Mining alpha/beta concepts as relevant bi-sets from transactional data

We are designing new data mining techniques on boolean contexts to identify a priori interesting bi-sets (i.e., sets of objects or transactions associated to sets of attributes or items). A typical impor- tant case concerns formal concept mining (i.e., maximal rectangles of true values or associated closed sets by means of the so-called Galois connec- tion). It has been applied with some success to, e.g., gene expression data analysis where objects denote biological situations and attributes denote gene expression properties. However in such real-life application domains, it turns out that the Galois association is a too strong one when considering intrinsically noisy data. It is clear that strong associations that would however accept a bounded number of exceptions would be extremely useful. We study the new pattern domain of fi=fl concepts, i.e., consistent maximal bi-sets with less than fi false values per row and less than fl false values per column. We provide a complete algorithm that computes all the fi=fl concepts based on the generation of concept unions pruned thanks to anti-monotonic constraints. An experimental validation on synthetic data is given. It illustrates that more relevant associations can be discovered in noisy data. We also discuss a practical application in molecular biology that illustrates an incomplete but quite useful extraction when all the concepts that are needed beforehand can not be discovered.