# 0.6 Nonparametric regression with wavelets

 Page 1 / 5

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] .

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers! By Mariah Hauptman By     By   