  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 effects 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. Information theoretic approaches are used to identify important
scale lenghts 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.
