Mining formal concepts using implications between items

Abstract

Formal Concept Analysis (FCA) provides a mathematical tool to analyze and discover concepts in Boolean datasets (i.e. Formal contexts). It does also provide a tool to analyze complex attributes by transforming them into Boolean ones (i.e. items) thanks to conceptual scaling. For instance, a numerical attribute whose values are {1, 2, 3} can be transformed to the set of items {≤ 1, ≤ 2, ≤ 3, ≥ 3, ≥ 2, ≥ 1} thanks to interordinal scaling. Such transformations allow us to use standard algorithms like Close-by-One (CbO) to look for concepts in complex datasets by leveraging a closure operator. However, these standard algorithms do not use the relationships between attributes to enumerate the concepts as for example the fact that ≤ 1 implies ≤ 2 and so on. For such, they can perform additional closure computations which substantially degrade their performance. We propose in this paper a generic algorithm, named CbOI for Close-by-One using Implications, to enumerate concepts in a formal context using the inherent implications between items provided as an input. We show that using the implications between items can reduce significantly the number of closure computations and hence the time effort spent to enumerate the whole set of concepts.