Formal concept analysis with hierarchically ordered attributes

Formal concept analysis is a method of exploratory data analysis that aims at the extraction of natural clusters from object–attribute data tables. The clusters, called formal concepts, are naturally interpreted as human-perceived concepts in a traditional sense and can be partially ordered by a subconcept–superconcept hierarchy. The hierarchical structure of formal concepts (so-called concept lattice) represents a structured information obtained automatically from the input data table. The present paper focuses on the analysis of input data with a predefined hierarchy on attributes extending thus the basic approach of formal concept analysis. The motivation of the present approach derives from the fact that very often, people (consciously or unconsciously) attach various importance to attributes which is then reflected in the conceptual classification based on these attributes. We define the notion of a formal concept respecting the attribute hierarchy. Formal concepts which do not respect the hierarchy are considered not relevant. Elimination of the non-relevant concepts leads to a reduced set of extracted concepts making the discovered structure of hidden concepts more comprehensible. We present basic formal results on our approach as well as illustrating examples.