Fix mistake in SHO parameter C_0

[tovrstra]

No description provided.

[RobbeBohy]

The definition Foreman-Mackey et al. use for the PSD is:

$$\sqrt{\frac{2}{\pi}} \frac{S_0 \omega_0^4}{(\omega^2 - \omega_0^2)^2 + \frac{\omega_0^2 \omega^2}{Q^2}}$$

If we plug in $\omega = 2 \pi f$ and the new definition $S_0 = \frac{C_0}{2}$, we still have a different prefactor as opposed to what is suggested by the ACID 1 documentation:

Now:

$$\sqrt{\frac{1}{2 \pi}} \frac{C_0 f_0^4}{(f^2 - f_0^2)^2 + \frac{f_0^2 f^2}{Q^2}}$$

ACID 1:

$$\frac{C_0 f_0^4}{(f^2 - f_0^2)^2 + \frac{f_0^2 f^2}{Q^2}}$$

Or is this difference exactly what is meant with the "unitary normalization convention for the Fourier transform"?

[RobbeBohy]

I just noticed an additional detail: the current version for the SHO ACF still uses $\tau$ instead of $\Delta_t$ for the $Q = \frac{1}{2}$ case.

[tovrstra]

There are some other differences too, where we should clarify things. The notation in ACID is such that C_0 is the integral of the autocorrelation function, which is convenient, but it deviates a bit from the convetions of the Lorentz model in STACIE. I'm fine with the difference, but it can be confusing. A more distinctive notation would help: C_0 here versus C_1 in STACIE is probably too similar and creates some expectations.

[tovrstra]

Indeed, the 1/sqrt(2 pi) is due to the difference in normalization. The unitary convention of the continuous Fourier transform is relatively uncommon, and mainly adds more notation, so I'd rather avoid it.

[tovrstra]

We could use $I_0$ instead of $C_0$. I refers to "integral", which is a bit similar to the STACIE papers and documentation, subcript zero hints at the zero frequency.

Another option is $\mathcal{I}$, but we never used this as a model parameter in our notation, so that's potentially confusing.

[RobbeBohy]

After thinking about this a bit more, perhaps $I_0$ could lead to confusion as well. In STACIE, the notation makes sense because we approximate the real $I$ via a model extrapolated to zero frequency, $I^{model} (f=0)$. For ACID, however, we never introduce such an approximation. We look directly at the zero‑frequency value of the kernel’s PSD, $C(f=0)$, rather than the zero‑frequency value of a model function.

For this reason, it seems least confusing to use the same symbol as the PSD but with a subscript 0. However, since our PSD is denoted $C(f)$, I understand the concern that this could be confused with the Lorentz $C_1$​. In that case, perhaps the most consistent solution would be to represent the PSD with a symbol other than $C$?

[tovrstra]

That can be a useful change. I've currently used uppercase $C$ because it is the FT of lowercase $c$, but pairing symbols this way for Fourier transforms is not that critical.

The most common notation is $K_{xx}(t)$ (or $R_{xx}(t)$) and $S_{xx}(\omega)$. I slightly prefer an ordinary frequency for interpretative purposes in STACIE, so using $S$ could cause some confusion.

[RobbeBohy]

Between the two ACF notations, I would choose $K_{xx} (t)$ to avoid any possible confusion with position coordinates. I agree that using ordinary frequencies is clearer for STACIE, but wouldn’t this already be sufficiently explicit if we simply always write $S_{xx} (f)$ instead of $S_{xx} (\omega)$?

[tovrstra]

I find it hard to come up with something that has no drawbacks.

Changing the notation of the ACF and PSD is a bit unpleasant because we have used the current notation already in published work. It would be better to keep the changes minimal, e.g. by only renaming the model parameters. I agree that $I_0$ is not ideal because the $I$ notation in the STACIE paper is for rescaled spectra, which is not relevant for the definitions of the kernels.

Something as mundane as a simple $A$ could work better. We could also take different symbols for the different models, e.g. $W$, $E$ and $H$ and give more descriptive names to the models themselves. We could still keep the zero subscript (smaller change, reference to zero frequency).

[RobbeBohy]

Thanks, looks good to me!

[tovrstra]

Thanks for checking. I'll proceed with preparing a 1.2.2 release on Zenodo, hopefully the last one in the 1.x series.