How can I quantify the phase/amplitude relationship between two signals?

I have two signals A and B. I want to show that high amplitude events in B are phase locked to oscillations in signal A. I have already identified candidate events signal B.

I estimate phase_of_A using the angle of the hilbert transform of A, and I estimate the envelope_of_B using the abs of the hilbert transform of B.

A scatter plot of the joint distribution shows a clear relationship between the two variables. Additionally if I randomly re-order the phase_of_A variable the structure is disrupted.

My current thinking is use Monte-Carlo methods to demonstrate that the real data is more structured than the shuffled data, I'm just not sure what parameter/statistic to compute on the joint distribution.

• What do the colors in you plot mean? And what form of "quantification" do you need? After all, the very existence of your plot shows you have a quantitative way to express the joint distribution (or an estimate thereof). How would a "clean" way differ from that?
– whuber
Dec 31 '12 at 16:44
• @whuber, the plot was generated in matlab using the jet colormap. Brighter colors indicate a higher density of observations. Now that I think about it because phase is a circular variable I can probably fit a a von-mises distribution to the data. Does that sound reasonable? Dec 31 '12 at 16:47
• I don't see how you can directly fit a von Mises distribution (which is univariate) to a bivariate dataset. Although you state that "brighter" colors indicate greater densities, I suspect that hue actually represents density, with red corresponding to high and blue to low. Such color-based maps do a poor job of conveying the information. For example, it is possible that this map is showing us a bivariate distribution in which the density varies along the x-axis but the conditional density in the y direction is constant.
– whuber
Dec 31 '12 at 17:21
• @whuber I re-wrote the question a bit, hopefully clarifying what i'm trying to do. Dec 31 '12 at 17:59

The first was to compute the circular-linear correlation between the two variables: phase_of_A and envelope_of_B. I did this for both the real and shuffled data sets.
The second measure was to compute the mean resultant vector. The angle was the phase_of_A and the length was envelope_of_B. These turned out to be quite sufficient for what I needed. I also did this analysis for shuffled data sets.