Abstract
Wavelet analysis has been found to be a powerful tool for the nonparametric estimation of spatially-variable objects. We discuss in detail wavelet methods in nonparametric regression, where the data are modelled as observations of a signal contaminated with additive Gaussian noise, and provide an extensive review of the vast literature of wavelet shrinkage and wavelet thresholding estimators developed to denoise such data. These estimators arise from a wide range of classical and empirical Bayes methods treating either individual or blocks of wavelet coefficients. We compare various estimators in an extensive simulation study on a variety of sample sizes, test functions, signal-to-noise ratios and wavelet filters. Because there is no single criterion that can adequately summarise the behaviour of an estimator, we use various criteria to measure performance in finite sample situations. Insight into the performance of these estimators is obtained from graphical outputs and numerical tables. In order to provide some hints of how these estimators should be used to analyse real data sets, a detailed practical step-by-step illustration of a wavelet denoising analysis on electrical consumption is provided. Matlab codes are provided so that all figures and tables in this paper can be reproduced.
Highlights
Nonparametric regression has been a fundamental tool in data analysis over the past two decades and is still an expanding area of ongoing research
Breiman & Peters (1992) designed a useful simulation study comparing some popular linear smoothing methods found in the literature on a variety of sample sizes, regression functions and signal-to-noise ratios using a number of different criterion to measure performance
Let R be the number of empirical wavelet coefficients that are not dropped by the thresholding procedure for a given sample
Summary
Nonparametric regression has been a fundamental tool in data analysis over the past two decades and is still an expanding area of ongoing research. Breiman & Peters (1992) designed a useful simulation study comparing some popular linear smoothing methods found in the literature on a variety of sample sizes, regression functions and signal-to-noise ratios using a number of different criterion to measure performance. If an estimate of a coefficient is sufficiently large in absolute value – that is, if it exceeds a predetermined threshold – the corresponding term in the empirical wavelet expansion is retained (or shrunk toward to zero by an amount equal to the threshold); otherwise it is omitted These papers study nonparametric regression from a minimax viewpoint, using some important function classes not previously considered in statistics when linear smoothers were studied (and have provided new viewpoints for understanding other nonparametric smoothers as well). We give a brief overview of some relevant material on the wavelet series expansion and a fast wavelet transform that we need later
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