound (join). The least upper bound of (Oi, Ai) and (Oj, Aj) is the concept with attributes Ai ∩ Aj; it has as its objects the union of Oi, Oj, and any additional objects that have all attributes in Ai ∩ Aj.
These meet and join operations satisfy the axioms defining a lattice. Conversely, any finite lattice may be generated as the concept lattice for some context. For, let L be a finite lattice, and form a context in which the objects and the attributes both correspond to elements of L. In this context, let object x have attribute y exactly when x and y are ordered as x ≤ y in the lattice. Then, the concept lattice of this context is isomorphic to L itself.[2] This construction may be interpreted as forming the Dedekind–MacNeille completion of L, which is known to produce an isomorphic lattice from any finite lattice.
[edit] Recovering the context from the Hasse diagram
The Hasse diagram of the concept lattice (also called, in formal concept analysis, a line diagram), encodes enough information to recover the original context from which it was formed. Each object of the context corresponds to a lattice element, the element with the minimal object set that contains that object, and with an attribute set consisting of all attributes of the object. Symmetrically, each attribute of the context corresponds to a lattice element, the one with the minimal attribute set containing that attribute, and with an object set consisting of all objects with that attribute. We may label the nodes of the Hasse diagram with the objects and attributes they correspond to; with this labeling, object x has attribute y if and only if there exists a monotonic path from x to y in the diagram.[3]
[edit] Efficient construction
Kuznetsov & Obiedkov (2001) survey the many algorithms that have been developed for constructing concept lattices. These algorithms vary in many details, but are in general based on the idea that each edge of the Hasse diagram of the concept lattice connects some concept C to the concept formed by the join of C with a single object. Thus, one can build up the concept lattice one concept at a time, by finding the neighbors in the Hasse diagram of known concepts, starting from the concept with an empty set of objects. The amount of time spent to traverse the entire concept lattice in this way is polynomial in the number of input objects and attributes per generated concept.
[edit] See also
[edit] Notes
[edit] References
- Ganter, Bernhard; Stumme, Gerd & Wille, Rudolf, eds. (2005), Formal Concept Analysis: Foundations and Applications, Lecture Notes in Artificial Intelligence, no. 3626, Springer-Verlag, ISBN 3-540-27891-5
- Ganter, Bernhard & Wille, Rudolf (1998), Formal Concept Analysis: Mathematical Foundations, Springer-Verlag, Berlin, ISBN 3-63311-62767-5. Translated by C. Franzke.
- Carpineto, Claudio & Romano, Giovanni (2004), Concept Data Analysis: Theory and Applications, Wiley, ISBN 978-0-470-85055-8.
- Kuznetsov, Sergei O. & Obiedkov, Sergei A. (2001), “Algorithms for the Construction of Concept Lattices and Their Diagram Graphs”, Principles of Data Mining and Knowledge Discovery, vol. 2168, Lecture Notes in Computer Science, Springer-Verlag, pp. 289–300, DOI 10.1007/3-540-44794-6_24.
- Wolff, Karl Erich (1994), “A first course in Formal Concept Analysis”, in F. Faulbaum, StatSoft '93, Gustav Fischer Verlag, pp. 429–438, <http://www.fbmn.fh-darmstadt.de/home/wolff/Publikationen/A_First_Course_in_Formal_Concept_Analysis.pdf>.
