Real nonparametric regression using complex wavelets

Summary.  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.