0.6 Nonparametric regression with wavelets

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In this section, we consider only real-valued wavelet functions that form an orthogonal basis, hence $\varphi \equiv \stackrel{˜}{\varphi }$ and $\psi \equiv \stackrel{˜}{\psi }$ . We saw in Orthogonal Bases from Multiresolution analysis and wavelets how a given function belonging to ${L}_{2}\left(\mathbb{R}\right)$ could be represented as a wavelet series. Here, we explain how to use a wavelet basis to construct a nonparametric estimator for the regression function $m$ in the model

${Y}_{i}=m\left({x}_{i}\right)+{ϵ}_{i},\phantom{\rule{0.277778em}{0ex}}i=1,...,n,\phantom{\rule{0.277778em}{0ex}}n={2}^{J},\phantom{\rule{0.277778em}{0ex}}J\in \mathbb{N}\phantom{\rule{3.33333pt}{0ex}},$

where ${x}_{i}=\frac{i}{n}$ are equispaced design points and the errors are i.i.d. Gaussian, ${ϵ}_{i}\phantom{\rule{3.33333pt}{0ex}}\sim \phantom{\rule{3.33333pt}{0ex}}N\left(0,{\sigma }_{ϵ}^{2}\right)$ .

A wavelet estimator can be linear or nonlinear . The linear wavelet estimator proceeds by projecting the data onto a coarse level space. This estimator is of a kernel-type, see "Linear smoothing with wavelets" . Another possibility for estimating $m$ is to detect which detail coefficients convey the important information about the function $m$ and to put equal to zero all the other coefficients. This yields a nonlinear wavelet estimator as described in "Nonlinear smoothing with wavelets" .

Linear smoothing with wavelets

Suppose we are given data ${\left({x}_{i},{Y}_{i}\right)}_{i=1}^{n}$ coming from the model [link] and an orthogonal wavelet basis generated by $\left\{\varphi ,\psi \right\}$ . The linear wavelet estimator proceeds by choosing a cutting level ${j}_{1}$ and represents an estimation of the projection of $m$ onto the space ${V}_{{j}_{1}}$ :

$\stackrel{^}{m}\left(x\right)=\sum _{k=0}^{{2}^{{j}_{0}}-1}{\stackrel{^}{s}}_{{j}_{0},k}{\varphi }_{{j}_{0},k}\left(x\right)+\sum _{j={j}_{0}}^{{j}_{1}-1}\sum _{k=0}^{{2}^{j}-1}{\stackrel{^}{d}}_{j,k}{\psi }_{j,k}\left(x\right)=\sum _{k}{\stackrel{^}{s}}_{{j}_{1},k}{\varphi }_{{j}_{1},k}\left(x\right),$

with ${j}_{0}$ the coarsest level in the decomposition, and where the so-called empirical coefficients are computed as

${\stackrel{^}{s}}_{j,k}=\frac{1}{n}\sum _{i=1}^{n}{Y}_{i}\phantom{\rule{4pt}{0ex}}{\varphi }_{jk}\left({x}_{i}\right)\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}{\stackrel{^}{d}}_{j,k}=\frac{1}{n}\sum _{i=1}^{n}{Y}_{i}\phantom{\rule{4pt}{0ex}}{\psi }_{jk}\left({x}_{i}\right)\phantom{\rule{3.33333pt}{0ex}}.$

The cutting level ${j}_{1}$ plays the role of a smoothing parameter: a small value of ${j}_{1}$ means that many detail coefficients are left out, and this may lead to oversmoothing. On the other hand, if ${j}_{1}$ is too large, too many coefficients will be kept, and some artificial bumps will probably remain in the estimation of $m\left(x\right)$ .

To see that the estimator [link] is of a kernel-type, consider first the projection of $m$ onto ${V}_{{j}_{1}}$ :

$\begin{array}{ccc}\hfill {\mathcal{P}}_{{V}_{{j}_{1}}}m\left(x\right)& =& \sum _{k}\left(\int \phantom{\rule{-0.166667em}{0ex}}m\left(y\right){\varphi }_{{j}_{1},k}\left(y\right)dy\right){\varphi }_{{j}_{1},k}\left(x\right)\hfill \\ & =& \int \phantom{\rule{-0.166667em}{0ex}}{K}_{{j}_{1}}\left(x,y\right)m\left(y\right)dy\phantom{\rule{3.33333pt}{0ex}},\hfill \end{array}$

where the (convolution) kernel ${K}_{{j}_{1}}\left(x,y\right)$ is given by

${K}_{{j}_{1}}\left(x,y\right)=\sum _{k}{\varphi }_{{j}_{1},k}\left(y\right){\varphi }_{{j}_{1},k}\left(x\right)\phantom{\rule{3.33333pt}{0ex}}.$

Härdle et al. [link] studied the approximation properties of this projection operator. In order to estimate [link] , Antoniadis et al. [link] proposed to take:

$\begin{array}{ccc}\hfill \stackrel{^}{{\mathcal{P}}_{{V}_{{j}_{1}}}}m\left(x\right)& =& \sum _{i=1}^{n}{Y}_{i}{\int }_{\left(i-1\right)/n}^{i/n}{K}_{{j}_{1}}\left(x,y\right)dy\hfill \\ & =& \sum _{k}\sum _{i=1}^{n}{Y}_{i}\left({\int }_{\left(i-1\right)/n}^{i/n}{\varphi }_{{j}_{1},k}\left(y\right)dy\right){\varphi }_{{j}_{1},k}\left(x\right)\phantom{\rule{3.33333pt}{0ex}}.\hfill \end{array}$

Approximating the last integral by $\frac{1}{n}{\varphi }_{{j}_{1},k}\left({x}_{i}\right)$ , we find back the estimator $\stackrel{^}{m}\left(x\right)$ in [link] .

By orthogonality of the wavelet transform and Parseval's equality, the ${L}_{2}-$ risk (or integrated mean square error IMSE) of a linear wavelet estimator is equal to the ${l}_{2}-$ risk of its wavelet coefficients:

$\begin{array}{ccc}\hfill \text{IMSE}=E{∥\stackrel{^}{m},-,m∥}_{{L}_{2}}^{2}& =& \sum _{k}E{\left[{\stackrel{^}{s}}_{{j}_{0},k}-{s}_{{j}_{0},k}^{\circ }\right]}^{2}+\sum _{j={j}_{0}}^{{j}_{1}-1}\sum _{k}E{\left[{\stackrel{^}{d}}_{jk}-{d}_{jk}^{\circ }\right]}^{2}\hfill \\ & +& \sum _{j={j}_{1}}^{\infty }\sum _{k}{d}_{jk}^{\circ \phantom{\rule{4pt}{0ex}}2}={S}_{1}+{S}_{2}+{S}_{3}\phantom{\rule{3.33333pt}{0ex}},\hfill \end{array}$

where

${s}_{jk}^{\circ }:=〈m,\phantom{\rule{0.166667em}{0ex}},,,{\varphi }_{jk}〉\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}{d}_{jk}^{\circ }=〈m,\phantom{\rule{0.166667em}{0ex}},,,{\psi }_{jk}〉$

are called `theoretical' coefficients in the regression context. The term ${S}_{1}+{S}_{2}$ in [link] constitutes the stochastic bias whereas ${S}_{3}$ is the deterministic bias. The optimal cutting level is such that these two bias are of the same order. If $m$ is $\beta -$ Hölder continuous, it is easy to see that the optimal cutting level is ${j}_{1}\left(n\right)=O\left({n}^{1/\left(1+2\beta \right)}\right)$ . The resulting optimal IMSE is of order ${n}^{-\frac{2\beta }{2\beta +1}}$ . In practice, cross-validation methods are often used to determine the optimal level ${j}_{1}$ [link] , [link] .

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