Formal concept analysis is a particular method of analysis of relational data. Also, formal concept analysis provides elaborate mathematical foundations for relational data. In the course of the last decade, several attempts appeared to extend formal concept analysis to data with graded (fuzzy) attributes. Among these attempts, an approach based on residuated implications plays an important role. This chapter presents an overview of foundations of formal concept analysis of data with graded attributes, with focus on the approach based on residuated implications and on its extensions and particular cases. Presented is an overview of both of the main parts of formal concept analysis, namely, concept lattices and attribute implications, and an overview of the underlying foundations and related methods. In addition to that, the chapter contains an overview of topics for future research.
Key Terms in this Chapter
Concept Lattice: A concept lattice of a formal context (table) is a collection of all formal concepts (conceptual clusters) equipped with a subconcept-superconcept hierarchy.
Non-Redundant Basis of Formal Context: A non-redundant basis of a formal context is a set of attribute implications with the following properties: (1) every attribute implication from is true in , i.e., true in degree , (2) any attribute implication is true in to a degree to which it follows from , (3) no proper subset of satisfies (1) and (2). In a sense, a non-redundant basis of is a minimal set of attribute implications which contains all the information about validity of attribute implications in .
Formal Context: Formal context is a triplet where and are finite sets of objects and attributes, respectively, and is a relation or fuzzy relation between and . A formal context can be represented by a table with rows and columns corresponding to objects and attributes, and table entries containing to degrees to which objects have attributes. In particular, if is an ordinary relation, table entries can contain only degrees and .
Formal Concept: A formal concept of a formal context (table) is a pair where and are collections (sets or fuzzy sets) of objects and attributes from and , respectively which satisfy that is the collection of all objects sharing all attributes from and is the collection of all attributes shared by all objects from .
Formal Concept Analysis: Formal concept analysis is a method of analysis of relational data. Two main outputs of formal concept analysis are a concept lattice, that is, a partially ordered collection of clusters, and a nonredundant basis of attribute implications, that is, a fully informative small set of particular attribute dependencies extracted from data.
Attribute Implication: An attribute implication is an expression such that and are collections (sets of fuzzy sets) of attributes. The basic meaning of being true in a formal context (table) is: every object from which has all attributes from has also all attributes from .