Package ‘biwavelet’

October 12, 2022

Type Package
Title Conduct Univariate and Bivariate Wavelet Analyses
Version 0.20.21
Date 2021-05-24
Author Tarik C. Gouhier, Aslak Grinsted, Viliam Simko
Maintainer Tarik C. Gouhier <tarik.gouhier@gmail.com>
Description This is a port of the WTC MATLAB package written by Aslak Grinsted
and the wavelet program written by Christopher Torrence and Gibert P.
Compo. This package can be used to perform univariate and bivariate
(cross-wavelet, wavelet coherence, wavelet clustering) analyses.
License GPL (>= 2)

URL https://github.com/tgouhier/biwavelet

BugReports https://github.com/tgouhier/biwavelet/issues
LazyData yes
LinkingTo Rcpp
Imports fields, foreach, Rcpp (>= 0.12.2)
Suggests testthat, knitr, rmarkdown, devtools
RoxygenNote 7.1.1
NeedsCompilation yes
Repository CRAN
Date/Publication 2021-05-26 05:10:10 UTC

R topics documented:
biwavelet-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
ar1.spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
ar1_ma0_sim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
arrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
arrow2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1
2 biwavelet-package

check.data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
check.datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
convolve2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
convolve2D_typeopen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
enviro.data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
get_minroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
MOTHERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
phase.plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
plot.biwavelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
pwtc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
rcpp_row_quantile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
rcpp_wt_bases_dog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
rcpp_wt_bases_morlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
rcpp_wt_bases_paul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
smooth.wavelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
wclust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
wdist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
wt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
wt.bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
wt.bases.dog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
wt.bases.morlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
wt.bases.paul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
wt.sig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
wtc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
wtc.sig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
wtc_sig_parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
xwt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Index 40

biwavelet-package Conduct Univariate and Bivariate Wavelet Analyses

Description
This is a port of the WTC MATLAB package written by Aslak Grinsted and the wavelet program
written by Christopher Torrence and Gibert P. Compo. This package can be used to perform uni-
variate and bivariate (cross-wavelet, wavelet coherence, wavelet clustering) wavelet analyses.

Details
As of biwavelet version 0.14, the bias-corrected wavelet and cross-wavelet spectra are automatically
computed and plotted by default using the methods described by Liu et al. (2007) and Veleda et
al. (2012). This correction is needed because the traditional approach for computing the power
spectrum (e.g., Torrence and Compo 1998) leads to an artificial and systematic reduction in power
at lower periods.
biwavelet-package 3

Author(s)
Tarik C. Gouhier
Maintainer: Tarik C. Gouhier <tarik.gouhier@gmail.com>
Code based on WTC MATLAB package written by Aslak Grinsted and the wavelet MATLAB
program written by Christopher Torrence and Gibert P. Compo.

References
Cazelles, B., M. Chavez, D. Berteaux, F. Menard, J. O. Vik, S. Jenouvrier, and N. C. Stenseth. 2008.
Wavelet analysis of ecological time series. Oecologia 156:287-304.
Grinsted, A., J. C. Moore, and S. Jevrejeva. 2004. Application of the cross wavelet transform and
wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics 11:561-566.
Liu, Y., X. San Liang, and R. H. Weisberg. 2007. Rectification of the Bias in the Wavelet Power
Spectrum. Journal of Atmospheric and Oceanic Technology 24:2093-2102.
Rouyer, T., J. M. Fromentin, F. Menard, B. Cazelles, K. Briand, R. Pianet, B. Planque, and N.
C. Stenseth. 2008. Complex interplays among population dynamics, environmental forcing, and
exploitation in fisheries. Proceedings of the National Academy of Sciences 105:5420-5425.
Rouyer, T., J. M. Fromentin, N. C. Stenseth, and B. Cazelles. 2008. Analysing multiple time series
and extending significance testing in wavelet analysis. Marine Ecology Progress Series 359:11-23.
Torrence, C., and G. P. Compo. 1998. A Practical Guide to Wavelet Analysis. Bulletin of the
American Meteorological Society 79:61-78.
Torrence, C., and P. J. Webster. 1998. The annual cycle of persistence in the El Nino/Southern
Oscillation. Quarterly Journal of the Royal Meteorological Society 124:1985-2004.
Veleda, D., R. Montagne, and M. Araujo. 2012. Cross-Wavelet Bias Corrected by Normalizing
Scales. Journal of Atmospheric and Oceanic Technology 29:1401-1408.

Examples
# As of biwavelet version 0.14, the bias-corrected wavelet and cross-wavelet spectra
# are automatically computed and plotted by default using the methods
# described by Liu et al. (2007) and Veleda et al. (2012). This correction
# is needed because the traditional approach for computing the power spectrum
# (e.g., Torrence and Compo 1998) leads to an artificial and systematic reduction
# in power at low periods.

# EXAMPLE OF BIAS CORRECTION:

require(biwavelet)
# Generate a synthetic time series 's' with the same power at three distinct periods
t1=sin(seq(from=0, to=2*5*pi, length=1000))
t2=sin(seq(from=0, to=2*15*pi, length=1000))
t3=sin(seq(from=0, to=2*40*pi, length=1000))
s=t1+t2+t3

# Compare non-corrected vs. corrected wavelet spectrum

wt1=wt(cbind(1:1000, s))
par(mfrow=c(1,2))
plot(wt1, type="power.corr.norm", main="Bias-corrected")
4 ar1.spectrum

plot(wt1, type="power.norm", main="Not-corrected")

# ADDITIONAL EXAMPLES
t1 <- cbind(1:100, rnorm(100))
t2 <- cbind(1:100, rnorm(100))

# Continuous wavelet transform

wt.t1 <- wt(t1)

# Plot power
# Make room to the right for the color bar
par(oma = c(0, 0, 0, 1), mar = c(5, 4, 4, 5) + 0.1)
plot(wt.t1, plot.cb=TRUE, plot.phase=FALSE)

# Compute cross-wavelet
xwt.t1t2 <- xwt(t1, t2)

# Plot cross wavelet power and phase difference (arrows)

plot(xwt.t1t2, plot.cb=TRUE)

# Wavelet coherence; nrands should be large (>= 1000)

wtc.t1t2=wtc(t1, t2, nrands=10)
# Plot wavelet coherence and phase difference (arrows)
# Make room to the right for the color bar
par(oma=c(0, 0, 0, 1), mar=c(5, 4, 4, 5) + 0.1)
plot(wtc.t1t2, plot.cb=TRUE)

# Perform wavelet clustering of three time series

t1=cbind(1:100, sin(seq(from=0, to=10*2*pi, length.out=100)))
t2=cbind(1:100, sin(seq(from=0, to=10*2*pi, length.out=100)+0.1*pi))
t3=cbind(1:100, rnorm(100))

# Compute wavelet spectra

wt.t1=wt(t1)
wt.t2=wt(t2)
wt.t3=wt(t3)

# Store all wavelet spectra into array

w.arr=array(NA, dim=c(3, NROW(wt.t1$wave), NCOL(wt.t1$wave)))
w.arr[1, , ]=wt.t1$wave
w.arr[2, , ]=wt.t2$wave
w.arr[3, , ]=wt.t3$wave

# Compute dissimilarity and distance matrices

w.arr.dis <- wclust(w.arr)
plot(hclust(w.arr.dis$dist.mat, method = "ward.D"), sub = "", main = "",
ylab = "Dissimilarity", hang = -1)

ar1.spectrum Power spectrum of a random red noise process

ar1_ma0_sim 5

Description
Generate the power spectrum of a random time series with a specific AR(1) coefficient.

Usage
ar1.spectrum(ar1, periods)

Arguments
ar1 First order coefficient desired.
periods Periods of the time series at which the spectrum should be computed.

Value
Returns the power spectrum as a vector of real numbers.

Author(s)
Tarik C. Gouhier (tarik.gouhier@gmail.com) Code based on WTC MATLAB package written by
Aslak Grinsted.

References
Cazelles, B., M. Chavez, D. Berteaux, F. Menard, J. O. Vik, S. Jenouvrier, and N. C. Stenseth. 2008.
Wavelet analysis of ecological time series. Oecologia 156:287-304.
Grinsted, A., J. C. Moore, and S. Jevrejeva. 2004. Application of the cross wavelet transform and
wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics 11:561-566.
Torrence, C., and G. P. Compo. 1998. A Practical Guide to Wavelet Analysis. Bulletin of the
American Meteorological Society 79:61-78.

Examples
p <- ar1.spectrum(0.5, 1:25)

ar1_ma0_sim Slightly faster arima.sim implementation which assumes AR(1) and

ma=0.

Description
Slightly faster arima.sim implementation which assumes AR(1) and ma=0.

Usage
ar1_ma0_sim(minroots, ar, n)
6 arrow

Arguments

minroots Output from get_minroots function.

ar The ’ar’ part of AR(1)
n Length of output series, before un-differencing. A strictly positive integer.

See Also

arima.sim

arrow Helper function for phase.plot (not exported)

Description

Helper function for phase.plot (not exported)

Usage

arrow(x, y, l = 0.1, w = 0.3 * l, alpha, col = "black")

Arguments

x X-coordinate of the arrow.

y Y-coordinate of the arrow.
l Length of the arrow.
w Width of the arrow.
alpha Angle of the arrow in radians (0 .. 2*pi).
col Color of the arrow.

Examples

plot.new()
arrow(0,0, alpha = 0)
arrow2 7

arrow2 This is an alternative helper function that plots arrows. It uses text()
to print a character using a default font. This way, it is possible to
render different types of arrows.

Description
This is an alternative helper function that plots arrows. It uses text() to print a character using a
default font. This way, it is possible to render different types of arrows.

Usage
arrow2(x, y, angle, size = 0.1, col = "black", chr = intToUtf8(10139))

Arguments
x X-coordinate of the arrow.
y Y-coordinate of the arrow.
angle Angle in radians.
size Similar to arrow.len parameter. Notice that we don’t need the arrow.lwd
anymore
col Color of the arrow.
chr Character representing the arrow. You should provide the character as escaped
UTF-8.

Author(s)
Viliam Simko

Examples
# Not run: arrow2(x[j], y[i], angle = phases[i, j],
# Not run: col = arrow.col, size = arrow.len)

check.data Check the format of time series

Description
Check the format of time series

Usage
check.data(y, x1 = NULL, x2 = NULL)
8 check.datum

Arguments
y Time series y in matrix format (n rows x 2 columns). The first column should
contain the time steps and the second column should contain the values.
x1 Time series x1 in matrix format (n rows x 2 columns). The first column should
contain the time steps and the second column should contain the values.
x2 Time series x2 in matrix format (n rows x 2 columns). The first column should
contain the time steps and the second column should contain the values.

Value
Returns a named list containing:

t Time steps
dt Size of a time step
n.obs Number of observations

Author(s)
Tarik C. Gouhier (tarik.gouhier@gmail.com)

References
Torrence, C., and G. P. Compo. 1998. A Practical Guide to Wavelet Analysis. Bulletin of the
American Meteorological Society 79:61-78.

Examples
t1 <- cbind(1:100, rnorm(100))
check.data(y = t1)

check.datum Helper function

Description
Helper function

Usage
check.datum(x)

Arguments
x matrix
convolve2D 9

Value
list(t, dt, n.obs)

Note
This function is not exported

convolve2D Fast column-wise convolution of a matrix

Description
Use the Fast Fourier Transform to perform convolutions between a sequence and each column of a
matrix.

Usage
convolve2D(x, y, conj = TRUE, type = c("circular", "open"))

Arguments
x M x n matrix.
y Numeric sequence of length N.
conj Logical; if TRUE, take the complex conjugate before back-transforming. TRUE is
used for usual convolution.
type Character; one of circular, open (beginning of word is ok).
For circular, the two sequences are treated as circular, i.e., periodic.
For open and filter, the sequences are padded with zeros (from left and right)
first; filter returns the middle sub-vector of open, namely, the result of running
a weighted mean of x with weights y.

Details
This is a corrupted version of convolve made by replacing fft with mvfft in a few places. It would
be nice to submit this to the R Developers for inclusion.

Value
M x n matrix

Note
This function was copied from waveslim to limit package dependencies.

Author(s)
Brandon Whitcher
10 enviro.data

convolve2D_typeopen Speed-optimized version of convolve2D

Description
Equivalent to convolve2D(x, y, type = "open"). The motivation for this function was that con-
volution is called many times in a loop from smooth.wavelet, always with the type = "open"
parameter.

Usage
convolve2D_typeopen(x, y)

Arguments
x M x n matrix.
y Numeric sequence of length N.

Author(s)
Viliam Simko

See Also
convolve2D

enviro.data Multivariate ENSO (MEI), NPGO, and PDO indices

Description
Monthly indices of ENSO, NPGO, and PDO from 1950 to 2009

Usage
data(enviro.data)

Format
A data frame with 720 observations on the following 6 variables.
month a numeric vector containing the month
year a numeric vector containing the year
date a numeric vecor containing the date
mei a numeric vector containing the MEI index
npgo a numeric vector containing the NPGO index
pdo a numeric vector containing the PDO index
get_minroots 11

Source

MEI: https://psl.noaa.gov/enso/mei/
NPGO: http://www.o3d.org/npgo/
PDO: http://research.jisao.washington.edu/pdo/

References

Di Lorenzo, E., N. Schneider, K. M. Cobb, P. J. S. Franks, K. Chhak, A. J. Miller, J. C. McWilliams,

S. J. Bograd, H. Arango, E. Curchitser, T. M. Powell, and P. Riviere. 2008. North Pacific Gyre
Oscillation links ocean climate and ecosystem change. Geophys. Res. Lett. 35:L08607.
Mantua, N. J., and S. R. Hare. 2002. The Pacific decadal oscillation. Journal of Oceanography
58:35-44.
Zhang, Y., J. M. Wallace, and D. S. Battisti. 1997. ENSO-like interdecadal variability: 1900-93.
Journal of Climate 10:1004-1020.

Examples
data(enviro.data)
head(enviro.data)

get_minroots Helper function (not exported)

Description

Helper function (not exported)

Usage

get_minroots(ar)

Arguments

ar The ’ar’ part of AR(1)

Value

double
12 phase.plot

MOTHERS Supported mother wavelets

Description
The list of supported mother wavelets is used in multiple places therefore, we provide it as a lazily
evaluated promise.

Usage
MOTHERS

Format
An object of class character of length 3.

phase.plot Plot phases with arrows

Description
Plot phases with arrows

Usage
phase.plot(
x,
y,
phases,
arrow.len = min(par()$pin[2]/30, par()$pin[1]/40),
arrow.col = "black",
arrow.lwd = arrow.len * 0.3
)

Arguments
x X-coordinates
y Y-coordinates
phases Phases
arrow.len Size of the arrows. Default is based on plotting region.
arrow.col Arrow line color.
arrow.lwd Width/thickness of arrows.
plot.biwavelet 13

Note
Arrows pointing to the right mean that x and y are in phase.
Arrows pointing to the left mean that x and y are in anti-phase.
Arrows pointing up mean that y leads x by π/2.
Arrows pointing down mean that x leads y by π/2.

Author(s)
Tarik C. Gouhier (tarik.gouhier@gmail.com)
Huidong Tian provided a much better implementation of the phase.plot function that allows for
more accurate phase arrows.
Original code based on WTC MATLAB package written by Aslak Grinsted.

Examples
# Not run: phase.plot(x, y, phases)

plot.biwavelet Plot biwavelet objects

Description
Plot biwavelet objects such as the cwt, cross-wavelet and wavelet coherence.

Usage
## S3 method for class 'biwavelet'
plot(
x,
ncol = 64,
fill.cols = NULL,
xlab = "Time",
ylab = "Period",
tol = 1,
plot.cb = FALSE,
plot.phase = FALSE,
type = "power.corr.norm",
plot.coi = TRUE,
lwd.coi = 1,
col.coi = "white",
lty.coi = 1,
alpha.coi = 0.5,
plot.sig = TRUE,
lwd.sig = 4,
14 plot.biwavelet

col.sig = "black",
lty.sig = 1,
bw = FALSE,
legend.loc = NULL,
legend.horiz = FALSE,
arrow.len = min(par()$pin[2]/30, par()$pin[1]/40),
arrow.lwd = arrow.len * 0.3,
arrow.cutoff = 0.8,
arrow.col = "black",
xlim = NULL,
ylim = NULL,
zlim = NULL,
xaxt = "s",
yaxt = "s",
form = "%Y",
...
)

Arguments
x biwavelet object generated by wt, xwt, or wtc.
ncol Number of colors to use.
fill.cols Vector of fill colors to be used. Users can specify color vectors using colorRampPalette
or brewer.pal from package RColorBrewer. Value NULL generates MATLAB’s
jet color palette.
xlab X-label of the figure.
ylab Y-label of the figure.
tol Tolerance level for significance contours. Sigificance contours will be drawn
around all regions of the spectrum where spectrum/percentile >= tol. If
strict ith percentile regions are desired, then tol must be set to 1.
plot.cb Plot color bar if TRUE.
plot.phase Plot phases with black arrows.
type Type of plot to create. Can be power to plot the power, power.corr to plot the
bias-corrected power, power.norm to plot the power normalized by the variance,
power.corr.norm to plot the bias-corrected power normalized by the variance,
wavelet to plot the wavelet coefficients, or phase to plot the phase.
plot.coi Plot cone of influence (COI) as a semi-transparent polygon if TRUE. Areas that
fall within the polygon can be affected by edge effects.
lwd.coi Line width of COI.
col.coi Color of COI.
lty.coi Line type of COI. Value 1 is for solide lines.
alpha.coi Transparency of COI. Range is 0 (full transparency) to 1 (no transparency).
plot.sig Plot contours for significance if TRUE.
lwd.sig Line width of significance contours.
plot.biwavelet 15

col.sig Color of significance contours.

lty.sig Line type of significance contours.
bw plot in black and white if TRUE.
legend.loc Legend location coordinates as defined by image.plot.
legend.horiz Plot a horizontal legend if TRUE.
arrow.len Size of the arrows. Default is based on plotting region.
arrow.lwd Width/thickness of arrows.
arrow.cutoff Cutoff value for plotting phase arrows. Phase arrows will be be plotted in regions
where the significance of the zvalues exceeds arrow.cutoff for wt and xwt
objects. For pwtc and wtc objects, phase arrows will be plotted in regions where
the rsq value exceeds arrow.cutoff. If the object being plotted does not have
a significance field, regions whose z-values exceed the arrow.cutoff quantile
will be plotted.
arrow.col Color of arrows.
xlim The x limits.
ylim The y limits.
zlim The z limits.
xaxt Add x-axis? Use n for none.
yaxt Add y-axis? Use n for none.
form Format to use to display dates on the x-axis. See Date for other valid formats.
... Other parameters.

Details
Arrows pointing to the right mean that x and y are in phase.
Arrows pointing to the left mean that x and y are in anti-phase.
Arrows pointing up mean that y leads x by π/2.
Arrows pointing down mean that x leads y by π/2.

Author(s)
Tarik C. Gouhier (tarik.gouhier@gmail.com) Code based on WTC MATLAB package written by
Aslak Grinsted.

References
Cazelles, B., M. Chavez, D. Berteaux, F. Menard, J. O. Vik, S. Jenouvrier, and N. C. Stenseth. 2008.
Wavelet analysis of ecological time series. Oecologia 156:287-304.
Grinsted, A., J. C. Moore, and S. Jevrejeva. 2004. Application of the cross wavelet transform and
wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics 11:561-566.
Torrence, C., and G. P. Compo. 1998. A Practical Guide to Wavelet Analysis. Bulletin of the
American Meteorological Society 79:61-78.
Liu, Y., X. San Liang, and R. H. Weisberg. 2007. Rectification of the Bias in the Wavelet Power
Spectrum. Journal of Atmospheric and Oceanic Technology 24:2093-2102.
16 plot.biwavelet

See Also

image.plot

Examples
t1 <- cbind(1:100, rnorm(100))
t2 <- cbind(1:100, rnorm(100))

# Continuous wavelet transform

wt.t1 <- wt(t1)

# Plot power
# Make room to the right for the color bar
par(oma = c(0, 0, 0, 1), mar = c(5, 4, 4, 5) + 0.1)
plot(wt.t1, plot.cb = TRUE, plot.phase = FALSE)

# Cross-wavelet transform
xwt.t1t2 <- xwt(t1, t2)

# Plot cross-wavelet
par(oma = c(0, 0, 0, 1), mar = c(5, 4, 4, 5) + 0.1)
plot(xwt.t1t2, plot.cb = TRUE)

# Example of bias-correction
t1 <- sin(seq(0, 2 * 5 * pi, length.out = 1000))
t2 <- sin(seq(0, 2 * 15 * pi, length.out = 1000))
t3 <- sin(seq(0, 2 * 40 * pi, length.out = 1000))

# This aggregate time series should have the same power

# at three distinct periods
s <- t1 + t2 + t3

# Compare plots to see bias-effect on CWT:

# biased power spectrum artificially
# reduces the power of higher-frequency fluctuations.
wt1 <- wt(cbind(1:1000, s))
par(mfrow = c(1,2))
plot(wt1, type = "power.corr.norm", main = "Bias-corrected")
plot(wt1, type = "power.norm", main = "Biased")

# Compare plots to see bias-effect on XWT:

# biased power spectrum artificially
# reduces the power of higher-frequency fluctuations.
x1 <- xwt(cbind(1:1000, s), cbind(1:1000, s))
par(mfrow = c(1,2))

plot(x1, type = "power.corr.norm", main = "Bias-corrected")

plot(x1, type = "power.norm", main = "Biased")
pwtc 17

pwtc Compute partial wavelet coherence

Description
Compute partial wavelet coherence between y and x1 by partialling out the effect of x2

Usage
pwtc(
y,
x1,
x2,
pad = TRUE,
dj = 1/12,
s0 = 2 * dt,
J1 = NULL,
max.scale = NULL,
mother = "morlet",
param = -1,
lag1 = NULL,
sig.level = 0.95,
sig.test = 0,
nrands = 300,
quiet = FALSE
)

Arguments
y Time series y in matrix format (n rows x 2 columns). The first column should
contain the time steps and the second column should contain the values.
x1 Time series x1 in matrix format (n rows x 2 columns). The first column should
contain the time steps and the second column should contain the values.
x2 Time series x2 whose effects should be partialled out in matrix format (n rows
x 2 columns). The first column should contain the time steps and the second
column should contain the values.
pad Pad the values will with zeros to increase the speed of the transform.
dj Spacing between successive scales.
s0 Smallest scale of the wavelet.
J1 Number of scales - 1.
max.scale Maximum scale. Computed automatically if left unspecified.
mother Type of mother wavelet function to use. Can be set to morlet, dog, or paul.
Significance testing is only available for morlet wavelet.
param Nondimensional parameter specific to the wavelet function.
18 pwtc

lag1 Vector containing the AR(1) coefficient of each time series.

sig.level Significance level.
sig.test Type of significance test. If set to 0, use a regular χ2 test. If set to 1, then
perform a time-average test. If set to 2, then do a scale-average test.
nrands Number of Monte Carlo randomizations.
quiet Do not display progress bar.

Value

Return a biwavelet object containing:

coi matrix containg cone of influence

wave matrix containing the cross-wavelet transform of y and x1
rsq matrix of partial wavelet coherence between y and x1 (with x2 partialled out)
phase matrix of phases between y and x1
period vector of periods
scale vector of scales
dt length of a time step
t vector of times
xaxis vector of values used to plot xaxis
s0 smallest scale of the wavelet
dj spacing between successive scales
y.sigma standard deviation of y
x1.sigma standard deviation of x1
mother mother wavelet used
type type of biwavelet object created (pwtc)
signif matrix containg sig.level percentiles of wavelet coherence based on the Monte
Carlo AR(1) time series

Note

The Monte Carlo randomizations can be extremely slow for large datasets. For instance, 1000
randomizations of a dataset consisting of 1000 samples will take ~30 minutes on a 2.66 GHz dual-
core Xeon processor.

Author(s)

Tarik C. Gouhier (tarik.gouhier@gmail.com) Code based on WTC MATLAB package written by

Aslak Grinsted.
rcpp_row_quantile 19

References
Aguiar-Conraria, L., and M. J. Soares. 2013. The Continuous Wavelet Transform: moving beyond
uni- and bivariate analysis. Journal of Economic Surveys In press.
Cazelles, B., M. Chavez, D. Berteaux, F. Menard, J. O. Vik, S. Jenouvrier, and N. C. Stenseth. 2008.
Wavelet analysis of ecological time series. Oecologia 156:287-304.
Grinsted, A., J. C. Moore, and S. Jevrejeva. 2004. Application of the cross wavelet transform and
wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics 11:561-566.
Ng, E. K. W., and J. C. L. Chan. 2012. Geophysical applications of partial wavelet coherence and
multiple wavelet coherence. Journal of Atmospheric and Oceanic Technology 29:1845-1853.
Torrence, C., and G. P. Compo. 1998. A Practical Guide to Wavelet Analysis. Bulletin of the
American Meteorological Society 79:61-78.
Torrence, C., and P. J. Webster. 1998. The annual cycle of persistence in the El Nino/Southern
Oscillation. Quarterly Journal of the Royal Meteorological Society 124:1985-2004.

Examples
y <- cbind(1:100, rnorm(100))
x1 <- cbind(1:100, rnorm(100))
x2 <- cbind(1:100, rnorm(100))

# Partial wavelet coherence of y and x1

pwtc.yx1 <- pwtc(y, x1, x2, nrands = 0)

# Partial wavelet coherence of y and x2

pwtc.yx2 <- pwtc(y, x2, x1, nrands = 0)

# Plot partial wavelet coherence and phase difference (arrows)

# Make room to the right for the color bar
par(mfrow = c(2,1), oma = c(4, 0, 0, 1),
mar = c(1, 4, 4, 5), mgp = c(1.5, 0.5, 0))

plot(pwtc.yx1, xlab = "", plot.cb = TRUE,

main = "Partial wavelet coherence of y and x1 | x2")

plot(pwtc.yx2, plot.cb = TRUE,

main = "Partial wavelet coherence of y and x2 | x1")

rcpp_row_quantile Row-wise quantile of a matrix

Description
This is a C++ speed-optimized version. It is equivalent to R version quantile(data, q, na.rm =
TRUE)
20 rcpp_wt_bases_dog

Usage
rcpp_row_quantile(data, q)

Arguments
data Numeric matrix whose row quantiles are wanted.
q Probability with value in [0,1]

Value
A vector of length nrows(data), where each element represents row quantile.

Author(s)
Viliam Simko

rcpp_wt_bases_dog Optimized "wt.bases.dog" function.

Description
This is a C++ version optimized for speed. Computes the wavelet as a function of Fourier frequency
for "dog" mother wavelet.

Usage
rcpp_wt_bases_dog(k, scale, param = -1L)

Arguments
k vector of frequencies at which to calculate the wavelet.
scale the wavelet scale.
param nondimensional parameter specific to the wavelet function.

Value
Returns a list containing:

daughter wavelet function

fourier.factor ratio of fourier period to scale
coi cone of influence
dof degrees of freedom for each point in wavelet power

Note
This c++ implementation is approx. 50
rcpp_wt_bases_morlet 21

Author(s)

Viliam Simko

rcpp_wt_bases_morlet Optimized "wt.bases.morlet" function.

Description

This si a C++ version optimized for speed. Computes the wavelet as a function of Fourier frequency
for "morlet" mother wavelet.

Usage

rcpp_wt_bases_morlet(k, scale, param = -1L)

Arguments

k vector of frequencies at which to calculate the wavelet.

scale the wavelet scale.
param nondimensional parameter specific to the wavelet function.

Value

Returns a list containing:

daughter wavelet function

fourier.factor ratio of fourier period to scale
coi cone of influence
dof degrees of freedom for each point in wavelet power

Note

This c++ implementation is approx. 60

Author(s)

Viliam Simko
22 rcpp_wt_bases_paul

rcpp_wt_bases_paul Optimized "wt.bases.paul" function.

Description

This si a C++ version optimized for speed. Computes the wavelet as a function of Fourier frequency
for "paul" mother wavelet.

Usage

rcpp_wt_bases_paul(k, scale, param = -1L)

Arguments

k vector of frequencies at which to calculate the wavelet.

scale the wavelet scale.
param nondimensional parameter specific to the wavelet function.

Value

Returns a list containing:

daughter wavelet function

fourier.factor ratio of fourier period to scale
coi cone of influence
dof degrees of freedom for each point in wavelet power

Note

This c++ implementation is approx. 59

Author(s)

Viliam Simko
smooth.wavelet 23

smooth.wavelet Smooth wavelet in both the time and scale domains

Description
The time smoothing uses a filter given by the absolute value of the wavelet function at each scale,
normalized to have a total weight of unity, which is a Gaussian function for the Morlet wavelet. The
scale smoothing is done with a boxcar function of width 0.6, which corresponds to the decorrelation
scale of the Morlet wavelet.

Usage
smooth.wavelet(wave, dt, dj, scale)

Arguments
wave wavelet coefficients
dt size of time steps
dj number of octaves per scale
scale wavelet scales

Value
Returns the smoothed wavelet.

Note
This function is used internally for computing wavelet coherence. It is only appropriate for the
morlet wavelet.

Author(s)
Tarik C. Gouhier (tarik.gouhier@gmail.com)
Code based on WTC MATLAB package written by Aslak Grinsted.

References
Torrence, C., and P. J. Webster. 1998. The annual cycle of persistence in the El Nino/Southern
Oscillation. Quarterly Journal of the Royal Meteorological Society 124:1985-2004.

Examples
# Not run: smooth.wt1 <- smooth.wavelet(wave, dt, dj, scale)
24 wclust

wclust Compute dissimilarity between multiple wavelet spectra

Description
Compute dissimilarity between multiple wavelet spectra

Usage
wclust(w.arr, quiet = FALSE)

Arguments
w.arr N x p x t array of wavelet spectra where N is the number of wavelet spectra to
be compared, p is the number of periods in each wavelet spectrum and t is the
number of time steps in each wavelet spectrum.
quiet Do not display progress bar.

Value
Returns a list containing:

diss.mat square dissimilarity matrix

dist.mat (lower triangular) distance matrix

Author(s)
Tarik C. Gouhier (tarik.gouhier@gmail.com)

References
Rouyer, T., J. M. Fromentin, F. Menard, B. Cazelles, K. Briand, R. Pianet, B. Planque, and N.
C. Stenseth. 2008. Complex interplays among population dynamics, environmental forcing, and
exploitation in fisheries. Proceedings of the National Academy of Sciences 105:5420-5425.
Rouyer, T., J. M. Fromentin, N. C. Stenseth, and B. Cazelles. 2008. Analysing multiple time series
and extending significance testing in wavelet analysis. Marine Ecology Progress Series 359:11-23.

Examples
t1 <- cbind(1:100, sin(seq(0, 10 * 2 * pi, length.out = 100)))
t2 <- cbind(1:100, sin(seq(0, 10 * 2 * pi, length.out = 100) + 0.1 * pi))
t3 <- cbind(1:100, rnorm(100)) # white noise

## Compute wavelet spectra

wt.t1 <- wt(t1)
wt.t2 <- wt(t2)
wt.t3 <- wt(t3)
wdist 25

## Store all wavelet spectra into array

w.arr <- array(dim = c(3, NROW(wt.t1$wave), NCOL(wt.t1$wave)))
w.arr[1, , ] <- wt.t1$wave
w.arr[2, , ] <- wt.t2$wave
w.arr[3, , ] <- wt.t3$wave

## Compute dissimilarity and distance matrices

w.arr.dis <- wclust(w.arr)
plot(hclust(w.arr.dis$dist.mat, method = "ward.D"),
sub = "", main = "", ylab = "Dissimilarity", hang = -1)

wdist Compute dissimilarity between two wavelet spectra

Description
Compute dissimilarity between two wavelet spectra

Usage
wdist(wt1, wt2, cutoff = 0.99)

Arguments
wt1 power, wave or rsq matrix from biwavelet object generated by wt, xwt, or wtc.
wt2 power, wave or rsq matrix from biwavelet object generated by wt, xwt, or wtc.
cutoff Cutoff value used to compute dissimilarity. Only orthogonal axes that contribute
more than 1-cutoff to the total covariance between the two wavelet spectra will
be used to compute their dissimilarity.

Value
Returns wavelet dissimilarity.

Author(s)
Tarik C. Gouhier (tarik.gouhier@gmail.com)

References
Rouyer, T., J. M. Fromentin, F. Menard, B. Cazelles, K. Briand, R. Pianet, B. Planque, and N.
C. Stenseth. 2008. Complex interplays among population dynamics, environmental forcing, and
exploitation in fisheries. Proceedings of the National Academy of Sciences 105:5420-5425.
Rouyer, T., J. M. Fromentin, N. C. Stenseth, and B. Cazelles. 2008. Analysing multiple time series
and extending significance testing in wavelet analysis. Marine Ecology Progress Series 359:11-23.
26 wt

Examples
t1 <- cbind(1:100, sin(seq(0, 10 * 2 * pi, length.out = 100)))
t2 <- cbind(1:100, sin(seq(0, 10 * 2 * pi, length.out = 100) + 0.1 * pi))

# Compute wavelet spectra

wt.t1 <- wt(t1)
wt.t2 <- wt(t2)

# Compute dissimilarity
wdist(wt.t1$wave, wt.t2$wave)

wt Compute wavelet transform

Description
Compute wavelet transform

Usage
wt(
d,
pad = TRUE,
dt = NULL,
dj = 1/12,
s0 = 2 * dt,
J1 = NULL,
max.scale = NULL,
mother = "morlet",
param = -1,
lag1 = NULL,
sig.level = 0.95,
sig.test = 0,
do.sig = TRUE,
arima.method = "CSS"
)

Arguments
d Time series in matrix format (n rows x 2 columns). The first column should
contain the time steps and the second column should contain the values.
pad Pad the values will with zeros to increase the speed of the transform.
dt Length of a time step.
dj Spacing between successive scales.
s0 Smallest scale of the wavelet.
wt 27

J1 Number of scales - 1.
max.scale Maximum scale. Computed automatically if left unspecified.
mother Type of mother wavelet function to use. Can be set to morlet, dog, or paul.
param Nondimensional parameter specific to the wavelet function.
lag1 AR(1) coefficient of time series used to test for significant patterns.
sig.level Significance level.
sig.test Type of significance test. If set to 0, use a regular χ2 test. If set to 1, then
perform a time-average test. If set to 2, then do a scale-average test.
do.sig Perform significance testing if TRUE.
arima.method Fitting method. This parameter is passed as the method Parameter to the arima
function.

Value
Returns a biwavelet object containing:
coi matrix containg cone of influence
wave matrix containing the wavelet transform
power matrix of power
power.corr matrix of bias-corrected power using the method described by Liu et al. (2007)
phase matrix of phases
period vector of periods
scale vector of scales
dt length of a time step
t vector of times
xaxis vector of values used to plot xaxis
s0 smallest scale of the wavelet
dj spacing between successive scales
sigma2 variance of time series
mother mother wavelet used
type type of biwavelet object created (wt)
signif matrix containg significance levels

Author(s)
Tarik C. Gouhier (tarik.gouhier@gmail.com)
Code based on wavelet MATLAB program written by Christopher Torrence and Gibert P. Compo.

References
Torrence, C., and G. P. Compo. 1998. A Practical Guide to Wavelet Analysis. Bulletin of the
American Meteorological Society 79:61-78.
Liu, Y., X. San Liang, and R. H. Weisberg. 2007. Rectification of the Bias in the Wavelet Power
Spectrum. Journal of Atmospheric and Oceanic Technology 24:2093-2102.
28 wt.bases

Examples
t1 <- cbind(1:100, rnorm(100))

## Continuous wavelet transform

wt.t1 <- wt(t1)

## Plot power
## Make room to the right for the color bar
par(oma = c(0, 0, 0, 1), mar = c(5, 4, 4, 5) + 0.1)
plot(wt.t1, plot.cb = TRUE, plot.phase = FALSE)

wt.bases Compute wavelet

Description
Computes the wavelet as a function of Fourier frequency.

Usage
wt.bases(mother = "morlet", ...)

Arguments
mother Type of mother wavelet function to use. Can be set to morlet, dog, or paul.
... See parameters k, scale and param in functions: wt.bases.morlet, wt.bases.paul
and wt.bases.dog

Value
Returns a list containing:
daughter wavelet function
fourier.factor ratio of fourier period to scale
coi cone of influence
dof degrees of freedom for each point in wavelet power

Author(s)
Tarik C. Gouhier (tarik.gouhier@gmail.com)
Code based on wavelet MATLAB program written by Christopher Torrence and Gibert P. Compo.

References
Torrence, C., and G. P. Compo. 1998. A Practical Guide to Wavelet Analysis. Bulletin of the
American Meteorological Society 79:61-78.
wt.bases.dog 29

Examples
# Not run: wb <- wt.bases(mother, k, scale[a1], param)

wt.bases.dog Helper method (not exported)

Description
Helper method (not exported)

Usage
wt.bases.dog(k, scale, param = -1)

Arguments
k Vector of frequencies at which to calculate the wavelet.
scale The wavelet scale.
param Nondimensional parameter specific to the wavelet function.

Value
Returns a list containing:

daughter wavelet function

fourier.factor ratio of fourier period to scale
coi cone of influence
dof degrees of freedom for each point in wavelet power

wt.bases.morlet Helper method (not exported)

Description
Helper method (not exported)

Usage
wt.bases.morlet(k, scale, param = -1)
30 wt.bases.paul

Arguments
k Vector of frequencies at which to calculate the wavelet.
scale The wavelet scale.
param Nondimensional parameter specific to the wavelet function.

Value
Returns a list containing:

daughter wavelet function

fourier.factor ratio of fourier period to scale
coi cone of influence
dof degrees of freedom for each point in wavelet power

wt.bases.paul Helper method (not exported)

Description
Helper method (not exported)

Usage
wt.bases.paul(k, scale, param = -1)

Arguments
k Vector of frequencies at which to calculate the wavelet.
scale The wavelet scale.
param Nondimensional parameter specific to the wavelet function.

Value
Returns a list containing:

daughter wavelet function

fourier.factor ratio of fourier period to scale
coi cone of influence
dof degrees of freedom for each point in wavelet power
wt.sig 31

wt.sig Determine significance of wavelet transform

Description
Determine significance of wavelet transform

Usage
wt.sig(
d,
dt,
scale,
sig.test = 0,
sig.level = 0.95,
dof = 2,
lag1 = NULL,
mother = "morlet",
param = -1,
sigma2 = NULL,
arima.method = "CSS"
)

Arguments
d Time series in matrix format (n rows x 2 columns). The first column should
contain the time steps and the second column should contain the values.
dt Length of a time step.
scale The wavelet scale.
sig.test Type of significance test. If set to 0, use a regular χ2 test. If set to 1, then
perform a time-average test. If set to 2, then do a scale-average test.
sig.level Significance level.
dof Degrees of freedom for each point in wavelet power.
lag1 AR(1) coefficient of time series used to test for significant patterns.
mother Type of mother wavelet function to use. Can be set to morlet, dog, or paul.
param Nondimensional parameter specific to the wavelet function.
sigma2 Variance of time series
arima.method Fitting method. This parameter is passed as the method Parameter to the arima
function.

Value
Returns a list containing:
signif vector containing significance level for each scale
signif vector of red-noise spectrum for each period
32 wtc

Author(s)
Tarik C. Gouhier (tarik.gouhier@gmail.com)
Code based on wavelet MATLAB program written by Christopher Torrence and Gibert P. Compo.

References
Torrence, C., and G. P. Compo. 1998. A Practical Guide to Wavelet Analysis. Bulletin of the
American Meteorological Society 79:61-78.

Examples
# Not run: wt.sig(d, dt, scale, sig.test, sig.level, lag1,
# dof = -1, mother = "morlet", sigma2 = 1)

wtc Compute wavelet coherence

Description
Compute wavelet coherence

Usage
wtc(
d1,
d2,
pad = TRUE,
dj = 1/12,
s0 = 2 * dt,
J1 = NULL,
max.scale = NULL,
mother = "morlet",
param = -1,
lag1 = NULL,
sig.level = 0.95,
sig.test = 0,
nrands = 300,
quiet = FALSE
)

Arguments
d1 Time series 1 in matrix format (n rows x 2 columns). The first column should
contain the time steps and the second column should contain the values.
d2 Time series 2 in matrix format (n rows x 2 columns). The first column should
contain the time steps and the second column should contain the values.
wtc 33

pad Pad the values will with zeros to increase the speed of the transform.
dj Spacing between successive scales.
s0 Smallest scale of the wavelet.
J1 Number of scales - 1.
max.scale Maximum scale. Computed automatically if left unspecified.
mother Type of mother wavelet function to use. Can be set to morlet, dog, or paul.
param Nondimensional parameter specific to the wavelet function.
lag1 Vector containing the AR(1) coefficient of each time series.
sig.level Significance level.
sig.test Type of significance test. If set to 0, use a regular χ2 test. If set to 1, then
perform a time-average test. If set to 2, then do a scale-average test.
nrands Number of Monte Carlo randomizations.
quiet Do not display progress bar.

Value
Return a biwavelet object containing:
coi matrix containg cone of influence
wave matrix containing the cross-wavelet transform
wave.corr matrix containing the bias-corrected cross-wavelet transform using the method
described by Veleda et al. (2012)
power matrix of power
power.corr matrix of bias-corrected cross-wavelet power using the method described by
Veleda et al. (2012)
rsq matrix of wavelet coherence
phase matrix of phases
period vector of periods
scale vector of scales
dt length of a time step
t vector of times
xaxis vector of values used to plot xaxis
s0 smallest scale of the wavelet
dj spacing between successive scales
d1.sigma standard deviation of time series 1
d2.sigma standard deviation of time series 2
mother mother wavelet used
type type of biwavelet object created (wtc)
signif matrix containing sig.level percentiles of wavelet coherence based on the
Monte Carlo AR(1) time series
34 wtc.sig

Note
The Monte Carlo randomizations can be extremely slow for large datasets. For instance, 1000
randomizations of a dataset consisting of 1000 samples will take ~30 minutes on a 2.66 GHz dual-
core Xeon processor.

Author(s)
Tarik C. Gouhier (tarik.gouhier@gmail.com)
Code based on WTC MATLAB package written by Aslak Grinsted.

References
Cazelles, B., M. Chavez, D. Berteaux, F. Menard, J. O. Vik, S. Jenouvrier, and N. C. Stenseth. 2008.
Wavelet analysis of ecological time series. Oecologia 156:287-304.
Grinsted, A., J. C. Moore, and S. Jevrejeva. 2004. Application of the cross wavelet transform and
wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics 11:561-566.
Torrence, C., and G. P. Compo. 1998. A Practical Guide to Wavelet Analysis. Bulletin of the
American Meteorological Society 79:61-78.
Torrence, C., and P. J. Webster. 1998. The annual cycle of persistence in the El Nino/Southern
Oscillation. Quarterly Journal of the Royal Meteorological Society 124:1985-2004.
Veleda, D., R. Montagne, and M. Araujo. 2012. Cross-Wavelet Bias Corrected by Normalizing
Scales. Journal of Atmospheric and Oceanic Technology 29:1401-1408.

Examples
t1 <- cbind(1:100, rnorm(100))
t2 <- cbind(1:100, rnorm(100))

## Wavelet coherence
wtc.t1t2 <- wtc(t1, t2, nrands = 10)

## Plot wavelet coherence and phase difference (arrows)

## Make room to the right for the color bar
par(oma = c(0, 0, 0, 1), mar = c(5, 4, 4, 5) + 0.1)
plot(wtc.t1t2, plot.cb = TRUE, plot.phase = TRUE)

wtc.sig Determine significance of wavelet coherence

Description
Determine significance of wavelet coherence
wtc.sig 35

Usage
wtc.sig(
nrands = 300,
lag1,
dt,
ntimesteps,
pad = TRUE,
dj = 1/12,
s0,
J1,
max.scale = NULL,
mother = "morlet",
sig.level = 0.95,
quiet = FALSE
)

Arguments
nrands Number of Monte Carlo randomizations.
lag1 Vector containing the AR(1) coefficient of each time series.
dt Length of a time step.
ntimesteps Number of time steps in time series.
pad Pad the values will with zeros to increase the speed of the transform.
dj Spacing between successive scales.
s0 Smallest scale of the wavelet.
J1 Number of scales - 1.
max.scale Maximum scale.
mother Type of mother wavelet function to use. Can be set to morlet, dog, or paul.
Significance testing is only available for morlet wavelet.
sig.level Significance level to compute.
quiet Do not display progress bar.

Value
Returns significance matrix containing the sig.level percentile of wavelet coherence at each time
step and scale.

Note
The Monte Carlo randomizations can be extremely slow for large datasets. For instance, 1000
randomizations of a dataset consisting of 1000 samples will take ~30 minutes on a 2.66 GHz dual-
core Xeon processor.

Author(s)
Tarik C. Gouhier (tarik.gouhier@gmail.com)
Code based on WTC MATLAB package written by Aslak Grinsted.
36 wtc_sig_parallel

References
Cazelles, B., M. Chavez, D. Berteaux, F. Menard, J. O. Vik, S. Jenouvrier, and N. C. Stenseth. 2008.
Wavelet analysis of ecological time series. Oecologia 156:287-304.
Grinsted, A., J. C. Moore, and S. Jevrejeva. 2004. Application of the cross wavelet transform and
wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics 11:561-566.
Torrence, C., and G. P. Compo. 1998. A Practical Guide to Wavelet Analysis. Bulletin of the
American Meteorological Society 79:61-78.
Torrence, C., and P. J. Webster. 1998. The annual cycle of persistence in the El Nino/Southern
Oscillation. Quarterly Journal of the Royal Meteorological Society 124:1985-2004.

Examples
# Not run: wtcsig <- wtc.sig(nrands, lag1 = c(d1.ar1, d2.ar1), dt,
# pad, dj, J1, s0, mother = "morlet")

wtc_sig_parallel Parallel wtc.sig

Description
Parallelized Monte Carlo simulation equivalent to wtc.sig.

Usage
wtc_sig_parallel(
nrands = 300,
lag1,
dt,
ntimesteps,
pad = TRUE,
dj = 1/12,
s0,
J1,
max.scale = NULL,
mother = "morlet",
sig.level = 0.95,
quiet = TRUE
)

Arguments
nrands Number of Monte Carlo randomizations.
lag1 Vector containing the AR(1) coefficient of each time series.
dt Length of a time step.
xwt 37

ntimesteps Number of time steps in time series.

pad Pad the values will with zeros to increase the speed of the transform.
dj Spacing between successive scales.
s0 Smallest scale of the wavelet.
J1 Number of scales - 1.
max.scale Maximum scale.
mother Type of mother wavelet function to use. Can be set to morlet, dog, or paul.
Significance testing is only available for morlet wavelet.
sig.level Significance level to compute.
quiet Do not display progress bar.

See Also
foreach
wtc.sig

Examples
# Not run: library(foreach)
# library(doParallel)
# cl <- makeCluster(4, outfile="") # number of cores. Notice 'outfile'
# registerDoParallel(cl)
# wtc_sig_parallel(your parameters go here)
# stopCluster(cl)

xwt Compute cross-wavelet

Description
Compute cross-wavelet

Usage
xwt(
d1,
d2,
pad = TRUE,
dj = 1/12,
s0 = 2 * dt,
J1 = NULL,
max.scale = NULL,
mother = "morlet",
param = -1,
38 xwt

lag1 = NULL,
sig.level = 0.95,
sig.test = 0,
arima.method = "CSS"
)

Arguments
d1 Time series 1 in matrix format (n rows x 2 columns). The first column should
contain the time steps and the second column should contain the values.
d2 Time series 2 in matrix format (n rows x 2 columns). The first column should
contain the time steps and the second column should contain the values.
pad Pad the values will with zeros to increase the speed of the transform.
dj Spacing between successive scales.
s0 Smallest scale of the wavelet.
J1 Number of scales - 1.
max.scale Maximum scale. Computed automatically if left unspecified.
mother Type of mother wavelet function to use. Can be set to morlet, dog, or paul.
Significance testing is only available for morlet wavelet.
param Nondimensional parameter specific to the wavelet function.
lag1 Vector containing the AR(1) coefficient of each time series.
sig.level Significance level.
sig.test Type of significance test. If set to 0, use a regular χ2 test. If set to 1, then
perform a time-average test. If set to 2, then do a scale-average test.
arima.method Fitting method. This parameter is passed as the method parameter to the arima
function.

Value
Returns a biwavelet object containing:
coi matrix containg cone of influence
wave matrix containing the cross-wavelet transform
wave.corr matrix containing the bias-corrected cross-wavelet transform using the method
described by Veleda et al. (2012)
power matrix of power
power.corr matrix of bias-corrected cross-wavelet power using the method described by
Veleda et al. (2012)
phase matrix of phases
period vector of periods
scale vector of scales
dt length of a time step
t vector of times
xwt 39

xaxis vector of values used to plot xaxis

s0 smallest scale of the wavelet
dj spacing between successive scales
d1.sigma standard deviation of time series 1
d2.sigma standard deviation of time series 2
mother mother wavelet used
type type of biwavelet object created (xwt)
signif matrix containg significance levels

Author(s)
Tarik C. Gouhier (tarik.gouhier@gmail.com) Code based on WTC MATLAB package written by
Aslak Grinsted.

References
Cazelles, B., M. Chavez, D. Berteaux, F. Menard, J. O. Vik, S. Jenouvrier, and N. C. Stenseth. 2008.
Wavelet analysis of ecological time series. Oecologia 156:287-304.
Grinsted, A., J. C. Moore, and S. Jevrejeva. 2004. Application of the cross wavelet transform and
wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics 11:561-566.
Torrence, C., and G. P. Compo. 1998. A Practical Guide to Wavelet Analysis. Bulletin of the
American Meteorological Society 79:61-78.
Torrence, C., and P. J. Webster. 1998. The annual cycle of persistence in the El Nino/Southern
Oscillation. Quarterly Journal of the Royal Meteorological Society 124:1985-2004.
Veleda, D., R. Montagne, and M. Araujo. 2012. Cross-Wavelet Bias Corrected by Normalizing
Scales. Journal of Atmospheric and Oceanic Technology 29:1401-1408.

Examples
t1 <- cbind(1:100, rnorm(100))
t2 <- cbind(1:100, rnorm(100))

# Compute Cross-wavelet
xwt.t1t2 <- xwt(t1, t2)
plot(xwt.t1t2, plot.cb = TRUE, plot.phase = TRUE,
main = "Plot cross-wavelet and phase difference (arrows)")

# Real data
data(enviro.data)

# Cross-wavelet of MEI and NPGO

xwt.mei.npgo <- xwt(subset(enviro.data, select = c("date", "mei")),
subset(enviro.data, select = c("date", "npgo")))

# Make room to the right for the color bar

par(oma = c(0, 0, 0, 1), mar = c(5, 4, 4, 5) + 0.1)
plot(xwt.mei.npgo, plot.cb = TRUE, plot.phase = TRUE)
Index

∗ coherence phase.plot, 6, 12
biwavelet-package, 2 plot.biwavelet, 13
∗ cross-wavelet pwtc, 15, 17, 18
biwavelet-package, 2
∗ datasets RColorBrewer, 14
MOTHERS, 12 rcpp_row_quantile, 19
∗ dataset rcpp_wt_bases_dog, 20
enviro.data, 10 rcpp_wt_bases_morlet, 21
∗ wavelet rcpp_wt_bases_paul, 22
biwavelet-package, 2
smooth.wavelet, 10, 23
ar1.spectrum, 4
ar1_ma0_sim, 5 text, 7
arima, 27, 31, 38
wclust, 24
arima.sim, 5, 6
wdist, 25
arrow, 6
wt, 14, 15, 25, 26, 27
arrow2, 7
wt.bases, 28
wt.bases.dog, 28, 29
biwavelet (biwavelet-package), 2
wt.bases.morlet, 28, 29
biwavelet-package, 2
wt.bases.paul, 28, 30
brewer.pal, 14
wt.sig, 31
check.data, 7 wtc, 14, 15, 25, 32, 33
check.datum, 8 wtc.sig, 34, 36, 37
colorRampPalette, 14 wtc_sig_parallel, 36
convolve2D, 9, 10
xwt, 14, 15, 25, 37, 39
convolve2D_typeopen, 10

Date, 15

enviro.data, 10

fft, 9
foreach, 37

get_minroots, 6, 11

image.plot, 15, 16

MOTHERS, 12
mvfft, 9

40