Multiscale methods for data on graphs and irregular

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 waveletlike 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 quantity 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 intended to be the basis of methods that might benefit from a multiscale representation of data either on graphs or for irregular spatial data.