Quantitative Concept Analysis

Abstract. Formal Concept Analysis (FCA) begins from a context, given as a bi-nary relation between some objects and some attributes, and derives a lattice of concepts, where each concept is given as a set of objects and a set of attributes, such that the first set consists of all objects that satisfy all attributes in the sec-ond, and vice versa. Many applications, though, provide contexts with quanti-tative information, telling not just whether an object satisfies an attribute, but also quantifying this satisfaction. Contexts in this form arise as rating matrices in recommender systems, as occurrence matrices in text analysis, as pixel inten-sity matrices in digital image processing, etc. Such applications have attracted a lot of attention, and several numeric extensions of FCA have been proposed. We propose the framework of proximity sets (proxets), which subsume partially or-dered sets (posets) as well as metric spaces. One feature of this approach is that it extracts from quantified contexts quantified concepts, and thus allows full use of the available information. Another feature is that the categorical approach allows analyzing any universal properties that the classical FCA and the new versions may have, and thus provides structural guidance for aligning and combining the approaches