WaveD

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Description

Performs statistical wavelet deconvolution using Meyer wavelet.

Usage

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WaveD(yobs, g = c(1, rep(0, (length(yobs) - 1))), MC = FALSE, SOFT = FALSE, F = find.j1(g, scale(yobs))[2], L = 3, deg = 3, eta = sqrt(6), thr = maxithresh(yobs, g, eta = eta), label = "WaveD")

Arguments

yobs

Sample of f*g + (Gaussian noise), a vector of dyadic length (i.e. 2^(J-1) where J is the largest resolution level). Here f is the target function, g is the convolution kernel.

g

Sample of g or g + (Gaussian noise), same length as yobs. The default is the Dirac mass at 0.

MC

Option to only return the (fast) translation-invariant WaveD estimate (MC=TRUE) as opposed to the full WaveD output (MC=FALSE, the default), as described below. MC=TRUE recommended for Monte Carlo simulation.

SOFT

if SOFT=TRUE, uses the soft thresholding policy as opposed to the hard (SOFT=FALSE, the default).

F

Finest resolution level; the default is the data-driven choice j1 (see Value below).

L

Lowest resolution level; the default is 3.

deg

The degree of the Meyer wavelet, either 1, 2, or 3 (the default).

eta

Tuning parameter of the maxiset threshold; default is √(6).

thr

A vector of length F-L+1, giving thresholds at each resolution levels L,L+1,...,F; default is maxiset threshold.

label

Auxiliary plotting parameter; do not change this.

Value

In the case that MC=TRUE, WaveD returns a vector consisting of the translation-invariant WaveD estimate. In the case that MC=FALSE (the default), WaveD returns a list with components

waved

translation invariant WaveD transform; in the case MC=TRUE this is all that is returned.

ordinary

ordinary WaveD transform

FWaveD

Forward WaveD Transform; see FWaveD.

w

alternate name for FWaveD

w.thr

thresholded version of w

IWaveD

Inverse WaveD Transform

iw

alternate name for IWaveD

s

estimate of the noise standard deviation

j1

estimate of optimal resolution level (for maxiset threshold).

F

Fine resolution level used (may be different to j1).

M

estimate of optimal Fourier frequency (for maxiset threshold).

thr

vector of thresholds used (default is maxiset threshold).

percent

percentage of thresholding per resolution level

noise

noise proxy, wavelet coefficients of the raw data at the largest resolution level, used for estimating noise features.

ps

P-value of the Shapiro-Wilk test for normality applied to the noise proxy.

residuals

wavelet coefficients that have been removed before fine level F.

Author(s)

Marc Raimondo and Michael Stewart

References

Cavalier, L. and Raimondo, M. (2007), ‘Wavelet deconvolution with noisy eigen-values’, IEEE Trans. Signal Process, Vol. 55(6), In the press.

Donoho, D. and Raimondo, M. (2004), ‘Translation invariant deconvolution in a periodic setting’, The International Journal of Wavelets, Multiresolution and Information Processing 14(1),415–423.

Johnstone, I., Kerkyacharian, G., Picard, D. and Raimondo, M. (2004), 'Wavelet deconvolution in a periodic setting', Journal of the Royal Statistical Society, Series B 66(3),547–573. with discussion pp.627–652.

Raimondo, M. and Stewart, M. (2007), ‘The WaveD Transform in R’, Journal of Statistical Software.

See Also

FWaveD

Examples

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library(waved)
data=waved.example(TRUE,FALSE)
doppler.wvd=WaveD(data$doppler.noisy,data$g)
summary(doppler.wvd)

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