Here Cross-Wavelet transforms (XWTs) are computed with the 'xwt' function of the R package 'biwavelet'.


The wavelet coefficients \(W_{s,x}(y_1,y_2)\) resulting from a XWT correspond to:

$$ W_{s,x}(y_1,y_2)=W_{s,x}(y_1)W_{s,x}(y_2) $$ Hence it is a multiplication of:

For more information about Cross-Wavelet Transforms, please refer to:

Torrence, Christopher, and Gilbert P. Compo. 1998. A Practical Guide to Wavelet Analysis." Bulletin of the American Meteorological Society 79 (1): 61-78.

Wavelet coherence

The 'rsq' coefficients \(R_{s,x}(y_1,y_2)\) resulting from a WTC correspond to:

$$ R_{s,x}(y_1,y_2)=\frac {S(s^{-1}W_{s,x}(y_1)W_{s,x}(y_2)))} {S(s^{-1}W_{s,x}(y_1))S(s^{-1}W_{s,x}(y_2))} $$ where S is a smoothing operator.

The wavelet coherence is comparable to a localized correlation coefficient.

For more information about Wavelet Coherence, please refer to:

Grinsted, Aslak, John C Moore, and Svetlana Jevrejeva. 2004. "Application of the Cross Wavelet Transform and Wavelet Coherence to Geophysical Time Series." Nonlinear Processes in Geophysics 11 (5/6): 561-66.