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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 ,


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] .

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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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