## 4 Formal Concept Analysis  
Formal Concept Analysis (FCA) [18] is a conceptual modeling paradigm that studies how objects can be hierarchically grouped together according to their common attributes. This grouping of objects actually represents their biclustering. This approach is based on the idea of preserving the object-attribute description of the similarity of clusters. The result of FCA algorithms is a conceptual lattice (lattice of formal concepts or Galois lattice) [18], which contains hierarchically related formal concepts, which are biclusters. Let \(G\) and \(M\) be sets, called sets of objects and their features, respectively, and \(I\subseteq G\times M\) be a relation. This relation is interpreted as follows: for \(g\in G\), \(m\in M\) takes place \(glm\) if the object \(g\) has the feature \(m\). The triple \(K=(G,\,M,\,I)\) is called the formal context. The context can be specified in the form of an object-attribute \((0,1)\)--matrix. In this matrix, objects correspond to rows and features-columns, and matrix elements encode the presence/absence of features in objects [44]. The Galois correspondence between the ordered sets \((2^{G},\,\subseteq)\) and \((2^{M},\,\subseteq)\) is given by the following pair of mappings (called Galois operators) for arbitrary \(A\subseteq G\) and \(B\subseteq G\) :  
\[ \begin{array}{l}A^{\uparrow}:\{m\in M|glm\text{ for all }g\in A\,\},\\ B^{\downarrow}:\{g\in G|glm\text{ for all }m\in B\,\}.\end{array}  \tag{5}\]  
A pair of sets \((A,\,B)\) such that \(A\subseteq G\), \(B\subseteq G\), \(A^{\uparrow}=B\), \(B^{\downarrow}=A\) is called a formal concept of context \(K\) with (formal) scope \(A\) and (formal) content \(B\). The sets \(A\) and \(B\) in a formal concept \((A,\,B)\) are called the extent and the intent, respectively [44]. Note that the formal definition of the concept in the form of pair \((A,\,B)\) is in accordance with the logico-philosophical tradition. For example, it is close to the idea of the concept in the Logic of Port-Royal ([PERSON], [PERSON], France, XVII century). If the context is represented as a \((0,1)\)--matrix, then the formal concept corresponds to its maximum submatrix filled with ones. As shown in [17; 18], the set of all formal context concepts \(K\) forms a complete lattice with the operations \(\wedge\) (intersection) and \(\wedge\) (union) :  
\[ \begin{array}{l}(A_{1},B_{1})\vee(A_{2},B_{2})\triangleq((B_{1}\cap B_{2})^ {\downarrow},B_{1}\cap B_{2}),\\ (A_{1},B_{1})\wedge(A_{2},B_{2})\triangleq(A_{1}\cap A_{2},(A_{1}\cap A_{2})^ {\uparrow}).\end{array}  \tag{6}\]  
This lattice, called the lattice of formal concepts (or the Galois lattice in French and Canadian literature), is denoted \(\mathfrak{B}(K)\). In addition to formal contexts, FCA explores the so-called multi-valued contexts in which features can take values from a certain set of values. Formally, a multivalued context is a quadruple \((G,\,M,\,W,\,I)\), where \(G,\,M,\) and \(W\) are sets (objects, features, and feature values, respectively), and \(I\) is a ternary relation \(I\subseteq G\times M\times W\), which specifies the value \(w\) of feature \(m\), and \((g,\,m,\,w)\in I\) and \((g,\,m,\,v)\in I\) implies \(w=v\). The representation of multi-valued contexts with two-valued ones is called scaling. Possible types of scaling are discussed in [18]. Let \(K=(G,\,M,\,I)\) be a formal context and \((A,\,B)\) be some formal concept. In this case, the stability index \(\sigma\) of the concept \((A,\,B)\) is determined by the expression \(\sigma(A,\,B)=\,\,\mid C(A,\,B)\mid/\,\,2\,\mid A\mid\), where \(C(A,\,B)\) is the union of subsets \(C\subseteq A\) such that \(C=B^{\prime}\). It is obvious that \(0\leq\sigma(A,\,B)\leq 1\). If the value \(\sigma_{\min}\in[0,\,1]\) is chosen, then the formal concept \((A,\,B)\) is called stable if \(\sigma(A,\,B)\geq\sigma_{\min}\). Biclusters, as well as dense and stable formal concepts, are used to form hypotheses in solving clustering problems [25].