CrossWavelet

CrossWavelet is a technique used in time‑series analysis to study the common features between two signals in both time and frequency domains. It builds on the continuous wavelet transform, which decomposes each series into a time‑frequency representation, and then calculates a joint measure of power and phase coherence. The resulting cross‑wavelet spectrum highlights regions where the two series exhibit similar oscillatory behaviour. Areas of high power in the cross‑wavelet display indicate co‑occurring patterns or synchrony, while phase arrows represent the relative lag between the series at each point.

The method is especially useful for non‑stationary data, where spectral characteristics change over time. Common applications include climatology, where researchers examine relationships between atmospheric variables such as sea‑surface temperature and precipitation, and in seismology, where fault‑loading signals are compared to after‑shock patterns. Financial analysts also use CrossWavelet to identify shared cycles in asset price movements or volatility indices.

Cross‑wavelet analysis can be implemented in several statistical software packages. In R, the 'Biwavelet' package provides functions such as `cwt()` for individual wavelet transforms and `waveletCross()` for computing the cross‑wavelet and its significance. MATLAB users can employ the `cwt` toolbox, and Python libraries such as PyWavelets or the `waveslim` module provide similar functionality. The technique’s primary output is a two‑dimensional plot where color intensity reflects cross‑power and directional arrows encode phase relationships. Accuracy of significance testing often relies on Monte Carlo simulations assuming red‑noise backgrounds.