This function computes the Windowed Scalogram Difference of two signals. The definition and details can be found in (Bolós et al. 2017).
wsd(signal1,
signal2,
dt = 1,
scaleparam = NULL,
windowrad = NULL,
rdist = NULL,
delta_t = NULL,
normalize = c("NO", "ENERGY", "MAX", "SCALE"),
refscale = NULL,
wname = c("MORLET", "DOG", "PAUL", "HAAR", "HAAR2"),
wparam = NULL,
waverad = NULL,
border_effects = c("BE", "INNER", "PER", "SYM"),
mc_nrand = 0,
commutative = TRUE,
wscnoise = 0.02,
compensation = 0,
energy_density = TRUE,
parallel = FALSE,
makefigure = TRUE,
time_values = NULL,
figureperiod = TRUE,
xlab = "Time",
ylab = NULL,
main = "-log2(WSD)",
zlim = NULL)A list with the following fields:
wsd: A matrix of size length(tcentral) x length(scales)
containing the values of the windowed scalogram differences at each scale and at each
time window.
tcentral: The vector of central times used in the computations of the
windowed scalograms.
scales: The vector of scales.
windowrad: Radius for the time windows of the windowed scalograms,
measured in dt.
rdist: The log-scale radius for the windows measured in suboctaves.
signif95: A logical matrix of size length(tcentral) x
length(scales). If TRUE, the corresponding point of the wsd matrix is in
the 95% significance.
signif05: A logical matrix of size length(tcentral) x
length(scales). If TRUE, the corresponding point of the wsd matrix is in
the 5% significance.
fourierfactor: A factor for converting scales into periods.
coi_maxscale: A vector of length length(tcentral) containing the
values of the maximum scale from which there are border effects for the respective
central time.
A vector containing the first signal.
A vector containing the second signal (its length should be equal to
that of signal1).
Numeric. The time step of the signals.
A vector of three elements with the minimum scale, the maximum scale and the number of suboctaves per octave for constructing power 2 scales (following Torrence and Compo 1998). If NULL, they are automatically constructed.
Integer. Time radius for the windows, measured in dt. By
default, it is set to \(ceiling(length(signal1) / 20)\).
Integer. Log-scale radius for the windows measured in suboctaves. By default, it is set to \(ceiling(length(scales) / 20)\).
Integer. Increment of time for the construction of windows central
times, measured in dt. By default, it is set to
\(ceiling(length(signal1) / 256)\).
String, equal to "NO", "ENERGY", "MAX" or "SCALE". If "ENERGY", signals are
divided by their respective energies. If "MAX", each signal is divided by the maximum
value attained by its scalogram. In these two cases, energy_density must be TRUE.
Finally, if "SCALE", each signal is divided by their scalogram value at scale
refscale.
Numeric. The reference scale for normalize.
A string, equal to "MORLET", "DOG", "PAUL", "HAAR" or "HAAR2". The difference between "HAAR" and "HAAR2" is that "HAAR2" is more accurate but slower.
The corresponding nondimensional parameter for the wavelet function (Morlet, DoG or Paul).
Numeric. The radius of the wavelet used in the computations for the cone of influence. If it is not specified, it is asumed to be \(\sqrt{2}\) for Morlet and DoG, \(1/\sqrt{2}\) for Paul and 0.5 for Haar.
String, equal to "BE", "INNER", "PER" or "SYM", which indicates how to manage the border effects which arise usually when a convolution is performed on finite-lenght signals.
"BE": With border effects, padding time series with zeroes.
"INNER": Normalized inner scalogram with security margin adapted for each different scale.
"PER": With border effects, using boundary wavelets (periodization of the original time series).
"SYM": With border effects, using a symmetric catenation of the original time series.
Integer. Number of Montecarlo simulations to be performed in order to determine the 95% and 5% significance contours.
Logical. If TRUE (default) the commutative windowed scalogram difference. Otherwise a non-commutative (but simpler) version is computed (see Bolós et al. 2017).
Numeric in \([0,1]\). If a (windowed) scalogram takes values close to zero, some problems may appear because we are considering relative differences. Specifically, we can get high relative differences that in fact are not relevant, or even divisions by zero.
If we consider absolute differences this would not happen but, on the other hand, using absolute differences is not appropriate for scalogram values not close to zero.
So, the parameter wscnoise stablishes a threshold for the scalogram values
above which a relative difference is computed, and below which a difference
proportional to the absolute difference is computed (the proportionality factor is
determined by requiring continuity).
Finally, wscnoise can be interpreted as the relative amplitude of the noise in
the scalograms and is chosen in order to make a relative (\(= 0\)), absolute
(\(= 1\)) or mix (in \((0,1)\)) difference between scalograms. Default value is
set to \(0.02\).
Numeric. It is an alternative to wscnoise for
preventing numerical errors or non-relevant high relative differences when scalogram
values are close to zero (see Bolós et al. 2017). It should be a non-negative
relatively small value.
Logical. If TRUE (default), divide the scalograms by the square
root of the scales for convert them into energy density. Note that it does not affect
the results if wscnoise \(= 0\).
Logical. If TRUE, it uses function parApply from package
parallel for the Montecarlo simulations. When FALSE (default) it uses the normal
apply function.
Logical. If TRUE (default), a figure with the WSD is plotted.
A numerical vector of length length(signal) containing custom
time values for the figure. If NULL (default), it will be computed starting at 0.
Logical. If TRUE (default), periods are used in the figure instead of scales.
A string giving a custom X axis label.
A string giving a custom Y axis label. If NULL (default) the Y label is
either "Scale" or "Period" depending on the value of figureperiod.
A string giving a custom main title for the figure.
A vector of length 2 with the limits for the z-axis (the color bar).
C. Torrence, G. P. Compo. A practical guide to wavelet analysis. B. Am. Meteorol. Soc. 79 (1998), 61–78.
V. J. Bolós, R. Benítez, R. Ferrer, R. Jammazi. The windowed scalogram difference: a novel wavelet tool for comparing time series. Appl. Math. Comput., 312 (2017), 49-65.
nt <- 1500
time <- 1:nt
sd_noise <- 0.2 #% In Bolós et al. 2017 Figure 1, sd_noise = 1.
signal1 <- rnorm(n = nt, mean = 0, sd = sd_noise) + sin(time / 10)
signal2 <- rnorm(n = nt, mean = 0, sd = sd_noise) + sin(time / 10)
signal2[500:1000] = signal2[500:1000] + sin((500:1000) / 2)
if (FALSE) {
wsd <- wsd(signal1 = signal1, signal2 = signal2)
}
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