| Title: | Functions for Computing Wavelet Filters, Wavelet Transforms and Multiresolution Analyses |
|---|---|
| Description: | Contains functions for computing and plotting discrete wavelet transforms (DWT) and maximal overlap discrete wavelet transforms (MODWT), as well as their inverses. Additionally, it contains functionality for computing and plotting wavelet transform filters that are used in the above decompositions as well as multiresolution analyses. |
| Authors: | Eric Aldrich <[email protected]> |
| Maintainer: | Eric Aldrich <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.3-0.2 |
| Built: | 2024-10-09 06:33:00 UTC |
| Source: | CRAN |
Aligns both wavelet (high pass) and scaling (low pass) coefficients of
objects of class dwt, modwt, dwpt, and
modwpt using phase shift values computed by
wt.filter.shift.
align(wt, coe=FALSE, inverse=FALSE)align(wt, coe=FALSE, inverse=FALSE)
wt |
An object of class |
coe |
Logical value indicating whether to use center of energy method in computing phase shifts. |
inverse |
Logical value indicating whether to shift wavelet and scaling coefficients of an aligned object back to their original positions. |
An object of class dwt, modwt, is characterized as
'unaligned' if the value in the aligned slot is FALSE.
Similarly, these objects are classified as 'aligned' if the value in
the aligned slot is TRUE. Thus, align will only
operate on an 'unaligned' wavelet transform object if inverse =
FALSE and on an 'aligned' wavelet transform object if inverse
= TRUE.
The argument coe is passed to the wt.filter.shift
function to determine what method to use for computing phase shifts
(see documentation for wt.filter.shift).
Either an aligned or unaligned object of the same class as wt
(see Details above).
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and compute DWT newX <- cbind(X1,X2) wt <- dwt(newX, n.levels=3, boundary="reflection", fast=FALSE) # align wt.aligned <- align(wt)# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and compute DWT newX <- cbind(X1,X2) wt <- dwt(newX, n.levels=3, boundary="reflection", fast=FALSE) # align wt.aligned <- align(wt)
Computes the discrete wavelet transform coefficients for a univariate or multivariate time series.
dwt(X, filter="la8", n.levels, boundary="periodic", fast=TRUE)dwt(X, filter="la8", n.levels, boundary="periodic", fast=TRUE)
X |
A univariate or multivariate time series. Numeric vectors, matrices and data frames are also accepted. |
filter |
Either a |
n.levels |
An integer specifying the level of the decomposition.
By default this is the value J such that the length of |
boundary |
A character string indicating which boundary method to
use. |
fast |
A logical flag which, if true, indicates that the pyramid algorithm is computed with an internal C function. Otherwise, only R code is used in all computations. |
The discrete wavelet transform is computed via the pyramid
algorithm, using pseudocode written by Percival and Walden (2000),
pp. 100-101. When boundary="periodic" the resulting wavelet and
scaling coefficients are computed without making changes to the
original series - the pyramid algorithm treats X as if it is
circular. However, when boundary="reflection" a call is made to
extend.series, resulting in a new series which is reflected to
twice the length of the original series. The wavelet and scaling
coefficients are then computed by using a periodic boundary condition
on the reflected sereis, resulting in twice as many wavelet and
scaling coefficients at each level.
Returns an object of class dwt, which is an S4 object with
slots
W |
A list with element |
V |
A list with element |
filter |
A |
level |
An integer value representing the level of wavelet decomposition. |
n.boundary |
A numeric vector indicating the number of boundary coefficients at each level of the decomposition. |
boundary |
A character string indicating the boundary method used in the decomposition. Valid values are "periodic" or "reflection". |
series |
The original time series, |
class.X |
A character string indicating the class of the input
series. Possible values are |
attr.X |
A list containing the attributes information of the
original time series, |
aligned |
A logical value indicating whether the wavelet and
scaling coefficients have been phase shifted so as to be aligned
with relevant time information from the original series. The value
of this slot is initially FALSE and can only be changed to TRUE via
the |
coe |
A logical value indicating whether the center of energy
method was used in phase alignement of the wavelet and scaling
coefficients. By default, this value is FALSE (and will always be
FALSE when |
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and compute DWT newX <- cbind(X1,X2) wt <- dwt(newX, n.levels=3, boundary="reflection", fast=FALSE)# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and compute DWT newX <- cbind(X1,X2) wt <- dwt(newX, n.levels=3, boundary="reflection", fast=FALSE)
An S4 object containing discrete wavelet transform information.
A list with element comprised of a matrix containing
the th level wavelet coefficients.
A list with element comprised of a matrix containing
the th level scaling coefficients.
A wt.filter object containing information for
the filter used in the decomposition. See help(wt.filter) for
details.
An integer value representing the level of wavelet decomposition.
A numeric vector indicating the number of boundary coefficients at each level of the decomposition.
A character string indicating the boundary method used in the decomposition. Valid values are "periodic" or "reflection".
The original time series, X, in matrix format.
A character string indicating the class of the input
series. Possible values are "ts", "mts",
"numeric", "matrix", or "data.frame".
A list containing the attributes information of the
original time series, X. This is useful if X is an
object of class ts or mts and it is desired to retain
relevant time information. If the original time series, X, is
a matrix or has no attributes, then attr.X is an empty list.
A logical value indicating whether the wavelet and
scaling coefficients have been phase shifted so as to be aligned
with relevant time information from the original series. The value
of this slot is initially FALSE and can only be changed to TRUE via
the align function, with the dwt object as input.
A logical value indicating whether the center of energy
method was used in phase alignement of the wavelet and scaling
coefficients. By default, this value is FALSE (and will always be
FALSE when aligned is FALSE) and will be set to true if the
dwt object is phase shifted via the align function and
center of energy method.
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
dwt,
modwt,
modwt-class,
wt.filter.
Implementation of DWT and MODWT forward and backward pyramid algorithms.
dwt.forward(V, filter) dwt.backward(W, V, filter) modwt.forward(V, filter, j) modwt.backward(W, V, filter, j)dwt.forward(V, filter) dwt.backward(W, V, filter) modwt.forward(V, filter, j) modwt.backward(W, V, filter, j)
W |
A vector of wavelet coefficients. |
V |
A vector of scaling coefficients. |
filter |
A |
j |
The level of wavelet and scaling coefficients to compute (for forward algorithm) or the level of wavelet and scaling coefficient inputs (for inverse algorithm). |
An implementation of the DWT and MODWT forward and backward
pyramid algorithms using pseudocode written by Percival and Walden
(2000), pp. 100-101, 177-178. These functions are intended primarily
as helper functions for the dwt, modwt, idwt and
imodwt functions.
dwt.forward and modwt.forward return a list of two
elements containing vectors of wavelet and scaling coefficients for
the subsequent level of analysis. dwt.backward and
modwt.backward return a vector of scaling coefficients for the
previous level of analysis.
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # compute the LA8 wavelet filter for both DWT and MODWT la8.dwt <- wt.filter() la8.modwt <- wt.filter(modwt=TRUE) # compute the DWT and MODWT level one wavelet and scaling coefficients wt.dwt <- dwt.forward(X1, la8.dwt) wt.modwt <- modwt.forward(X2, la8.modwt, 1) # compute the original series with the level one coefficients newX.dwt <- dwt.backward(wt.dwt$W, wt.dwt$V, la8.dwt) newX.modwt <- modwt.backward(wt.modwt$W, wt.modwt$V, la8.modwt, 1)# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # compute the LA8 wavelet filter for both DWT and MODWT la8.dwt <- wt.filter() la8.modwt <- wt.filter(modwt=TRUE) # compute the DWT and MODWT level one wavelet and scaling coefficients wt.dwt <- dwt.forward(X1, la8.dwt) wt.modwt <- modwt.forward(X2, la8.modwt, 1) # compute the original series with the level one coefficients newX.dwt <- dwt.backward(wt.dwt$W, wt.dwt$V, la8.dwt) newX.modwt <- modwt.backward(wt.modwt$W, wt.modwt$V, la8.modwt, 1)
Electrocardiogram measurements for an individual with arhythmia. Observations were made over an 11.37 second interval at a rate of 180 samples per second. For more detailed information regarding the collection of this data, see Percival and Walden (2000), p. 125.
data(ecg)data(ecg)
A time series object containing 2048 observations.
http://www.ms.washington.edu/~s530/data.html
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press, sec. 4.10.
Extends a univariate or multivariate time series beyond its original length.
extend.series(X, method="reflection", length="double", n, j)extend.series(X, method="reflection", length="double", n, j)
X |
A univariate or multivariate time series. Numeric vectors, matrices and data frames are accepted. |
method |
A character string indicating which extension method to
use. Possible values are |
length |
A character string indicating how to determine the
length of the extended series. Possible values are
|
n |
An integer value specifying the length of the extended
series. This argument is only relevant when |
j |
An integer value specifying the power of two of which the
length of the extended series should be a multiple. This argument is
only relevant for |
The original time series, X, is extended to a new length
as determined by length: if length = "arbitrary", a value
for the argument n must be specified in order to extend the
series to length n; if length = "powerof2" the series
will be extended to the nearst multiple of , where j
must be specified as an argument to the function; if length =
"double" the series will be extended to twice its original length.
Once the length of the extended series is determined, the actual
extension is performed using one of five methods: if method =
"periodic" a periodic extension is made by concatenating the series
with itself (without changing the order of the values); if
method = "reflection" an extension is made by concatenating
the series with a reflection (reverse ordering) of itself; if
method = "zero" the series is padded with zeros; if
method = "mean" the series is padded with its mean; if
method = "reflection.inverse" an extension is made by
concatenating the series with a reflection (reverse ordering) of
itself that is further reflected of the horizontal axis
where is the last value of the observed series.
In the case that the series is extended beyond twice it's original
length, the "periodic" and "reflection" methods are
repetetively applied so that every subseries of length (the
length of the original series) is either a periodic or reflected
extension of any other adjacent values. Similarly, the
"reflection.inverse" method is repetetively applied so that
every values are a periodic extension of any adjacent
values.
The extended time series, in its original class.
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and extend using reflection method newX <- cbind(X1,X2) newX.ext1 <- extend.series(newX, length="arbitrary", n=64) plot.ts(newX.ext1) # apply another extension method newX.ext2 <- extend.series(newX, method="reflection.inverse", length="powerof2", j=6) plot.ts(newX.ext2)# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and extend using reflection method newX <- cbind(X1,X2) newX.ext1 <- extend.series(newX, length="arbitrary", n=64) plot.ts(newX.ext1) # apply another extension method newX.ext2 <- extend.series(newX, method="reflection.inverse", length="powerof2", j=6) plot.ts(newX.ext2)
Plots multiple DWT Wavelet or Scaling Filters similar to Figure 108 in Wavelet Methods for Time Series Analysis by Percival and Walden (2000).
figure108.wt.filter(filter.objects, level = 1, l = NULL, wavelet = TRUE)figure108.wt.filter(filter.objects, level = 1, l = NULL, wavelet = TRUE)
filter.objects |
List containing 'wt.filter' objects, character strings specifying a wavelet filter, or numeric vectors of wavelet coefficients. The list can contain a combination of 'wt.filter' objects, character strings, and numeric vectors. If only one filter is to be plotted, a single 'wt.filter' object, character string, or numeric vector may be supplied. See 'help(wt.filter)' for acceptable filter names. |
level |
If |
l |
Single integer representing the right hand limit of the
horizontal axis. If unspecified, it will default to the length
of the filter of greatest length given in
|
wavelet |
A logical flag indicating whether to plot the wavelet (high pass) or scaling (low pass) filter. |
The plotting space available for each filter is dictated by the value of greatest magnitude of all the filters plotted. The vertical plotting space for each level will then be 2 times the absolute value of this magnitude.
The filters are successively plotted in the order given in
filter.object, where the first filter in filter.object
is drawn at the top of the plot region, and the successive filters
are plotted below.
Kelvin Ma, [email protected]
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# Plotting the LA8 Wavelet Filter filter <- wt.filter() figure108.wt.filter(filter) # Alternatively figure108.wt.filter("la8") # Plotting the Haar, D4, D6 Wavelet Filters figure108.wt.filter(list("haar", "d4", "d6")) # Plotting the Haar, D4, D6 Scaling Filters figure108.wt.filter(list("haar", "d4", "d6"), wavelet = FALSE) # Alternatively haar <- wt.filter("haar") d6 <- wt.filter("d6") figure108.wt.filter(list(haar, "d4", d6), wavelet = FALSE) # Adding an "made up" filter (represented by c(1,-1,1,-1) figure108.wt.filter(list(haar, "d4", d6, c(1,-1,1,-1)), wavelet = FALSE)# Plotting the LA8 Wavelet Filter filter <- wt.filter() figure108.wt.filter(filter) # Alternatively figure108.wt.filter("la8") # Plotting the Haar, D4, D6 Wavelet Filters figure108.wt.filter(list("haar", "d4", "d6")) # Plotting the Haar, D4, D6 Scaling Filters figure108.wt.filter(list("haar", "d4", "d6"), wavelet = FALSE) # Alternatively haar <- wt.filter("haar") d6 <- wt.filter("d6") figure108.wt.filter(list(haar, "d4", d6), wavelet = FALSE) # Adding an "made up" filter (represented by c(1,-1,1,-1) figure108.wt.filter(list(haar, "d4", d6, c(1,-1,1,-1)), wavelet = FALSE)
Plots a DWT Wavelet or Scaling Filter for specific levels similar to Figure 98 in Wavelet Methods for Time Series Analysis by Percival and Walden (2000).
figure98.wt.filter(filter, levels = NULL, wavelet = TRUE, y.normalize = TRUE)figure98.wt.filter(filter, levels = NULL, wavelet = TRUE, y.normalize = TRUE)
filter |
Either a 'wt.filter' object, a character string specifying a wavelet filter, or a numeric vector of wavelet coefficients. See 'help(wt.filter)' for acceptable filter names. |
levels |
Number or vector indicating levels of filter to plot. See Details. |
wavelet |
A logical flag indicating whether to plot the wavelet (high pass) or scaling (low pass) filter. |
y.normalize |
A logical flag indicating whether to vertically normalize each level of the filter to the plotting space available for each level of the filter. |
If a single number is specified for levels, then the filter
of levels 1 through levels will be plotted.
Otherwise, a vector will specify which levels of the wavelet
coefficients will be plotted. If levels is not defined, it
will default to the vector 1:7.
The plotting space available for each level of the filter is dictated by the value of greatest magnitude of all the levels of the filter plotted. The vertical plotting space for each level will then be 2 times the absolute value of this magnitude.
The label 'L sub j' on the horizontal axis varies from level to level, but 'L sub j' indicates the length of a filter at level j. The filter vector at a given level j is indexed from 0 to 'L sub j' - 1. 'L sub j' is defined by formula 96a in Wavelet Methods for Time Series Analysis by Percival and Walden (2000).
Kelvin Ma, [email protected]
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# Plotting LA8 Wavelet Filter Coefficients Levels 1 through 7. filter <- wt.filter() figure98.wt.filter(filter) # Alternatively figure98.wt.filter("la8") # Plotting D4 Scaling Filter Coefficients Levels 1, 3, and 5 and not # vertically normalizing each level to its plotting region. figure98.wt.filter("d4", levels = c(1,3,5), wavelet = FALSE, y.normalize = FALSE)# Plotting LA8 Wavelet Filter Coefficients Levels 1 through 7. filter <- wt.filter() figure98.wt.filter(filter) # Alternatively figure98.wt.filter("la8") # Plotting D4 Scaling Filter Coefficients Levels 1, 3, and 5 and not # vertically normalizing each level to its plotting region. figure98.wt.filter("d4", levels = c(1,3,5), wavelet = FALSE, y.normalize = FALSE)
Computes the inverse discrete wavelet transform for a discrete wavelet transform that was obtained from a univariate or multivariate time series.
idwt(wt, fast=TRUE)idwt(wt, fast=TRUE)
wt |
A |
fast |
A logical flag which, if true, indicates that the inverse pyramid algorithm is computed with an internal C function. Otherwise, only R code is used in all computations. |
The inverse discrete wavelet transform is computed via the inverse pyramid algorithm, using pseudocode written by Percival and Walden (2000), p. 101.
An object with class and attributes equivalent to the original
series that was used to compute the DWT. In general, the output will
be equivalent to the original series (i.e. X = idwt(dwt(X))), however
when thresholding or shrinkage methods are used on the dwt
object, the output of idwt may differ from the original series.
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and compute DWT newX <- cbind(X1,X2) wt <- dwt(newX, n.levels=3, boundary="reflection") # compute the inverse DWT invX <- idwt(wt) # compare newX invX# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and compute DWT newX <- cbind(X1,X2) wt <- dwt(newX, n.levels=3, boundary="reflection") # compute the inverse DWT invX <- idwt(wt) # compare newX invX
Computes the inverse maximal overlap discrete wavelet transform for a maximal overlap discrete wavelet transform that was obtained from a univariate or multivariate time series.
imodwt(wt, fast=TRUE)imodwt(wt, fast=TRUE)
wt |
A |
fast |
A logical flag which, if true, indicates that the inverse pyramid algorithm is computed with an internal C function. Otherwise, only R code is used in all computations. |
The inverse discrete wavelet transform is computed via the inverse pyramid algorithm, using pseudocode written by Percival and Walden (2000), p. 101.
An object with class and attributes equivalent to the original
series that was used to compute the MODWT. In general, the output will
be equivalent to the original series (i.e. X = imodwt(modwt(X))), however
when thresholding or shrinkage methods are used on the modwt
object, the output of imodwt may differ from the original series.
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and compute MODWT newX <- cbind(X1,X2) wt <- modwt(newX, n.levels=3, boundary="reflection") # compute the inverse MODWT invX <- imodwt(wt) # compare newX invX# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and compute MODWT newX <- cbind(X1,X2) wt <- modwt(newX, n.levels=3, boundary="reflection") # compute the inverse MODWT invX <- imodwt(wt) # compare newX invX
Computes the maximal overlap discrete wavelet transform coefficients for a univariate or multivariate time series.
modwt(X, filter="la8", n.levels, boundary="periodic", fast=TRUE)modwt(X, filter="la8", n.levels, boundary="periodic", fast=TRUE)
X |
A univariate or multivariate time series. Numeric vectors, matrices and data frames are also accepted. |
filter |
Either a |
n.levels |
An integer specifying the level of the decomposition.
By default this is the value J such that the length of |
boundary |
A character string indicating which boundary method to
use. |
fast |
A logical flag which, if true, indicates that the pyramid algorithm is computed with an internal C function. Otherwise, only R code is used in all computations. |
The maximal overlap discrete wavelet transform is computed via
the pyramid algorithm, using pseudocode written by Percival and Walden
(2000), p. 178. When boundary="periodic" the resulting wavelet
and scaling coefficients are computed without making changes to the
original series - the pyramid algorithm treats X as if it is
circular. However, when boundary="reflection" a call is made to
extend.series, resulting in a new series which is reflected to
twice the length of the original series. The wavelet and scaling
coefficients are then computed by using a periodic boundary condition
on the reflected sereis, resulting in twice as many wavelet and
scaling coefficients at each level.
Returns an object of class modwt, which is an S4 object with
slots
W |
A list with element |
V |
A list with element |
filter |
A |
level |
An integer value representing the level of wavelet decomposition. |
n.boundary |
A numeric vector indicating the number of boundary coefficients at each level of the decomposition. |
boundary |
A character string indicating the boundary method used in the decomposition. Valid values are "periodic" or "reflection". |
series |
The original time series, |
class.X |
A character string indicating the class of the input
series. Possible values are |
attr.X |
A list containing the attributes information of the
original time series, |
aligned |
A logical value indicating whether the wavelet and
scaling coefficients have been phase shifted so as to be aligned
with relevant time information from the original series. The value
of this slot is initially FALSE and can only be changed to TRUE via
the |
coe |
A logical value indicating whether the center of energy
method was used in phase alignement of the wavelet and scaling
coefficients. By default, this value is FALSE (and will always be
FALSE when |
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and compute DWT newX <- cbind(X1,X2) wt <- dwt(newX, n.level=3, boundary="reflection", fast=FALSE)# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and compute DWT newX <- cbind(X1,X2) wt <- dwt(newX, n.level=3, boundary="reflection", fast=FALSE)
An S4 object containing maximal overlap discrete wavelet transform information.
A list with element comprised of a matrix containing
the th level wavelet coefficients.
A list with element comprised of a matrix containing
the th level scaling coefficients.
A wt.filter object containing information for
the filter used in the decomposition. See help(wt.filter) for
details.
An integer value representing the level of wavelet decomposition.
A numeric vector indicating the number of boundary coefficients at each level of the decomposition.
A character string indicating the boundary method used in the decomposition. Valid values are "periodic" or "reflection".
The original time series, X, in matrix format.
A character string indicating the class of the input
series. Possible values are "ts", "mts",
"numeric", "matrix", or "data.frame".
A list containing the attributes information of the
original time series, X. This is useful if X is an
object of class ts or mts and it is desired to retain
relevant time information. If the original time series, X, is
a matrix or has no attributes, then attr.X is an empty list.
A logical value indicating whether the wavelet and
scaling coefficients have been phase shifted so as to be aligned
with relevant time information from the original series. The value
of this slot is initially FALSE and can only be changed to TRUE via
the align function, with the modwt object as input.
A logical value indicating whether the center of energy
method was used in phase alignement of the wavelet and scaling
coefficients. By default, this value is FALSE (and will always be
FALSE when aligned is FALSE) and will be set to true if the
modwt object is phase shifted via the align function
and center of energy method.
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
dwt,
dwt-class,
modwt,
wt.filter.
Computes the multiresolution analysis for a univariate or multivariate time series.
mra(X, filter="la8", n.levels, boundary="periodic", fast=TRUE, method="dwt")mra(X, filter="la8", n.levels, boundary="periodic", fast=TRUE, method="dwt")
X |
A univariate or multivariate time series. Numeric vectors, matrices and data frames are also accepted. |
filter |
Either a |
n.levels |
An integer specifying the level of the decomposition.
By default this is the value J such that the length of |
boundary |
A character string indicating which boundary method to
use. |
fast |
A logical flag which, if true, indicates that the pyramid algorithm is computed with an internal C function. Otherwise, only R code is used in all computations. |
method |
A character string, taking values "dwt" or "modwt", that indicates which type of transform to use when computing the MRA. |
Returns an object of class mra, which is an S4 object with
slots
D |
A list with element |
S |
A list with element |
filter |
A |
level |
An integer value representing the level of wavelet decomposition. |
boundary |
A character string indicating the boundary method used in the wavelet decomposition. Valid values are "periodic" or "reflection". |
series |
The original time series, |
class.X |
A character string indicating the class of the input
series. Possible values are |
attr.X |
A list containing the attributes information of the
original time series, |
method |
A character string indicating which type of wavelet decomposition was performed (either "dwt" or "modwt"). |
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and compute MRA newX <- cbind(X1,X2) mra.out <- mra(newX, n.levels=3, boundary="reflection")# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and compute MRA newX <- cbind(X1,X2) mra.out <- mra(newX, n.levels=3, boundary="reflection")
An S4 object containing multiresolution analysis information.
A list with element comprised of a matrix containing
the th level wavelet detail.
A list with element comprised of a matrix containing
the th level wavelet smooths.
A wt.filter object containing information for
the filter used in the decomposition. See help(wt.filter) for
details.
An integer value representing the level of wavelet decomposition.
A character string indicating the boundary method used in the wavelet decomposition. Valid values are "periodic" or "reflection".
The original time series, X, in matrix format.
A character string indicating the class of the input
series. Possible values are "ts", "mts",
"numeric", "matrix", or "data.frame".
A list containing the attributes information of the
original time series, X. This is useful if X is an
object of class ts or mts and it is desired to retain
relevant time information. If the original time series, X, is
a matrix or has no attributes, then attr.X is an empty list.
A character string indicating which type of wavelet decomposition was performed (either "dwt" or "modwt").
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
Yearly minimum water level of the Nile River starting in 622 A.D. and ending in 1284 A.D. For more information see Percival and Walden (2000), p. 190.
data(nile)data(nile)
A time series object containing 683 observations.
http://www.ms.washington.edu/~s530/data.html
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press, sec. 5.9.
Measurements of vertical ocean sheer, obtained by dropping an instrument veritcally into the ocean and recording observations at 0.1 meter intervals. Starting depth for the series is 350.0 meters and ending depth is 1037.4 meters. For more detailed information regarding the collection of this data, see Percival and Walden (2000), p. 193.
data(ocean)data(ocean)
A time series object containing 6875 observations.
http://www.ms.washington.edu/~s530/data.html
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press, sec. 5.10.
Plot DWT wavelet and scaling coefficients.
## S3 method for class 'dwt' plot(x, levels = NULL, draw.boundary = FALSE, type = "stack", col.plot = "black", col.boundary = "red", X.xtick.at = NULL, X.ytick.at = NULL, Stack.xtick.at = NULL, Stack.ytick.at = NULL, X.xlab = "t", y.rlabs = TRUE, plot.X = TRUE, plot.W = TRUE, plot.V = TRUE, ...)## S3 method for class 'dwt' plot(x, levels = NULL, draw.boundary = FALSE, type = "stack", col.plot = "black", col.boundary = "red", X.xtick.at = NULL, X.ytick.at = NULL, Stack.xtick.at = NULL, Stack.ytick.at = NULL, X.xlab = "t", y.rlabs = TRUE, plot.X = TRUE, plot.W = TRUE, plot.V = TRUE, ...)
x |
An object of class |
levels |
Number, vector, or list of two vectors indicating range of levels to plot. See details. |
draw.boundary |
Logical value indicating whether to draw boundary coefficients. |
type |
Type of plot to draw. Currently only "stack" is implemented. |
col.plot |
Color of wavelet and scaling coefficients. |
col.boundary |
Color of boundary coefficient lines. |
X.xtick.at |
Vector specifying the extreme tick mark locations
and the number of intervals between those extreme tick marks on the
horizontal axis of the plot of the original times series. This
vector takes the form similar to par("xaxp"), and is defaulted to
par("xaxp") when |
X.ytick.at |
Vector specifying the extreme tick mark locations
and the number of intervals between those extreme tick marks on the
vertical axis of the plot of the original times series. This vector
takes the form similar to par("yaxp"), and is defaulted to
par("yaxp") when |
Stack.xtick.at |
Vector of form similar to |
Stack.ytick.at |
Vector of form similar to |
X.xlab |
String specifying the label of the horizontal axis of the plot of the original time series. |
y.rlabs |
Logical flag indicating whether to draw the vertical labels on the right vertical axis of the stacked plot. These labels indicate the number of boundary coefficients to the right of the right boundary line, and the level of decomposition of the wavelet (or scaling) coefficients. |
plot.X |
Logical flag indicating whether to draw original time series. |
plot.W |
Logical flag indicating whether to draw the wavelet coefficients in the stacked plot. |
plot.V |
Logical flag indicating whether to draw the scaling coefficients in the stacked plot. |
... |
Additional paramters that are acceptable arguments to the
generic |
plot.dwt plots the DWT wavelet and scaling coefficients. The
type parameter species the type of plot, which is currently
only stack. If the DWT object is defined for multiple time series,
only the data pertaining to the first time series of the DWT object
is plotted. Thus, only the wavelet coefficients and scaling
coefficients of the first time series of the DWT object will be
plotted.
If a single number is specified for levels, then the wavelet
coefficients of levels 1 through levels will be plotted.
Otherwise, a vector or the first element of a list will specify
which levels of the wavelet coefficients will be plotted.
Unless specified in the second element of a list, only one level
of scaling coefficients will be plotted and this level is equal
to the highest level of the wavelet coefficients plotted.
For each respective axis, the distance between a tick mark on the
plot of the original time series is equivalent to the distance
between a tick mark on the stacked plot. Thus, when altering the
relative spacing of the tick marks on the plot of the original time
series using X.xtick.at or X.ytick.at, the tick marks
of the stacked plot are automatically adjusted. If the plot of the
original time series is not drawn, then the user can alter the
spacing of the tick marks using Stack.xtick.at and
Stack.ytick.at for the horizontal and vertical axes,
respectively.
One of plot.W or plot.V must be TRUE.
Kelvin Ma, [email protected]
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
X <- rnorm(2048) dwtobj <- dwt(X) # Plotting wavelet coefficients of levels 1 through 6 and scaling # coefficients of level 6. plot.dwt(dwtobj, levels = 6) # Plotting wavelet coefficients of levels 1, 3, 5, and scaling # coefficients of levels 4 and 5, and green boundary coefficient lines. plot.dwt(dwtobj, levels = list(c(1,3,5),c(4,5)), draw.boundary = TRUE, col.boundary = "green") # Plotting wavelet coefficients of level 1 through 6 and not plotting # any scaling coefficients. plot.dwt(dwtobj, levels = 6, plot.V = FALSE)X <- rnorm(2048) dwtobj <- dwt(X) # Plotting wavelet coefficients of levels 1 through 6 and scaling # coefficients of level 6. plot.dwt(dwtobj, levels = 6) # Plotting wavelet coefficients of levels 1, 3, 5, and scaling # coefficients of levels 4 and 5, and green boundary coefficient lines. plot.dwt(dwtobj, levels = list(c(1,3,5),c(4,5)), draw.boundary = TRUE, col.boundary = "green") # Plotting wavelet coefficients of level 1 through 6 and not plotting # any scaling coefficients. plot.dwt(dwtobj, levels = 6, plot.V = FALSE)
Plot wavelet and scaling coefficients of multiple DWT objects.
## S3 method for class 'multiple' plot.dwt(x, levels = NULL, ylim = NULL, draw.dashed.lines = TRUE, draw.level.labels = TRUE, col = c("red","blue"), ...)## S3 method for class 'multiple' plot.dwt(x, levels = NULL, ylim = NULL, draw.dashed.lines = TRUE, draw.level.labels = TRUE, col = c("red","blue"), ...)
x |
A list of |
levels |
Number, vector, or list of two vectors indicating range of levels to plot. See details. |
ylim |
Vector specifying the lower and upper limits of the vertical y-axis of each plot. |
draw.dashed.lines |
Boolean indicating whether dashed lines should be drawn between levels of wavelet and scaling coefficients. |
draw.level.labels |
Boolean indicating whether the labels for the levels of wavelet and scaling coefficients should be drawn. |
col |
Vector of length 2 for the alternating colors of the coefficients to be drawn. The colors alternate by level of the wavelet or scaling coefficients drawn. The first element of the vector is the color of the coefficients of the first level drawn. |
... |
Additional paramters that are acceptable arguments to the
generic |
If a single number is specified for levels, then the wavelet
coefficients of levels 1 through levels will be plotted.
Otherwise, a vector or the first element of a list will specify
which levels of the wavelet coefficients will be plotted.
Unless specified in the second element of a list, only one level
of scaling coefficients will be plotted and this level is equal
to the highest level of the wavelet coefficients plotted. If a DWT
object is defined for multiple time series, only the data pertaining
to the first time series of the DWT object is plotted. Thus, only
the wavelet coefficients and scaling coefficients of the first time
series of the DWT objects will be plotted.
For each dwtobject in the list of x,
plot.dwt.multiple takes the coefficients of the dwt
object and concatenates wavelet coefficients and scaling
coefficients by levels specified in levels. The wavelet
coefficients will always be plotted preceding the scaling
coefficients.
This function allows users to visually examine differences in the
DWT transform of a time series using different filters (different
dwt objects).
For an example, see Figure 126 of Wavelet Methods for Time Series Analysis by Percival and Walden (2000).
Kelvin Ma, [email protected]
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
X <- rnorm(2048) # Create DWT Object of X with the "la8" filter. dwtobj1 <- dwt(X, filter = "la8") # Create DWT Object of X with the "d4" filter. dwtobj2 <- dwt(X, filter = "d4") # Create DWT Object of X with the "haar" filter dwtobj3 <- dwt(X, filter = "haar") # Create DWT Object of X with the "c6" filter dwtobj4 <- dwt(X, filter = "c6") #Create list of dwt objects dwtlist <- list(dwtobj1, dwtobj2, dwtobj3, dwtobj4) # Plot the dwt objects and the wavelet coefficients of level 1 through 6 # and the scaling coefficients of level 6. The first level drawn will # be purple and the next level drawn will be gold. plot.dwt.multiple(dwtlist, levels = 6, col = c("purple", "gold")) # Plot the dwt objects and the wavelet coefficients of level 1, 3, and 5 # and scaling coefficients of level 2, and 4. plot.dwt.multiple(dwtlist, levels = list(c(1,3,5), c(2,4)))X <- rnorm(2048) # Create DWT Object of X with the "la8" filter. dwtobj1 <- dwt(X, filter = "la8") # Create DWT Object of X with the "d4" filter. dwtobj2 <- dwt(X, filter = "d4") # Create DWT Object of X with the "haar" filter dwtobj3 <- dwt(X, filter = "haar") # Create DWT Object of X with the "c6" filter dwtobj4 <- dwt(X, filter = "c6") #Create list of dwt objects dwtlist <- list(dwtobj1, dwtobj2, dwtobj3, dwtobj4) # Plot the dwt objects and the wavelet coefficients of level 1 through 6 # and the scaling coefficients of level 6. The first level drawn will # be purple and the next level drawn will be gold. plot.dwt.multiple(dwtlist, levels = 6, col = c("purple", "gold")) # Plot the dwt objects and the wavelet coefficients of level 1, 3, and 5 # and scaling coefficients of level 2, and 4. plot.dwt.multiple(dwtlist, levels = list(c(1,3,5), c(2,4)))
Plot MODWT wavelet and scaling coefficients.
## S3 method for class 'modwt' plot(x, levels = NULL, draw.boundary = FALSE, type = "stack", col.plot = "black", col.boundary = "red", X.xtick.at = NULL, X.ytick.at = NULL, Stack.xtick.at = NULL, Stack.ytick.at = NULL, X.xlab = "t", y.rlabs = TRUE, plot.X = TRUE, plot.W = TRUE, plot.V = TRUE, ...)## S3 method for class 'modwt' plot(x, levels = NULL, draw.boundary = FALSE, type = "stack", col.plot = "black", col.boundary = "red", X.xtick.at = NULL, X.ytick.at = NULL, Stack.xtick.at = NULL, Stack.ytick.at = NULL, X.xlab = "t", y.rlabs = TRUE, plot.X = TRUE, plot.W = TRUE, plot.V = TRUE, ...)
x |
An object of class |
levels |
Number, vector, or list of two vectors indicating range of levels to plot. See details. |
draw.boundary |
Logical value indicating whether to draw boundary coefficients. |
type |
Type of plot to draw. Currently only "stack" is implemented. |
col.plot |
Color of wavelet and scaling coefficients. |
col.boundary |
Color of boundary coefficient lines. |
X.xtick.at |
Vector specifying the extreme tick mark locations
and the number of intervals between those extreme tick marks on the
horizontal axis of the plot of the original times series. This
vector takes the form similar to par("xaxp"), and is defaulted to
par("xaxp") when |
X.ytick.at |
Vector specifying the extreme tick mark locations
and the number of intervals between those extreme tick marks on the
vertical axis of the plot of the original times series. This vector
takes the form similar to par("yaxp"), and is defaulted to
par("yaxp") when |
Stack.xtick.at |
Vector of form similar to |
Stack.ytick.at |
Vector of form similar to |
X.xlab |
String specifying the label of the horizontal axis of the plot of the original time series. |
y.rlabs |
Logical flag indicating whether to draw the vertical labels on the right vertical axis of the stacked plot. These labels indicate the number of boundary coefficients to the right of the right boundary line, and the level of decomposition of the wavelet (or scaling) coefficients. |
plot.X |
Logical flag indicating whether to draw original time series. |
plot.W |
Logical flag indicating whether to draw the wavelet coefficients in the stacked plot. |
plot.V |
Logical flag indicating whether to draw the scaling coefficients in the stacked plot. |
... |
Additional paramters that are acceptable arguments to the
generic |
plot.modwt plots the MODWT wavelet and scaling coefficients.
The type parameter species the type of plot, which is currently
only stack. If the MODWT object is defined for multiple time series,
only the data pertaining to the first time series of the MODWT object
is plotted. Thus, only the wavelet coefficients and scaling
coefficients of the first time series of the MODWT object will be
plotted. If a single number is specified for levels, then the
wavelet coefficients of levels 1 through levels will be
plotted. Otherwise, a vector or the first element of a list will
specify which levels of the wavelet coefficients will be
plotted. Unless specified in the second element of a list, only one
level of scaling coefficients will be plotted and this level is equal
to the highest level of the wavelet coefficients plotted.
For each respective axis, the distance between a tick mark on the plot
of the original time series is equivalent to the distance between a
tick mark on the stacked plot. Thus, when altering the relative
spacing of the tick marks on the plot of the original time series
using X.xtick.at or X.ytick.at, the tick marks of the
stacked plot are automatically adjusted. If the plot of the original
time series is not drawn, then the user can alter the spacing of the
tick marks using Stack.xtick.at and Stack.ytick.at for
the horizontal and vertical axes, respectively.
One of plot.W or plot.V must be TRUE.
Kelvin Ma, [email protected]
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
X <- rnorm(2048) modwtobj <- modwt(X) # Plotting wavelet coefficients at levels 1 through 6 and scaling # coefficients at level 6. plot.modwt(modwtobj, levels = 6) # Plotting wavelet coefficients of levels 1, 3, 5, and scaling # coefficients of levels 4 and 5, and green boundary coefficients # lines. plot.modwt(modwtobj, levels = list(c(1,3,5),c(4,5)), draw.boundary = TRUE, col.boundary = "green") # Plotting wavelet coefficients at level 1 through 6 and not plotting # any scaling coefficients. plot.modwt(modwtobj, levels = 6, plot.V = FALSE)X <- rnorm(2048) modwtobj <- modwt(X) # Plotting wavelet coefficients at levels 1 through 6 and scaling # coefficients at level 6. plot.modwt(modwtobj, levels = 6) # Plotting wavelet coefficients of levels 1, 3, 5, and scaling # coefficients of levels 4 and 5, and green boundary coefficients # lines. plot.modwt(modwtobj, levels = list(c(1,3,5),c(4,5)), draw.boundary = TRUE, col.boundary = "green") # Plotting wavelet coefficients at level 1 through 6 and not plotting # any scaling coefficients. plot.modwt(modwtobj, levels = 6, plot.V = FALSE)
Finds circular shift of DWT scaling coefficients at a specified level, for a specified filter length.
scalingshift.dwt(L, j, N = NULL)scalingshift.dwt(L, j, N = NULL)
L |
Length of wavelet transform filter used. |
j |
Level of DWT scaling coefficients to be shifted. |
N |
Length of original time series sample. |
This function computes the circular shift associated with a wavelet
transform filter of length L and level j. L must
be of even length.
Each coefficient's index will be circularly shifted forward by the
value outputted. For example, if we have a vector of DWT scaling
coefficients (with NAs inserted) with indices 1,2,3,4,5,6,7,8 and the
value outputted from the function scalingshift.dwt is 2, the
new order of the vector of aligned DWT wavelet coefficients would be
7,8,1,2,3,4,5,6.
If N is specified, the function will output the "shift" Modulo
N. Because the shift is circular, the value outputted when N is
specified is equivalent to the case when N is not specified.
This function calculates the absolute value of vjH in formulas (114c) in Wavelet Methods for Time Series Analysis by Percival and Walden (2000). (And optionally vjH Modulo N)
shift |
Circular shift for specified level of DWT scaling coefficients for a specified filter length. |
Kelvin Ma, [email protected]
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# Finding the circular shift for a wavelet transform filter of length 8, # pertaining to DWT Scaling Coefficients of level 5. scalingshift.dwt(8, 5) # If the sample size of the original time series is of length 1024. scalingshift.dwt(8, 5, N = 1024)# Finding the circular shift for a wavelet transform filter of length 8, # pertaining to DWT Scaling Coefficients of level 5. scalingshift.dwt(8, 5) # If the sample size of the original time series is of length 1024. scalingshift.dwt(8, 5, N = 1024)
Plots squared gain function for a wavelet filter.
squaredgain.wt.filter(filter, level = 1, N = NULL, draw.bands = TRUE, wavelet = TRUE)squaredgain.wt.filter(filter, level = 1, N = NULL, draw.bands = TRUE, wavelet = TRUE)
filter |
Either a 'wt.filter' object, a character string specifying a wavelet filter, or a numeric vector of wavelet coefficients. See 'help(wt.filter)' for acceptable filter names. |
level |
Level of wavelet filter. Applicable only if a character
string is supplied in |
N |
Length of filter vector after padding the wavelet filter with zeroes. The zeroes are inserted after the filter values. |
draw.bands |
Draws dashed lines marking the lower and upper ends of the nominal pass-band for the wavelet filter. |
wavelet |
A logical flag indicating whether to plot the squared gain function for the wavelet (high pass) or scaling (low pass) filter. |
If N is not specified, and the filter supplied is of length
less than 1024, then the filter vector will be padded to a length of
1024. Otherwise, if the filter supplied is of length greater than
1024, then the filter vector will be padded to the first 'power of 2'
that is greater than the length of the filter supplied.
Kelvin Ma, [email protected]
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# Plotting the squared gain function of LA8 Wavelet Filter Coefficients. filter <- wt.filter("la8") squaredgain.wt.filter(filter) # Plotting the squared gain function of LA8 Scaling Filter Coefficients. squaredgain.wt.filter(filter, wavelet = FALSE) # Plotting the squared gain function of Haar Wavelet Filter Coefficients # without supplying a filter object. squaredgain.wt.filter("haar")# Plotting the squared gain function of LA8 Wavelet Filter Coefficients. filter <- wt.filter("la8") squaredgain.wt.filter(filter) # Plotting the squared gain function of LA8 Scaling Filter Coefficients. squaredgain.wt.filter(filter, wavelet = FALSE) # Plotting the squared gain function of Haar Wavelet Filter Coefficients # without supplying a filter object. squaredgain.wt.filter("haar")
Plot an object as a stacked set of plots, plotting each column using the same x-axis and plotting the y-values at the same scale with an offset equal to the magnitude of each column vector.
stackplot(x, y = NULL, type = "h", axes.labels = FALSE, xlab = "", ylab = "", y.llabs = NULL, y.rlabs = NULL, draw.divides = TRUE, xtick.at = NULL, ytick.at = NULL, col = "black", main = "")stackplot(x, y = NULL, type = "h", axes.labels = FALSE, xlab = "", ylab = "", y.llabs = NULL, y.rlabs = NULL, draw.divides = TRUE, xtick.at = NULL, ytick.at = NULL, col = "black", main = "")
x |
Specifies the coordinates of the points to be
plotted. Alternatively, |
y |
If |
type |
Type of plots to be drawn. Currently only |
axes.labels |
Boolean representing whether or not to display the numeric labels of the horizontal and vertical axes. |
xlab |
String indicating the label for the horizontal x-axis. |
ylab |
String indicating the label for the vertical y-axis. |
y.llabs |
Vector or list of labels to be placed on left side of plot. |
y.rlabs |
Vector or list of labels to be placed on right side of plot. |
draw.divides |
Logical value indicating whether to draw lines to separate each subplot in the stackplot. |
xtick.at |
Vector specifying the tick mark locations on the horizontal axis of the plot. Defaulted to par("xaxp"). |
ytick.at |
Vector specifying the tick mark locations on the vertical axis of the plot. Defaulted to par("yaxp"). |
col |
String representing color of plot. |
main |
String representing the title of the plot. |
Each object specified by x or y will be
coerced into a list, and each object of the list will be coerced into
a matrix using the function as.matrix. Each column of a matrix
will represent the vertical y-values for each individual plot in the
stacked plot. Horizontal x-values are discussed below.
stackplot will plot a matrix of values as a series of
stacked plots, where the individual columns are plotted using
the same x and y scales, but offset by the magnitude (= abs(max-min))
of the column.
If the object to be plotted is a Time Series object, the start
and end values of the object will represent the first and last
horizontal x-values of the points to be plotted. The Time Series
will be coerced into matrix.
The limits of the horizontal x-axis and vertical y-axis will be dictated by the data in the first object of the list.
If the object to be plotted is a list of matrices, and x is not
specified, the horizontal x-values of the points to be plotted for a
given matrix will default to the rownames of that matrix. The rownames
can be either numeric or character. If the first matrix
of the list does not have rownames, then the horizontal x-values for
each matrix (and its corresponding plot) will default to range from 1
to the length of the columns of the respective matrix.
If the first matrix of the list does have specified rownames, then the horizontal x-values assume the values of the rownames of the respective matrix. However, if another matrix of the list does not have specified rownames, then its horizontal x-values will default to range from 1 to the length of the columns of that respective matrix.
The user can specify the type of plot (currently only "h" and "l") to
be drawn for every element of the list in x or y using
the vector type. If the Nth element (N>1) in type is NA
and the Nth element of the list in x or y exists, then
the type of plot for the Nth element of the list will be equal to the
type of plot for the N-1th element of the list. Similarly for
col, which specifies the colors for each element of the list in
x or y.
The labels y.llabs and y.rlabs will be placed at the
"zero" lines of each individual plot starting from the bottom working
up to the top.
Kelvin Ma, [email protected]
plot.dwt and plot.modwt
A <- array(c(1,2,3,4), c(3,4)) # Plotting the vectors c(1,2,3), c(4,1,2), c(3,4,1), and c(2,3,4) in a # stacked plot with 4 separate plots). The x-values will be 1:3. stackplot(A) #Plotting A with numeric labels on the axis and making the plot "red". #Also labeling the x-axis with "X-Label", and the y-axis with "Y-Label". stackplot(A, axes.labels = TRUE, xlab = "X-Label", ylab = "Y-Label") timeSeries <- ts(A) # Plotting the Time Series created by coercing the array A into a Time # Series. Observe that this plot similar to the plot in the first example. stackplot(timeSeries) lfig <- 2:11 hfig <- 1:10 # Plotting a stacked plot of two types: histogram and lines. Where the # line joins the points (1,2) and (10,11), and the histogram is similar to # the histogram formed by entering plot(1:10, type = "h"). The line will # be red and the histogram will be yellow. stackplot(list(lfig, hfig), type = c("l", "h"), col = c("red", "yellow"))A <- array(c(1,2,3,4), c(3,4)) # Plotting the vectors c(1,2,3), c(4,1,2), c(3,4,1), and c(2,3,4) in a # stacked plot with 4 separate plots). The x-values will be 1:3. stackplot(A) #Plotting A with numeric labels on the axis and making the plot "red". #Also labeling the x-axis with "X-Label", and the y-axis with "Y-Label". stackplot(A, axes.labels = TRUE, xlab = "X-Label", ylab = "Y-Label") timeSeries <- ts(A) # Plotting the Time Series created by coercing the array A into a Time # Series. Observe that this plot similar to the plot in the first example. stackplot(timeSeries) lfig <- 2:11 hfig <- 1:10 # Plotting a stacked plot of two types: histogram and lines. Where the # line joins the points (1,2) and (10,11), and the histogram is similar to # the histogram formed by entering plot(1:10, type = "h"). The line will # be red and the histogram will be yellow. stackplot(list(lfig, hfig), type = c("l", "h"), col = c("red", "yellow"))
Wwater level measurements from a stilling well on the open coast in Crescent Bay, CA. Observations were recorded every six minutes, from the beginning of 1980 to the end of 1991. For more detailed information regarding the collection of this data, see Percival and Walden (2000), p. 185.
data(subtidal)data(subtidal)
A time series object containing 8746 observations.
http://www.ms.washington.edu/~s530/data.html
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press, sec. 5.8.
Finds circular shift of DWT wavelet coefficients at a specified level, for a specified filter length.
waveletshift.dwt(L, j, N = NULL)waveletshift.dwt(L, j, N = NULL)
L |
Length of wavelet transform filter used. |
j |
Level of DWT wavelet coefficients to be shifted. |
N |
Length of original time series sample. |
This function computes the circular shift associated with a wavelet
transform filter of length L and level j. L must
be of even length.
Each coefficient's index will be circularly shifted forward by the
value outputted. For example, if we have a vector of DWT wavelet
coefficients (with NAs inserted) with indices 1,2,3,4,5,6,7,8 and the
value outputted from the function waveletshift.dwt is 2, the
new order of the vector of aligned DWT wavelet coefficients would be
7,8,1,2,3,4,5,6.
If N is specified, the function will output the "shift" Modulo
N. Because the shift is circular, the value outputted when N is
specified is equivalent to the case when N is not specified.
This function calculates the absolute value of vjG in formulas (114c) in Wavelet Methods for Time Series Analysis by Percival and Walden (2000). (And optionally vjG Modulo N)
shift |
Circular shift for specified level of DWT wavelet coefficients for a specified filter length. |
Kelvin Ma, [email protected]
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# Finding the circular shift for a wavelet transform filter of length 8, # pertaining to DWT Wavelet Coefficients of level 5. waveletshift.dwt(8, 5) # If the sample size of the original time series is of length 1024. waveletshift.dwt(8, 5, N = 1024)# Finding the circular shift for a wavelet transform filter of length 8, # pertaining to DWT Wavelet Coefficients of level 5. waveletshift.dwt(8, 5) # If the sample size of the original time series is of length 1024. waveletshift.dwt(8, 5, N = 1024)
Generates a wavelet transform filter.
wt.filter(filter="la8", modwt=FALSE, level=1)wt.filter(filter="la8", modwt=FALSE, level=1)
filter |
A character string indicating which wavelet transform filter to compute or a numeric vector of wavelet (high pass) filter coefficients (not scaling (low pass) coefficients). If a numeric vector is supplied, the length must be even. |
modwt |
A logical value indicating whether to compute the maximal overlap discrete wavelet transform filter. |
level |
An integer value indicating the level of the wavelet filter to compute. |
The character strings currently supported are derived from one of four
classes of wavelet transform filters: Daubechies, Least Asymetric,
Best Localized and Coiflet. The prefixes for filters of these classes
are d, la, bl and c,
respectively. Following the prefix, the filter name consists of an
integer indicating length. Supported lengths are as follows:
2,4,6,8,10,12,14,16,18,20.
8,10,12,14,16,18,20.
14,18,20.
6,12,18,24,30.
Thus, to obtain the Daubechies wavelet transform filter of length 4,
the character string "d4" can be passed to
wt.filter.
This naming convention has one exception: the Daubechies wavelet
transform filter of length 2 is denoted by haar instead of
d2.
Returns an object of class wt.filter, which is an S4 object
with slots
L |
An integer representing the length of the wavelet and scaling filters. |
h |
A numeric vector of wavelet filter coefficients. |
g |
A numeric vector of scaling filter coefficients. |
wt.class |
A character string indicating the class of the wavelet
transform filter. Possible values are |
wt.name |
A character string indicating the name of the wavlet
filter as listed in the Details section, above. If the
|
transform |
A character string indicating whether the resulting
wavelet transform object contains DWT or MODWT coefficients. Possible
values are |
The notation h and g for wavelet and scaling
coefficients, respectively, follows Percival and Walden (2000). In
other texts and articles the reverse notation is often adopted.
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
wt.filter("la14") wt.filter(1:10, modwt=TRUE)wt.filter("la14") wt.filter(1:10, modwt=TRUE)
An S4 object containing wavelet transform filter information.
An integer representing the length of the wavelet and scaling filters.
An integer representing the level of the wavelet filter.
A numeric vector of wavelet filter coefficients.
A numeric vector of scaling filter coefficients.
A character string indicating the class of the wavelet
transform filter. Possible values are "Daubechies",
"Least Asymetric", "Best Localized", and
"Coiflet". If the wt.filter object is generated from a
numeric vector of wavelet coefficients, wt.class is
"none".
A character string indicating the name of the wavlet
filter as listed in the Details section, above. If the
wt.filter object is generated from a numeric vector of
wavelet coefficients, wt.name is "none".
A character string indicating whether the resulting
wavelet transform object contains DWT or MODWT coefficients. Possible
values are "dwt" and "modwt".
The notation h and g for wavelet and scaling
coefficients, respectively, follows Percival and Walden (2000). In
other texts and articles the reverse notation is often adopted.
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
Generates an equivalent filter for a specified wavelet transform filter and level..
wt.filter.equivalent(wt.filter, J)wt.filter.equivalent(wt.filter, J)
wt.filter |
A |
J |
Level of equivalent filter to compute. |
A wavelet transform of level can be computed by iterating the
pyramid algorithm times with a level 1 wavelet filter, or by
simply implementing the pyramid once with the level equivalent
filter. The equivalent filter is obtained by iteratively filtering
the level 1 wavelet filter and scaling filters with upsampled versions
of themselves. For details regarding the computation, see Percival
and Walden (2000), eqs. 95e and 96e.
Returns the original wt.filter object with modifications made
to the wavelet and scaling filter coefficients, filter length and
filter level.
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
wf <- wt.filter("la14") wt.filter.equivalent(wf, 3)wf <- wt.filter("la14") wt.filter.equivalent(wf, 3)
Computes the quadrature mirror filter of a series of even length.
wt.filter.qmf(x, inverse=FALSE)wt.filter.qmf(x, inverse=FALSE)
x |
A numeric vector of even length |
inverse |
A logical flag indicating whether to compute the inverse quadrature mirror filter. |
The quadrature mirror filter is computed as outlined in Percival and Walden, page 75. Specifically, the forward QMF makes use of a wavelet filter to obtain a scaling filter and the inverse QMF does the opposite.
A numeric vector which contains the values of x in reverse
order, with the values of the resulting vector at even indices being
negated in the case that inverse=FALSE and the values at odd
indices being negated in the case that inverse=TRUE.
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
# obtain both the la8 wavelet and scaling filters la8 <- wt.filter("la8") # now compare output of qmf with values in wt.filter object wt.filter.qmf(la8@h) la8@g# obtain both the la8 wavelet and scaling filters la8 <- wt.filter("la8") # now compare output of qmf with values in wt.filter object wt.filter.qmf(la8@h) la8@g
Computes phase shifts for wavelet transform coefficients corresponding to a particular filter.
wt.filter.shift(filter, J, wavelet=TRUE, coe=FALSE, modwt=FALSE)wt.filter.shift(filter, J, wavelet=TRUE, coe=FALSE, modwt=FALSE)
filter |
A |
J |
A vector of positive integers indicating levels for which to compute phase shifts. |
wavelet |
A logical flag indicating whether to compute the wavelet (high pass) or scaling (low pass) phase shift(s). |
coe |
A logical value indicating whether to use the center of energy method (see Percival and Walden 2000, page 118) to compute the phase shift(s). |
modwt |
A logical value indicating whether to compute MODWT phase shift(s). |
For wavelet filters of class 'Least Assymetric' or 'Coiflet', the
default method for calculating phase shifts is outlined on pages
112-114 and page 124 of Percival and Walden 2000. For wavelet filters of
class 'Best Localized', the default shifts are given on page 119 of
Percival and Walden 2000. For the Haar filter, both the level
wavelet and scaling phase shifts are half the length of the level
wavelet and scaling filters and the phase shifts for the D(4)
filter are determined by specifying in equations (114a)
and (114b) of Percival and Walden 2000.
For all other filters, the center of energy method is the default
method for computing phase shifts (thus rendering the coe
argument irrelevant). If coe = TRUE, then the center of energy
method is used regardless of filter class.
By default the DWT phase shifts are computed, using the MODWT phase
shifts and the methods outlined on pages 115-116 of Percival and
Walden 2000. If modwt = TRUE, then only the MODWT phase shifts
are computed.
A vector of phase shifts, expressed in absolute value. Each element
corresponds to the equivalent element in J.
Eric Aldrich. [email protected].
Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.
wt.filter.shift("la14", J=1:6)wt.filter.shift("la14", J=1:6)