Formal concept analysis and linguistic hedges

Abstract

This paper presents an application of linguistic hedges to formal concept analysis of data with fuzzy attributes. Formal concept analysis aims at extraction of particular (bi-)clusters, called formal concepts, from data. The clusters link collections of objects (extents) and attributes (intents), and have a clear interpretation due to a simple verbal description of the concept-forming operators. We insert linguistic hedges such as ‘very’ or ‘extremely’ in the description of the operators. In this way, linguistic hedges become parameters for formal concept analysis that control the number of clusters extracted from data. Namely, as we show theoretically as well as experimentally, stronger hedges result in a smaller number of clusters. The new concept-forming operators form Galois-like connections. We study their properties and axiomatize them. Then, we show that a concept lattice with hedges, i.e. the set of all formal concepts of the new operators is indeed a complete lattice which is isomorphic to a particular ordinary concept lattice. We describe the isomorphism and its inverse. These mappings serve as translation procedures. As a consequence, we obtain a theorem characterizing the structure of concept lattices with hedges which generalizes the well-known main theorem of ordinary concept lattices. The isomorphism and its inverse enable us to compute a concept lattice with hedges using algorithms for ordinary concept lattices. We demonstrate by experiments that when selecting various hedges from the strongest to weaker hedges, the reduction in size of the corresponding concept lattices is smooth. From a broader perspective, we argue that linguistic hedges represent mathematically and computationally a feasible way to parameterize methods for knowledge extraction from data that enable one to emphasize or to suppress extracted patterns while keeping their interpretation.