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Unsupervised discretization is a crucial step in many knowledge discoverytasks. The state-of-the-art method for one-dimensional data infers locallyadaptive histograms using the minimum description length (MDL) principle, butthe multi-dimensional case is far less studied: current methods consider thedimensions one at a time (if not independently), which result indiscretizations based on rectangular cells of adaptive size. Unfortunately,this approach is unable to adequately characterize dependencies amongdimensions and/or results in discretizations consisting of more cells (or bins)than is desirable.  To address this problem, we propose an expressive model class that allows forfar more flexible partitions of two-dimensional data. We extend the state ofthe art for the one-dimensional case to obtain a model selection problem basedon the normalized maximum likelihood, a form of refined MDL. As the flexibilityof our model class comes at the cost of a vast search space, we introduce aheuristic algorithm, named PALM, which Partitions each dimension ALternatelyand then Merges neighboring regions, all using the MDL principle. Experimentson synthetic data show that PALM 1) accurately reveals ground truth partitionsthat are within the model class (i.e., the search space), given a large enoughsample size; 2) approximates well a wide range of partitions outside the modelclass; 3) converges, in contrast to the state-of-the-art multivariatediscretization method IPD. Finally, we apply our algorithm to three spatialdatasets, and we demonstrate that, compared to kernel density estimation (KDE),our algorithm not only reveals more detailed density changes, but also fitsunseen data better, as measured by the log-likelihood.

Unsupervised Discretization by Two-dimensional MDL-based Histogram