# WaveletTransform: Morlet wavelet transformation of a time series In WaveletComp: Computational Wavelet Analysis

## Description

It computes the Morlet wavelet transformation of a given time series, subject to criteria concerning: the time and frequency resolution, an (optional) lower and/or upper Fourier period.

The output is further processed by higher-order functions `wt`, `WaveletCoherency` and `wc`, and can be retrieved from `analyze.wavelet` and `analyze.coherency`.

The name and layout were inspired by a similar function developed by Huidong Tian and Bernard Cazelles (archived R package `WaveletCo`).

## Usage

 ```1 2``` ```WaveletTransform(x, dt = 1, dj = 1/20, lowerPeriod = 2*dt, upperPeriod = floor(length(x)*dt/3)) ```

## Arguments

 `x` the time series to be analyzed
 `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`.

## Value

A list of class `analyze.wavelet` with the following elements:

 `Wave` complex wavelet transform of the series `Phase` phases `Ampl` amplitudes
 `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)
 `Power` wavelet power in the time/frequency domain
 `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)

## Author(s)

Angi Roesch and Harald Schmidbauer; credits are also due to Huidong Tian, and Bernard Cazelles.

## References

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

Carmona R., Hwang W.-L., and Torresani B., 1998. Practical Time Frequency Analysis. Gabor and Wavelet Transforms with an Implementation in S. Academic Press, San Diego.

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 Y., Liang X.S., and Weisberg R.H., 2007. Rectification of the Bias in the Wavelet Power Spectrum. Journal of Atmospheric and Oceanic Technology 24, 2093–2102.

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.

`wt`, `WaveletCoherency`, `wc`, `analyze.wavelet`, `analyze.coherency`