Fahdvfm

2022-11-12

Find a regression to this : $a\equiv t\phantom{\rule{0.444em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}\mathrm{\Delta }\right)$, $a$ and $\mathrm{\Delta }$ are the unknowns constants.

meexeniexia17h

It sounds like problem would benefit from Fourier analysis more than any sort of regression model.
Given that the events are discrete, you could use$\sum _{j=1}^{n}{e}^{\frac{2\pi i{t}_{j}}{p}}$ to get the "Fourier transform" $F\left(p\right)$ of your samples, where p is the period, or "tick interval", you're testing for. Even with some random dispersion, if your points are more or less regularly spaced at integer multiples of an interval $\mathrm{\Delta }$, you'd see a spike in magnitude at $p=\mathrm{\Delta }$, with smaller spikes at $p=\frac{\mathrm{\Delta }}{2},\frac{\mathrm{\Delta }}{3},\frac{\mathrm{\Delta }}{4}$, etc.
As an added bonus, since $F\left(p\right)$ will be a complex number, the argument of $F\left(\mathrm{\Delta }\right)$ at the peak magnitude would also give you an estimate of the value of $a$.
One thing to watch out for is that you shouldn't try to test values of $p$ greater than the total length of the music itself, because then the values will tend toward a maximum just due to the angles of the samples being squished close to 0, and that's not the kind of spike you're looking for.

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