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Note: I don't have formal training in statistics, so please excuse any mistake I've made in my question.

I have two arrays, each with the time-series data of a binary variable, like this:

A: 001101011010100100101001...
B: 100101000101001100101101...

The time-series are of length $N$ (they can vary). (Edit: Thanks to @Mai for pointing out that I've said "time-series" here, but I only care about their "spatial" correlations, so the samples for each time-step can be treated as independent samples.)

The goal is to determine whether $A$ and $B$ correlate (negative or positive) with one another or not, but to limit the chance of a false positive to $0.0001\%$ (for example). This obviously means raising the probability of a false negative - and that's fine for my use case. Also, to be clear, I don't care about the strength of the correlation - only whether or not they correlate at all.

A simple approach would be to look at the difference between $P(A|B)$ and $P(A)$. If there is a difference, then they correlate. This obviously will give a lot of false positives, since the time-series are of length $N$, which is finite.

So I had the idea that I'd get the absolute value of the difference between $P(A|B)$ and $P(A)$, and then subtract the maximum absolute difference that we'd "expect" to see between those values in $99.9999\%$ of experiments between two independent binary variables with $P(A)$ and $P(B)$ where the time-series are of length $N$. As far as I can see, that would cause the probability of a false positive to be $0.0001\%$.

Is this a reasonable approach? Is there a simpler way? I'm actually dealing with trillions of pairs of binary variables, and so performance is a consideration. If I went with the above approach, then I think I'd need to use a combination of normal and poisson approximations to the binomial distribution to be able to handle the full range of $N$ and $p$ values. Thanks!

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I think you can do this with a chi-squared test or Fishers exact test!

Just collapse your data to the needed format:

enter image description here

With the chi-squared test you can calculate the probability that your 2 measurements are independent or not. I think this is what you want to know.

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  • $\begingroup$ I think the fact that they are time series implies they are not sequences of independent trials. Not sure. $\endgroup$ Commented Nov 13, 2019 at 14:22
  • $\begingroup$ Thanks! This seems like just what I'm after! @Mai I have updated the post to clarify that the samples can be treated independently - thanks! $\endgroup$
    – Mifa
    Commented Nov 13, 2019 at 18:14

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