A continuous wavelet transform module for Python. Includes a collection of routines for wavelet transform and statistical analysis via FFT algorithm. Most recently cross-wavelet tranforms, wavelet coherence tests and plotting functions were added to the module.

This module requires NumPy and SciPy.

The sample scripts (sample.py, sample_xwt.py) illustrate the use of the wavelet and inverse wavelet transforms, cross-wavelet transform and wavelet transform coherence. Results are plotted in figures similar to the sample images.

This module is based on routines provided by C. Torrence and G. P. Compo available at http://paos.colorado.edu/research/wavelets/, on routines provided by A. Grinsted, J. Moore and S. Jevrejeva available at http://noc.ac.uk/using-science/crosswavelet-wavelet-coherence, and on routines provided by A. Brazhe available at http://cell.biophys.msu.ru/static/swan/.

This software is released under a BSD-style open source license. Please read the license file for furter information. This routine is provided as is without any express or implied warranties whatsoever.

Download the code and run the below line within the top level folder.

python setup.py install

There is an errata page at the wavelet website maintaned at the Program in Atmospheric and Oceanic Sciences, University of Colorado, Boulder, Colorado, which is accessible throught the link http://paos.colorado.edu/research/wavelets/errata.html

A Practical Guide to Wavelet Analysis

Christopher Torrence and Gilbert P. Compo

Program in Atmospheric and Oceanic Sciences, University of Colorado, Boulder, Colorado

Errata

• Figure 3: N/(2 sigma^2) should just be N/sigma^2.
• Equation (17), left-hand side: Factor of 1/2 should be removed.
• Table 1, DOG, Psi-hat (third column, bottom row): Should be a minus sign in front of the equation.
• Sec 3f, last paragraph: Plugging N=506, dt=1/4 yr, s0=2dt, and dj=0.125 into Eqn (10) actually gives J=64, not J=56 as stated in the text. However, in Figure 1b, the scales are only plotted out to J=56 since the power is so low at larger scales.

Additional information

Table 3: Cross-wavelet significance levels, from Eqn.(30)-(31). (DOF = degrees of freedom)

Significance level	Real wavelet (1 DOF)	Complex wavelet (2 DOF)

   0.10        |        1.595         |          3.214
   0.05        |        2.182         |          3.999
   0.01        |        3.604         |          5.767

We would like to thank Christopher Torrence, Gilbert P. Compo, Aslak Grinsted, John Moore, Svetlana Jevrejevaand and Alexey Brazhe for their code and also Jack Ireland and Renaud Dussurget for their attentive eyes, feedback and debugging.

Nabil Freij, Sebastian Krieger, Alexey Brazhe, Christopher Torrence, Gilbert P. Compo and contributors.

1. Mallat, S. (2008). A wavelet tour of signal processing: The sparse way. Academic Press, 2008, 805.
2. Addison, P. S. (2002). The illustrated wavelet transform handbook: introductory theory and applications in science, engineering, medicine and finance. IOP Publishing. http://dx.doi.org/10.1201/9781420033397.
3. Torrence, C. and Compo, G. P. (1998). A Practical Guide to Wavelet Analysis. Bulletin of the American Meteorological Society, American Meteorological Society, 1998, 79, 61-78. <http://dx.doi.org/10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2>.
4. Torrence, C. and Webster, P. J. (1999). Interdecadal changes in the ENSO-Monsoon system, Journal of Climate, 12(8), 2679-2690. <http://dx.doi.org/10.1175/1520-0442(1999)012<2679:ICITEM>2.0.CO;2>.
5. Grinsted, A.; Moore, J. C. & Jevrejeva, S. (2004). Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics, 11, 561-566. http://dx.doi.org/10.5194/npg-11-561-2004.
6. 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(12), 2093-2102. http://dx.doi.org/10.1175/2007JTECHO511.1.