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:
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: