Real nonparametric regression using complex wavelets

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 (DWT) of data consisting of a signal corrupted by noise; shrink or remove the wavelet coefficients to remove the noise; and then invert the DWT to form an estimate of the true underlying curve. Various authors have proposed increasingly sophisticated methods of doing this using real-valued wavelets. Complex-valued wavelets exist, but are rarely used. We propose two new shrinkage techniques which use complex-valued wavelets. Extensive simulations show that our methods almost always give significantly more accurate estimates when compared with methods based on real-valued wavelets. Further, one of our methods is both simpler and dramatically faster than its competitors. In an attempt to understand the excellent performance of this latter method we present a new risk bound on its hard thresholded coefficients.