Package 'waved'

Title: Wavelet Deconvolution
Description: Makes available code necessary to reproduce figures and tables in papers on the WaveD method for wavelet deconvolution of noisy signals as presented in The WaveD Transform in R, Journal of Statistical Software Volume 21, No. 3, 2007.
Authors: Marc Raimondo <[email protected]> and Michael Stewart <[email protected]>
Maintainer: Michael Stewart <[email protected]>
License: GPL
Version: 1.3
Built: 2024-10-12 02:38:16 UTC
Source: https://github.com/cran/waved

Help Index

FWaveD

Description

Computes the Forward WaveD Transform.

Usage

FWaveD(y,g=1,L=3,deg=3,F=(log2(length(y))-1),thr=rep(0,log2(length(y))),SOFT=FALSE)

Arguments

y

Sample of f∗gf*g + (Gaussian noise), a vector of dyadic length (i.e. 2J−12^{J-1} where J is the largest resolution level). Here f is the target function, g is the convolution kernel.
g

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

Lowest resolution level; the default is 3.
deg

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

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

A vector of length F−L+1F-L+1, giving thresholds at each resolution levels L,L+1,…,FL,L+1,\ldots,F; default is maxiset threshold.
SOFT

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

Value

Returns a vector of wavelet coefficients of length n (the same length as y), the last n/2n/2 entries are wavelet coefficients at resolution level J−1J-1, where J=log⁡2(n)J = \log_2(n); the n/4n/4 entries before that are the wavelet coefficients at resolution level J−2J-2, and so on until level L. In addition the 2L2^L entries are scaling coefficients at coarse level C=LC=L.

References

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. (2006), ‘The WaveD Transform in R’, preprint, School and Mathematics and Statistics, University of Sydney.

See Also

WaveD

Examples

library(waved)
data=waved.example(TRUE,FALSE)
lidar.w=FWaveD(data$lidar.blur,data$g)

WaveD

Description

Performs statistical wavelet deconvolution using Meyer wavelet.

Usage

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∗gf*g + (Gaussian noise), a vector of dyadic length (i.e. 2J−12^{J-1} where J is the largest resolution level). Here f is the target function, g is the convolution kernel.
g

Sample of gg or gg + (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\sqrt6.
thr

A vector of length F−L+1F-L+1, giving thresholds at each resolution levels L,L+1,…,FL,L+1,\ldots,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

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

WaveD examples

Description

Generate data sets and figures to illustrate the WaveD function.

Usage

waved.example(pr = TRUE, gr=TRUE)

Arguments

pr

If pr=TRUE (default) uses the same parameters as in the reference paper below. If pr=FALSE user level parameter specifications.

gr

If gr=TRUE (default) text and graphical displays are provided.

Value

lidar.noisy

Noisy blurred LIDAR signal (Gaussian noise)
lidar.noisyT

Noisy blurred LIDAR signal (Student $t_2$ noise)
doppler.noisy

Noisy blurred Doppler signal (Gaussian noise)
doppler.noisyT

Noisy blurred Doppler signal (Student $t_2$ noise)
lidar.blur

Blurred LIDAR signal
doppler.blur

Blurred Doppler signal
t

Rime vector scaled to [0,1]
n

Sample size
g

Convolution kernel
lidar

LIDAR signal
doppler

Doppler signal.
seed

Used in set.seed
sigma

Noise standard deviation.
g.noisy

Convolution kernel plus Gaussian noise.
g.noisyT

Convolution kernel plus Student $t_2$ noise.
dip

Degree of Ill-posedness.
k.scale

Scale of the convolution kernel

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

data=waved.example(TRUE,FALSE)