基于二维的MDL直方图的无监督离散化
Unsupervised Discretization by Two-dimensional MDL-based Histogram
摘要：
无监督离散化是许多知识发现任务中的关键步骤。一维数据的最新方法使用最小描述长度（MDL）原则推断局部自适应直方图，但多维情况下研究得少得多：当前方法一次考虑一个维度（如果不是独立的），这导致基于自适应大小的矩形单元格的离散化。不幸的是，这种方法无法充分表征维度之间的依赖性和/或导致由比期望的更多的单元（或箱）组成的离散化。为了解决这个问题，我们提出了一个表达模型类，允许更灵活的二维的【形】数据分区。我们扩展了一维情况下的最新技术水平，以获得基于归一化最大似然的模型选择问题，这是一种改进的MDL形式。由于我们的模型类的灵活性是以巨大的搜索空间为代价的，我们引入了一个启发式算法，命名为PALM，分区每个维度交替，然后合并相邻区域，所有使用MDL原则。对合成数据的实验表明，PALM 1）准确地揭示了模型类内的地面实况分区（即，搜索空间）; 2）在给定足够大的样本大小的情况下，很好地近似模型类之外的大范围的划分; 3）与现有技术的多元离散化方法IPD相比，收敛。最后，我们将我们的算法应用于三个空间数据集，我们证明，与核密度估计（KDE）相比，我们的算法不仅揭示了更详细的密度变化，而且更好地拟合了看不见的数据，如对数似然所测量的。
英文摘要：
Unsupervised discretization is a crucial step in many knowledge discovery tasks. The state-of-the-art method for one-dimensional data infers locally adaptive histograms using the minimum description length (MDL) principle, but the multi-dimensional case is far less studied: current methods consider the dimensions one at a time (if not independently), which result in discretizations based on rectangular cells of adaptive size. Unfortunately, this approach is unable to adequately characterize dependencies among dimensions 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 for far more flexible partitions of two-dimensional data. We extend the state of the art for the one-dimensional case to obtain a model selection problem based on the normalized maximum likelihood, a form of refined MDL. As the flexibility of our model class comes at the cost of a vast search space, we introduce a heuristic algorithm, named PALM, which Partitions each dimension ALternately and then Merges neighboring regions, all using the MDL principle. Experiments on synthetic data show that PALM 1) accurately reveals ground truth partitions that are within the model class (i.e., the search space), given a large enough sample size; 2) approximates well a wide range of partitions outside the model class; 3) converges, in contrast to the state-of-the-art multivariate discretization method IPD. Finally, we apply our algorithm to three spatial datasets, and we demonstrate that, compared to kernel density estimation (KDE), our algorithm not only reveals more detailed density changes, but also fits unseen data better, as measured by the log-likelihood.
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