wc: Cross-wavelet transformation, wavelet coherence computation,... In WaveletComp: Computational Wavelet Analysis

Description

This function provides Morlet cross-wavelet transformation results of the given two time series, performed within the lower-order functions WaveletCoherency and WaveletTransform subject to criteria concerning the time and frequency resolution, an (optional) lower and/or upper Fourier period, and a variety of filtering methods for the coherence computation. It performs a simulation algorithm to test against a specified alternative hypothesis, which can be chosen from a variety of options, and provides p-values. The selected model will be fitted to the data and simulated according to estimated parameters in order to provide surrogate time series. This function is called by function analyze.coherency.

The name and parts of the layout were inspired by a similar function developed by Huidong Tian and Bernard Cazelles (archived R package WaveletCo). The major part of the code for the computation of the cone of influence and the code for Fourier-randomized surrogate time series have been adopted from Huidong Tian. The implementation of a choice of filtering windows for the computation of the wavelet coherence was inspired by Luis Aguiar-Conraria and Maria Joana Soares (GWPackage).

Usage

 1 2 3 4 5 6 wc(x, y, start = 1, dt = 1, dj = 1/20, lowerPeriod = 2*dt, upperPeriod = floor(length(x)/3)*dt, window.type.t = 1, window.type.s = 1, window.size.t = 5, window.size.s = 1/4, make.pval = TRUE, method = "white.noise", params = NULL, n.sim = 100, save.sim = FALSE)

Arguments

 x the time series x to be analyzed y the time series y to be analyzed (of the same length as x
 start starting point in time (for the computation of the cone of influence). Default: start = 1.
 dt time resolution, i.e. sampling resolution in the time domain, 1/dt = number of observations per time unit. For example: a natural choice of dt in case of hourly data is dt = 1/24, resulting in one time unit equaling one day. This is also the time unit in which periods are measured. If dt = 1, the time interval between two consecutive observations will equal one time unit. Default: 1. dj frequency resolution, i.e. sampling resolution in the frequency domain, 1/dj = number of suboctaves (voices per octave). Default: 1/20.
 lowerPeriod lower Fourier period (measured in time units determined by dt, see the explanations concerning dt) for wavelet decomposition. If dt = 1, the minimum admissible value is 2. Default: 2*dt. upperPeriod upper Fourier period (measured in time units determined by dt, see the explanations concerning dt) for wavelet decomposition. Default: (floor of one third of time series length)*dt.
window.type.t

type of window for smoothing in time direction; select from:

 0 ("none") : no smoothing in time direction 1 ("bar") : Bartlett 2 ("tri") : Triangular (Non-Bartlett) 3 ("box") : Boxcar (Rectangular, Dirichlet) 4 ("han") : Hanning 5 ("ham") : Hamming 6 ("bla") : Blackman

Default: 1 = "bar".

window.type.s

type of window for smoothing in scale (period) direction; select from:

 0 ("none") : no smoothing in scale (period) direction 1 ("bar") : Bartlett 2 ("tri") : Triangular (Non-Bartlett) 3 ("box") : Boxcar (Rectangular, Dirichlet) 4 ("han") : Hanning 5 ("ham") : Hamming 6 ("bla") : Blackman

Default: 1 = "bar".

window.size.t

size of the window used for smoothing in time direction, measured in time units determined by dt, see the explanations concerning dt. Default: 5, which together with dt = 1 defines a window of length 5*(1/dt) = 5, equaling 5 observations (observation epochs). Windows of even-numbered sizes are extended by 1.

window.size.s

size of the window used for smoothing in scale (period) direction in units of 1/dj. Default: 1/4, which together with dj = 1/20 defines a window of length (1/4)*(1/dj) = 5. Windows of even-numbered sizes are extended by 1.

 make.pval Compute p-values? Logical. Default: TRUE.
method

the method of generating surrogate time series; select from:

 "white.noise" : white noise "shuffle" : shuffling the given time series "Fourier.rand" : time series with a similar spectrum "AR" : AR(p) "ARIMA" : ARIMA(p,0,q)

Default: "white.noise".

params

a list of assignments between methods (AR, and ARIMA) and lists of parameter values applying to surrogates. Default: NULL.

Default includes two lists named AR and ARIMA:

• AR = list(...), a list containing one single element:

 p : AR order. Default: 1.
• ARIMA = list(...), a list of six elements:

 p : AR order. Default: 1. q : MA order. Default: 1. include.mean : Include a mean/intercept term? Default: TRUE. sd.fac : magnification factor to boost the residual standard deviation. Default: 1. trim : Simulate trimmed data? Default: FALSE. trim.prop : high/low trimming proportion. Default: 0.01.
 n.sim number of simulations. Default: 100.
 save.sim Shall simulations be saved on the output list? Logical. Default: FALSE.

Value

A list with the following elements:

 Wave.xy (complex-valued) cross-wavelet transform (analogous to Fourier cross-frequency spectrum, and to the covariance in statistics) Angle phase difference, i.e. phase lead of x over y (= phase.x-phase.y) sWave.xy smoothed (complex-valued) cross-wavelet transform sAngle phase difference, i.e. phase lead of x over y, affected by smoothing
 Power.xy cross-wavelet power (analogous to Fourier cross-frequency power spectrum) Power.xy.avg average cross-wavelet power in the frequency domain (averages over time) Power.xy.pval p-values of cross-wavelet power Power.xy.avg.pval p-values of average cross-wavelet power
 Coherency (complex-valued) wavelet coherency of series x over series y in the time/frequency domain, affected by smoothing (analogous to Fourier coherency, and to the coefficient of correlation in statistics) Coherence wavelet coherence (analogous to Fourier coherence, and to the coefficient of determination in statistics (affected by smoothing) Coherence.avg average wavelet coherence in the frequency domain (averages across time) Coherence.pval p-values of wavelet coherence Coherence.avg.pval p-values of average wavelet coherence
 Wave.x, Wave.y (complex-valued) wavelet transforms of series x and y Phase.x, Phase.y phases of series x and y Ampl.x, Ampl.y amplitudes of series x and y Power.x, Power.y wavelet power of series x and y Power.x.avg, Power.y.avg average wavelet power of series x and y, averages across time Power.x.pval, Power.y.pval p-values of wavelet power of series x and y Power.x.avg.pval, Power.y.avg.pval p-values of average wavelet power of series x and y sPower.x, sPower.y smoothed wavelet power of series x and y
 Period the Fourier periods (measured in time units determined by dt, see the explanations concerning dt) Scale the scales (the Fourier periods divided by the Fourier factor)
 coi.1, coi.2 borders of the region where the wavelet transforms are not influenced by edge effects (cone of influence). The coordinates of the borders are expressed in terms of internal axes axis.1 and axis.2.
 nc number of columns = number of observations = number of observation epochs; "epoch" meaning point in time nr number of rows = number of scales (Fourier periods)
 axis.1 tick levels corresponding to the time steps used for (cross-)wavelet transformation: 1, 1+dt, 1+2dt, .... The default time axis in plot functions provided by WaveletComp is determined by observation epochs, however; "epoch" meaning point in time. axis.2 tick levels corresponding to the log of Fourier periods: log2(Period). This determines the period axis in plot functions provided by WaveletComp.
 series.sim a data frame of the series simulated as surrogates for the (detrended) time series (if both make.pval = TRUE and save.sim = TRUE.)

Author(s)

Angi Roesch and Harald Schmidbauer; credits are also due to Huidong Tian, Bernard Cazelles, Luis Aguiar-Conraria, and Maria Joana Soares.

References

Aguiar-Conraria L., and Soares M.J., 2011. Business cycle synchronization and the Euro: A wavelet analysis. Journal of Macroeconomics 33 (3), 477–489.

Aguiar-Conraria L., and Soares M.J., 2011. The Continuous Wavelet Transform: A Primer. NIPE Working Paper Series 16/2011.

Aguiar-Conraria L., and Soares M.J., 2012. GWPackage. Available at https://sites.google.com/site/aguiarconraria/joanasoares-wavelets; accessed September 4, 2013.

Cazelles B., Chavez M., Berteaux, D., Menard F., Vik J.O., Jenouvrier S., and Stenseth N.C., 2008. Wavelet analysis of ecological time series. Oecologia 156, 287–304.

Liu P.C., 1994. Wavelet spectrum analysis and ocean wind waves. In: Foufoula-Georgiou E., and Kumar P., (eds.), Wavelets in Geophysics, Academic Press, San Diego, 151–166.

Tian, H., and Cazelles, B., 2012. WaveletCo. Available at https://cran.r-project.org/src/contrib/Archive/WaveletCo/, archived April 2013; accessed July 26, 2013.

Torrence C., and Compo G.P., 1998. A practical guide to wavelet analysis. Bulletin of the American Meteorological Society 79 (1), 61–78.

Veleda D., Montagne R., and Araujo M., 2012. Cross-Wavelet Bias Corrected by Normalizing Scales. Journal of Atmospheric and Oceanic Technology 29, 1401–1408.

WaveletComp documentation built on May 2, 2019, 6:33 a.m.