Delta-Closure Structure for Studying Data Distribution

In this paper, we revisit pattern mining and study the distribution underlying a binary dataset thanks to the closure structure which is based on passkeys, i.e., minimum generators in equivalence classes robust to noise. We introduce $\Delta$-closedness, a generalization of the closure operator, where $\Delta$ measures how a closed set differs from its upper neighbors in the partial order induced by closure. A $\Delta$-class of equivalence includes minimum and maximum elements and allows us to characterize the distribution underlying the data. Moreover, the set of $\Delta$-classes of equivalence can be partitioned into the so-called $\Delta$-closure structure. In particular, a $\Delta$-class of equivalence with a high level demonstrates correlations among many attributes, which are supported by more observations when $\Delta$ is large. In the experiments, we study the $\Delta$-closure structure of several real-world datasets and show that this structure is very stable for large $\Delta$ and does not substantially depend on the data sampling used for the analysis.