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In this section, we consider only real-valued wavelet functions that form an orthogonal basis, hence ϕ ϕ ˜ and ψ ψ ˜ . We saw in Orthogonal Bases from Multiresolution analysis and wavelets how a given function belonging to L 2 ( R ) 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 ( x i ) + ϵ i , i = 1 , ... , n , n = 2 J , J N ,

where x i = i n are equispaced design points and the errors are i.i.d. Gaussian, ϵ i N ( 0 , σ ϵ 2 ) .

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 ( x i , Y i ) i = 1 n coming from the model [link] and an orthogonal wavelet basis generated by { ϕ , ψ } . 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 :

m ^ ( x ) = k = 0 2 j 0 - 1 s ^ j 0 , k ϕ j 0 , k ( x ) + j = j 0 j 1 - 1 k = 0 2 j - 1 d ^ j , k ψ j , k ( x ) = k s ^ j 1 , k ϕ j 1 , k ( x ) ,

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

s ^ j , k = 1 n i = 1 n Y i ϕ j k ( x i ) and d ^ j , k = 1 n i = 1 n Y i ψ j k ( x i ) .

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 ( x ) .

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

P V j 1 m ( x ) = k m ( y ) ϕ j 1 , k ( y ) d y ϕ j 1 , k ( x ) = K j 1 ( x , y ) m ( y ) d y ,

where the (convolution) kernel K j 1 ( x , y ) is given by

K j 1 ( x , y ) = k ϕ j 1 , k ( y ) ϕ j 1 , k ( x ) .

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

P V j 1 ^ m ( x ) = i = 1 n Y i ( i - 1 ) / n i / n K j 1 ( x , y ) d y = k i = 1 n Y i ( i - 1 ) / n i / n ϕ j 1 , k ( y ) d y ϕ j 1 , k ( x ) .

Approximating the last integral by 1 n ϕ j 1 , k ( x i ) , we find back the estimator m ^ ( x ) 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:

IMSE = E m ^ - m L 2 2 = k E [ s ^ j 0 , k - s j 0 , k ] 2 + j = j 0 j 1 - 1 k E [ d ^ j k - d j k ] 2 + j = j 1 k d j k 2 = S 1 + S 2 + S 3 ,

where

s j k : = m , ϕ j k and d j k = m , ψ j k

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 β - Hölder continuous, it is easy to see that the optimal cutting level is j 1 ( n ) = O ( n 1 / ( 1 + 2 β ) ) . The resulting optimal IMSE is of order n - 2 β 2 β + 1 . In practice, cross-validation methods are often used to determine the optimal level j 1 [link] , [link] .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
da
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Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, An introduction to wavelet analysis. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10566/1.3
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