Real nonparametric regression using complex wavelets

Stuart Barber

doi:10.1111/j.1467-9868.2004.B5604.x

Real nonparametric regression using complex wavelets

2004-10-13

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DOI：10.1111/j.1467-9868.2004.B5604.x

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Abstract

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Wavelet shrinkage is an effective nonparametric regression technique, especially when the underlying curve has irregular features such as spikes or discontinuities. The basic idea is simple: take the discrete wavelet transform of data consisting of a signal corrupted by noise; shrink or remove the wavelet coefficients to remove the noise; then invert the discrete wavelet transform to form an estimate of the true underlying curve. Various researchers have proposed increasingly sophisticated methods of doing this by using real-valued wavelets. Complex-valued wavelets exist but are rarely used. We propose two new complex-valued wavelet shrinkage techniques: one based on multiwavelet style shrinkage and the other using Bayesian methods. Extensive simulations show that our methods almost always give significantly more accurate estimates than methods based on real-valued wavelets. Further, our multiwavelet style shrinkage method is both simpler and dramatically faster than its competitors. To understand the excellent performance of this method we present a new risk bound on its hard thresholded coefficients.

Keywords:

complex normal distribution

complex-valued wavelets

curve estimation

Empirical Bayes method

multiwavelets

wavelet shrinkage

Real nonparametric regression using complex wavelets

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