## Biwavelet

The biwavelet R package is a port of the WTC MATLAB program 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 wavelet analyses. Wavelet analyses are resolved in the time and frequency domains, and thus ideal for identifying changes over time in the contribution of each frequency (or period) of a time series.

**Importantly, biwavelet version 0.14 or later computes and plots the bias-corrected wavelet and cross-wavelet power spectrum 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. To demonstrate this bias, we can construct a time series by summing three sinusoidal waves each characterized by the same power at a different period:

```
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))

timeseries=t1+t2+t3

The wavelet spectrum of the time series should show peaks of identical power at each of the three dominant periods. However, the traditional approach leads to a consistent reduction in power at low periodicities:

`wt1=wt(cbind(1:1000, s)) `

par(mfrow=c(1,2))

plot(wt1, type="power.corr.norm", main="Bias-corrected wavelet power")

plot(wt1, type="power.norm", main="Biased wavelet power")

This bias also affects the cross-wavelet, which can be used to determine the time- and frequency-resolved relationship between two time series. The new version of biwavelet implements the bias-correction developed by Veleda et al. (2012). The bias does not affect the wavelet coherence, however. For instance, MEI and NPGO undergo coherent fluctuations at periodicities of ~64-150 months (5-12 years) between 1966-2010, with peaks in NPGO leading peaks in MEI by one quarter of a cycle. For help interpreting these results, please see the references below and the biwavelet manual.

The biwavelet package can also be used to compute the (dis)similarity between pairs of time series based on their wavelet spectrum:

To download the development version of the package, visit the download page or type the following at the R command line:

`install.packages("biwavelet", repos="http://R-Forge.R-project.org")`

To download the release version of the package on CRAN, type the following at the R command line:

`install.packages("biwavelet")`

## 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. -
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 Niño/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.