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Suppose I have two sequences of measurements, $x_1[n]$ and $x_2[n]$ for $0 \le n \le N-1$.

How do I determine if there is a causal relationship between the two?

My first thought was, well... I can just find the correlation between $x_1$ and $x_2$. But almost immediately I was struck by correlation is not causation.

OK, so in Python, we can generate a uniform noise signal x1 and then filter it to get x2. Clearly, in this case, x2 is causally related to x1.

import numpy as np
import scipy.signal as signal

Q = 50
N =1000
x1 = np.random.uniform(-1,1,N)
b,a = signal.iirfilter(1,[2*500*(1-1/(2*Q))/44100,2*500*(1+1/(2*Q))/44100])
x2 = signal.lfilter(b,a,noise1)

x1

The noise signal x1.

x2

The filtered version of x1, x2.

Unfortunately, the correlation of the two doesn't tell us much:

Correlation of x1 and x2.

Next, I thought about using coherence:

$$ C_{x_1x_2}(\omega) = \frac{|G_{x_1x_2}(\omega)|^2}{G_{x_1x_1}(\omega) G_{x_2x_2}(\omega)} $$

where $G_{ab}$ is the cross-spectral density between $a$ and $b$.

However, this just seems to tell me that the two are causally related... it doesn't tell me the direction of the causation.

Coherences in both directions.

The figure above shows what should be obvious: $C_{x_1x_2}(\omega) = C_{x_2x_1}(\omega)$.

So... how do I determine the direction of the causation?


For example, suppose I want to know whether $x_1$ caused $x_2$, or vice versa.

So either $$ x_1 = h \star x_2 $$ or $$ x_2 = g \star x_1 $$ where $\star$ is convolution, and $h$ and $g$ are the impulse responses of LTI systems. Presumably, $g$ is the inverse of $h$ so that $g \star h = 1$.

Is it possible that both $g$ and $h$ are stable and causal?

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    $\begingroup$ Is this even possible in general? If x1 = x2 + 1, you could say that x2 came first and caused x1, or x1 came first and caused x2 = x1 - 1. Maybe, if you can establish that x1 = f(x2) where f is non-invertible, then x2 had to come first? $\endgroup$
    – MBaz
    Commented Mar 3, 2021 at 22:26
  • $\begingroup$ watching this. I could try to give a response, but I feel that there will be people better able to do. $\endgroup$
    – Bob
    Commented Mar 4, 2021 at 9:54
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    $\begingroup$ @MBaz Right, but in your example there are no dynamics. I'm wondering if one can make some assumptions about the dynamics relating the two measurements. See my update. $\endgroup$
    – Peter K.
    Commented Mar 4, 2021 at 13:32
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    $\begingroup$ If $h$ introduces a delay, then $g$ would have to be acausal, right? (or vice versa). $\endgroup$
    – MBaz
    Commented Mar 4, 2021 at 18:33
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    $\begingroup$ @MBaz Not really! I've read quite a bit. This long post by Michael Nielsen is quite interesting. $\endgroup$
    – Peter K.
    Commented Oct 15, 2021 at 19:28

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