  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.
