Generating a Condensed Representation for Association Rules

Association rule extraction from operational datasets often produces several tens of thousands, and even millions, of association rules. Moreover, many of these rules are redundant and thus useless. Using a semantic based on the closure of the Galois connection, we define a condensed representation for association rules. This representation is characterized by frequent closed itemsets and their generators. It contains the non-redundant association rules having minimal antecedent and maximal consequent, called min-max association rules. We think that these rules are the most relevant since they are the most general non-redundant association rules. Furthermore, this representation is a basis, i.e., a generating set for all association rules, their supports and their confidences, and all of them can be retrieved needless accessing the data. We introduce algorithms for extracting this basis and for reconstructing all association rules. Results of experiments carried out on real datasets show the usefulness of this approach. In order to generate this basis when an algorithm for extracting frequent itemsets—such as Apriori for instance—is used, we also present an algorithm for deriving frequent closed itemsets and their generators from frequent itemsets without using the dataset.     

Phase Diagrams of the Systems CuBiS<Subscript>2</Subscript>-LnBiS<Subscript>3</Subscript> (Ln = La,Nd)

Phase relations in the CuBiS₂-LaBiS₃ and CuBiS₂-NdBiS₃ systems were studied for the first time using physicochemical analysis, and the T-x phase diagrams of these systems were constructed. Both systems contain quaternary compounds with the Cu₂LnBi₃S₇ stoichiometry, which melt incongruently at 815 and 910 K, respectively. Cu₂LaBi₃S₇ and Cu₂NdBi₃S₇ are isostructural with one another and crystallize in orthorhombic symmetry. The lattice parameters of Cu₂LaBi₃S₇ are a = 14.66 Å, b = 21.60 Å, and c = 4.12 Å, and those of Cu₂NdBi₃S₇ are a = 14.60 Å, b = 21.56 Å, and c = 4.10 Å.