# How to interpret the arrows output by wcoherence

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Kimberly on 28 Nov 2023
Answered: Abhimenyu on 8 Dec 2023
I understand that the arrows in a wcoherence plot need to be interpreted with care, but I am wondering how to interpret them even with care.
In the El Nino example found here ( https://www.mathworks.com/help/wavelet/ug/compare-time-frequency-content-in-signals-with-wavelet-coherence.html ), the authors wrote "The plot also shows that there is an approximate 3/8-to-1/2 cycle delay between the two time series at those periods." I am working through the example and have reproduced the plot. I understand how the authors converted 3/8 to 1/2 cycles to years. I would like to know how the authors quantified the cycle delay (3/8 to 1/2). Thank you.

Abhimenyu on 8 Dec 2023
Hi Kimberly,
I understand that you are trying to interpret the wavelet coherence plot. The arrows in the wcoherence plot indicate the phase relationship between the two signals at each time and frequency (or period) point. The authors of the El Nino example quantified the cycle delay by using the phase information from the wavelet cross-spectrum, which is the complex-valued output of the wcoherence function. The phase angle of the cross-spectrum is the same as the phase lag between the signals.
The direction of the arrows in high coherence regions can help to understand the phase relationship. For example, if the arrows are pointing to the right, it indicates that the two signals are in phase (0 delay), and if they are pointing to the left, it indicates an anti-phase relationship (delay of 1/2 cycle). A 1/4 cycle lag in a signal at a particular frequency is indicated by an arrow pointing vertically. Hence, an arrow pointing in north-east direction (between 0 and 1/4 cycle lag direction) corresponds to 1/8 cycle delay (half of 1/4) and similarly an arrow pointing in north-west direction (between 1/4 and 1/2 cycle lag direction) corresponds to 3/8 cycle delay (adding half of 1/4 to 1/4). The delays move in an anti-clockwise fashion.
In the “El-Nino example”, the arrows in the plot majorly point in north-west and anti-phase direction which shows that there is an approximate 3/8-to-1/2 cycle delay between the two time series at those periods.
Please refer to the below mentioned MATLAB documentation links to understand more on “wcoherence function and Compare Time-Frequency Content in Signals with Wavelet Coherence” respectively:
I hope this helps to resolve the query.
Thanks,
Abhimenyu

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