Deterministic/stochastic wavelet decomposition for recovery of signal from noisy data

In a series of recent articles on nonparametric regression, Donoho and Johnstone developed wavelet-shrinkage methods for recovering unknown piecewise-smooth deterministic signals from noisy data. Wavelet shrinkage based on the Bayesian approach involves specifying a prior distribution on the wavelet coefficients, which is usually assumed to have a distribution with zero mean. There is no a priori reason why all prior means should be 0; indeed, one can imagine certain types of signals in which this is not a good choice of model. In this article, we take an empirical Bayes approach in which we propose an estimator for the prior mean that is "plugged into" the Bayesian shrinkage formulas. Another way we are more general than previous work is that we assume that the underlying signal is composed of a piecewise-smooth deterministic part plus a zero-mean stochastic part: that is, the signal may contain a reasonably large number of nonzero wavelet coefficients. Our goal is to predict this signal from noisy data. We also develop a new estimator for the noise variance based on a geostatistical method that considers the behavior of the variogram near the origin. Simulation studies show that our method (DecompShrink) outperforms the well-known VisuShrink and SureShrink methods for recovering a wide variety of signals. Moreover. it is insensitive to the choice of the lowest-scale cut-off parameter, which is typically not the case for other wavelet-shrinkage methods.