Version: 1.3
License: GPL-2 | GPL-3 [expanded from: GPL]
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.
Title: Wavelet Deconvolution
Author: Marc Raimondo <marcr@maths.usyd.edu.au> and Michael Stewart <michael.stewart@sydney.edu.au>
Maintainer: Michael Stewart <michael.stewart@sydney.edu.au>
URL: https://www.jstatsoft.org/v21/i02
NeedsCompilation: no
Packaged: 2024-02-14 07:34:41 UTC; michaels
Repository: CRAN
Date/Publication: 2024-02-14 08:01:26 UTC

Blurr Signal

Description

Compute the convolution of $f$ and $g$ in the periodic setting.

Usage

BlurSignal(f, g)

Arguments

f

A sample of $f$.

g

A sampel of $g$

Value

Returns the convolution of $f$ and $g$.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

 
x=1:10
y=1:10 
BlurSignal(x,y)

Forward Wavelet Transform (translation invariant).

Description

Compute the Forward Wavelet Transform of a signal $f$ for the Meyer wavelet (translation invariant).

Usage

FWT_TI(f_fft, psyJ_fft)

Arguments

f_fft

vector of the Fourier coefficient of $f$.

psyJ_fft

vector of the Fourier coefficient of the Meyer wavelet.

Value

vector of wavelet coefficients (non-ordered).

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

 psyJ_fft=wavelet_YM(4,10,3);
 f_fft=fft(sin(2*pi*seq(0,1,le=1024)))
 FWT_TI(f_fft,psyJ_fft)

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*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.

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+1, giving thresholds at each resolution levels L,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/2 entries are wavelet coefficients at resolution level J-1, where J = \log_2(n); the n/4 entries before that are the wavelet coefficients at resolution level J-2, and so on until level L. In addition the 2^L entries are scaling coefficients at coarse level C=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)

Hard Threshold

Description

Apply hard threshold.

Usage

HardThresh(y, t)

Arguments

y

vector

t

threshold

Value

vector $y$ thresholded: entries below $t$ are replaced by zeros.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

HardThresh(1:5,2)

Inverse Forward Wavelet Transform (translation invariant).

Description

Compute the Inverse Forward Wavelet Transform of a signal $f$ for the Meyer wavelet (translation invariant).

Usage

IFWT_TI(f_fft, psyJ_fft, lev, thr, nn, SOFT = FALSE)

Arguments

f_fft

vector of the Fourier coefficient of $f$

psyJ_fft

vector of the Fourier coefficient of the Meyer wavelet.

lev

resolution level

thr

threshold (has lentgh=1)

nn

sample size

SOFT

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

Value

Inverse Forward Wavelet Transform of a signal $f$, after thresholding.

Author(s)

Marc Raimondo

References

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

See Also

WaveD, ~~~

Examples

psyJ_fft=wavelet_YM(4,10,3);
f_fft=fft(sin(2*pi*seq(0,1,le=1024)));
IFWT_TI(f_fft, psyJ_fft, 4, 0, 1024)

Computes the Inverse WaveD transform

Description

Computes the Inverse WaveD transform based on a vector of wavelet coefficients.

Usage

IWaveD(w, C = 3, deg = 3, F = log2(length(w)))

Arguments

w

vector of wavelet coefficents, must be of dyadic length; typically returned by the function FWaveD

C

coarse resolution level

deg

degree of the Meyer wavelet

F

fine resolution level

Value

Returns a vector of the same length as w, giving the inverse wavelet transform.

Author(s)

Marc Raimondo

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)  # lidar.blur is lidar*g 
IWaveD(lidar.w)               # same as lidar

Meyer wavelet window

Description

Auxiliary window function for Meyer wavelets.

Usage

MeyerWindow(xi, deg)

Arguments

xi

Abscissa values for window evaluation

deg

The degree of the Meyer wavelet, either 1, 2, or 3

Value

a sampel vector of the window function for Meyer wavelets.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

  plot(seq(0,1,le=1000),MeyerWindow(seq(0,1,le=1000),3),type='l')

Maxiset threshold

Description

Compute the maxiset threshold for WaveD fit.

Usage

MultiThresh1(s, g, L, eta)

Arguments

s

noise standard deviation

g

Sample of g or g + (Gaussian noise).

L

Lowest resolution level.

eta

Tuning parameter of the maxiset threshold.

Value

vector of thresholds

Author(s)

Marc Raimondo

References

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

See Also

WaveD,

Examples


MultiThresh1(.1, sin(2*pi*seq(0,1,le=1024)), 3, sqrt(2))

Phase matrix

Description

compute phase matrix, auxilliary function compute wavelet coefficients in the Fourier Domain

Usage

PhaseC(l, j)

Arguments

l

Fourier frequency (integer)

j

resolution level (integer)

Value

Matrix of phases.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

PhaseC(3,4)

Soft Threshold

Description

Apply soft threshold.

Usage

SoftThresh(y, t)

Arguments

y

vector

t

threshold

Value

vector $y$ thresholded: entries below $t$ are replaced by shrunken versions.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

SoftThresh(1:5,2)

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*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 \sqrt6.

thr

A vector of length F-L+1, giving thresholds at each resolution levels L,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 projection, coarse level.

Description

Compute WaveD projection of $f$, coarse level.

Usage

WaveDjC(y_fft, f2fft, j)

Arguments

y_fft

Fourier transform of $f$.

f2fft

Fourier transform of the wavelet.

j

Resolution level.

Value

Vector: WaveD projection of $f$, coarse resolution level.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples


waveJ0_fft=scaling_YM(3,10,3);
WaveDjC(fft(sin(2*pi*seq(0,1,le=1024))),waveJ0_fft,3)

WaveD projection, details.

Description

Compute WaveD projection of $f$, details.

Usage

WaveDjD(y_fft, f2fft, j)

Arguments

y_fft

Fourier transform of $f$.

f2fft

Fourier transform of the wavelet.

j

Resolution level.

Value

Vector: WaveD projection of $f$, details.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples


waveJ0_fft=wavelet_YM(5,10,3);
WaveDjD(fft(sin(2*pi*seq(0,1,le=1024))),waveJ0_fft,3)

WaveD projection, fine resolution level.

Description

Compute WaveD projection of $f$, fine resolution level.

Usage

WaveDjF(f1fft, f2fft, j)

Arguments

f1fft

Fourier transform of $f$.

f2fft

Fourier transform of the wavelet.

j

Resolution level.

Value

Vector: WaveD projection of $f$, fine resolution level.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples


waveJ0_fft=fine_YM(9,10,3);
WaveDjF(fft(sin(2*pi*seq(0,1,le=1024))),waveJ0_fft,3)

Dyadic band

Description

Returns a vector of integers 2^j+1,\ldots,2^{j+1}.

Usage

dyad(j)

Arguments

j

Resolution Level

Value

Returns a vector of integers 2^j+1,...,2^{j+1}.

Author(s)

Marc Raimondo

See Also

FWaveD

Examples

library(waved)
data=waved.example(TRUE,FALSE)
lidar.w=FWaveD(data$lidar.blur,data$g)
lidar.w[dyad(7)]

Lexicographic ordering (dyadic)

Description

return the index of a wavelet coefficient using dyadic lexicographic ordering

Usage

dyadjk(j, k)

Arguments

j

Resolution level (integer)

k

Location parameter (0,1,...,2^j-1)

Value

Returns an integer giving the index position of the wavelet coefficient w_{j,k} in a vector of wavelet coefficients.

Author(s)

Marc Raimondo

References

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

See Also

FWaveD

Examples

print(dyadjk(5,4))

Shift Fourier frequencies

Description

rearranges the outputs of fft by moving the zero-frequency component to the center of the array

Usage

fftshift(y)

Arguments

y

A vector

Value

Rearranged version of the vector $y$

Author(s)

Marc Raimondo

References

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

See Also

fft

Examples

 print(fftshift(1:5))

Fine resolution level for WaveD fit

Description

Find the optimal Fourier frequency and resolution level for WaveD fit

Usage

find.j1(g, sigma)

Arguments

g

vector (convolution kenel)

sigma

noise standard deviation

Value

M

Fourier frequency

j1

Resolution level

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

library(waved)
data=waved.example(TRUE,FALSE)
find.j1(data$g,data$sigma)

Find positive entries

Description

Find positive entries in a vector

Usage

findONE(x)

Arguments

x

vector

Value

A vector of indices where $x$ has positive values.

Author(s)

Marc Raimondo

References

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

See Also

findZERO

Examples

findONE(-5:5)

Find negative entries

Description

Find negative entries in a vector

Usage

findZERO(x)

Arguments

x

vector

Value

A vector of indices where $x$ has negative values.

Author(s)

Marc Raimondo

References

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

See Also

findONE

Examples

findZERO(-5:5)

~~function to do ... ~~

Description

generate Fourier transform of the Meyer wavelet function in the periodic setting at fine resolution level.

Usage

fine_YM(j, j_max, deg)

Arguments

j

Resolution level (positive integer)

j_max

Maximum resolution level (positive integer)

deg

Degree of Meyer wavelet (1,2,3)

Value

Fourier transform of the Meyer wavelet function at resolution level $j$

Author(s)

Marc Raimondo

References

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

See Also

WaveD,

Examples

fine_YM(9,10,3)

Make Doppler signal

Description

Generate Doppler signal.

Usage

make.doppler(n)

Arguments

n

sample size

Value

a vector of size $n$

Author(s)

Marc Raimondo

References

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

See Also

waved.example

Examples

plot(seq(0,1,le=1000),make.doppler(1000),type='l')

Make LIDAR signal

Description

Generate artificial LIDAR signal.

Usage

make.lidar(n)

Arguments

n

sample size

Value

a vector of size $n$

Author(s)

Marc Raimondo

References

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

See Also

waved.example

Examples

plot(seq(0,1,le=1000),make.lidar(1000),type='l')

Returns maxiset threshold

Description

Returns maxiset threshold for specified resolution levels.

Usage

maxithresh(data, g, L = 2, F = (log2(length(data)) - 1), eta = sqrt(6))

Arguments

data

data to be processed

g

sample of g

L

Lowest resolution level

F

Finest resolution level

eta

constant describing tail of noise distribution

Value

Returns maxiset threshold for coarse resolution level equal to L, and wavelet resolution levels L,...,F.

Author(s)

Marc Raimondo

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)
maxithresh(data$lidar.noisy,data$g,L=3,F=7)

Create Multi-resolution Plot

Description

Depicts wavelet coefficients according to time and resolution level.

Usage

multires(wcUntrimmed,lowest=3,coarse=3,highestplot=NULL,descending=FALSE,sc = 1)

Arguments

wcUntrimmed

Vector of wavelet coefficients; must be of dyadic length.

lowest

Lowest resolution level.

coarse

Coarse resolution level; same as lowest.

highestplot

Highest resolution level.

descending

logical indicating whether resolutions are depicted with highest at the top of the plot (FALSE, the default), or at the bottom (TRUE).

sc

graphical scaling parameter of heights of lines representing wavelet coefficients; default is 1.

Value

Depicts wavelet coefficients according to time (horizontal axis) and resolution level lowest, lowest+1,...,highestplot.

Author(s)

Marc Raimondo and Michael Stewart

References

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. (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,F=7)
multires(lidar.w,lo=3,hi=7)

Meyer scaling function (Fourier domain).

Description

Compute Meyer Scaling function in the Fourier domain.

Usage

phyHAT(x, deg)

Arguments

x

Abscissa (frequency) values for evaluation.

deg

The degree of the Meyer wavelet, either 1, 2, or 3

Value

Meyer scaling function at frequency $x$.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

plot(seq(-2,2,le=1000),abs(phyHAT(seq(-2,2,le=1000),3)),type='l')

Plot wvd objects

Description

Plot wvd objects

Usage

## S3 method for class 'wvd'
plot(x,...)

Arguments

x

A list created by the WaveD function

Value

Graphical output only.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

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

Plot spectrum

Description

This function plots the log spectrum and the optimal noise threshold which is used in WaveD fit.

Usage

plotspec(g, s)

Arguments

g

Sample of g or g + (Gaussian noise).

s

Noise standard deviation.

Value

Graphical output only.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

plotspec(sin(2*pi*seq(0,1,le=1024)),0.01)

Projection onto $F_j$

Description

Compute the projection of $f$ onto $F_j$ (fine resolution)).

Usage

projFj(beta, n, deg)

Arguments

beta

vector of wavelet coefficients of $f$

n

sample size

deg

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

Value

the projection of $f$ onto $F_j$

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

 plot(projFj(rnorm(1024),1024,3))

Projection onto $V_j$

Description

Compute the projection of $f$ onto $V_j$.

Usage

projVj(beta, n, deg)

Arguments

beta

vector of wavelet coefficients of $f$

n

sample size

deg

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

Value

the projection of $f$ onto $V_j$

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

 plot(projVj(rnorm(512),1024,3))

Projection onto $W_j$

Description

Compute the projection of $f$ onto $W_j$ (details).

Usage

projWj(beta, n, deg)

Arguments

beta

vector of wavelet coefficients of $f$

n

sample size

deg

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

Value

the projection of $f$ onto $W_j$

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

 plot(projWj(rnorm(512),1024,3))

Meyer wavelet function (Fourier domain).

Description

Compute Meyer wavelet function in the Fourier domain.

Usage

psyHAT(x, deg)

Arguments

x

Abscissa (frequency) values for evaluation.

deg

The degree of the Meyer wavelet, either 1, 2, or 3

Value

Meyer scaling function at frequency $x$.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

plot(seq(-2,2,le=1000),abs(psyHAT(seq(-2,2,le=1000),3)),type='l')

Rotate matrix 90 degrees

Description

Rotate matrix 90 degrees

Usage

rot90(x)

Arguments

x

A square matrix.

Value

90 degrees rotation of $x$.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

 rot90(1:5)

Estimates standard deviation of noise

Description

Estimates standard deviation of noise in the nonparametric signal+(Gaussian noise) regression model. Input vector must be of dyadic length and assumes a regular grid.

Usage

scale(yobs, L=3, deg=3)

Arguments

yobs

a vector of dyadic length representing signal+(Gaussian noise)

L

lowest resolution level

deg

degree of Meyer wavelet

Value

Returns a positive estimate of the standard deviation of noise in the nonparametric regression model.

Author(s)

Marc Raimondo

References

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)
scale(data$lidar.noisy)

Meyer scaling function, Fourier domain.

Description

Generate Fourier transform of the Meyer scaling function in the periodic setting at fine resolution level.

Usage

scaling_YM(j, j_max, deg)

Arguments

j

Resolution level (positive integer)

j_max

Maximum resolution level (positive integer)

deg

Degree of Meyer wavelet (1,2,3)

Value

Fourier transform of the Meyer scaling function at resolution level $j$

Author(s)

Marc Raimondo

References

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

See Also

WaveD,

Examples

scaling_YM(9,10,3)

Plot spectrum

Description

This function plots the log spectrum of $g$.

Usage

speczoom(y_test, fenetre)

Arguments

y_test

Sample vector of $g$.

fenetre

Window (a positive integer).

Value

log-spectrum of $g$.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

speczoom(sin(2*pi*seq(0,1,le=1024)),200)

Optimal stoping time

Description

Compute the stoping time in the Fourier domain using noisy egein values.

Usage

stoptime(g, sigma)

Arguments

g

A sample of the convolution kernel $g$.

sigma

Noise standard deviation.

Value

M

estimate of optimal Fourier frequency.

j1

estimate of optimal resolution level.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

stoptime(log(abs(sin(2*pi*seq(0,1,le=1024)))),1)

Summary of wvd objects

Description

Provide a summary of wvd objects

Usage

## S3 method for class 'wvd'
summary(object,...)

Arguments

object

A list created by the WaveD function

Value

Text output only.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

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

Show threshold effects

Description

Provide a summary of threshold effects in WaveD fit.

Usage

threshsum(w.res, L = 3, F = (log2(length(w.res)) - 1))

Arguments

w.res

A vector of wavelet coefficients

L

Low resolution level

F

Fine resolution level

Value

A vector of length F-L+1, with ONES and ZEROS. The ZEROS show that no coefficient remains at the corresponding resolution level.

Author(s)

Marc Raimondo

References

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

See Also

WaveD

Examples

library(waved)
data=waved.example(TRUE,FALSE)
doppler.wvd=WaveD(data$doppler.noisy,data$g)
threshsum(doppler.wvd$w,3,8)

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)

Meyer wavelet function, Fourier domain.

Description

Generate Fourier transform of the Meyer scaling function in the periodic setting at fine resolution level.

Usage

wavelet_YM(j, j_max, deg)

Arguments

j

Resolution level (positive integer)

j_max

Maximum resolution level (positive integer)

deg

Degree of Meyer wavelet (1,2,3)

Value

Fourier transform of the Meyer wavelet function at resolution level $j$

Author(s)

Marc Raimondo

References

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

See Also

WaveD,

Examples

wavelet_YM(5,10,3)