Package 'wavethresh'

Description

Performs 1, 2 and 3D real and complex-valued wavelet transforms, nondecimated transforms, wavelet packet transforms, nondecimated wavelet packet transforms, multiple wavelet transforms, complex-valued wavelet transforms, wavelet shrinkage for various kinds of data, locally stationary wavelet time series, nonstationary multiscale transfer function modeling, density estimation.

Details

Index of help topics:

AutoBasis               Run Coifman-Wickerhauser best basis algorithm
                        on wavelet packet object
AvBasis                 Basis averaging ("inversion")
AvBasis.wst             Perform basis averaging for (packet-ordered)
                        non-decimated wavelet transform.
AvBasis.wst2D           Perform basis averaging for (packet-ordered) 2D
                        non-decimated wavelet transform.
BAYES.THR               Bayesian wavelet thresholding.
BMdiscr                 Subsidiary routine for makewpstDO function
BabyECG                 Physiological data time series.
BabySS                  Physiological data time series.
Best1DCols              Extract the best (one-dimensional) nondecimated
                        WP packets
CWCV                    C Wavelet Cross-validation
CWavDE                  Simple wavelet density estimator with hard
                        thresholding
CanUseMoreThanOneColor
                        Deprecated function
Chires5                 Subsid routine for denproj (calcs scaling
                        function coefs without cov)
Chires6                 Subsid routine for denproj (calcs scaling
                        function coefs with cov)
ConvertMessage          Print out a text message about an object which
                        is from old version of WaveThresh
Crsswav                 Wrapper to C code version of rsswav
Cthreshold              Calls C code to threshold wd class object.
DJ.EX                   Produce Donoho and Johnstone test functions
FullWaveletCV           Perform whole wavelet cross-validation in C
                        code
GenW                    Generate (inverse) discrete wavelet transform
                        matrix.
GetRSSWST               Computes estimate of error for function
                        estimate.
HaarConcat              Generate a concatenated Haar MA process
HaarMA                  Generate Haar MA processes.
InvBasis                Generic basis inversion for libraries
InvBasis.wp             Invert a wp library representation with a
                        particular basis spec
InvBasis.wst            Invert a wst library representation with a
                        basis specification
IsEarly                 Generic function to detect whether object is
                        from an early version
IsEarly.default         Detects whether object is from an earlier
                        version of WaveThresh
IsEarly.wd              Function to detect whether a wd object is from
                        WaveThresh2 or not
IsPowerOfTwo            Decides whether vector elements are integral
                        powers of two (returns NA if not).
LSWsim                  Simulate arbitrary locally stationary wavelet
                        process.
LocalSpec               Compute Nason and Silverman smoothed wavelet
                        periodogram.
LocalSpec.wd            Compute Nason and Silverman raw or smoothed
                        wavelet periodogram.
LocalSpec.wst           Obsolete function (use ewspec)
MaNoVe                  Make Node Vector (using Coifman-Wickerhauser
                        best-basis type algorithm)
MaNoVe.wp               Make Node Vector (using Coifman-Wickerhauser
                        best-basis type algorithm) on wavelet packet
                        object
MaNoVe.wst              Make Node Vector (using Coifman-Wickerhauser
                        best-basis type algorithm) on nondecimated
                        wavelet transform object
PsiJ                    Compute discrete autocorrelation wavelets.
PsiJmat                 Compute discrete autocorrelation wavelets but
                        return result in matrix form.
Psiname                 Return a PsiJ list object style name.
ScalingFunction         Compute scaling functions on internally
                        predefined grid
Shannon.entropy         Compute Shannon entropy
TOgetthrda1             Subsidiary routines for Ogden and Parzen's
                        wavelet shrinkage methods
TOthreshda1             Data analytic wavelet thresholding routine
TOthreshda2             Data analytic wavelet thresholding routine
WTEnv                   Environment that exists to store intermediate
                        calculations for re-use within the same R
                        session.
WaveletCV               Wavelet cross-validation
Whistory                Obsolete function supposedly detailed history
                        of object
Whistory.wst            Obsolete function: as Whistory, but for wst
                        objects
accessC                 Get "detail" (mother wavelet) coefficients data
                        from wavelet object
accessC.mwd             Get Smoothed Data from Wavelet Structure
accessC.wd              Get smoothed data from wavelet object (wd)
accessC.wp              Warning function when trying to access smooths
                        from wavelet packet object (wp).
accessC.wst             Get smoothed data from packet ordered
                        non-decimated wavelet object (wst)
accessD                 Get "detail" (mother wavelet) coefficients data
                        from wavelet object
accessD.mwd             Get wavelet coefficients from multiple wavelet
                        structure (mwd).
accessD.wd              Get detail (mother wavelet) coefficients from
                        wavelet object (wd).
accessD.wd3D            Get wavelet coefficients from 3D wavelet object
accessD.wp              Obtain whole resolution level of wavelet packet
                        coefficients from a wavelet packet object (wp).
accessD.wpst            Get coefficients from a non-decimated wavelet
                        packet object (wpst) in time order.
accessD.wst             Get mother wavelet coefficients from a packet
                        ordered non-decimated wavelet object (wst).
accessc                 Get variance information from irregularly
                        spaced wavelet decomposition object.
addpkt                  Add a wavelet packet box to an already set up
                        time-frequency plot
av.basis                Perform basis averaging for wst class object
basisplot               Generic basis plot function
basisplot.BP            Plot time-frequency plane and basis slots
                        associated with basis object
basisplot.wp            Function to graphically select a wavelet packet
                        basis
bestm                   Function called by makewpstRO to identify which
                        packets are individually good for correlating
                        with a response
c2to4                   Take integer, represent in binary, then think
                        of and return that representation in base 4
checkmyews              Check a LSW spectrum through repeated
                        simulation and empirical averages
cns                     Create new zeroed spectrum.
compare.filters         Compares two filters.
compgrot                Compute empirical shift for time ordered
                        non-decimated transforms.
compress                Compress objects
compress.default        Do "zero" run-length encoding compression of a
                        vector of numbers.
compress.imwd           Compress a (thresholded) imwd class object by
                        removing zeroes.
conbar                  Performs inverse DWT reconstruction step
convert                 Convert one type of wavelet object into
                        another.
convert.wd              Convert a non-decimated wd object into a wst
                        object.
convert.wst             Convert a non-decimated wst object into a wd
                        object.
cthresh                 Estimate real signal using complex-valued
                        wavelets
dclaw                   Claw distribution
dencvwd                 Calculate variances of wavlet coefficients of a
                        p.d.f.
denplot                 Calculate plotting information for a density
                        estimate.
denproj                 Calculate empirical scaling function
                        coefficients of a p.d.f.
denwd                   Wavelet decomposition of empirical scaling
                        function coefficients of a p.d.f.
denwr                   Wavelet reconstruction for density estimation.
dof                     Compute number of non-zero coefficients in wd
                        object
doppler                 Evaluate the Donoho and Johnstone Doppler
                        signal.
draw                    Draw wavelets or scaling functions.
draw.default            Draw picture of a wavelet or scaling function.
draw.imwd               Draw mother wavelet associated with an imwd
                        object.
draw.imwdc              Draw mother wavelet associated with an imwdc
                        object.
draw.mwd                Draws a wavelet or scaling function used to
                        compute an 'mwd' object
draw.wd                 Draw mother wavelet or scaling function
                        associated with wd object.
draw.wp                 Draw wavelet packet associated with a wp
                        object.
draw.wst                Draw mother wavelet or scaling function
                        associated with wst object.
drawbox                 Draw a shaded coloured box
drawwp.default          Subsidiary routine that actually computes
                        wavelet packet values
ewspec                  Compute evolutionary wavelet spectrum estimate.
example.1               Compute and return piecewise polynomial
                        coordinates.
filter.select           Provide wavelet filter coefficients.
find.parameters         Find estimates of prior parameters
first.last              Build a first/last database for wavelet
                        transforms.
first.last.dh           Build special first/last database for some
                        wavelet density functions
firstdot                Return the location of the first period
                        character within a character string (for a
                        vector of strings of arbitrary length).
getarrvec               Compute and return weaving permutation for
                        conversion from wst objects to wd class
                        objects.
getpacket               Get a packet of coefficients from a wavelet
                        object
getpacket.wp            Get packet of coefficients from a wavelet
                        packet object (wp).
getpacket.wpst          Get packet of coefficients from a non-decimated
                        wavelet packet object (wpst).
getpacket.wst           Get packet of coefficients from a packet
                        ordered non-decimated wavelet object (wst).
getpacket.wst2D         Get packet of coefficients from a
                        two-dimensional non-decimated wavelet object
                        (wst2D).
griddata objects        Data interpolated to a grid objects.
guyrot                  Cyclically rotate elements of a vector
image.wd                Produce image representation of nondecimated
                        wavelet transform
image.wst               Produce image representation of a wst class
                        object
imwd                    Two-dimensional wavelet transform
                        (decomposition).
imwd.object             Two-dimensional wavelet decomposition objects.
imwdc.object            Two-dimensional compressed wavelet
                        decomposition objects.
imwr                    Inverse two-dimensional wavelet transform.
imwr.imwd               Inverse two-dimensional discrete wavelet
                        transform.
imwr.imwdc              Inverse two-dimensional discrete wavelet
                        transform.
ipd                     Inductance plethysmography data.
ipndacw                 Compute inner product matrix of discrete
                        non-decimated autocorrelation wavelets.
irregwd                 Irregular wavelet transform (decomposition).
irregwd.objects         Irregular wavelet decomposition objects.
l2norm                  Compute L2 distance between two vectors of
                        numbers.
lennon                  John Lennon image.
levarr                  Subsidiary routine that generates a particular
                        permutation
linfnorm                Compute L infinity distance between two vectors
                        of numbers.
logabs                  Take the logarithm of the squares of the
                        argument
lt.to.name              Convert desired level and orientation into code
                        used by imwd
madmad                  Compute square of median absolute deviation
                        (mad) function.
make.dwwt               Compute diagonal of the matrix WWT
makegrid                Interpolate data to a grid.
makewpstDO              Help page for a function
makewpstRO              Make a wavelet packet regression object from a
                        dependent and independent time series variable.
mfilter.select          Provide filter coefficients for multiple
                        wavelets.
mfirst.last             Build a first/last database for multiple
                        wavelet transforms.
modernise               Generic function to upgrade a V2 WaveThresh
                        object to V4
modernise.wd            Modernise a wd class object
mpostfilter             Multiwavelet postfilter
mprefilter              Multiwavelet prefilter
mwd                     Discrete multiple wavelet transform
                        (decomposition).
mwd.object              Multiple wavelet decomposition object (1D)
mwr                     Multiple discrete wavelet transform
                        (reconstruction).
newsure                 Version of sure that acts as subsidiary for
                        threshold.irregwd
nlevelsWT               Returns number of scale (resolution) levels.
nlevelsWT.default       Returns number of levels associated with an
                        object
nullevels               Set whole resolution levels of coefficients
                        equal to zero.
nullevels.imwd          Sets whole resolution levels of coefficients
                        equal to zero in a imwd object.
nullevels.wd            Sets whole resolution levels of coefficients
                        equal to zero in a wd object.
nullevels.wst           Sets whole resolution levels of coefficients
                        equal to zero in a wst object.
numtonv                 Convert an index number into a node vector
                        object.
nv.object               Node vector objects.
plot.imwd               Draw a picture of the 2D wavelet coefficients
                        using image
plot.irregwd            Plot variance factors of wavelet transform
                        coefficients for irregularly spaced wavelet
                        transform object
plot.mwd                Use plot on an mwd object.
plot.nvwp               Depict wavelet packet basis specfication
plot.wd                 Plot wavelet transform coefficients.
plot.wp                 Plot wavelet packet transform coefficients
plot.wst                Plot packet-ordered non-decimated wavelet
                        transform coefficients.
plot.wst2D              Plot packet-ordered 2D non-decimated wavelet
                        coefficients.
plotdenwd               Plot the wavelet coefficients of a p.d.f.
plotpkt                 Sets up a high level plot ready to show the
                        time-frequency plane and wavelet packet basis
                        slots
print.BP                Print top best basis information for BP class
                        object
print.imwd              Print out information about an imwd object in
                        readable form.
print.imwdc             Print out information about an imwdc object in
                        readable form.
print.mwd               Use print() on a mwd object.
print.nv                Print a node vector object, also used by
                        several other functions to obtain packet list
                        information
print.nvwp              Print a wavelet packet node vector object, also
                        used by several other functions to obtain
                        packet list information
print.w2d               Print method for printing w2d class objects
print.w2m               Print a w2m class object
print.wd                Print out information about an wd object in
                        readable form.
print.wd3D              Print out information about an wd3D object in a
                        readable form.
print.wp                Print out information about an wd object in
                        readable form.
print.wpst              Prints out basic information about a wpst class
                        object
print.wpstCL            Prints some information about a wpstCL object
print.wpstDO            Print information about a wpstDO class object
print.wpstRO            Print a wpstRO class object
print.wst               Print out information about an wst object in
                        readable form.
print.wst2D             Print out information about an wst2d object in
                        a readable form.
putC                    Put smoothed data (father wavelet) coefficients
                        into wavelet structure
putC.mwd                Put smoothed data into wavelet structure
putC.wd                 Puts a whole resolution level of father wavelet
                        coeffients into wd wavelet object.
putC.wp                 Warning function when trying to insert father
                        wavelet coefficients into wavelet packet object
                        (wp).
putC.wst                Puts a whole resolution level of father wavelet
                        coeffients into wst wavelet object.
putD                    Put mother wavelet coefficients into wavelet
                        structure
putD.mwd                Put wavelet coefficients into multiple wavelet
                        structure
putD.wd                 Puts a whole resolution level of mother wavelet
                        coeffients into wd wavelet object.
putD.wd3D               Put wavelet coefficient array into a 3D wavelet
                        object
putD.wp                 Puts a whole resolution level of wavelet packet
                        coeffients into wp wavelet object.
putD.wst                Puts a whole resolution level of mother wavelet
                        coeffients into wst wavelet object.
putDwd3Dcheck           Check argument list for putD.wd3D
putpacket               Insert a packet of coefficients into a wavelet
                        object.
putpacket.wp            Inserts a packet of coefficients into a wavelet
                        packet object (wp).
putpacket.wst           Put a packet of coefficients into a packet
                        ordered non-decimated wavelet object (wst).
putpacket.wst2D         Replace packet of coefficients in a
                        two-dimensional non-decimated wavelet object
                        (wst2D).
rcov                    Computes robust estimate of covariance matrix
rfft                    Real Fast Fourier transform
rfftinv                 Inverse real FFT, inverse of rfft
rfftwt                  Weight a Fourier series sequence by a set of
                        weights
rm.det                  Set coarse levels of a wavelets on the interval
                        transform object to zero
rmget                   Search for existing ipndacw matrices.
rmname                  Return a ipndacw matrix style name.
rotateback              Cyclically shift a vector one place to the
                        right
rsswav                  Compute mean residual sum of squares for odd
                        prediction of even ordinates and vice versa
simchirp                Compute and return simulated chirp function.
ssq                     Compute sum of squares difference between two
                        vectors
summary.imwd            Print out some basic information associated
                        with an imwd object
summary.imwdc           Print out some basic information associated
                        with an imwdc object
summary.mwd             Use summary() on a mwd object.
summary.wd              Print out some basic information associated
                        with a wd object
summary.wd3D            Print out some basic information associated
                        with a wd3D object
summary.wp              Print out some basic information associated
                        with a wp object
summary.wpst            Print out some basic information associated
                        with a wpst object
summary.wst             Print out some basic information associated
                        with a wst object
summary.wst2D           Print out some basic information associated
                        with a wst2D object
support                 Returns support of compactly supported
                        wavelets.
sure                    Computes the minimum of the SURE thresholding
                        function
teddy                   Picture of a teddy bear's picnic.
test.dataCT             Test functions for wavelet regression and
                        thresholding
threshold               Threshold coefficients
threshold.imwd          Threshold two-dimensional wavelet decomposition
                        object
threshold.imwdc         Threshold two-dimensional compressed wavelet
                        decomposition object
threshold.irregwd       hold irregularly spaced wavelet decomposition
                        object
threshold.mwd           Use threshold on an mwd object.
threshold.wd            Threshold (DWT) wavelet decomposition object
threshold.wd3D          Threshold 3D DWT object
threshold.wp            Threshold wavelet packet decomposition object
threshold.wst           Threshold (NDWT) packet-ordered non-decimated
                        wavelet decomposition object
tpwd                    Tensor product 2D wavelet transform
tpwr                    Inverse tensor product 2D wavelet transform.
uncompress              Uncompress objects
uncompress.default      Undo zero run-length encoding for a vector.
uncompress.imwdc        Uncompress an imwdc class object
wavegrow                Interactive graphical tool to grow a wavelet
                        synthesis
wavethresh-package      Wavelets Statistics and Transforms
wd                      Wavelet transform (decomposition).
wd.dh                   Compute specialized wavelet transform for
                        density estimation
wd.int                  Computes "wavelets on the interval" transform
wd.object               Wavelet decomposition objects
wd3D                    Three-dimensional discrete wavelet transform
wd3D.object             Three-dimensional wavelet object
wp                      Wavelet packet transform.
wp.object               Wavelet Packet decomposition objects.
wpst                    Non-decimated wavelet packet transform.
wpst2discr              Reshape/reformat packet coefficients into a
                        multivariate data set
wpst2m                  Converts a nondecimated wavelet packet object
                        to a (large) matrix with packets stored as
                        columns
wpstCLASS               Predict values using new time series values via
                        a non-decimated wavelet packet discrimination
                        object.
wpstREGR                Construct data frame using new time series
                        using information from a previously constructed
                        wpstRO object
wr                      Wavelet reconstruction (inverse DWT).
wr.int                  Computes inverse "wavelets on the interval"
                        transform.
wr.mwd                  Multiple wavelet reconstruction for mwd objects
wr.wd                   Wavelet reconstruction for wd class objects
                        (inverse discrete wavelet transform).
wr3D                    Inverse DWT for 3D DWT object.
wst                     Packet-ordered non-decimated wavelet transform.
wst.object              (Packet ordered) Nondecimated wavelet transform
                        decomposition objects.
wst2D                   (Packet-ordered) 2D non-decimated wavelet
                        transform.
wst2D.object            (Packet ordered) Two-dimensional nondecimated
                        wavelet transform decomposition objects.
wstCV                   Performs two-fold cross-validation estimation
                        using packet-ordered non-decimated wavelet
                        transforms and one, global, threshold.
wstCVl                  Performs two-fold cross-validation estimation
                        using packet-ordered non-decimated wavelet
                        transforms and a (vector) level-dependent
                        threshold.
wvcvlrss                Computes estimate of error for function
                        estimate.
wvmoments               Compute moments of wavelets or scaling function
wvrelease               Prints out the release number of the WaveThresh
                        package

See book or individual help pages for main functions. For example, wd for the one-dimensional discrete wavelet transform.

Author(s)

Guy Nason [aut, cre], Stuart Barber [ctb], Tim Downie [ctb], Piotr Frylewicz [ctb], Arne Kovac [ctb], Todd Ogden [ctb], Bernard Silverman [ctb]

Maintainer: Guy Nason <[email protected]>

References

Nason, G.P. (2008) Wavelet methods in Statistics with R. Springer, New York. Book URL.

See Also

ewspec, imwd, threshold, wd, wst

Examples

#
# See examples in individual help pages
#

Description

This function gets information from the c component of an irregwd.objects an irregularly spaced wavelet decomposition object.

Note that this function is not the same as accessC which obtains father wavelet coefficients from an wd class object.

Usage

accessc(irregwd.structure, level, boundary=FALSE)

Arguments

Details

The irregwd function produces a irregular wavelet decomposition (reconstruction) structure.

The c component is stored in a similar way to the C and D vectors which store the father and mother wavelet coefficients respectively. Hence to access the information the accessc function plays a similar role to accessC and accessD functions.

Value

A vector of the extracted data.

RELEASE

Version 3.9.4 Code Copyright Arne Kovac 1997. Help Copyright Guy Nason 2004.

Author(s)

G P Nason

See Also

irregwd, irregwd.objects, threshold.irregwd,makegrid, plot.irregwd

Examples

#
# Most users will not need to use this function. However, see the main
# examples for the irregular wavelet denoising in the examples for
# makegrid.
#

Description

This generic function extracts detail from various types of wavelet objects. It extracts and returns a whole resolution level of coefficients. To obtain individual packets from relevant transforms use the getpacket() series of functions. This function is generic.

Particular methods exist. For objects of class:

wd

use the accessC.wd method

wp

use the accessC.wp method

wst

use the accessC.wst method

See individual method help pages for operation and examples.

Usage

accessC(...)

Arguments

Value

A vector coefficients representing the detail coefficients for the requested resolution level.

RELEASE

Version 3.5.3 Copyright Guy Nason 1994

Author(s)

G P Nason

See Also

accessC.wd, accessC.wp,accessC.wst,accessD

Description

The smoothed and original data from a multiple wavelet decomposition structure, mwd.object ect (e.g. returned from mwd) are packed into a single matrix in that structure. TRUE his function extracts the data corresponding to a particular resolution level.

Usage

## S3 method for class 'mwd'
accessC(mwd, level = nlevelsWT(mwd), ...)

Arguments

Details

The mwd function produces a wavelet decomposition structure.

For decomposition, the top level contains the original data, and subsequent lower levels contain the successively smoothed data. So if there are mwd$filter$npsi*2^m original data points (mwd$filter$npsi is the multiplicity of wavelets), there will be m+1 levels indexed 0,1,...,m. So

accessC.mwd(Mwd, level=m)

pulls out the original data, as does

accessC.mwd(mwd)

To get hold of lower levels just specify the level that you're interested in, e.g.

accessC.mwd(mwd, level=2)

Gets hold of the second level.

The need for this function is a consequence of the pyramidal structure of Mallat's algorithm and the memory efficiency gain achieved by storing the pyramid as a linear matrix of coefficients. AccessC obtains information about where the smoothed data appears from the fl.dbase component of mwd, in particular the array fl.dbase$first.last.c which gives a complete specification of index numbers and offsets for mwd$C.

Note also that this function only gets information from mwd class objects. To put coefficients into mwd structures you have to use the putC.mwd function.

See Downie and Silverman, 1998.

Value

A matrix with mwd$filter$npsi rows containing the extracted data of all the coefficients at that level.

RELEASE

Version 3.9.6 (Although Copyright Tim Downie 1995-6.)

Author(s)

G P Nason

See Also

accessD.mwd, draw.mwd, mfirst.last, mfilter.select, mwd, mwd.object, mwr, plot.mwd, print.mwd, putC.mwd, putD.mwd, summary.mwd, threshold.mwd, wd

Examples

#
# Get the 3rd level of smoothed data from a decomposition
#
dat <- rnorm(32)
accessC.mwd(mwd(dat), level=3)

Description

The smoothed and original data from a wavelet decomposition structure (returned from wd) are packed into a single vector in that structure. This function extracts the data corresponding to a particular resolution level.

Usage

## S3 method for class 'wd'
accessC(wd, level = nlevelsWT(wd), boundary=FALSE, aspect, ...)

Arguments

Details

The wd (wr.wd) function produces a wavelet decomposition (reconstruction) structure.

For decomposition, the top level contains the original data, and subsequent lower levels contain the successively smoothed data. So if there are $2^m$ original data points, there will be m+1 levels indexed 0,1,...,m. So

accessC.wd(wdobj, level=m)

pulls out the original data, as does

accessC.wd(wdobj)

To get hold of lower levels just specify the level that you're interested in, e.g.

accessC.wd(wdobj, level=2)

gets hold of the second level.

For reconstruction, the top level contains the ultimate step in the Mallat pyramid reconstruction algorithm, lower levels are intermediate steps.

The need for this function is a consequence of the pyramidal structure of Mallat's algorithm and the memory efficiency gain achieved by storing the pyramid as a linear vector. AccessC obtains information about where the smoothed data appears from the fl.dbase component of an wd.object, in particular the array fl.dbase$first.last.c which gives a complete specification of index numbers and offsets for wd.object$C.

Note that this function is method for the generic function accessC. When the wd.object is definitely a wd class object then you only need use the generic version of this function.

Note that this function only gets information from wd class objects. To insert coefficients etc. into wd structures you have to use the putC function (or more precisely, the putC.wd method).

Value

A vector of the extracted data.

RELEASE

Version 3.5.3 Copyright Guy Nason 1994

Author(s)

G P Nason

References

Mallat, S. G. (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11, 674–693.

Nason, G. P. and Silverman, B. W. (1994). The discrete wavelet transform in S. Journal of Computational and Graphical Statistics, 3, 163–191.

See Also

wr, wd, accessD, accessD.wd, filter.select, threshold, putC.wd, putD.wd.

Examples

## Get the 3rd level of smoothed data from a decomposition
dat <- rnorm(64)
accessC(wd(dat), level=3)

Description

There are no real smooths to access in a wp wavelet packet object. This function returns an error message. To obtain coefficients from a wavelet packet object you should use the getpacket collection of functions.

Usage

## S3 method for class 'wp'
accessC(wp, ...)

Arguments

Value

An error message!

RELEASE

Version 3.5.3 Copyright Guy Nason 1994

Author(s)

G P Nason

See Also

getpacket

Description

The smoothed data from a packet ordered non-decimated wavelet object (returned from wst) are stored in a matrix. This function extracts all the coefficients corresponding to a particular resolution level.

Usage

## S3 method for class 'wst'
accessC(wst, level, aspect, ...)

Arguments

Details

The wst function performs a packet-ordered non-decimated wavelet transform. This function extracts all the father wavelet coefficients at a particular resolution level specified by level.

Note that coefficients returned by this function are in packet order. They can be used as is but for many applications it might be more useful to deal with the coefficients in packets: see the function getpacket.wst for further details.

Value

A vector of the extracted data.

Author(s)

G P Nason

References

Nason, G. P. and Silverman, B. W. (1994). The discrete wavelet transform in S. Journal of Computational and Graphical Statistics, 3, 163–191.

See Also

wst, wst.object, accessC, getpacket.wst

Examples

#
# Get the 3rd level of smoothed data from a decomposition
#
dat <- rnorm(64)
accessC(wst(dat), level=3)

Description

This generic function extracts detail from various types of wavelet objects. It extracts and returns a whole resolution level of coefficients. To obtain individual packets from relevant transforms use the getpacket() series of functions. This function is generic.

Particular methods exist. For objects of class:

wd

use the accessD.wd method

wd3D

use the accessD.wd3D method

wp

use the accessD.wp method

wpst

use the accessD.wpst method

wst

use the accessD.wst method

See individual method help pages for operation and examples.

Usage

accessD(...)

Arguments

Value

A vector coefficients representing the detail coefficients for the requested resolution level.

RELEASE

Version 3.5.3 Copyright Guy Nason 1994

Author(s)

G P Nason

See Also

accessD.wd, accessD.wp,accessD.wst,accessC

Description

The wavelet coefficients from a multiple wavelet decomposition structure, mwd.object, (e.g. returned from mwd) are packed into a single matrix in that structure. This function extracts the coefficients corresponding to a particular resolution level.

Usage

## S3 method for class 'mwd'
accessD(mwd, level, ...)

Arguments

Details

The mwd function produces a multiple wavelet decomposition object .

The need for this function is a consequence of the pyramidal structure of Mallats algorithm and the memory efficiency gain achieved by storing the pyramid as a linear matrix. AccessD obtains information about where the coefficients appear from the fl.dbase component of mwd, in particular the array fl.dbase$first.last.d which gives a complete specification of index numbers and offsets for mwd$D.

Note that this function and accessC only work on objects of class mwd to extract coefficients. You have to use putD.mwd to insert wavelet coefficients into a mwd object.

See Downie and Silverman, 1998.

Value

A matrix with mwd$filter$npsi rows containing the extracted coefficients.

RELEASE

Tim Downie 1995-6

Author(s)

G P Nason

See Also

accessD.mwd, draw.mwd, mfirst.last, mfilter.select, mwd, mwd.object, plot.mwd, print.mwd, putC.mwd, putD.mwd, summary.mwd, threshold.mwd, wd, wr.mwd

Examples

#
# Get the 3rd level of smoothed data from a decomposition
#
data(ipd)
accessD.mwd(mwd(ipd), level=3)

Description

This function extracts and returns a vector of mother wavelet coefficients, corresponding to a particular resolution level, from a wd wavelet decomposition object.

The pyramid of coefficients in a wavelet decomposition (returned from the wd function, say) are packed into a single vector in WaveThresh.

Usage

## S3 method for class 'wd'
accessD(wd, level, boundary=FALSE, aspect="Identity", ...)

Arguments

Details

The need for this function is a consequence of the pyramidal structure of Mallat's algorithm and the memory efficiency gain achieved by storing the pyramid as a linear vector. AccessD obtains information about where the smoothed data appears from the fl.dbase component of an wd object, in particular the array

fl.dbase$first.last.d

which gives a complete specification of index numbers and offsets for

wd.object$D.

Note that this function is a method for the generic function accessD.

Note also that this function only retrieves information from wd class objects. To insert coefficients into wd objects you have to use the putD function (or more precisely, the putD.wd method).

Value

A vector containing the mother wavelet coefficients at the required resolution level (the coefficients might have been modified depending on the value of the aspect argument).

RELEASE

Version 3.5.3 Copyright Guy Nason 1994

Author(s)

G P Nason

References

Mallat, S. G. (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence. 11, 674–693.

Nason, G. P. and Silverman, B. W. (1994). The discrete wavelet transform in S. Journal of Computational and Graphical Statistics, 3, 163–191

See Also

wr, wd, accessD, filter.select, threshold

Examples

#
# Get the 4th resolution level of wavelet coefficients. 
#
dat <- rnorm(128)
accessD(wd(dat), level=4)

Description

This function extracts and returns arrays of wavelet coefficients, corresponding to a particular resolution level, from a wd wavelet decomposition object.

The pyramid of coefficients in a wavelet decomposition (returned from the wd3D function, say) are packed into a single array in WaveThresh3.

Usage

## S3 method for class 'wd3D'
accessD(obj, level = nlevelsWT(obj)-1, block, ...)

Arguments

Details

The need for this function is a consequence of the pyramidal structure of Mallat's algorithm and the memory efficiency gain achieved by storing the pyramid as a array.

Note that this functiOn is a method for the generic function accessD.

Value

If the block is missing then a list is returned containing all the sub-blocks of coefficients for the specificed resolution level.

Otherwise the block character string specifies which sub-block of coefficients to return.

RELEASE

Version 3.9.6 Copyright Guy Nason 1997

Author(s)

G P Nason

See Also

link{accessD}, link{print.wd3D}, link{putD.wd3D}, link{putDwd3Dcheck}, link{summary.wd3D}, link{threshold.wd3D}, link{wd3D}, link{wd3D object}, link{wr3D}.

Examples

#
# Generate some test data
#
a <- array(rnorm(8*8*8), dim=c(8,8,8))
#
# Perform the 3D DWT
#
awd3D <- wd3D(a)
#
# How many levels does this object have?
#
nlevelsWT(awd3D)
# [1] 3
#
# So conceivably we could access levels 0, 1 or 2.
#
# Ok. Let's get the level 1 HGH sub-block coefficients:
#
accessD(awd3D, level=1, block="HGH")
#
#, , 1
#           [,1]      [,2] 
#[1,]  0.8359289 1.3596832
#[2,] -0.1771688 0.2987303
#
#, , 2
#           [,1]       [,2] 
#[1,] -1.2633313 1.00221652
#[2,] -0.3004413 0.04728019
#
# This was a 3D array of dimension size 2 (8 -> 4 -> 2, level 3, 2 and then 1)
#
#
# Let's do the same call except this time don't specify the block arg.
#
alllev1 <- accessD(awd3D, level=1)
#
# This new object should be a list containing all the subblocks at this level.
# What are the components?
#
names(alllev1)
#[1] "GHH" "HGH" "GGH" "HHG" "GHG" "HGG" "GGG"
#
# O.k. Let's look at HGH again
#
alllev1$HGH
#
#, , 1
#           [,1]      [,2] 
#[1,]  0.8359289 1.3596832
#[2,] -0.1771688 0.2987303
#
#, , 2
#           [,1]       [,2] 
#[1,] -1.2633313 1.00221652
#[2,] -0.3004413 0.04728019
#
# Same as before.
#

Description

Get a whole resolution level's worth of coefficients from a wp wavelet packet object. To obtain packets of coefficients from a wavelet packet object you should use the getpacket collection of functions.

Usage

## S3 method for class 'wp'
accessD(wp, level, ...)

Arguments

.

Details

The wavelet packet coefficients are actually stored in a straightforward manner in a matrix component of a wp object so it would not be too difficult to extract whole resolution levels yourself. However, this routine makes it easier to do.

Value

A vector containing the coefficients that you wanted to extract.

RELEASE

Version 3.5.3 Copyright Guy Nason 1994

Author(s)

G P Nason

See Also

accessD, getpacket

Description

The coefficients from a non-decimated wavelet packet object, wpst, are stored in a particular order in the wpst component of the wpstobj object. This function extracts all the coefficients corresponding to a particular wavelet packet in time order.

Usage

## S3 method for class 'wpst'
accessD(wpst, level, index, ...)

Arguments

Details

The wpst function performs a non-decimated wavelet packet transform. This function extracts the coefficients at a particular resolution level specified by level in time order.

It is possible to extract the individual packets (before interweaving, i.e. the direct result of multiple applications of the packet operators) by using the getpacket.wpst function.

Author(s)

G P Nason

References

Nason, G.P., Sapatinas, T. and Sawczenko, A. Statistical modelling using undecimated wavelet transforms.

See Also

wpst, wpst.object, accessD, getpacket.wpst

Examples

#
# Get the 4th level of coefficients from a decomposition
#
dat <- rnorm(128)
accessD(wpst(dat), level=4, index=3)

Description

The mother wavelet coefficients from a packet ordered non-decimated wavelet object, wst, are stored in a matrix. This function extracts all the coefficients corresponding to a particular resolution level.

Usage

## S3 method for class 'wst'
accessD(wst, level, aspect = "Identity", ...)

Arguments

Details

The wst function performs a packet-ordered non-decimated wavelet transform. This function extracts all the mother wavelet coefficients at a particular resolution level specified by level.

Note that coefficients returned by this function are in packet order. They can be used as is but for many applications it might be more useful to deal with the coefficients in packets: see the function getpacket.wst for further details.

Note that all the coefficients here are those of mother wavelets. The non-decimated transform efficiently computes all possible shifts of the discrete wavelet transform computed by wd.

Value

A vector of the extracted coefficients.

Author(s)

G P Nason

References

Nason, G.P. and Silverman, B.W. The stationary wavelet transform and some statistical applications.

See Also

wst, wst.object, accessD, getpacket.wst

Examples

#
# Get the 4th level of mother wavelet coefficients from a decomposition
#
dat <- rnorm(128)
accessD(wst(dat), level=4)

Description

This function assumes that a high-level plot has already been set up using plotpkt. Given that this function plots a wavelet packet box at a given level, packet index and with particular shading and color and optionally plotting a sequence of coefficients at that location rather than a shaded box.

Usage

addpkt(level, index, density, col, yvals)

Arguments

Details

Description says all

Value

Nothing

Author(s)

G P Nason

See Also

basisplot,basisplot.BP, basisplot.wp, plotpkt, plot.nvwp

Description

Runs the Coifman-Wickerhauser best basis algorithm on a wavelet packet object. Packets not in the basis are replaced by vectors of NAs. Superceded by the MaNoVe functions.

Details

Superceded by the MaNoVe functions (which run in C code).

Value

A wp class object which contains the select basis. All packets that are not in the basis get replaced by vectors of NAs.

Author(s)

G P Nason

See Also

MaNoVe

Description

Note: that this function is not for direct user use. This function is a helper routine for the AvBasis.wst function which is the one that should be used by users.

This function works by recursion, essentially it merges the current levels C coefficients from one packet shift with its associated D coefficients, does the same for the other packet shift and then averages the two reconstructions to provide the C coefficients for the next level up.

Usage

av.basis(wst, level, ix1, ix2, filter)

Arguments

Details

Description says all, see help page for AvBasis.wst.

Value

Returns the average basis reconstruction of a wst.object.

Author(s)

G P Nason

See Also

AvBasis, AvBasis.wst, conbar, rotateback, getpacket

Description

Average of whole collection of basis functions.

This function is generic.

Particular methods exist. For the wst class object this generic function uses AvBasis.wst. In the future we hope to add methods for wp and wpst class objects.

Usage

AvBasis(...)

Arguments

Details

See individual method help pages for operation and examples.

Value

A vector containing the average of the representation over all bases.

RELEASE

Version 3.6.0 Copyright Guy Nason 1995

Author(s)

G P Nason

See Also

wst, wst.object, AvBasis.wst

Description

Perform basis averaging for (packet-ordered) non-decimated wavelet transform.

Usage

## S3 method for class 'wst'
AvBasis(wst, Ccode=TRUE, ...)

Arguments

Details

The packet-ordered non-decimated wavelet transform computed by wst computes the coefficients of an input vector with respect to a library of all shifts of wavelet basis functions at all scales. Here "all shifts" means all integral shifts with respect to the finest scale coefficients, and "all scales" means all dyadic scales from 0 (the coarsest) to J-1 (the finest) where 2^J = n where n is the number of data points of the input vector. As such the packet-ordered non-decimated wavelet transform contains a library of all possible shifted wavelet bases.

Basis selection It is possible to select a particular basis and invert that particular representation. In WaveThresh a basis is selected by creating a nv (node.vector) class object which identifies the basis. Then the function InvBasis takes the wavelet representation and the node.vector and inverts the representation with respect to the selected basis. The two functions MaNoVe and numtonv create a node.vector: the first by using a Coifman-Wickerhauser minimum entropy best-basis algorithm and the second by basis index.

Basis averaging. Rather than select a basis it is often useful to preserve information from all of the bases. For examples, in curve estimation, after thresholding a wavelet representation the coefficients are coefficients of an estimate of the truth with respect to all of the shifted basis functions. Rather than select one of them we can average over all estimates. This sometimes gives a better curve estimate and can, for examples, get rid of Gibbs effects. See Coifman and Donoho (1995) for more information on how to do curve estimation using the packet ordered non-decimated wavelet transform, thresholding and basis averaging.

Further it might seem that inverting each wavelet transform and averaging might be a computationally expensive operation: since each wavelet inversion costs order $n$ operations and there are n different bases and so you might think that the overall order is $n^2$. It turns out that since many of the coarser scale basis functions are duplicated between bases there is redundancy in the non-decimated transform. Coifman and Donoho's TI-denoising algorithm makes use of this redundancy which results in an algorithm which only takes order $n \log n$ operations.

For an examples of denoising using the packet-ordered non-decimated wavelet transform and basis averaging see Johnstone and Silverman, 1997. The WaveThresh implementation of the basis averaging algorithm is to be found in Nason and Silverman, 1995

Value

A vector containing the average of the wavelet representation over all the basis functions. The length of the vector is 2^nlev where nlev is the number of levels in the input wst object.

RELEASE

Version 3.6.0 Copyright Guy Nason 1995

Author(s)

G P Nason

See Also

av.basis, wst, wst.object, MaNoVe, numtonv, InvBasis, wavegrow

Examples

#
# Generate some test data
#
test.data <- example.1()$y
#
# Now take the packet-ordered non-decimated wavelet transform
#
tdwst <- wst(test.data)
#
# Now "invert" it using basis averaging
#
tdwstAB <- AvBasis(tdwst)
#
# Let's compare it to the original
#
sum( (tdwstAB - test.data)^2)
#
# [1] 9.819351e-17 
#
# Very small. They're essentially same.
#
# See the threshold.wst help page for an
# an examples of using basis averaging in curve estimation.

Description

Perform basis averaging for (packet-ordered) 2D non-decimated wavelet transform.

Usage

## S3 method for class 'wst2D'
AvBasis(wst2D, ...)

Arguments

Details

The packet-ordered 2D non-decimated wavelet transform computed by wst2D computes the coefficients of an input matrix with respect to a library of all shifts of wavelet basis functions at all scales. Here "all shifts" means all integral shifts with respect to the finest scale coefficients with shifts in both the horizontal and vertical directions, and "all scales" means all dyadic scales from 0 (the coarsest) to J-1 (the finest) where 2^J = n where n is the dimension of the input matrix. As such the packet-ordered 2D non-decimated wavelet transform contains a library of all possible shifted wavelet bases.

Basis averaging. Rather than select a basis it is often useful to preserve information from all of the bases. For examples, in curve estimation, after thresholding, the coefficients are coefficients of an estimate of the truth with respect to all of the shifted basis functions. Rather than select one of them we can average over all estimates. This sometimes gives a better curve estimate and can, for examples, get rid of Gibbs effects. See Coifman and Donoho (1995) for more information on how to do curve estimation using the packet ordered non-decimated wavelet transform, thresholding and basis averaging. See Lang et al. (1995) for further details of surface/image estimation using the 2D non-decimated DWT.

Value

A square matrix of dimension $2^nlevelsWT$ containing the average-basis “reconstruction” of the wst2D object.

RELEASE

Version 3.9 Copyright Guy Nason 1998

Author(s)

G P Nason

See Also

wst2D, wst2D.object

Examples

#
# Generate some test data
#
#test.data <- matrix(rnorm(16), 4,4)
#
# Now take the 2D packet ordered DWT 
#
#tdwst2D <- wst2D(test.data)
#
# Now "invert" it using basis averaging
#
#tdwstAB <- AvBasis(tdwst2D)
#
# Let's compare it to the original
#
#sum( (tdwstAB - test.data)^2)
#
# [1] 1.61215e-17
#
# Very small. They're essentially same.
#

Description

Two linked medical time series containing 2048 observations sampled every 16 seconds recorded from 21:17:59 to 06:27:18. Both these time series were recorded from the same 66 day old infant by Prof. Peter Fleming, Dr Andrew Sawczenko and Jeanine Young of the Institute of Child Health, Royal Hospital for Sick Children, Bristol. BabyECG, is a record of the infant's heart rate (in beats per minute). BabySS is a record of the infant's sleep state on a scale of 1 to 4 as determined by a trained expert monitoring EEG (brain) and EOG (eye-movement). The sleep state codes are 1=quiet sleep, 2=between quiet and active sleep, 3=active sleep, 4=awake.

Format

The BabyECG time series is a nice examples of a non-stationary time series whose spectral (time-scale) properties vary over time. The function ewspec can be used to anaylse this time series to inspect the variation in the power of the series over time and scales.

The BabySS time series is a useful independent time series that can be associated with changing power in the BabyECG series. See the discussion in Nason, von Sachs and Kroisandt.

RELEASE

Version 3.9 Copyright Guy Nason 1998

SEE ALSO

ewspec

Author(s)

G P Nason

Source

Institute of Child Health, Royal Hospital for Sick Children, Bristol.

References

Nason, G.P., von Sachs, R. and Kroisandt, G. (1998). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. Technical Report, Department of Mathematics University of Bristol/ Fachbereich Mathematik, Kaiserslautern.

Examples

data(BabyECG)
data(BabySS)
#
# Plot the BabyECG data with BabySS overlaid
#

# Note the following code does some clever scaling to get the two

# time series overlaid.

#

myhrs <- c(22, 23, 24, 25, 26, 27, 28, 29, 30)

mylab <- c("22", "23", "00", "01", "02", "03", "04", "05", "06")

initsecs <- 59 + 60 * (17 + 60 * 21)

mysecs <- (myhrs * 3600)

secsat <- (mysecs - initsecs)/16

mxy <- max(BabyECG)

mny <- min(BabyECG)

ro <- range(BabySS)

no <- ((mxy - mny) * (BabySS - ro[1]))/(ro[2] - ro[1]) + mny

rc <- 0:4

nc <- ((mxy - mny) * (rc - ro[1]))/(ro[2] - ro[1]) + mny

## Not run: plot(1:length(BabyECG), BabyECG, xaxt = "n", type = "l", xlab = 
	"Time (hours)", ylab = "Heart rate (beats per minute)")
## End(Not run)

## Not run: lines(1:length(BabyECG), no, lty = 3)

## Not run: axis(1, at = secsat, labels = mylab)

## Not run: axis(4, at = nc, labels = as.character(rc))

#

# Sleep state is the right hand axis

#
#

Description

Two linked medical time series containing 2048 observations sampled every 16 seconds recorded from 21:17:59 to 06:27:18. Both these time series were recorded from the same 66 day old infant by Prof. Peter Fleming, Dr Andrew Sawczenko and Jeanine Young of the Institute of Child Health, Royal Hospital for Sick Children, Bristol. BabyECG, is a record of the infant's heart rate (in beats per minute). BabySS is a record of the infant's sleep state on a scale of 1 to 4 as determined by a trained expert monitoring EEG (brain) and EOG (eye-movement). The sleep state codes are 1=quiet sleep, 2=between quiet and active sleep, 3=active sleep, 4=awake.

Format

The BabyECG time series is a nice examples of a non-stationary time series whose spectral (time-scale) properties vary over time. The function ewspec can be used to anaylse this time series to inspect the variation in the power of the series over time and scales.

The BabySS time series is a useful independent time series that can be associated with changing power in the BabyECG series. See the discussion in Nason, von Sachs and Kroisandt.

RELEASE

Version 3.9 Copyright Guy Nason 1998

SEE ALSO

ewspec

Author(s)

G P Nason

Source

Institute of Child Health, Royal Hospital for Sick Children, Bristol.

References

Nason, G.P., von Sachs, R. and Kroisandt, G. (1998). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. Technical Report, Department of Mathematics University of Bristol/ Fachbereich Mathematik, Kaiserslautern.

Examples

data(BabyECG)
data(BabySS)
#

# Plot the BabyECG data with BabySS overlaid

#

# Note the following code does some clever scaling to get the two

# time series overlaid.

#

myhrs <- c(22, 23, 24, 25, 26, 27, 28, 29, 30)

mylab <- c("22", "23", "00", "01", "02", "03", "04", "05", "06")

initsecs <- 59 + 60 * (17 + 60 * 21)

mysecs <- (myhrs * 3600)

secsat <- (mysecs - initsecs)/16

mxy <- max(BabyECG)

mny <- min(BabyECG)

ro <- range(BabySS)

no <- ((mxy - mny) * (BabySS - ro[1]))/(ro[2] - ro[1]) + mny

rc <- 0:4

nc <- ((mxy - mny) * (rc - ro[1]))/(ro[2] - ro[1]) + mny

## Not run: plot(1:length(BabyECG), BabyECG, xaxt = "n", type = "l", xlab = 
	"Time (hours)", ylab = "Heart rate (beats per minute)")
## End(Not run)

## Not run: lines(1:length(BabyECG), no, lty = 3)

## Not run: axis(1, at = secsat, labels = mylab)

## Not run: axis(4, at = nc, labels = as.character(rc))

#

# Sleep state is the right hand axis

#
#

Description

Plots a representation of a time-frequency plane and then plots the locations, and sometimes time series representations of coefficients, for the packets in the basis.

Usage

basisplot(x, ...)

Arguments

Details

Description says all

Value

Nothing, usually

Author(s)

G P Nason

See Also

basisplot.BP, basisplot.wp

Description

The x objects store basis information obtained through the makewpstDO object. This function plots where the basis packets are on the time frequency plane.

Usage

## S3 method for class 'BP'
basisplot(x, num=min(10, length(BP$level)), ...)

Arguments

Details

Description says all

Value

Nothing of note

Author(s)

G P Nason

See Also

makewpstDO,Best1DCols

Examples

#
# See example in help for \code{\link{makewpstDO}}
#

Description

Note, one or two (depending on the state of draw.mode) graphics windows with mouse-clickable interfaces have to open to use this function.

Graphically select a wavelet packet basis associated with a wavelet packet object. Left-click selects packets, right click exits the routine.

Usage

## S3 method for class 'wp'
basisplot(x, draw.mode=FALSE, ...)

Arguments

Details

A wavelet packet basis described in WaveThresh using the node vector object (class from MaNoVe.wp) which for wavelet packets is nvwp. This function takes a wp.object object and graphically depicts all possible basis function locations. The user is then invited to click on different packets, these change colour. When finished, the user right clicks on the graphic and the selected basis is returned.

Note that the routine does not check to see whether the basis is legal. You have to do this. A legal basis can select packets from different levels, however you can't select packets that both cover the same packet index, however every packet index has to be covered.

A better function would check basis legality!

Value

An object of class nvwp which contains the specification for the basis.

Author(s)

G P Nason

See Also

addpkt, InvBasis, MaNoVe.wp, plotpkt, wp

Description

This function carries out Bayesian wavelet thresholding of noisy data, using the BayesThresh method of Abramovich, Sapatinas, & Silverman (1998).

Usage

BAYES.THR(data, alpha = 0.5, beta = 1, filter.number = 8, family =
"DaubLeAsymm", bc = "periodic", dev = var, j0 = 5, plotfn = FALSE)

Arguments

Details

A mixture prior consisting of a zero-mean normal distribution and a point mass at zero is placed on each wavelet coefficient. The empirical coefficients are then calculated and the priors updated to give posterior distributions for each coefficient. The thresholded value of each coefficient is the median of that coefficient's posterior distribution. See Abramovich, Sapatinas, & Silverman (1998) for more details of the procedure; the help page for threshold.wd has more information about wavelet thresholding in general.

The function wave.band uses the same priors to compute posterior credible intervals for the regression function, using the method described by Barber, Nason, & Silverman (2001).

Value

A vector containing the thresholded estimate of the function from which the data was drawn.

RELEASE

3.9.5 Code by Fanis Sapatinas/Felix Abramovich Documentation by Stuart Barber

Author(s)

G P Nason

See Also

threshold.wd, wd

Examples

#
# Generate some noisy test data and plot it.
#
blocks.data <- DJ.EX(n=512, noisy=TRUE)$blocks
#
# Now try BAYES.THR with the default parameters.
#
blocks.thr <- BAYES.THR(blocks.data, plotfn=TRUE)
#
# The default wavelet is Daubechies' least asymmetric wavelet 
# with 8 vanishing moments; quite a smooth wavelet. Since the 
# flat sections are still rather noisy, try Haar wavelets:
# 
blocks.thr <- BAYES.THR(blocks.data, plotfn=TRUE, filter.number=1,
        family = "DaubExPhase") 
#
# To show the importance of a sensible prior, consider alpha = 4, 
# beta = 1 (which implies a smoother prior than the default). 
#
blocks.thr <- BAYES.THR(blocks.data, plotfn=TRUE, filter.number=1,
        family = "DaubExPhase", alpha=4, beta=1)
#
# Here, the extreme values of the function are being smoothed towards zero.
#

Description

This function takes the whole set of nondecimated wavelet packets and selects those packets that correlate best with the "response" groups. The idea is to reduce the large dimensionality (number of packets) into something more manageable which can then be fed into a proper discriminator.

Usage

Best1DCols(w2d, mincor=    0.69999999999999996)

Arguments

Details

This function is not intended for direct user use. In this function, the w2d object contains a matrix where each column contains the coefficients of a single packet from a non-decimated wavelet packet transform. The number of rows of the matrix is the same as the original time series and hence each column can be correlated with a separate groups variable that contains the group membership of a separate variable which changes over time. Those packet columns that have correlation greater than the mincor value are extracted and returned in the BasisMatrix item of the returned list.

Value

A list with the following components:

Author(s)

G P Nason

See Also

makewpstDO,wpst2discr

Description

This function is used when you have a huge number of packets where you want to identify which ones are, individually, candidates for the good prediction of a response

Usage

bestm(w2mobj, y, percentage = 50)

Arguments

Details

This function naively addresses a very common problem. The object w2mobj contains a huge number of variables which might shed some light on the response object y. The problem is that the dimensionality of w2mobj is larger than that of the length of the series y.

The solution here is to choose a large, but not huge, subset of the variables that might be potentially useful in correlating with y, discard the rest, and return the "best" or preselected variables. Then the dimensionality is reduced and more sophisticated methods can be used to perform better quality modelling of the response y on the packets in w2mobj.

Value

A list of class w2m with the following components:

Author(s)

G P Nason

See Also

makewpstRO

Description

Function actually performs discrimination on reduced variable set supplied to it from Best1DCols function.

Usage

BMdiscr(BP)

Arguments

Details

Not intended for direct user use

Value

Returns a list of objects: essentially the input argument BP and the return value from a call to the lda function which performs the discrimination operation.

Author(s)

G P Nason

See Also

Best1DCols,makewpstDO

Description

Not designed, or really useful, for casual user use!

For example: take the integer 5. In binary this is 101. Then, this representation in base 4 is 16+1 =17.

This function is used by accessD.wpst to help it access coefficients.

Usage

c2to4(index)

Arguments

Details

Description says all

Value

The converted number

Author(s)

G P Nason

See Also

accessD.wpst

Examples

c2to4(5)

Description

Not used any more. This function used to interrogate the display device to see whether more than one color could be used. The function is set to return true whether of not the display device actually has this capability. It is used in the plot.wp function.

Usage

CanUseMoreThanOneColor()

Details

Description says it all.

Value

This function always returns TRUE

Author(s)

G P Nason

See Also

plot.wp

Description

Given a LSW spectrum this function simulates nsim realizations, estimates the spectrum, and then averages the results. The large sample averages should converge to the original spectrum.

Usage

checkmyews(spec, nsim=10)

Arguments

Value

A LSW spectrum obtained as the average of nsim simulations from the spec spectrum.

Author(s)

G P Nason

See Also

cns,LSWsim, ewspec

Description

A subsidiary routine for denproj. Not intended for direct user use.

Usage

Chires5(x, tau=1, J, filter.number=10, family="DaubLeAsymm", nT=20)

Arguments

Details

As description

Value

A list with the following components:

Author(s)

David Herrick

See Also

Chires6,denproj

Description

Function is essentially the same as Chires5 but also returns covariances between coefficients. A subsidiary routine for denproj. Not intended for direct user use.

Usage

Chires6(x, tau=1, J, filter.number=10, family="DaubLeAsymm", nT=20)

Arguments

Details

As description

Value

A list with the following components:

Author(s)

David Herrick

See Also

Chires6,denproj

Description

Part of a two-stage function suite designed to simulate locally stationary wavelet processes in conjunction with the LSWsim function.

Usage

cns(n, filter.number=1, family="DaubExPhase")

Arguments

Details

This simple routine merely computes the time-ordered non-decimated wavelet transform of a zero vector of the same length as the eventual simulated series that you wish to produce.

If you look at this routine you will see that it is extremely simple. First, it checks to see whether the n that you supplied is a power of two. If it is then it creates a zero vector of that length. This is then non-decimated wavelet transformed with the appropriate wavelet.

The output can then be processed and then finally supplied to LSWsim for process simulation.

Value

An object of class: wd, and, in fact, of the non-decimated variety. All wavelet coefficients of this are zero.

Author(s)

G P Nason

See Also

LSWsim, ewspec

Description

Compares two filters (such as those returned from filter.select). This function returns TRUE is they are the same otherwise returns FALSE.

Usage

compare.filters(f1,f2)

Arguments

Details

A very simple function. It only needs to check that the family and filter.number components of the filter are the same.

Value

If f1 and f2 are the same the function returns TRUE, otherwise it returns FALSE.

RELEASE

Version 3.9 Copyright Guy Nason 1998

Author(s)

G P Nason

See Also

filter.select.

Examples

#
# Create three filters!
#
filt1 <- filter.select(4, family="DaubExPhase")
filt2 <- filter.select(3, family="DaubExPhase")
filt3 <- filter.select(4, family="DaubLeAsymm")
#
# Now let us see if they are the same...
#
compare.filters(filt1, filt2)
# [1] FALSE
compare.filters(filt1, filt3)
# [1] FALSE
compare.filters(filt2, filt3)
# [1] FALSE
#
# Nope, (what a surprise) they weren't. How about
#
compare.filters(filt1, filt1)
# [1] TRUE
#
# Yes, they were the same!

Description

Computes the empirical shift required for time-ordered non-decimated transform coefficients to bring them into time order.

Usage

compgrot(J, filter.number, family)

Arguments

Details

Time-ordered non-decimated transform coefficients when raw are not in exact time alignment due to the phase of the underlying wavelet. This function returns the shifts that are necessary to apply to each resolution level in the transform to bring back each set of time-ordered coefficients into time alignment. Note that the shifts returned are approximate shifts which work for any Daubechies wavelet. More accurate shifts can be computed using detailed knowledge of the particular wavelet used.

Each shift is "to the left". I.e. higher indexed coefficients should take the place of lower-indexed coefficients. Periodic boundaries are assumed.

This realignment is mentioned in Walden and Contreras Cristan, (1997) and Nason, Sapatinas and Sawczenko, (1998).

Value

A vector containing the shifts that need to be applied to each scale level to return them to the correct time alignment.

There are J entries in the vector. The first entry corresponds to the shift required for the finest level coefficients (i.e. level J-1) and the last entry corresponds to the coarsest level (i.e. level 0). Entry j corresponds to the shift required for scale level J-j.

RELEASE

Version 3.6 Copyright Guy Nason 1997

Note

GROT was the shop started by Reginald Perrin. Unfortunately, GROT stands for "Guy ROTation".

Author(s)

G P Nason

See Also

wst, wst.object, wpst, wpst.object.

Examples

#
# Let's see how the resolution levels have to be shifted 
#
compgrot(4, filter.number=10, family="DaubExPhase")
#[1]  2  6 15 31
#
# In other words. Scale level 3 needs to be shifted two units.
# Scale level 2 needs to be shifted 6 units
# Scale level 1 needs to be shifted 15 units
# Scale level 0 needs to be shifted 31 units.

Description

Compress objects.

This function is generic.

Particular methods exist. For the imwd class object this generic function uses compress.imwd. There is a default compression method: compress.default that works on vectors.

Usage

compress(...)

Arguments

Details

See individual method help pages for operation and examples

Value

A compressed version of the input.

RELEASE

Version 2.0 Copyright Guy Nason 1993

Author(s)

G P Nason

See Also

compress.default, compress.imwd, imwd, imwd.object, imwdc.object, threshold.imwd

Description

Efficiently compress a vector containing many zeroes.

Usage

## Default S3 method:
compress(v, verbose=FALSE,...)

Arguments

Details

Images are large objects. Thresholded 2d wavelet objects (imwd) are also large, but many of their elements are zero. compress.default takes a vector, decides whether compression is necessary and if it is makes an object of class compressed containing the nonzero elements and their position in the original vector.

The decision whether to compress the vector or not depends on two things, first the number of non-zero elements in the vector (r, say), and second the length of the vector (n, say). Since the position and value of the non-zero elements is stored we will need to store 2r values for the non-zero elements. So compression takes place if 2r < n.

This function is the default method for the generic function compress. It can be invoked by calling compress for an object of the appropriate class, or directly by calling compress.default regardless of the class of the object.

Value

An object of class compressed if compression took place, otherwise a an object of class uncompressed.

RELEASE

Version 3.5.3 Copyright Guy Nason 1994

Note

Sometimes the compressed object can be larger than the original. This usually only happens for small objects, so it doesn't really matter.

Author(s)

G P Nason

See Also

compress, imwd, threshold.imwd, uncompress

Examples

#
# Compress a vector with lots of zeroes
#
compress(c(rep(0,100),99))
#$position:
#[1] 101
#
#$values:
#[1] 99
#
#$original.length:
#[1] 101
#
#attr(, "class"):
#[1] "compressed"
#
# Try to compress a vector with not many zeroes
#
compress(1:10)
#$vector:
#[1]  1 2  3  4  5  6  7  8  9 10
#
#attr(, "class"):
#[1] "uncompressed"
#
#

Description

Compress a (thresholded) imwd class object by removing zeroes.

Usage

## S3 method for class 'imwd'
compress(x, verbose=FALSE, ...)

Arguments

Details

Thresholded imwd objects are usually very large and contain many zero elements. This function compresses these objects into smaller imwd objects by using the compress.default function which removing the zeroes. This function is a method for the generic function compress for class imwd objects. It can be invoked by calling compress for an object of the appropriate class, or directly by calling compress.imwd regardless of the class of the object

Value

An object of type "imwdc" representing the compressed imwd object.

RELEASE

Version 3.5.3 Copyright Guy Nason 1994

Author(s)

G P Nason

See Also

compress, compress.default, imwd, imwd.object, imwdc.object, threshold.imwd.

Examples

#
# The user shouldn't need to use this function directly as the
# \code{\link{threshold.imwd}} function calls it
# automatically.
#

Description

Wrapper to the C function conbar which is the main function in WaveThresh to do filter convolution/reconstruction with data. Although users use the wr function to perform a complete inverse discrete wavelet transform (DWT) this function repeatedly uses the conbar routine, once for each level to reconstruct the next finest level. The C conbar routine is possibly the most frequently utilized by WaveThresh.

Usage

conbar(c.in, d.in, filter)

Arguments

Details

The wr function performs the inverse wavelet transform on an wd.object class object.

Internally, the wr function uses the C conbar function. Other functions also make use of conbar and some R functions also would benefit from using the fast C code of the conbar reconstruction hence this WaveThresh function. Some of the other functions that use conbar are listed in the SEE ALSO section. Many other functions call C code that then uses the C version of conbar.

Value

A vector containing the reconstructed coefficients.

Author(s)

G P Nason

See Also

av.basis InvBasis.wp wr

Examples

#
# Let's generate some test data, just some 32 normal variates.
#
v <- rnorm(32)
#
# Now take the wavelet transform with default filter arguments (which
# are filter.number=10, family="DaubLeAsymm")
#
vwd <- wd(v)
#
# Now, let's take an arbitrary level, say 2, and reconstruct level 3
# scaling function coefficients
#
c.in <- accessC(vwd, lev=2)
d.in <- accessD(vwd, lev=2)
#
conbar(c.in, d.in, filter.select(filter.number=10, family="DaubLeAsymm"))
#[1] -0.50368115  0.04738620 -0.90331807  1.08497622  0.90490528  0.06252717
#[7]  2.55894899 -1.26067508
#
# Ok, this was the pure reconstruction from using only level 2 information.
#
# Let's check this against the "original" level 3 coefficients (which get
# stored on the decomposition step in wd)
#
accessC(vwd, lev=3)
#[1] -0.50368115  0.04738620 -0.90331807  1.08497622  0.90490528  0.06252717
#[7]  2.55894899 -1.26067508
#
# Yep, the same numbers!
#

Description

Convert one type of wavelet object into another.

This function is generic.

Particular methods exist:

convert.wd is used to convert non-decimated wd objects into wst objects. convert.wst is used to convert wst objects into non-decimated wd objects.

Usage

convert(...)

Arguments

Details

See individual method help pages for operation and examples.

Value

An object containing the converted representation.

RELEASE

Version 3.6.0 Copyright Guy Nason 1995

Author(s)

G P Nason

See Also

convert.wd, convert.wst, wst, wst.object, wst, wst.object.

Description

Convert a time-ordered non-decimated wavelet transform object into a packet-ordered non-decimated wavelet transform object.

Usage

## S3 method for class 'wd'
convert(wd, ...)

Arguments

Details

In WaveThresh3 a non-decimated wavelet transform can be ordered in two different ways: as a time-ordered or packet-ordered representation. The coefficients in the two objects are exactly the same it is just their internal representation and ordering which is different. The two different representations are useful in different situations. The packet-ordering is useful for curve estimation applications and the time-ordering is useful for time series applications.

See Nason, Sapatinas and Sawczenko, 1998 for further details on ordering and weaving.

Note that the input object must be of the non-decimated type. In other words the type component of the input object must BE "station". Once the input object has been converted the output can be used with any of the functions suitable for the wst.object.

The getarrvec function actually computes the permutation to weave coefficients from one ordering to another.

Value

An object of class wst containing exactly the same information as the input object but ordered differently as a packet-ordered object.

RELEASE

Version 3.6 Copyright Guy Nason 1997

Author(s)

G P Nason

See Also

convert, getarrvec, levarr, wd, wd.object, wst, wst.object.

Examples

#
# Generate a sequence of 32 random normals (say) and take their
# \code{time-ordered non-decimated wavelet transform}
#
myrand <- wd(rnorm(32), type="station")
#
# Print out the result (to verify the class and type of the object)
#
#myrand
#Class 'wd' : Discrete Wavelet Transform Object:
#       ~~  : List with 8 components with names
#              C D nlevelsWT fl.dbase filter type bc date 
#
#$ C and $ D are LONG coefficient vectors !
#
#Created on : Tue Sep 29 12:17:53 1998 
#Type of decomposition:  station 
#
#summary(.):
#----------
#Levels:  5 
#Length of original:  32 
#Filter was:  Daub cmpct on least asymm N=10 
#Boundary handling:  periodic 
#Transform type:  station 
#Date:  Tue Sep 29 12:17:53 1998 
#
# Yep, the myrand object is of class: \code{\link{wd.object}}.
#
# Now let's convert it to class \code{\link{wst}}. The object
# gets returned and, as usual in S, is printed.
#
convert(myrand)
#Class 'wst' : Stationary Wavelet Transform Object:
#       ~~~  : List with 5 components with names
#              wp Carray nlevelsWT filter date 
#
#$wp and $Carray are the coefficient matrices
#
#Created on : Tue Sep 29 12:17:53 1998 
#
#summary(.):
#----------
#Levels:  5 
#Length of original:  32 
#Filter was:  Daub cmpct on least asymm N=10 
#Date:  Tue Sep 29 12:17:53 1998 
#
# Yes. The returned object is of class \code{\link{wst.object}}.
# I.e. it has been converted successfully.

Description

Convert a packed-ordered non-decimated wavelet transform object into a time-ordered non-decimated wavelet transform object.

Usage

## S3 method for class 'wst'
convert(wst, ...)

Arguments

Details

In WaveThresh3 a non-decimated wavelet transform can be ordered in two different ways: as a time-ordered or packet-ordered representation. The coefficients in the two objects are exactly the same it is just their internal representation and ordering which is different. The two different representations are useful in different situations. The packet-ordering is useful for curve estimation applications and the time-ordering is useful for time series applications.

See Nason, Sapatinas and Sawczenko, 1998 for further details on ordering and weaving.

Note that the input object must be of the non-decimated type. In other words the type component of the input object must be "station". Once the input object has been converted the output can be used with any of the functions suitable for the wd.object.

The actual weaving permutation for shuffling coefficients from one representation to another is achieved by the getarrvec function.

Value

An object of class wd containing exactly the same information as the input object but ordered differently as a packet-ordered object.

RELEASE

Version 3.6 Copyright Guy Nason 1997

Author(s)

G P Nason

See Also

convert, getarrvec, levarr, wd, wd.object, wst, wst.object.

Examples

#
# Generate a sequence of 32 random normals (say) and take their
# \code{packed-ordered non-decimated wavelet transform}
#
myrand <- wst(rnorm(32))
#
# Print out the result (to verify the class and type of the object)
#
#myrand
#Class 'wst' : Stationary Wavelet Transform Object:
#       ~~~  : List with 8 components with names
#              wp Carray nlevelsWT filter date 
#
#$WP and $Carray are the coefficient matrices

#
#Created on : Tue Sep 29 12:29:45 1998 
#
#summary(.):
#----------
#Levels:  5 
#Length of original:  32 
#Filter was:  Daub cmpct on least asymm N=10 
#Boundary handling:  periodic 

#Date:  Tue Sep 29 12:29:45 1998 
#
# Yep, the myrand object is of class: \code{\link{wst.object}}.
#
# Now let's convert it to class \code{\link{wd}}. The object
# gets returned and, as usual in S, is printed.
#
convert(myrand)
#Class 'wd' : Discrete Wavelet Transform Object:
#       ~~  : List with 8 components with names
#              C D nlevelsWT fl.dbase filter type bc date 
#
#$ C and $ D are LONG coefficient vectors !
#
#Created on : Tue Sep 29 12:29:45 1998 
#Type of decomposition:  station 
#
#summary(.):
#----------
#Levels:  5 
#Length of original:  32 
#Filter was:  Daub cmpct on least asymm N=10 
#Boundary handling:  periodic 
#Transform type:  station 
#Date:  Tue Sep 29 12:29:45 1998 
#
# The returned object is of class \code{\link{wd}} with a
# type of "station".
# I.e. it has been converted successfully.

Description

Print out text message about an object being from an old version of WaveThresh.

Usage

ConvertMessage()

Arguments

None

Details

Description says all!

Value

None

Author(s)

G P Nason

See Also

IsEarly.default,IsEarly

Description

Crsswav is called by WaveletCV which is itself called by threshold.wd to carry out its cross-validation policy.

Usage

Crsswav(noisy, value = 1, filter.number = 10, family = "DaubLeAsymm",
    thresh.type = "hard", ll = 3)

Arguments

Details

Description says all

Value

Same value as for rsswav

Author(s)

G P Nason

See Also

rsswav, WaveletCV

Description

Implements the multiwavelet style and empirical Bayes shrinkage procedures described in Barber & Nason (2004)

Usage

cthresh(data, j0 = 3, dwwt, dev = madmad, rule = "hard", 
    filter.number = 3.1, family = "LinaMayrand", plotfn = FALSE,  
    TI = FALSE, details = FALSE, policy = "mws", code = "NAG", tol = 0.01)

Arguments

Details

If a real-valued signal is decomposed using a complex-valued wavelet (like the Lina-Mayrand wavelets supplied by filter.select), then the wavelet coefficients are also complex-valued. Wavelet shrinkage can still be used to estimate the signal, by asking the question "which coefficients are small (and represent noise) and which are large (and represent signal)?" Two methods of determining which coefficients are small and which are large are proposed by Barber & Nason (2004). One is "multiwavelet style" thresholding (similar to that in Downie & Silverman (1998) where the coefficients are treated like the coefficients of a multiwavelet). Here, the "size" of the wavelet coefficient is determined as modulus of a standardised version of the coefficient. The standardisation is by the square root of the covariance matrix of the coefficient. A Bayesian method is to place a mixture prior on each coefficient. The prior has two components: a bivariate normal and a point mass at (0,0). The parameters are determined by an empirical Bayes argument and then the prior is updated by the data.

Value

Either a vector containing the estimated signal (if details = FALSE), or a list with the following components:

RELEASE

Part of the CThresh addon to WaveThresh. Copyright Stuart Barber and Guy Nason 2004.

Note

The estimates returned by cthresh have an imaginary component. In practice, this component is usually negligible.

Author(s)

Stuart Barber

See Also

filter.select, find.parameters, make.dwwt, test.dataCT, and the undocumented functions in CThresh.

Examples

#
# Make up some noisy data
#
y <- example.1()$y
ynoise <- y + rnorm(512, sd=0.1)
#
# Do complex-valued wavelet shrinkage with decimated wavelets
#
est1 <- cthresh(ynoise, TI=FALSE)
#
# Do complex-valued wavelet shrinkage with nondecimated wavelets
#
est2 <- cthresh(ynoise, TI=TRUE)
#
#
#
plot(1:512, y, lty=2, type="l")
lines(1:512, est1, col=2)
lines(1:512, est2, col=3)

Description

A routine that calls a C code function to do thresholding. This is really a test routine to call a C thresholding function (Cthreshold) and the user is advised to use the R based generic thresholding function threshold and/or its methods as they contain a wider range of thresholding options.

Usage

Cthreshold(wd, thresh.type = "soft", value = 0, levels = 3:(nlevelsWT(wd) - 1))

Arguments

Details

For general use it is recommended to use the threshold functions as they have a wider variety of options and also work for more complex varieties of wavelet transforms (i.e. non-decimated, complex-valued, etc). However, in the right, limited, situation this function can be useful. This function directly calls the C thresholding function Cthreshold(). The C function is used by routines that operate on behalf of the function that carries out two-fold cross validation in C (CWCV) which is also accessible using the policy="cv" option too threshold.wd

This function can be used by the user. It might be a bit faster than threshold.wd but mostly because it is simpler and does less checking than threshold.wd.

Value

A wd.object class object, but containing thresholded coefficients.

Author(s)

G P Nason

See Also

threshold

Examples

#
# See copious examples in the help to threshold.wd
#

Description

This function implements the density estimator with hard thresholding described by Hall, P. and Patil, P. (1995) Formulae for mean integrated squared error of nonlinear wavelet-based density estimators, Ann. Statist., 23, 905-928.

Usage

CWavDE(x, Jmax, threshold=0, nout=100, primary.resolution=1, filter.number=10,
	family="DaubLeAsymm", verbose=0, SF=NULL, WV=NULL)

Arguments

Details

As the description.

Value

A list containing the following components:

Author(s)

G P Nason

Examples

#
# Let's generate a bi-modal artificial set of data.
#
x <- c( rnorm(100), rnorm(100, 10))
#
# Now perform simple wavelet density estimate
#
wde <- CWavDE(x, Jmax=10, threshold=1)
#
# Plot results
#
## Not run: plot(wde$x, wde$y, type="l")

Description

Two-fold wavelet shrinkage cross-validation (in C)

Usage

CWCV(ynoise, ll, x = 1:length(ynoise), filter.number = 10, family = 
	"DaubLeAsymm", thresh.type = "soft", tol = 0.01,
	maxits=500, verbose = 0, 
	plot.it = TRUE, interptype = "noise")

Arguments

Details

Compute the two-fold cross-validated wavelet shrunk estimate given the noisy data ynoise according to the description given in Nason, 1996.

You must specify a primary resolution given by ll. This must be specified individually on each data set and can itself be estimated using cross-validation (although I haven't written the code to do this).

Note. The two-fold cross-validation method performs very badly if the input data is correlated. In this case I would advise using the methods proposed in Donoho and Johnstone, 1995 or Johnstone and Silverman, 1997 which can be carried out in WaveThresh using the threshold function using the policy="sure" option.

Value

A list with the following components

RELEASE

Version 3.0 Copyright Guy Nason 1994

Note

Plots of the universal and cross-validated shrunk estimates might be plotted if plot.it=TRUE.

Author(s)

G P Nason

See Also

threshold. threshold.wd.

Examples

#
# This function is best used via the policy="cv" option in
# the threshold.wd function.
# See examples there.
#

Description

Random generation, density and cumulative probability for the claw distribution.

Usage

rclaw(n)
dclaw(x)
pclaw(q)

Arguments

Details

The claw distribution is a normal mixture distribution, introduced in Marron & Wand (1992). Marron, J.S. & Wand, M.P. (1992). Exact Mean Integrated Squared Error. Ann. Stat., 20, 712–736.

Value

Random samples (rclaw), density (dclaw) or probability (pclaw) of the claw distribution.

Author(s)

David Herrick

Examples

# Plot the claw density on the interval [-3,3]
x <- seq(from=-3, to=3, length=500)
## Not run: plot(x, dclaw(x), type="l")

Description

Calculates the variances of the empirical wavelet coefficients by performing a 2D wavelet decomposition on the covariance matrix of the empirical scaling function coefficients of the probability density function.

Usage

dencvwd(hrproj, filter.number=hrproj$filter$filter.number,
    family=hrproj$filter$family, type="wavelet", bc="zero",
    firstk=hrproj$klim, RetFather=TRUE, verbose=FALSE)

Arguments

Details

This function is basically imwd adapted to handle zero boundary conditions, except that only the variances are returned, i.e. the diagonals of the covariance matrices produced. Note that this code is not very efficient. The full covariance matrices of all levels of coefficients are calculated, and then the diagonals are extracted.

Value

An object of class wd.object, but the contents are not a standard wavelet transform, ie the object is used to hold other information which organisationally is arranged like a wavelet tranform, ie variances of coefficients.

Author(s)

David Herrick

See Also

denproj,imwd

Examples

# Simulate data from the claw density, find the 
# empirical scaling function coefficients and covariances and then decompose
# both to give wavelet coefficients and their variances.

data <- rclaw(100)
datahr <- denproj(data, J=8, filter.number=2,family="DaubExPhase", covar=TRUE)
data.wd <- denwd(datahr)
## Not run: plotdenwd(data.wd, top.level=(datahr$res$J-1))

datavar <- dencvwd(datahr)
## Not run: plotdenwd(datavar, top.level=(datahr$res$J-1))

Description

Calculates plotting information for a wavelet density estimate from high level scaling function coefficients.

Usage

denplot(wr, coef, nT=20, lims, n=50)

Arguments

Details

The density estimate is evaluated at n points between the values in lims. This function can be used to plot the empirical scaling function density estimate by entering wr=coef$coef, but since the empirical coefficients are usually found at some very high resolution, such a plot will be very noisy and not very informative. This function will be of much more use as and when thresholding function are included in this density estimation package.

Value

A list with components:

Author(s)

David Herrick

See Also

denproj,rclaw

Examples

# Simulate data from the claw density and find the 
# empirical scaling function coefficients at a lowish resolution and plot
# the resulting density estimate
data <- rclaw(100)
datahr <- denproj(data, J=3, filter.number=2,family="DaubExPhase")
datapl <- denplot(datahr$coef, datahr, lims=c(-3,3), n=1000)
## Not run: plot(datapl, type="l")

Description

Calculates empirical scaling function coefficients of the probability density function from sample of data from that density, usually at some "high" resoloution.

Usage

denproj(x, tau=1, J, filter.number=10, family="DaubLeAsymm", covar=FALSE, nT=20)

Arguments

Details

This projection of data onto a high resolution wavelet space is described in detail in Chapter 3 of Herrick (2000). The maximum and minimum values of k for which the empirical scaling function coefficient is non-zero are determined and the coefficients calculated for all k between these limits as sum(phiJk(xi))/n. The scaling functions are evaluated at the data points efficiently, using the Daubechies-Lagarias algorithm (Daubechies & Lagarias (1992)). Coded kindly by Brani Vidakovic.

Herrick, D.R.M. (2000) Wavelet Methods for Curve and Surface Estimation. PhD Thesis, University of Bristol.

Daubechies, I. & Lagarias, J.C. (1992). Two-Scale Difference Equations II. Local Regularity, Infinite Products of Matrices and Fractals. SIAM Journal on Mathematical Analysis, 24(4), 1031–1079.

Value

A list with components:

Author(s)

David Herrick

See Also

Chires5, Chires6, denwd, denwr

Examples

# Simulate data from the claw density and find the 
# empirical scaling function coefficients
data <- rclaw(100)
datahr <- denproj(data, J=8, filter.number=4,family="DaubLeAsymm")

Description

Performs wavelet decomposition on the empirical scaling function coefficients of the probability density function.

Usage

denwd(coef)

Arguments

Details

The empirical scaling function coefficients are decomposed using the DWT with zero boundary conditions.

Value

An object of class wd.object

Author(s)

David Herrick

See Also

denproj,plotdenwd,wd, denwr

Examples

# Simulate data from the claw density, find the empirical
# scaling function coefficients, decompose them and plot
# the resulting wavelet coefficients
 
data <- rclaw(100)
datahr <- denproj(data, J=8, filter.number=2,family="DaubExPhase")
data.wd <- denwd(datahr)
## Not run: plotdenwd(data.wd, top.level=(datahr$res$J-1))

Description

Performs wavelet reconstruction for density estimation.

Usage

denwr(wd, start.level=0, verbose=FALSE, bc=wd$bc,
    return.object=FALSE, filter.number=wd$filter$filter.number,
    family=wd$filter$family)

Arguments

Details

This is the same as wr.wd, except that it can handle zero boundary conditions.

Value

Either a vector containing the top level reconstruction or an object of class wd.object containing the results of the reconstruction.

Author(s)

David Herrick

Description

Function to produce the blocks, bumps, Doppler and heavisine functions described by Donoho and Johnstone (1994).

Usage

DJ.EX(n=1024, signal=7, rsnr=7, noisy=FALSE, plotfn=FALSE)

Arguments

Details

The Donoho and Johnstone test functions were designed to reproduce various features to be found in real world signals such as jump discontinuities (blocks), spikes (the NMR-like bumps), varying frequency behaviour (Doppler) and jumps/spikes in smooth signals (heavisine). These functions are most often used for testing wavelet shrinkage methods and comparing them to other nonparametric regression techniques. (Donoho, D.L. and Johnstone, I.M. (1994), Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81, 425–455).

Another version of the Doppler function can be found in the standalone doppler function.

Another function for this purpose is the Piecewise Polynomial created in Nason and Silverman (1994) an encapsulated in WaveThresh by example.1 (Nason, G.P. and Silverman, B.W. (1994) The discrete wavelet transform in S, J. Comput. Graph. Statist., 3, 163–191.

NOTE: This function might not give exactly the same function values as the equivalent function in WaveLab

Value

A list with four components: blocks, bumps, heavi and doppler containing the sampled signal values for the four types of Donoho and Johnstone test functions. Each of these are deemed to be sampled on an equally spaced grid from 0 to 1.

Author(s)

Theofanis Sapatinas

See Also

doppler,example.1, threshold, wd

Examples

#
# Show a picture of the four test functions with the default args
#
## Not run: DJ.EX(plotfn=TRUE)

Description

Compute number of non-zero coefficients in wd object

Usage

dof(wd)

Arguments

Details

Very simple function that counts the number of non-zero coefficients in a wd class object.

Value

An integer that represents the number of non-zero coefficients in the input wd object.

RELEASE

Version 3.0 Copyright Guy Nason 1994

Author(s)

G P Nason

See Also

wd, wd.object, threshold, threshold.wd.

Examples

#
# Let's generate some purely random numbers!!
#
myrandom <- rnorm(512)
#
# Take the discrete wavelet transform
#
myrandomWD <- wd(myrandom)
#
# How many coefficients are non-zero?
#
dof(myrandomWD)
# [1] 512
#
# All of them were nonzero!
#
# Threshold it
#
myrandomWDT <- threshold(myrandomWD, policy="universal")
#
# Now lets see how many are nonzero
#
dof(myrandomWDT)
# [1] 8
#
# Wow so 504 of the coefficients were set to zero! Spooky!
#

Description

This function evaluates and returns the Doppler signal from Donoho and Johnstone, (1994).

Usage

doppler(t)

Arguments

Details

This function evaluates and returns the Doppler signal from Donoho and Johnstone, (1994). (Donoho, D.L. and Johnstone, I.M. (1994), Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81, 425–455).

Another version of this function can be found in DJ.EX.

Value

A vector of the same length as the input vector containing the Doppler signal at t

Author(s)

Arne Kovac

See Also

DJ.EX

Examples

#
# Evalute the Doppler signal at 100 arbitrarily spaced points.
#
tt <- sort(runif(100))
dopp <- doppler(tt)
## Not run: plot(tt, dopp, type="l")

Description

Draws the mother wavelet or scaling function associated with an object.

This function is generic.

Particular methods exist. The following functions are used for the following objects:

imwd.object

the draw.imwd function is used.

imwdc.object

the draw.imwdc function is used.

wd.object

the draw.wd function is used.

wp.object

the draw.wp function is used.

wst.object

the draw.wst function is used.

All of the above method functions use the draw.default function which is the function which actually does the drawing.

Usage

draw(...)

Arguments

Details

See individual method help pages for operation and examples.

Value

If the plot.it argument is supplied then the draw functions tend to return the coordinates of what they were meant to draw and don't actually draw anything.

RELEASE

Version 2 Copyright Guy Nason 1993

Author(s)

G P Nason

See Also

draw.default, draw.imwd, draw.imwdc, draw.wd, draw.wp, draw.wst, imwd.object, imwdc.object, wd.object, wp.object, wst.object.

Description

This function can produce pictures of one- or two-dimensional wavelets or scaling functions at various levels of resolution.

Usage

## Default S3 method:
draw(filter.number = 10, family = "DaubLeAsymm", resolution = 8192,
    verbose = FALSE, plot.it = TRUE, main = "Wavelet Picture", sub = zwd$
    filter$name, xlab = "x", ylab = "psi", dimension = 1, twodplot
     = persp, enhance = TRUE, efactor = 0.05, scaling.function = FALSE, 
	type="l",
    ...)

Arguments

Details

The algorithm underlying this function produces a picture of the wavelet or scaling function by first building a wavelet decomposition object of the correct size (i.e. correct resolution) and setting all entries equal to zero. Then one coefficient at a carefully selected resolution level is set to one and the decomposition is inverted to obtain a picture of the wavelet.

Value

If plot.it=FALSE then usually a list containing coordinates of the object that would have been plotted is returned. This can be useful if you don't want S-Plus to do the plotting or you wish to use the coordinates of the wavelets for other purposes.

RELEASE

Version 3.5.3 Copyright Guy Nason 1994

Note

A plot is produced of the wavelet or scaling function if plot.it=TRUE.

Author(s)

G P Nason

See Also

filter.select, ScalingFunction,wd, wd.object, wr, wr.wd.

Examples

#
# First make sure that your favourite graphics device is open
# otherwise S-Plus will complain.
#
# Let's draw a one-dimensional Daubechies least-asymmetric wavelet
# N=10
#
## Not run: draw.default(filter.number=10, family="DaubLeAsymm")
#
# Wow. What a great picture!
#
# Now how about a one-dimensional Daubechies extremal-phase scaling function
# with N=2
#
## Not run: draw.default(filter.number=2, family="DaubExPhase")
#
# Excellent! Now how about a two-dimensional Daubechies least-asymmetric
# N=6 wavelet
#
# N.b. we'll also reduce the resolution down a bit. If we used the default
# resolution of 8192 this would be probably too much for most computers.
#
## Not run: draw.default(filter.number=6, family="DaubLeAsymm", dimension=2, res=256)
#
# What a pretty picture!

Description

This function draws the mother wavelet associated with an imwd.object — a two-dimensional wavelet decomposition.

Usage

## S3 method for class 'imwd'
draw(wd, resolution=128, ...)

Arguments

Details

This function extracts the filter component from the imwd object (which is constructed using the filter.select function) to decide which wavelet to draw. Once decided the draw.default function is used to actually do the drawing.

Value

If the plot.it argument is set to TRUE then nothing is returned. Otherwise, as with draw.default, the coordinates of what would have been plotted are returned.

RELEASE

Version 2 Copyright Guy Nason 1993

Note

If the plot.it argument isTRUE (which it is by default) a plot of the mother wavelet or scaling function is plotted on the active graphics device.

Author(s)

G P Nason

See Also

filter.select, imwd.object, draw.default.

Examples

#
# Let's use the lennon test image
#
data(lennon)
## Not run: image(lennon)
#
# Now let's do the 2D discrete wavelet transform using Daubechies'
# least-asymmetric wavelet N=6
#
lwd <- imwd(lennon, filter.number=6)
#
# And now draw the wavelet that did this transform
#
## Not run: draw(lwd)
#
# A nice little two-dimensional wavelet!
#

Description

This function draws the mother wavelet associated with an imwdc.object — a compressed two-dimensional wavelet decomposition.

Usage

## S3 method for class 'imwdc'
draw(wd, resolution=128, ...)

Arguments

Details

This function extracts the filter component from the imwd object (which is constructed using the filter.select function) to decide which wavelet to draw. Once decided the draw.default function is used to actually do the drawing.

Value

If the plot.it argument is set to TRUE then nothing is returned. Otherwise, as with draw.default, the coordinates of what would have been plotted are returned.

RELEASE

Version 2 Copyright Guy Nason 1993

Note

If the plot.it argument is TRUE (which it is by default) a plot of the mother wavelet or scaling function is plotted on the active graphics device.

Author(s)

G P Nason

See Also

filter.select, imwdc.object, draw.default.

Examples

#
# Let's use the lennon test image
#
data(lennon)
## Not run: image(lennon)
#
# Now let's do the 2D discrete wavelet transform using Daubechies'
# least-asymmetric wavelet N=6
#
lwd <- imwd(lennon, filter.number=6)
#
# Now let's threshold the 2D DWT
# The resultant class of object is imwdc object.
#
lwdT <- threshold(lwd)
#
# And now draw the wavelet that did this transform
#
## Not run: draw(lwdT)
#
# A nice little two-dimensional wavelet!
#

Description

Draws picture of one wavelet or scaling function associated with the multiple wavelet decomposition object. mwd.object.

Usage

## S3 method for class 'mwd'
draw(mwd, phi = 0, psi = 0, return.funct = FALSE, ...)

Arguments

Details

It is usual to specify just one of phi and psi. IF neither phi nor psi are specified then phi=1 is the default. An error is generated if both phi=0 and psi=0 or if both are nonzero.

Value

If the return.funct argument is set to TRUE then the function values in the plot are returned otherwise NULL is returned.

RELEASE

Version 3.9.6 (Although Copyright Tim Downie 1995-6).

Note

If the return.funct argument is FALSE a plot of the mother wavelet or scaling function is plotted on the active graphics device.

Author(s)

G P Nason

See Also

accessC.mwd, accessD.mwd, mfirst.last, mfilter.select, mwd, mwd.object, mwr, plot.mwd, print.mwd, putC.mwd, putD.mwd, summary.mwd, threshold.mwd, wd, wr.mwd.

Examples

#
# Do a multiple wavelet decomposition on vector: ynoise
#
ynoise <- rnorm(512, sd = 0.1)
ymwd <- mwd(ynoise,filter.type="Geronimo")
#
# Draw a picture of the second Geronimo wavelet.
#
## Not run: draw(ymwd,psi=2)
#
#

Description

This function draws the mother wavelet or scaling function associated with a wd.object.

Usage

## S3 method for class 'wd'
draw(wd, ...)

Arguments

Details

This function extracts the filter component from the wd object (which is constructed using the filter.select function) to decide which wavelet to draw. Once decided the draw.default function is used to actually do the drawing.

Value

If the plot.it argument is set to TRUE then nothing is returned. Otherwise, as with draw.default, the coordinates of what would have been plotted are returned.

RELEASE

Version 2 Copyright Guy Nason 1993

Note

If the plot.it argument is TRUE (which it is by default) a plot of the mother wavelet or scaling function is plotted on the active graphics device.

Author(s)

G P Nason

See Also

filter.select, wd.object, draw.default.

Examples

#
# Generate some test data
#
test.data <- example.1()$y
## Not run: ts.plot(test.data)
#
# Now do the discrete wavelet transform of the data using the Daubechies
# least-asymmetric wavelet N=10 (the default arguments in
# wd).
#
tdwd <- wd(test.data)
#
# What happens if we try to draw this new tdwd object?
#
## Not run: draw(tdwd)
#
# We get a picture of the wavelet that did the transform
#

Description

This function draws a wavelet packet associated with a wp.object.

Usage

## S3 method for class 'wp'
draw(wp, level, index, plot.it=TRUE, main, sub, xlab, ylab, ...)

Arguments

Details

This function extracts the filter component from the wp object (which is constructed using the filter.select function) to decide which wavelet packet family to draw. Once decided the drawwp.default function is used to actually do the drawing.

Value

If the plot.it argument is set to TRUE then nothing is returned. Otherwise, if plot.it is set to FALSE the coordinates of what would have been plotted are returned.

RELEASE

Version 3.9.6 Copyright Guy Nason 1998

Note

If the plot.it argument is TRUE (which it is by default) a plot of the appropriate wavelet packet is plotted on the active graphics device.

Author(s)

G P Nason

See Also

filter.select, wp, wp.object, drawwp.default.

Examples

#
# Generate some test data
#
test.data <- example.1()$y
## Not run: ts.plot(test.data)
#
# Now do the wavelet packet transform of the data using the Daubechies
# least-asymmetric wavelet N=10 (the default arguments in
# wp).
#
tdwp <- wp(test.data)
#
# What happens if we try to draw this new tdwp object?
#
## Not run: draw(tdwd, level=4, index=12)

Description

This function draws the mother wavelet or scaling function associated with a wst.object.

Usage

## S3 method for class 'wst'
draw(wst, ...)

Arguments

Details

This function extracts the filter component from the wst object (which is constructed using the filter.select function) to decide which wavelet packet family to draw. Once decided the draw.default function is used to actually do the drawing.

Value

If the plot.it argument is set to TRUE then nothing is returned. Otherwise, Otherwise, as with draw.default, the coordinates of what would have been plotted are returned.

RELEASE

Version 3.6 Copyright Guy Nason 1995

Note

If the plot.it argument is TRUE (which it is by default) a plot of the appropriate wavelet packet is plotted on the active graphics device.

Author(s)

G P Nason

See Also

filter.select, wst.object, draw.default.

Examples

#
# Generate some test data
#
test.data <- example.1()$y
## Not run: ts.plot(test.data)
#
# Now do the \code{packet-ordered non-decimated DWT}  of the data using the Daubechies
# least-asymmetric wavelet N=10 (the default arguments in \code{\link{wst}}).
#
tdwst <- wst(test.data)
#
# What happens if we try to draw this new tdwst object?
#
## Not run: draw(tdwst)
#
# We get a picture of the wavelet that did the transform
#

Description

Simply draws a box with bottom left corner at (x,y), or width w and height h with shading of density and colour of col.

Usage

drawbox(x,y,w,h,density,col)

Arguments

Details

Description says all

Value

None

Author(s)

G P Nason

See Also

addpkt

Description

Function computes the values of a given wavelet packet on a discrete grid.

Usage

drawwp.default(level, index, filter.number = 10, family = "DaubLeAsymm",
    resolution = 64 * 2^level)

Arguments

Details

Function works by computing a wavelet packet transform of a zero vector. Then inserting a single one somewhere in the desired packet, and then inverts the transform.

Value

A vector containing the "y" values of the required wavelet packet.

Author(s)

G P Nason

See Also

draw.wp,InvBasis, nlevelsWT, putpacket, wp

Description

This function computes the evolutionary wavelet spectrum (EWS) estimate from a time series (or non-decimated wavelet transform of a time series). The estimate is computed by taking the non-decimated wavelet transform of the time series data, taking its modulus; smoothing using TI-wavelet shrinkage and then correction for the redundancy caused by use of the non-decimated wavelet transform. Options below beginning with smooth. are passed directly to the TI-wavelet shrinkage routines.

Usage

ewspec(x, filter.number = 10, family = "DaubLeAsymm",
    UseLocalSpec = TRUE, DoSWT = TRUE, WPsmooth = TRUE,
    verbose = FALSE, smooth.filter.number = 10,
    smooth.family = "DaubLeAsymm",
    smooth.levels = 3:(nlevelsWT(WPwst) - 1), smooth.dev = madmad,
    smooth.policy = "LSuniversal", smooth.value = 0, smooth.by.level = FALSE,
    smooth.type = "soft", smooth.verbose = FALSE,
    smooth.cvtol = 0.01, smooth.cvnorm = l2norm,
    smooth.transform = I, smooth.inverse = I)

Arguments

Details

This function computes an estimate of the evolutionary wavelet spectrum of a time series according to the paper by Nason, von Sachs and Kroisandt. The function works as follows:

1

The non-decimated wavelet transform of the series is computed.

2

The squared modulus of the non-decimated wavelet transform is computed (this is the raw wavelet periodogram, which is returned).

3

The squared modulus is smoothed using TI-wavelet shrinkage.

4

The smoothed coefficients are corrected using the inverse of the inner product matrix of the discrete non-decimated autocorrelation wavelets (produced using the ipndacw function).

To display the EWS use the plotfunction on the S component, see the examples below.

It is possible to supply the non-decimated wavelet transform of the time series and set DoSWT=F or to supply the squared modulus of the non-decimated wavelet transform using LocalSpec and setting UseLocalSpec=F. This facility saves time because the function is then only used for smoothing and correction.

Value

A list with the following components:

RELEASE

Version 3.9 Copyright Guy Nason 1998

Author(s)

G P Nason

References

Nason, G.P., von Sachs, R. and Kroisandt, G. (1998). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. Technical Report, Department of Mathematics University of Bristol/ Fachbereich Mathematik, Kaiserslautern.

See Also

Baby Data, filter.select, ipndacw, LocalSpec, threshold wd wd.object

Examples

#
# Apply the EWS estimate function to the baby data
#

Description

This function computes and returns the coordinates of the piecewise polynomial described by Nason and Silverman, 1994. This function is a useful test function for evaluating wavelet shrinkage methodology as it contains smooth parts, a discontinuity and it is periodic.

(Nason, G.P. and Silverman, B.W. (1994) The discrete wavelet transform in S, J. Comput. Graph. Statist., 3, 163–191.)

Usage

example.1()

Arguments

None

Details

This function computes and returns the x and y coordinates of the piecewise polynomial function described in Nason and Silverman, 1994. The formula for the piecewise polynomial (which is piecewise cubic) is given in Nason and Silverman, 1994.

The piecewise polynomial returned is a discrete sample on 512 equally spaced points between 0 and 1 (including 0 but excluding 1).

The Donoho and Johnstone test functions can be generated using the DJ.EX function.

Value

A list with two components:

Author(s)

G P Nason

See Also

DJ.EX

Examples

#
# Generate the piecewise polynomial 
#
test.data <- example.1()$y
## Not run: ts.plot(test.data)

Description

This function stores the filter coefficients necessary for doing a discrete wavelet transform (and its inverse), including complex-valued compactly supported wavelets.

Usage

filter.select(filter.number, family="DaubLeAsymm", constant=1)

Arguments

Details

This function contains at least three types of filter. Two types can be selected with family set to DaubExPhase, these wavelets are the Haar wavelet (selected by filter.number=1 within this family) and Daubechies “extremal phase” wavelets selected by filter.numbers ranging from 2 to 10). Setting family to DaubLeAsymm gives you Daubechies least asymmetric wavelets, but here the filter number ranges from 4 to 10. For Daubechies wavelets, filter.number corresponds to the N of that paper, the wavelets become more regular as the filter.number increases, but they are all of compact support.

With family equal to “Coiflets” the function supports filter numbers ranging from 1 to 5. Coiflets are wavelets where the scaling function also has vanishing moments.

With family equal to “LinaMayrand”, the function returns complex-valued Daubechies wavelets. For odd numbers of vanishing moments, there are symmetric complex-valued wavelets i this family, and for five or more vanishing moments there are multiple distinct complex-valued wavelets, distinguished by their (arbitrary) solution number. At present, Lina-Mayrand wavelets 3.1, 4.1, 5.1, 5.2, 5.3, and 5.4 are available in WaveThresh.

Setting family equal to “Lawton” chooses complex-valued wavelets. The only wavelet available is the one with “filter.number” equal to 3.

With family equal to “LittlewoodPaley” the Littlewood-Paley wavelet is used. The scaling function is also the same as (or at least proportional to, depending on your normalization) that of the Shannon scaling function, so its an approximation to the Shannon wavelet transform. The “filter.number” argument has a special meaning for the Littlewood-Paley wavelets: it does not represent vanishing moments here. Instead, it controls the number of filter taps in the quadrature mirror filter: typically longer values are better, up to the length of the series. Increasing it higher than the length of the series does not usually have much effect. Note: extreme caution should be taken with the Littlewood-Paley wavelet. This implementation is pure time-domain and as such can only be thought of as an approximation to a complete Shannon/LP implementation. For example, in actuality the wavelets are NOT finite impluse response filters like with Daubechies wavelets. This means that it is possible for an infinite number of Littlewood Paley wavelet coefficients to be nonzero. However, computers can not store an infinite number of coefficients and some will be lost. This is most noticeable with functions with discontinuities and other homogeneities but it can also happen with some smooth functions. A way to check how "bad" is can be is to transform your desired function followed immediately by the inverse transform and compare the original with the resultant sequence.

The function compare.filters can be used to compare two filters.

Value

Alist is returned with four components describing the filter:

RELEASE

Version 3.5.3 Copyright Guy Nason 1994, This version originally part of the cthresh release which was merged into wavethresh in Oct 2012. Original cthresh version due to Stuart Barber

Note

The (Daubechies) filter coefficients should always sum to sqrt(2). This is a useful check on their validity.

Author(s)

Stuart Barber and G P Nason

See Also

wd, wr, wr.wd, accessC, accessD, compare.filters, imwd, imwr, threshold, draw.

Examples

#This function is usually called by others.
#However, on occasion you may wish to look at the coefficients themselves.

#
# look at the filter coefficients for N=4 (by default Daubechies'
# least-asymmetric wavelets.)
#
filter.select(4)
#$H:
#[1] -0.07576571 -0.02963553  0.49761867  0.80373875  0.29785780
#[6] -0.09921954 -0.01260397  0.03222310
#
#$G:
#NULL
#
#$name:
#[1] "Daub cmpct on least asymm N=4"
#
#$family:
#[1] "DaubLeAsymm"
#
#$filter.number:
#[1] 4

Description

Estimate the prior parameters for the complex empirical Bayes shrinkage procedure.

Usage

find.parameters(data.wd, dwwt, j0, code, tol, Sigma)

Arguments

Details

The complex empirical Bayes (CEB) shrinkage procedure described by Barber & Nason (2004) places independent mixture priors on each complex-valued wavelet coefficient. This routine finds marginal maximum likelihood estimates of the prior parameters. If the NAG library is available, routine E04JYF is used otherwise the search is done using optimize (in R) or nlminb (in S-plus). In the latter case, the likelihood values should be computed externally using the C code supplied as part of the CThresh package - although a pure R / S-plus version is available, it is very slow. This function will not usually be called directly by the user, but is called from within cthresh.

Value

A list with the following components:

RELEASE

Part of the CThresh addon to WaveThresh. Copyright Stuart Barber and Guy Nason 2004.

Note

There may be warning messages from the NAG routine E04JYF. If the indicator variable IFAIL is equal to 5, 6, 7, or 8, then a solution has been found but there is doubt over the convergence. For IFAIL = 5, it is likely that the correct solution has been found, while IFAIL = 8 means that you should have little confidence in the parameter estimates. For more details, see the NAG software documentation available online at http://www.nag.co.uk/numeric/fl/manual19/pdf/E04/e04jyf_fl19.pdf

Author(s)

Stuart Barber

See Also

cthresh

Description

This function is not intended for user use, but is used by various functions involved in computing and displaying wavelet transforms. It basically constructs "bookeeping" vectors that WaveThresh uses for working out where coefficient vectors begin and end.

Usage

first.last(LengthH, DataLength, type, bc="periodic", current.scale=0)

Arguments

Details

Suppose you begin with $2^m=2048$ coefficients. At the next level you would expect 1024 smoothed data coefficients, and 1024 wavelet coefficients, and if bc="periodic" this is indeed what happens. However, if bc="symmetric" you actually need more than 1024 (as the wavelets extend over the edges). The first last database keeps track of where all these "extras" appear and also where they are located in the packed vectors C and D of pyramidal coefficients within wavelet structures.

For examples, given a first.last.c row of

$-2 3 20$

The actual coefficients would be

$c_{-2}, c_{-1}, c_{0}, c_{1}, c_{2}, c_{3}$