1. Wavelet Coherence & Cross-Wavelet Transform

2. Squared Coherency

3.  = 0.1, 0.05, 0.01

4. Wavelet Coherence
R2 looks like definition of correlation coefficient
(localized correlation coefficient in frq-time space)
𝑹 𝒏 𝟐 𝐬 = 𝑺 𝒔 −𝟏 𝑾 𝒏 𝑿𝒀 𝒔 𝟐 𝑺 𝒔 −𝟏 𝑾 𝒏 𝑿 𝒔 𝟐 ∗𝑺 𝒔 −𝟏 𝑾 𝒏 𝒀 𝒔 𝟐
Calculate Coherence from Wavelets of each time series
S is smoothing operator, in time and frequency
W X , W Y are the wavelets for each time series
W XY is the cross wavelet
Similar to a correlation coefficient varying in time/frequency
Get 95% confidence level using chi-squared
s-1 is used to convert to energy density (s = scale, n = time index)

5. Example

6. Example
U, m/s

7. Wavelet Coherence
Period, days

8. Phase of Coherence Phase arrows pointing: right: in-phase
left: anti-phase
down: X leading Y by 90°
up: Y leading X by 90°
𝝓 𝒏 𝐬 = 𝐭𝐚𝐧 −𝟏 𝒊𝒎𝒂𝒈 𝑺 𝒔 −𝟏 𝑾 𝒏 𝑿𝒀 𝒔 𝒓𝒆𝒂𝒍 𝑺 𝒔 −𝟏 𝑾 𝒏 𝑿𝒀 𝒔

9. Coherence and Phase
Days in 2010
right: in-phase; left: anti-phase; down: X leading Y by 90°;
up: Y leading X by 90°

10. Cross-Wavelet Transform
The Cross-Wavelet Transform of two time series xn and yn indicates common power and relative phase in time-frequency space
The Cross-Wavelet Transform is given by:
W XY = W X W Y*; * is complex conjugate
|W XY | = cross-wavelet power
arg (W XY ) = relative phase between xn and yn

12. Wavelet Coherence & Cross-Wavelet Tips
Normally distributed histograms
Consider transforming the time series, if the probability density functions of the time series are far from Gaussian.
Know what to expect from the results, could find statistically significant links by chance
Caution with results within the COI
If significant coherence, phase lag should vary slowly and be consistent
The significance level of the WTC has to be determined using Monte Carlo methods

13. References
Grinsted, A., Moore, J.C., Jevrejeva, S. (2004) Application of the cross wavelet transform and wavelet coherence to geophysical time series, Nonlin. Processes Geophys., 11, 561–566, doi: /npg  [pdf]
Jevrejeva, S., Moore, J.C.,  Grinsted, A. (2003) Influence of the Arctic Oscillation and El Niño-Southern Oscillation (ENSO) on ice conditions in the Baltic Sea: The wavelet approach, J. Geophys. Res., 108(D21), 4677, doi: /2003JD003417 [pdf]
Torrence, C., Compo, G.P. (1998) A practical guide to wavelet analysis, Bull. Am. Meteorol. Soc., 79, 61–78 [pdf]
Torrence, C., Webster, P. (1999)  Interdecadal changes in the ESNO-Monsoon System, J.Clim., 12, 2679–2690