Multivariate nonparametric regression using lifting

For regularly spaced one-dimensional data wavelet shrinkage has proven to be a compelling method for nonparametric function estimation. We argue that this is not the case for irregularly spaced data in two or higher dimensions. This article develops three methods for the multiscale analysis of irregularly spaced data based on the recently developed lifting paradigm by “lifting one coefficient at a time”. The concept of scale still exists within these transforms but as a continuous quantity rather than dyadic levels. We develop empirical Bayes methods that take account of the continuous nature of the scale. We apply our new methods to the problems of estimation of krill density and rail arrival delays. We demonstrate good performance in a simulation study on new two-dimensional analogues of the well-known Blocks, Bumps, Doppler and Heavisine and a new piecewise linear function called maartenfunc.