2.2 Definition of a Formal Concept

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A formal concept of a binary context is the pair (A,B), such that f A = B and g B = A. We call A the extent and B the intent of the concept (A,B). If A1 B1 and A2 B2 are two concepts, A1 B1 is called a subconcept of A2 B2 , provided that A1 A2 B2 B1 . In this case, A2 B2 is a superconcept of A1 B1 and it is written A1 B1 < A2 B2 . The relation < is called the hierarchical order relation of the concepts. The set of all concepts of (G, M, I) ordered in this way is called the concept lattice of the Context (G, M, I). Formal Concept Analysis (FCA) is used for deriving conceptual structures from data. These structures can be graphically represented as conceptual hierarchies, allowing the analysis of complex structures and the discovery of dependencies within the data. Formal concept analysis is based on the philosophical understanding that a concept is constituted by two parts: its extent, which consists of all objects belonging to the concept, and its intent, which

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Conceptual Data Classification

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Table 23.2 Formal context K. A G1 G2 G3 G4 G5 1 1 0 0 0 B 1 1 1 1 0 C 0 0 1 1 1 D 0 0 0 1 1

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comprises all attributes shared by those objects. One of the main objectives of this method is to visualize the data in the form of concept lattices. Let K be the formal context presented in Table 23.2. Then Figure 23.1 represents the structured set of concepts. A formal concept has been defined and introduced by different scientific communities in the world, starting in 1948 with Riguet [14] under the name of maximal rectangle. In graph theory, a maximal bipartite graph was exploited in the thesis of Le Than in 1986 [15] in a database to introduce a new kind of dependencies called Iso-dependencies , independently, Jaoua et al. introduced difunctional dependencies as the most suitable name for iso-dependencies in a database [16]. The most recent mathematical studies about formal concept analysis have been done by Ganter and Wille. More details may be found in [9 11]. What is remarkable is that concepts are increasingly used in several areas in real-life applications: text analysis, machine learning, databases, data mining, software decomposition, reasoning and pattern recognition. A complete conjunctive query and its associated answer in a database is no more nor less than a concept (i.e. an element in a lattice of concepts). Its generality and simplicity is very attractive. We almost may find a bridge between any computer science application and concepts. Combined with other methods for mapping any kind of data into a binary context, it gives an elegant base for data mining.

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2.3 Galois Connection

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A Galois connection is a conceptual learning structure used to extract new knowledge from an existing context (database). The context is represented by a binary relation. We can decompose the context into

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{} {G1,G2,G3,G4,G5} {B} {G1,G2,G3,G4} {C} {G3,G4,G5}

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{A,B} {G1,G2}

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{B,C} {G3,G4}

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{C,D} {G4,G5}

{B,C,D} {G4}

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{A,B,C,D} {}

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Figure 23.1 Concept lattice of the context K.

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Mathematical Foundations

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h (B)

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f (A)

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Figure 23.2 A Galois connection. a set of concepts and we can build a hierarchy of concepts also known as a Galois Lattice . A pair (f (A), h (B)) of maps is called a Galois connection if and only if A h B B f A , as we can see in Figure 23.2. It is also known that f A = fhf A and h B = hfh B .

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2.4 Optimal Concept or Rectangle

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A binary relation can be decomposed into a minimal set of optimal rectangles (or optimal concepts).

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2.4.1 Definition 1: Rectangle

Let R be a binary relation defined from E to F. A rectangle of R is a pair of sets (A,B) such that A E B F and A B R. A is the domain of the rectangle and B is the co-domain or range [11]. A rectangle is maximal if and only if it is a concept. A binary relation can be represented by different sets of rectangles. In order to save storage space, the gain function W(R) defined in Section 2.4.2 is important for the selection of a minimal significant set of maximal rectangles representing the relation. In the next section, we use interchangeably the word concept and maximal rectangle . We also introduce the definition of an optimal rectangle (i.e. optimal concept). Our conviction is that intelligence does not keep in mind the whole lattice structure of concepts, but only a few concepts covering all the data context. As a matter of fact, this thinking process is efficient because it optimizes the quantity of data kept in the main memory. Here, we propose a method for such coverage. But in the future, we may propose a variety of other methods perhaps more efficient.

2.4.2 Definition 2: Gain in Storage Space (Economy)

The gain in storage space W (R) of binary relation R is given by: W R = r/dc r d + c (23.3)

where r is the cardinality of R (i.e. the number of pairs in binary relation R), d is the cardinality of the domain of R and c is the cardinality of the range of R. Note that the quantity r/dc provides a measure of the density of the relation R. The quantity r d + c is a measure of the economy of information.