wavCWT {wmtsa} | R Documentation |
The continuous wavelet transform (CWT) is a highly redundant transformation of a real-valued or complex-valued function f(x), mapping it from the time domain to the so-called time-scale domain. Loosely, speaking the CWT coefficients are proportional to the variability of a function at a given time and scale.
The CWT is defined by a complex correlation of a scaled and time-shifted mother wavelet with a function f(x). Let psi(x) be a real- or complex-valued function representing a mother wavelet, i.e. a function which meets the standard mathematical criteria for a wavelet and one that can be used to generate all other wavelets within the same family. Let Conj{psi(w)} be the complex conjugate of psi(w). The CWT of f(x) is defined as
W(a,b)=1/sqrt(a) Int{f(x) psi((x-b)/a)} dx
for (a,b) in R and a > 0, where a is the scale of the wavelet and b is the shift of the wavelet in time. It can be shown that the above complex correlation maintains a duality with the Fourier transform defined by the relation
sqrt{a} * F(w) * Conj{Psi(a * w)}
where F() is the Fourier transform of f(x) and w is the frequency in radians. This function calculates the CWT in the Fourier domain followed by an inverse Fourier transform.
wavCWT(x, scale.range=deltat(x) * c(1, length(x)), n.scale=100, wavelet="gaussian2", shift=5, variance=1)
x |
a vector containing a uniformly-sampled real-valued time series. The time series
may be of class class rts , ts , cts , or signalSeries , or be a numeric
vector. |
n.scale |
the number of scales to evaluate over the scale.range . Default: 100 . |
scale.range |
a two-element vector containing the range of scales over which to evaluate the CWT.
The smallest specified scale must be greater than or
equal to the sampling.interval of the time series.
Default: deltat(x) * c(1, length(x)) . |
shift |
numeric value representing the frequency shift to
use for the Morlet wavelet filter. Default: 5 . |
variance |
if the wavelet filter
is of type "gaussian1" or
"gaussian2" then this parameter
represents the variance of the Gaussian PDF used to
scale the corresponding filters. Default: 1 . |
wavelet |
a character string denoting the wavelet filter to use in
calculating the CWT. Choices are "haar", "gaussian1", "gaussian2", and "morlet",
where gaussian1 and gaussian2 represent the first and second derivatives of a
Gaussian PDF. Default: "gaussian2" . |
an object of class wavCWT
.
TRUE
, the phase of the CWT is plotted. Default: FALSE
."time"
."log2(units)"
where units
are the units of the time
series if available. If units
are not available, "log2(scale)"
is used.power.stretch=0
, then image is
transformed with log(abs(x)+1)
. Otherwise, the image is
transformed with (abs(x))
^power.stretch
. Default: 0.5
."image"
and
"persp"
which plot the CWT as an image or as a meshed perspective plot, respectively. The perspective
plot resamples the data to contain a maximum of grid.size
rows and columns.NULL
(no zoom).100
.TRUE
, a new plot is forced with the frame()
command.
D. B. Percival and A. T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 2000.
## calculate the CWT of the sunspots series using ## a Mexican hat wavelet (gaussian2) sunspots.cwt <- wavCWT(sunspots) ## print the result print(sunspots.cwt) ## plot an image of the modulus of the CWT and the ## time series plot(sunspots.cwt, series=TRUE) ## plot a coarse-scale wire-frame perspective of ## the CWT plot(sunspots.cwt, type="persp")