Real nonparametric regression using complex wavelets
Stuart Barber & Guy P. Nason
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 complex-valued wavelet shrinkage techniques:
one based on multiwavelet style
shrinkage (CMWS) 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, one of our methods, CMWS, 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.
Some key words:
Complex-valued wavelets; Complex normal distribution;
Curve estimation; Empirical Bayes; Multiwavelets; Wavelet shrinkage.
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