How to calculate phase synchrony?

21

Hm, after a bit of poking around, I find Pearson’s correlation coefficient might be interesting.

If the data sets have a perfect linear correlation, the coefficient is 1.0. (Perfect negative correlation is -1.0.) 0 is completely decorrelated. “Perfect correlation” does not mean “equal” – x[i] - xMean eliminates DC offset, and the standard deviation product accounts for amplitude differences.

(
f = { |x, y|
	var xMean = x.mean,
	yMean = y.mean,
	momentSum = 0, xDev = 0, yDev = 0;
	
	x.size.do { |i|
		var xDiff = (x[i] - xMean),
		yDiff = (y[i] - yMean);
		momentSum = momentSum + (xDiff * yDiff);
		xDev = xDev + xDiff.squared;
		yDev = yDev + yDiff.squared;
	};
	xDev = (xDev / (x.size - 1)).sqrt;
	yDev = (yDev / (x.size - 1)).sqrt;  // if y has extra items, they're ignored
	
	momentSum / ((x.size - 1) * xDev * yDev)
};
)

f.(x, x);  // equal datasets
-> 1.0

f.(x, x * 3 + 0.5);  // in phase but different amplitude + DC
-> 1.0

f.(x, x.rotate(20));  // out of phase
-> 0.5555702330196

Edit: The formula is sigma[i = 0…n-1]((x[i] - xMean) * (y[i] - yMean)) / ((n-1) * stdev(x) * stdev(y)) … (x[i] - xMean) matches x - xMean in my original attempt, meaning that the top of the fraction is actually a sum(x * y) with an adjustment for DC. So I’d guessed that part right. What we were missing was the correction for amplitude, which Pearson provides with the product of standard deviations.

hjh

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