Fahdvfm

2022-11-12

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

meexeniexia17h

Beginner2022-11-13Added 18 answers

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" $F(p)$ 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 $F(p)$ will be a complex number, the argument of $F(\mathrm{\Delta})$ 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.

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" $F(p)$ 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 $F(p)$ will be a complex number, the argument of $F(\mathrm{\Delta})$ 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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