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Multiscale methods for data on graphs and irregular multidimensional situations
by Martin Jansen, Guy Nason and Bernard Silverman
For regularly spaced one-dimensional data, wavelet shrinkage has proven to be a compelling method
for nonparametric function estimation. We create three new multiscale methods that provide wavelet-
like transforms for both data arising on graphs and for irregularly spaced spatial data in more than
one dimension. The concept of scale still exists within these transforms but as a continuous quan-
tity rather than dyadic levels. Further, we adapt recent empirical Bayesian shrinkage techniques to
enable us to perform multiscale shrinkage for function estimation both on graphs and for irregular
spatial data. We demonstrate that our methods perform very well when compared to several other
methods for spatial regression for both real and simulated data. Although our article concentrates
on multiscale shrinkage (regression) we present our new ‘wavelet transforms’ as generic tools in-
tended to be the basis of methods that might benefit from a multiscale representation of data either
on graphs or for irregular spatial data.
Key words: Graph; Irregular data; Lifting; Wavelets; Wavelet shrinkage
Full text of the paper (pdf),
which appeared in the Journal of the Royal Statistical Society, B (2009).
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