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Wavelet methods

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This overview article motivates the use of wavelets in statistics and introduces the basic mathematics behind the construction of wavelets. Topics covered include the continuous and discrete wavelet transforms, multiresolution analysis, and the non‐decimated wavelet transform. We describe the basic mechanics of nonparametric function estimation via wavelets, emphasizing the concepts of sparsity and thresholding. A simple proof of the mean‐square consistency of the wavelet estimator is also included. The article ends with two special topics: function estimation with unbalanced Haar wavelets and variance stabilization via the Haar–Fisz transformation. WIREs Comp Stat 2010 2 654–667 DOI: 10.1002/wics.124

Figure 1.

Top‐left: simulated noisy signal. Top‐right: reconstruction via adaptive weights smoothing (black) and true signal (red, also in other plots). Bottom‐left: reconstruction via Taut string methodology. Bottom‐right: reconstruction via unbalanced Haar wavelets.

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Figure 2.

Daubechies' extremal phase wavelets with different numbers n = N − 1 of vanishing moments.

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Figure 3.

Top: Discrete wavelet transform of noisy vector of Figure 1. Bottom: its non‐decimated wavelet transform. Both using ‘DaubExPhase 2’ wavelets.

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Figure 4.

Top: squared daily log‐returns (normalized so that overall variance is 1) of the Dow Jones Industrial Average index on 2048 trading days ending March 10, 2010. Middle plot: logged. Bottom plot: Haar–Fisz'ed.

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Applications of Computational Statistics > Signal and Image Processing and Coding
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Computational Intensive Statistical Methods > Density Estimation and Curve Fitting
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