A tensor provides a concise way to codify the interdependence of complex data. Treating a tensor as a d-way array, each entry records the interaction between the different indices. Clustering provides a way to parse the complexity of the data into more readily understandable information. Clustering methods are heavily dependent on the algorithm of choice, as well as the chosen hyperparameters of the algorithm. However, their sensitivity to data scales is largely unknown.

In this work, we apply the discrete wavelet transform to analyze the effects of coarse-graining on clustering tensor data. We are particularly interested in understanding how scale affects clustering of the Earth’s climate system. The discrete wavelet transform allows classification of the Earth’s climate across a multitude of spatial-temporal scales. The discrete wavelet transform is used to produce an ensemble of classification estimates, as opposed to a single classification. Each element of the ensemble is a clustering at a different spatial-temporal scale. Information theoretic approaches are used to identify important scale lengths in clustering the L15 Climate Datset. We also discover a sub-collection of the ensemble that spans the majority of the variance observed, allowing for efficient consensus clustering techniques that can be used to identify climate biomes.

张量提供了一种简洁的方法来整理复杂数据的相互依赖性。将张量视为d向数组，每个条目记录不同索引之间的交互。群集提供了一种将数据的复杂性解析为更容易理解的信息的方法。聚类方法在很大程度上取决于所选择的算法以及该算法的所选超参数。但是，它们对数据规模的敏感性很大程度上未知。

在这项工作中，我们应用离散小波变换来分析粗粒度对聚类张量数据的影响。我们特别想了解规模如何影响地球气候系统的聚集。离散小波变换可以在多个时空尺度上对地球的气候进行分类。与单一分类相反，离散小波变换用于产生分类估计的整体。集合中的每个元素都是不同时空尺度上的聚类。信息理论方法用于识别L15气候数据集聚类中重要的尺度长度。我们还发现了涵盖整个观察到的大部分方差的整体子集，