Wavelet shrinkage (WaveShrink) is a relatively new technique for nonparametric function estimation that has been shown to have asymptotic near-optimality properties over a wide class of functions. As originally formulated by Donoho and Johnstone, WaveShrink assumes equally spaced data. Because so many statistical applications (e.g., scatterplot smoothing) naturally involve unequally spaced data, we investigate in this paper how WaveShrink can be adapted to handle such data. Focusing on the Haar wavelet, we propose four approaches that extend the Haar wavelet transform to the unequally spaced case. Each approach is formulated in terms of continuous wavelet basis functions applied to a piecewise constant interpolation of the observed data, and each approach leads to wavelet coefficients that can be computed via a matrix transform of the original data. For each approach, we propose a practical way of adapting WaveShrink. We compare the four approaches in a Monte Carlo study and find them to be quite comparable in performance. The computationally simplest approach (isometric wavelets) has an appealing justification in terms of a weighted mean square error criterion and readily generalizes to wavelets of higher order than the Haar.

Many nonparametric regression procedures are based on "subset selection": they choose a subset of carriers from a large or even infinite set, and then determine the coefficients of the chosen carriers by least squares. Procedures which can be cast in this framework include Projection Pursuit, Turbo, Mars, and Matching Pursuit. Recently, considerable attention has been given to "ensemble estimators" which combine least squares estimates obtained from multiple subsets of carriers. In the parametric regression setting, such ensemble estimators have been shown to improve on the accuracy of subset selection procedures in some situations. In this paper we compare subset selection estimators and ensemble estimators in the context of wavelet de-noising. We present simulation results demonstrating that a certain class of ensemble wavelet estimators, based on the concept of "cycle spinning", are significantly more accurate than subset selection methods. These advantages hold even when the subset selection procedures can rely on an oracle to select the optimal number of carriers. We compute ideal thresholds for translation invariant wavelet shrinkage and investigate other approaches to ensemble wavelet estimation.

In this article, we combine Donoho and Johnstone's wavelet shrinkage denoising technique (known as WaveShrink) with Breiman's non-negative garrote. We show that the non-negative garrote shrinkage estimate enjoys the same asymptotic convergence rate as the hard and the soft shrinkage estimates. Simulations are used to demonstrate that garrote shrinkage offers advantages over both hard shrinkage (generally smaller mean-square-error and less sensitivity to small perturbations in the data) and soft shrinkage (generally smaller bias and overall mean-square-error). The minimax thresholds for the non-negative garrote are derived and the threshold selection procedure based on Stein's unbiased risk estimate (SURE) is studied. We also propose a threshold selection procedure based on combining Coifman and Donoho's cycle-spinning and SURE. The procedure is called SPINSURE. We use examples to show that SPINSURE is more stable than SURE: smaller standard deviation and smaller range.

We study the problem of estimating the log‐spectrum of a stationary Gaussian time series by thresholding the empirical wavelet coefficients. We propose the use of thresholds tj,n depending on sample size n, wavelet basis ψ and resolution level j. At fine resolution levels (j = 1, 2, ...) we propose t j,n = αj log nwhere {αj} are level‐dependent constants and at coarse levels (j≫ 1) t j,n = (π/√3)(log n)1/2.The purpose of this thresholding level is to make the reconstructed log‐spectrum as nearly noise‐free as possible. In addition to being pleasant from a visual point of view, the noise‐free character leads to attractive theoretical properties over a wide range of smoothness assumptions. Previous proposals set much smaller thresholds and did not enjoy these properties.

We study the following heteroscedastic nonparametric regression model:y i = f(t i ) + oe(t i )z iwhere fz i g is independent identically distributed random noise with z i ? N(0; 1)and oe2(t) is the variance function. We want to estimate f . We extend Donoho andJohnstone's wavelet shrinkage technique (known as WaveShrink) to this model. We addressthe issue of variance estimation and propose a procedure based on non-decimatedwavelet transform and running MAD (Median Absolute Deviation ...

Donoho and Johnstone's (1994) WaveShrink procedure has proven valu- able for signal de-noising and non-parametric regression. WaveShrink has very broad asymptotic near-optimality properties. In this paper, we introduce a new shrinkage scheme, firm, which generalizes the hard and soft shrinkage proposed by Donoho and Johnstone (1994). We derive minimax thresholds and provide for- mulas for computing the pointwise variance, bias, and risk for WaveShrink with firm shrinkage. We study the properties of the shrinkage functions, and demon- strate that firm shrinkage offers advantages over both hard shrinkage (uniformly smaller risk and less sensitivity to small perturbations in the data) and soft shrink- age (smaller bias and overall L2 risk). Software is provided to reproduce all results in this paper.

Donoho & Johnstone's WaveShrink procedure has proved valuable for function estimation and nonparametric regression. WaveShrink is based on the principle of shrinking wavelet coefficients towards zero to remove noise. WaveShrink has very broad asymptotic near-optimality properties and achieves the optimal risk to within a factor of log n. In this paper, we derive computationally efficient formulae for computing the exact bias, variance and L(2) risk of WaveShrink estimates in finite sample situations. We use these formulae to understand the behaviour of WaveShrink estimators; construct approximate confidence intervals and bias estimates for WaveShrink; and compute ideal thresholds for a given function. We show that hard shrinkage has smaller bias but larger variance than soft shrinkage, and that significantly smaller thresholds should be used for soft shrinkage. We also compute minimax thresholds for WaveShrink estimators and demonstrate that the minimax thresholds can nearly achieve the ideal rank for a range of functions.

We present the application of wavelet-based de-noising techniques to the demodulation of digital communication signals. These techniques are related to those originally devised by Donoho and Johnstone (see Biometrika, v.81, p.455, 1994, and IEEE Trans. Info. Theory, vol.41, p.613, 1995) for minimum L/sub 2/-risk signal reconstruction. The important minimax property of wavelet de-noising filters yields significant noise reduction while retaining the essential signal features. However, the two problems are quite different, i.e., the objective is not signal reconstruction but rather minimizing bit error rate. This generally results in different design rules for optimizing performance. We show that wavelet de-noising can be effectively used for data demodulation and we propose simple design rules for its implementation.

In a series of papers, Donoho and Johnstone develop a powerful theory based on wavelets for extracting non-smooth signals from noisy data. Several nonlinear smoothing algorithms are presented which provide high performance for removing Gaussian noise from a wide range of spatially inhomogeneous signals. However, like other methods based on the linear wavelet transform, these algorithms are very sensitive to certain types of non-Gaussian noise, such as outliers. In this paper, we develop outlier resistant wavelet transforms. In these transforms, outliers and outlier patches are localized to just a few scales. By using the outlier resistant wavelet transform, we improve upon the Donoho and Johnstone nonlinear signal extraction methods. The outlier resistant wavelet algorithms are included with the 'S+WAVELETS' object-oriented toolkit for wavelet analysis.

We study the problem of estimating the spectral density of a stationary Gaussian timeseries. We use an orthogonal wavelet system whose members are periodic functions andhave a finite number of non-zero Fourier coefficients -- periodized Meyer wavelets. Weapply shrinkage rules to the empirical wavelet coefficients. We show that estimates basedon thresholds t j;n = j log n for certain j , with n the sample size, have near-optimal L 2convergence rates, over any Besov class in a wide range. ...

Donoho and Johnstone's wavelet shrinkage denoising technique (known as WaveShrink)consists three steps: (1) transform data into wavelet domain; (2) shrink the wavelet coefficients;and (3) transform the shrunk coefficients back. The choice of shrinkage functionand thresholds in step (2) plays an important role for WaveShrink both theoreticallyand in practice. In this paper, we discuss the issue of threshold selection in WaveShrink.We first review the threshold selection procedure based...

Donoho and Johnstone's WaveShrink procedure has proven valuable for signal de-noisingand non-parametric regression. WaveShrink has very broad asymptotic near-optimalityproperties. In this paper, we introduce a new shrinkage scheme, semisoft, which generalizesthe hard and soft shrinkage proposed by Donoho and Johnstone. We derive minimaxthresholds and provide formulas for computing the pointwise variance, bias, and risk forWaveShrink with semisoft shrinkage. We study the properties of the...