Autocorrelation feature extractor

[hughrawlinson]

Would require a minor bump.

There's a weird performance thing for v8, the code looks bad but actually it's there to trigger a compiler optimization. Happy to explain if someone's interested.

It includes tests, but the tests make me nervous, would love to try on more signals.

Someone should definitely have a look at the output and compare it with other implementations. It looks v suspicious to me, particularly seeing as the last half are zeroes.

[jakubfiala]

Hey @hughrawlinson, really sorry for getting round to this so late. I had a look at your impl and compared it with Matlab's xcorr, and an implementation using the IFFT method @nevosegal mentioned. The results are almost exactly the same at the beginning, the small differences might be because of f32/f64.

But the zeroes definitely shouldn't be there - I'm testing with a 128-sample sine wave, whose positive xcorr should be gradually approaching zero until the last sample.

Also, I think usually developers will expect a full xcorr with both negative and positive lags - that way it can be used for linear prediction and other things.

I have an implementation of the IFFT method we could potentially use, unless we're worried about the performance implications. Happy to chat about this privately if that's easier :)

[jakubfiala]

thanks a lot for doing this btw :)

[hughrawlinson]

I'd love to use your implementation! Will review a PR any time :) As long as the performance is fine for realtime, I'm sold. I had trouble implementing this in realtime, there was a v8 issue where a deop was triggered so I had to get around that to make it run fast enough.

[hughrawlinson]

It didn't even occur to me to do negative lags! Will get on it :)

[hughrawlinson]

@jakubfiala this passes tests now. Accurate to 0.0000018530201888518416 on the sin test (once I added a zero at the start). I think since it's so close, it's reasonable to call this done (pending feedback, obviously).

The one remaining mystery I have is in the source formula:

There were lots of equations and implementations all over the internet that I looked for - I initially wanted to implement a generic 'correlation of two signals' function, but couldn't find a formula that I could understand. There were some generic implementations that seemed parameterizable, and I didn't particularly want to let users configure which correlation equation was to be used under the hood, nor configure further lower level stuff. I found the above formula in these lecture notes at the very end. I implemented the extractor based on that, and I understand it all except the term dt. Obviously the tests pass regardless of that term, which I guess is just 1 in this case - but I would love to know what it is. I'm sure it's related to the term ΔT, further up the notes, but I'm not sure what the relationship is. Clearly, I have a lot of reading to do.

[nevosegal]

Sorry I didn't join the conversation earlier. The term dt just means that we're integrating this over t. It's not something you multiply against. Note that this equation is for continuous signals - it looks a bit different for discrete signals.

In any case, regarding the benchmarking Jakub did (comparing time-domain and freq-domain autocorrelation): Even if there isn't big difference in computation speed (although there should be - O(N^2) vs Nlog(N)) I still believe doing that it in the frequency domain makes more sense as we already computed all of the components needed for it. The way I understand it, we only need to perform an IFFT on the power spectrum (magnitude spectrum ^ 2) to get the autocorrelation result.

This is just an optimisation so it shouldn't block the PR - thanks a lot for working on this @hughrawlinson !!

[hughrawlinson]

Hey,

I tried implementing it based on IFFT like you suggested, using fftjs. I didn't quite get the result I expected.

Expected result:

Actual result:

Forgetting about the normalization errors - the second one is upside down, and cropped. I can fix the fact that the length is off by zero padding I would guess. Any ideas?

[nevosegal]

Oh. I think fftjs always return half of the FFT result to be more efficient, and that might not be good for this use case... I'll check it out.

[nevosegal]

@hughrawlinson What are you comparing the result to? Autocorrelation output from librosa?

[hughrawlinson]

@nevosegal I'm comparing against an autocorrelation result that @jakubfiala gave me from an implementation of autocorrelation that he did.

[hughrawlinson]

Our long nightmare is over, GitHub Copilot implemented autocorrelation for us lol

function mean(x) {
	var sum = 0;
	for (var i = 0; i < x.length; i++) {
		sum += x[i];
	}
	return sum / x.length;
}

function autocorrelation(x, y) {
	var n = x.length;
	var meanX = mean(x);
	var meanY = mean(y);
	var sum = 0;
	for (var i = 0; i < n; i++) {
		sum += (x[i] - meanX) * (y[i] - meanY);
	}
	return sum / (n - 1);
}