Cross-Wavelet Transform
Also known as: XWT, Cross-spectrum wavelet
The cross-wavelet transform (XWT) is a bivariate extension of the continuous wavelet transform that measures the joint time-frequency representation of two signals. Introduced by Torrence and Compo (1998) and applied extensively by Grinsted, Moore, and Jevrejeva (2004) to geophysical data, XWT reveals where two signals share common spectral power and the phase relationship between them at each time-frequency point. This is the natural generalization of classical cross-spectral analysis to the time-varying domain.
Read the full method
Sign in with a free account to read this section.
Method map
The neighbourhood of related methods — select a node to explore.
When to use it
Use the cross-wavelet transform when you need to analyze interaction between two time series with time-varying frequency content. It is ideal for climate research (coupling between atmospheric and oceanic modes), geophysics (seismic wave interactions), economics (analyzing relationships between financial indicators), and any domain where understanding phase-locked behavior matters. XWT is superior to static correlation when relationships change over time or across frequencies.
Strengths & limitations
- Reveals both timing and frequency of signal interactions, showing where and when two signals are coupled
- Phase information reveals which signal leads or lags the other at each time-frequency region
- Can identify intermittent or frequency-specific relationships missed by global correlation
- Naturally handles non-stationary signals where relationships change over time
- Provides intuitive visual representations (time-frequency contour plots) of signal interactions
- Computationally more expensive than univariate methods—requires two CWT computations plus multiplication
- Results are highly sensitive to wavelet choice and parameter selection; different choices produce different patterns
- Interpretation of cross-wavelet magnitude is ambiguous when one signal has low power—high XWT power requires both signals to have significant power
- Statistical significance testing is complex due to non-orthogonal time-frequency atoms; standard assumptions do not apply
Frequently asked
What does high cross-wavelet power mean?
High XWT power means the two signals have significant oscillatory activity at that time-frequency point AND that activity is coherent between the signals. However, this requires both signals to have substantial individual power—consult univariate spectrograms to avoid misinterpretation.
How do I interpret the phase of the cross-wavelet transform?
Phase angle indicates the lead-lag relationship: zero phase means perfect in-phase oscillation, pi/2 means one signal leads the other by a quarter cycle, etc. Use arrows on the spectrogram: pointing right = signal x leads, left = signal y leads.
Can I test for statistical significance of XWT?
Yes, but it is more complex than univariate tests. Use Monte Carlo simulation against the null hypothesis of independent white or AR(1) noise. Grinsted's implementation (2004) includes these tests.
Why do different wavelets give different XWT results?
Wavelets have different time-frequency trade-offs. Narrower wavelets (e.g., Morlet) provide better frequency resolution but worse temporal localization. Use results that are robust across wavelet choices.
What is the difference between cross-wavelet transform and wavelet coherence?
XWT measures absolute joint power and phase. Wavelet coherence normalizes XWT by the individual wavelet powers, showing correlation independent of amplitude changes. Use XWT for absolute relationships, coherence for correlation-like measures.
Sources
- Torrence, C., & Compo, G. P. (1998). A practical guide to wavelet analysis. Bulletin of the American Meteorological Society, 79(1), 61–78. DOI: 10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2 ↗
- Torrence, C., & Webster, P. J. (1999). Interdecadal changes in the ENSO–monsoon system. Journal of Climate, 12(8), 2679–2690. DOI: 10.1175/1520-0442(1999)012<2679:icitem>2.0.co;2 ↗
- 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(5–6), 561–566. DOI: 10.5194/npg-11-561-2004 ↗
How to cite this page
ScholarGate. (2026, June 3). Cross-Wavelet Transform. ScholarGate. https://scholargate.app/en/time-series/cross-wavelet-transform
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Wavelet CoherenceTime Series↔ compare