Pearson correlation coefficient for complex time series

Question

I would like to know how to calculate the Pearson correlation coefficient for two complex time series.

Do we simply do

Or there is something else?

import numpy as np
    
R = lambda x,y: ((x-x.mean())*(y-y.mean())).sum()/(np.sqrt(((x-x.mean())**2).sum())*np.sqrt(((y-y.mean())**2).sum()))

x,y = np.loadtxt("data.txt",dtype=np.complex128).T

ri = R(x,y)
ri = (ri*ri.conj()).real
print(ri)

File in https://file.io/KzwvQFx8XsXQ

That denominator doesn't look right. You've only got a single sum there. To make it a bit more readable, i suggest just using numpy.linalg.norm 3 times here. — Mikael Öhman
– Mikael Öhman, Commented Apr 8, 2023 at 0:05
and that the sum is in the wrong spot. R = lambda x,y: np.linalg.norm((x-x.mean())*(y-y.mean())) / (np.linalg.norm(x-x.mean())*np.linalg.norm(y-y.mean())) — Mikael Öhman
– Mikael Öhman, Commented Apr 8, 2023 at 0:10
You are right, I copied the code wrongly. The numerator should be a sum instead of a norm, I believe. — iury simoes-sousa
– iury simoes-sousa, Commented Apr 8, 2023 at 2:13

Kat Kononov · Accepted Answer · 2023-10-24 13:04:57Z

1

Keep in mind that complex time series can be correlated not only by a linear scale factor, as is the case for real time series, but also by a linear phase rotation or phase reflection. That results in a complex correlation coefficient.

import numpy as np
def corr_complex_rotational(x, y):
    x_bar = x - np.mean(x)
    y_bar = y - np.mean(y)
    Rxy = x_bar @ y_bar.T.conj() / (len(x)-1)
    Rxx = x_bar @ x_bar.T.conj() / (len(x)-1)
    Ryy = y_bar @ y_bar.T.conj() / (len(x)-1)
    Rxx, Ryy = Rxx.real, Ryy.real  # they are real but carry a 0j that we don't need
    rho = Rxy / np.sqrt(Rxx * Ryy)
    return rho

I recommend consulting the book "Statistical Signal Processing of Complex-Valued Data: The Theory of Improper and Noncircular Signals" by Schreier and Scharf, 2010.

answered Oct 24, 2023 at 13:04

Kat Kononov

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