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

 Page 5 / 5
${R}_{n}\left(V,p\right)=\underset{{T}_{n}}{inf}\underset{m\in V}{sup}E{∥{T}_{n},-,m∥}_{p}^{p},$

where the infimum is taken over all measurable estimators ${T}_{n}$ of $m.$ Similarly, we define the linear ${L}_{p}-$ minimax risk as

${R}_{n}^{\text{lin}}\left(V,p\right)=\underset{{T}_{n}^{\text{lin}}}{inf}\underset{m\in V}{sup}E{∥{T}_{n}^{\text{lin}},-,m∥}_{p}^{p},$

where the infimum is now taken over all linear estimators ${T}_{n}^{\text{lin}}.$ Obviously, ${R}_{n}^{\text{lin}}\left(V,p\right)\ge {R}_{n}\left(V,p\right).$ We first state some definitions.

The sequences $\left\{{a}_{n}\right\}$ and $\left\{{b}_{n}\right\}$ are said to be asymptotically equivalent and are noted ${a}_{n}\sim {b}_{n}$ if the ratio ${a}_{n}/{b}_{n}$ is bounded away from zero and $\infty$ as $n\to \infty .$
The sequence ${a}_{n}$ is called optimal rate of convergence , (or minimax rate of convergence ) on the class $V$ for the ${L}_{p}-$ risk if ${a}_{n}\sim {R}_{n}{\left(V,p\right)}^{1/p}$ . We say that an estimator ${m}_{n}$ of $m$ attains the optimal rate of convergence if $\underset{m\in V}{sup}E{∥{m}_{n},-,m∥}_{p}^{p}\sim {R}_{n}\left(V,p\right).$

In order to fix the idea, we consider only the ${L}_{2}-$ risk in the remaining part of this section, thus $p:=2$ .

In [link] , [link] , the authors found that the optimal rate of convergence attainable by an estimator when the underlying function belongs to the Sobolev class ${W}_{q}^{s}$ is ${a}_{n}={n}^{\frac{-s}{2s+1}}$ , hence ${R}_{n}\left(V,2\right)={n}^{\frac{-2s}{2s+1}}$ . We saw in "Linear smoothing with wavelets" that linear wavelet estimators attain the optimal rate for $s-$ Hölder function in case of the ${L}_{2}-$ risk (also called `IMSE'). For a Sobolev class ${W}_{q}^{s}$ , the same result holds provided that $q\ge 2$ . More precisely, we have the two following situations.

1. If $q\ge 2,$ we are in the so-called homogeneous zone. In this zone of spatial homogeneity, linear estimators can attain the optimal rate of convergence ${n}^{-s/\left(2s+1\right)}.$
2. If $q<2,$ we are in the non-homogeneous zone, where linear estimators do not attain the optimal rate of convergence. Instead, we have:
${R}_{n}^{\text{lin}}\left(V,2\right)/{R}_{n}\left(V,2\right)\to \infty ,\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}n\to \infty .$

The second result is due to the spatial variability of functions in Sobolev spaces with small index $q$ . Linear estimators are based on the idea of spatial homogeneity of the function and hence do perform poorly in the presence of non-homogeneous functions. In contrast, even if $q<2$ , the SureShrink estimator attains the minimax rate [link] . The same type of results holds for more general Besov spaces, see for example [link] , Chapter 10.

We just saw that a nonlinear wavelet estimator is able to estimate in an optimal way functions ofinhomogeneous regularity. However, it may not be sufficient to know that for $m$ belonging to a given space, the estimator performs well. Indeed, in general we do not know which space the function belongs to. Hence it is ofgreat interest to consider a scale of function classes and to look for an estimator that attains simultaneously the best rates of convergence across the whole scale. For example, the ${L}_{q}-$ Sobolev scale is a set of Sobolev function classes ${W}_{q}^{s}\left(C\right)$ indexed by the parameters $s$ and $C$ , see [link] for the definition of a Sobolev class. We now formalize the notion of an adaptive estimator.

Let $A$ be a given set and let $\left\{{\mathcal{F}}_{\alpha },\alpha \in A\right\}$ be the scale of functional classes ${\mathcal{F}}_{\alpha }$ indexed by $\alpha \in A.$ Denote ${R}_{n}\left(\alpha ,p\right)$ the minimax risk over ${\mathcal{F}}_{\alpha }$ for the ${L}_{p}-$ loss:

${R}_{n}\left(\alpha ,p\right)=\underset{{\stackrel{^}{m}}_{n}}{inf}\underset{m\in {\mathcal{F}}_{\alpha }}{sup}E{∥{\stackrel{^}{m}}_{n},-,m∥}_{p}^{p}.$
The estimator ${m}_{n}^{*}$ is called rate adaptive for the ${L}_{p}-$ loss and the scale of classes ${\mathcal{F}}_{\alpha },\alpha \in A$ if for any $\alpha \in A$ there exists ${c}_{\alpha }>0$ such that
$\underset{m\in {\mathcal{F}}_{\alpha }}{sup}E{∥{m}_{n}^{*},-,m∥}_{p}^{p}\le {c}_{\alpha }{R}_{n}\left(\alpha ,p\right)\phantom{\rule{0.222222em}{0ex}}\forall n\ge 1.$

The estimator ${m}_{n}^{*}$ is called adaptive up to a logarithmic factor for the ${L}_{p}-$ loss and the scale of classes ${\mathcal{F}}_{\alpha },\alpha \in A$ if for any $\alpha \in A$ there exist ${c}_{\alpha }>0$ and $\gamma ={\gamma }_{\alpha }>0$ such that

$\underset{m\in {\mathcal{F}}_{\alpha }}{sup}E{∥{m}_{n}^{*},-,m∥}_{p}^{p}\le {c}_{\alpha }{\left(logn\right)}^{\gamma }{R}_{n}\left(\alpha ,p\right)\phantom{\rule{0.222222em}{0ex}}\forall n\ge 1.$

Thus, adaptive estimators have an optimal rate of convergence and behave as if they know in advance in which class the function to be estimated lies.

The VisuShrink procedure is adaptive up to a logarithmic factor for the ${L}_{2}-$ loss over every Besov, Hölder and Sobolev class that is contained in $C\left[0,1\right]$ , see Theorem 1.2 in [link] . The SureShrink estimator does better: it is adaptive for the ${L}_{2}-$ loss, for a large scale of Besov, Hölder and Sobolev classes, see Theorem 1 in [link] .

## Conclusion

In this chapter, we saw the basic properties of standard wavelet theory and explained how these are related to the construction of wavelet regression estimators.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
Got questions? Join the online conversation and get instant answers!