The Haar Wavelet Transform of a Dendrogram – I

While there is a very long tradition of approximating a data array by projecting row or column vectors into a lower dimensional subspace the direct approximation of a data matrix through smoothing is less common. Applications of data array smoothing include visualization; filtering of less relevant, and thus harder to interpret, values; and as a means towards compression. Wavelet smoothing or regression is a term applied to data filtering in wavelet space, followed by data reconstruction. Due to boundaries, and invariance of rows and columns, applying a wavelet transform to a data array is very problematic, unlike applying a wavelet transform to a two-dimensional pixelated image. We develop a new wavelet transform for application to data arrays. This is based on prior hierarchical clustering, which takes internal data structure and interrelationships into account. We motivate and describe the integrated clustering and wavelet transform in this work, and discuss its use for data array smoothing, and for approximating a dendrogram by “collapsing” clusters. In a companion paper (Paper II) we explore background theory and address the question as to how this new post-processing analysis of hierarchical clustering is indeed a wavelet transform.