# 0.6 Nonparametric regression with wavelets  (Page 3/5)

 Page 3 / 5

The choice of the primary resolution level in nonlinear wavelet estimation has the same importance as the choice of a particular kernel in local polynomial estimation, i.e., it is not the most important factor. It is common practice to take ${j}_{0}=2$ or ${j}_{0}=3$ , although a cross-validation determination is of course possible [link] .

The selection of a threshold value is much more crucial. If it is chosen too large, the thresholding operation will kill too many coefficients. Too few coefficients will then be included in the reconstruction, resulting in an oversmoothed estimator. Conversely, a small threshold value will allow many coefficients to be included in the reconstruction, giving a rough, or undersmoothed estimator. A proper choice of the threshold involves thus a careful balance between smoothness and closeness of fit.

In case of an orthogonal transform and i.i.d. white noise, the same threshold can be applied to all detail coefficients, since the errors in the wavelet domain are still i.i.d. white noise. However, if the errors are stationary but correlated, or if the transform is biorthogonal, a level-dependent threshold is necessary to obtain optimal results [link] , [link] . Finally, in the irregular setting, a level and location dependent threshold must be utilized.

Many efforts have been devoted to propose methods for selecting the threshold. We now review some of the procedures encountered in the literature.

## Universal threshold

The most simple method to find a threshold whose value is supported by some statistical arguments, is probably to use the so-called universal threshold' [link] , [link]

${t}_{\text{univ}}={\sigma }_{d}\sqrt{2logn}\phantom{\rule{3.33333pt}{0ex}},$

where the only quantity to be estimated is ${\sigma }_{d}^{2}$ , which constitutes the variance of the empirical wavelet coefficients. In case of an orthogonal transform, ${\sigma }_{d}={\sigma }_{ϵ}/\sqrt{n}$ .

In a wavelet transform, the detail coefficients at fine scales are, with a small fraction of exception, essentially pure noise. This is the reason why Donoho and Johnstone proposed in [link] to estimate ${\sigma }_{d}$ in a robust way using the median absolute deviation from the median (MAD) of ${\stackrel{^}{d}}_{J-1}$ :

${\stackrel{^}{\sigma }}_{d}=\frac{\text{median}\left(\left|{\stackrel{^}{d}}_{J-1},-,\text{median},\left({\stackrel{^}{d}}_{J-1}\right)\right|\right)}{0.6745}\phantom{\rule{3.33333pt}{0ex}}.$

If the universal threshold is used in conjunction with soft thresholding, the resulting estimator possesses a noise-free property: with a high probability,an appropriate interpolation of $\left\{\stackrel{^}{m}\left({x}_{i}\right)\right\}$ produces an estimator which is at least as smooth as the function $m$ , see Theorem 1.1 in [link] . Hence the reconstruction is of good visual quality, so that Donoho andJohnstone called the procedure VisuShrink' [link] . Although simple, this estimator enjoys a near-minimax adaptivity property, see "Adaptivity of wavelet estimator" . However, this near-optimality is an asymptotic one: for small sample size ${t}_{\text{univ}}$ may be too large, leading to a poor mean square error.

## Oracle inequality

Consider the soft-thresholded detail coefficients ${\stackrel{^}{d}}^{t}$ . Another approach to find an optimal threshold is to look at the ${l}_{2}-$ risk

$\mathcal{R}\left({\stackrel{^}{d}}^{t},d\right)=E\sum _{\left(j,k\right)}{\left({\stackrel{^}{d}}_{jk}^{t}-{d}_{jk}\right)}^{2}=E{∥{\stackrel{^}{d}}^{t},-,d∥}_{{l}_{2}}^{2}\phantom{\rule{3.33333pt}{0ex}},$

and to relate this risk with the one of an ideal risk ${\mathcal{R}}_{\text{ideal}}$ . The ideal risk is the risk obtained if an oracle tells us exactly which coefficients to keep or to kill.

what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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