-neighborhood wavelet shrinkage
Introduction
In the present paper, we consider a new Bayesian model as a solution to the classical nonparametric regression problem where , are equispaced sampling points, and the random errors are i.i.d. normal, with zero mean and variance . The interest is to estimate the function using the observations . After applying a linear and orthogonal wavelet transformation, the equation in (1) becomes where and are the wavelet coefficients (at resolution and position ) corresponding to , and , respectively. Due to the whitening property of wavelet transforms (Flandrin, 1992) many existing methods assume independence of the coefficients, and omit the double indices to work with a generic model To estimate in model (2) Bayesian shrinkage rules have been proposed in the literature by many authors. The observed wavelet coefficients are replaced with their shrunk version representing a Bayes estimator of . Most of the signals encountered in practical applications have (at each resolution level) empirical distributions of detail wavelet coefficients centered around and peaked at zero (Mallat, 1989). A range of models complying with Mallat’s observation have been considered in the literature. The traditional Bayesian models consider the prior distribution on the wavelet coefficient as where is a point mass at zero, is a unimodal distribution symmetric about 0, and is a fixed parameter in [0,1], usually level dependent, that controls the amount of shrinkage for values of close to 0. This type of model was considered by Abramovich et al. (1998), Vidakovic (1998), Vidakovic and Ruggeri (2001), and Johnstone and Silverman (2005), among others.
On the other hand, many authors argued that shrinkage performance can be improved by considering the neighborhoods of wavelet coefficients (blocks, parent–child relations, cones of influence, etc.). These authors report improvement over the coefficient-by-coefficient or diagonal methods. Examples include block thresholding methods by Hall et al., 1997, Hall et al., 1998, Hall et al., 1999, Cai, 1999, Cai, 2002, and Cai and Silverman (2001) where wavelet coefficients are thresholded based on block sums of squares. Abramovich et al. (2002) and De Canditiis and Vidakovic (2004) consider Bayesian block shrinkage methods allowing for dependence between the wavelet coefficients. Wang and Wood (2006) considered a Bayesian block shrinkage approach based directly on the block sum of squares. Sendur and Selesnick (2002) and Fryzlewicz (2007), among others, use parent–child neighboring relation to improve on shrinkage performance. In Fryzlewicz (2007), the coupling of wavelet coefficients from different levels leads to a bivariate model in which the energy, under appropriate assumptions, is -distributed.
In this paper the neighboring structure is enhanced by looking simultaneously at two neighboring coefficients at the same level and their common parental coefficient from the level in the wavelet ordering given by the parametrization , . This leads to a joint “energy” distributed as noncentral , in which the Bayesian model accounts for the noncentrality parameter and leads to simple and fast shrinkage rules. The idea of considering neighboring and parental coefficients in the denoising procedure has been used in signal and particularly in image denoising algorithms to improve the performance and visual appearance of the procedures. One example is the tree-based wavelet thresholding estimator; see for example Autin (2008) and Autin et al. (2011). Another popular method that incorporates neighboring structure is the Hidden Markov Tree (HMT) model which is explored for example by Crouse et al. (1998) and Romberg et al. (2001).
The paper is organized as follows. Section 2 introduces the model, then derives and discusses the properties of the shrinkage rule. Section 3 explains the elicitation of hyperparameters via the empirical Bayes method. Section 4 contains simulations and comparisons to existing methods in terms of the average mean squared error and Section 5 contains an application of the method to real-world data. Conclusions and discussion are provided in Section 6.
Section snippets
Model and estimation
The idea of our method is to estimate wavelet coefficients by forming a variable composed of two neighboring and their parental wavelet coefficient. Our model is based on the sum of the energy of this “family” or “clique”. We will call it -neighborhood motivated by the geometric shape of the neighborhood. This also motivates the name of the induced shrinkage methodology: -neighborhood wavelet shrinkage (LNWS). The idea of using a variable as a “thresholding statistics” was
Eliciting the hyperparameters
The described model and hence the Bayes estimators depend on hyperparameters that have to be specified. Purely subjective elicitation is only possible when considerable knowledge about the underlying signal is available. We followed the empirical Bayes paradigm in this paper; Johnstone and Silverman (2005), Clyde and George, 1999, Clyde and George, 2000, and Abramovich et al. (2002) estimate the hyperparameters by the marginal maximum likelihood (MLII) method in the wavelet denoising context.
Simulations and comparisons
In this section we discuss the performance of the proposed estimator (12) and compare it to established methods from the literature considering block-type wavelet denoising. In our simulations four standard test functions (Blocks, Bumps, Doppler, Heavisine) were considered (Donoho and Johnstone, 1994). The functions were rescaled such that the added noise produced preassigned signal-to-noise ratio (SNR), as standardly done. The test functions were simulated at , and 1024 equally spaced
Application to inductance plethysmography data
In this section we apply the proposed LNWS method to a real-world data set from anesthesiology generated by inductance plethysmography. The recordings were made by the Department of Anesthesia at the Bristol Royal Infirmary and represent measure of flow of air during breathing. The measurements are the output voltage of the inductance plethysmograph over time. The data set is popular in the wavelet denoising literature and was used as an example by several authors; for example Nason (1996),
Conclusions
In this article we proposed a wavelet shrinkage method based on a neighborhood of wavelet coefficients, which includes two neighboring and a parental coefficient. We called the methodology -neighborhood wavelet shrinkage, motivated by the shape of the considered neighborhood. A Bayesian model was formulated on the total energy of the coefficients in the neighborhood, and different Bayes estimators of the mean energy were derived and explored. Shrinkage of the neighboring wavelet coefficients
Acknowledgments
We are grateful to the editor and three anonymous referees for the comments that improved the manuscript.
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