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I've been using the Lomb-Scargle and String Length Minimization methods to find the periods of various astronomical objects with unevenly spaced data. I wanted to see if I could find any patterns between the two that could determine why a particular method was more accurate in one case but not another.

Unfortunately, I'm not having much luck. In pretty much every case, String Length is always more accurate. In only one case did the Lomb-Scargle and String Length find the exact same period.

There don't seem to be any patterns between the number of data points or the time between observations. I've looked at different objects including different variable stars, and asteroids. etc. I did notice that String-Length was able to find the actual period for objects that have two minima and two maxima (like asteroids and eclipsing binaries), but the period found by the Lomb-Scargle had to be doubled.

What determines how accurate each of these methods is? So far, I don't how I could ever recommend somebody uses Lomb-Scargle when it's much less accurate in comparison to String Length Minimization in 99% of cases.

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    $\begingroup$ Can you indicate how you measure the accuracy of the methods? $\endgroup$
    – ProfRob
    Dec 5, 2022 at 19:35
  • $\begingroup$ @ProfRob I didn't do any crazy in-depth analysis. I just ran the same data set through both period-finding methods, and then compared the found period to the "known" period of the object found in the literature. $\endgroup$ Dec 5, 2022 at 21:30

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I think they are just not the best period-finding algorithms. They are too simple. For high signal-to-nose data with SINGLE periods they seem to work ok, but the problem is always with low-quality data and multi-periodic sources. Take a look at this paper, which compares different period-finding algorithms.

My advice is the following: just use multiple algorithms, incorporating bootstrap or something similar, and consider median values if you are studying large data sets. If you have a few lightcurves, you may use the algorithms' output and check manually. You can also try to look in the literature which algorithm performs better on the objects you are looking for, but you will never get one 100% accurate.

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I do think the previous response contains almost all you need to know, so I won't write everything, only a little bit of additional information I did find useful at least in my case.

There are two main types of periodograms used for finding periods of variable stars - trigonometric (based on Fourier transform, Long-Scargle here is one of the simpler examples, the simplest one is basic DFT, and the most advanced/accurate is currently FastChi2).

The other used technique is phase-folding, to which string length minimization does belong. Currently, the most accurate one belonging to that class does utilize conditional entropy instead of string length.

Phase-folding periodograms are usually able to find the period with an error close to $\frac{10^{-5}}{d}$ while the precision of trigonometric ones usually doesn't exceed $\frac{10^{-4}}{d}$.

Why are trigonometric periodograms more popular, than the phase-folding ones despite having worse both accuracy and precision? Here is only one reason for that - speed.

I haven't had the opportunity to benchmark an optimized periodogram based on conditional entropy (which seems to be quite fast for its class), but usually, the trigonometric ones can calculate periods of at least ~100 stars per second per CPU core on relatively modern hardware with SSD drives (it does depend mostly on max frequency, the number is stated for max period of $\frac{12}{d}$, and the growth of time is usually slightly superlinear).

Searches of large photometry databases (like OGLE one, containing ~1 billion stars) does usually take a lot of time and computing power. I've heard about a recent search for BLAPs in OGLE database taking ~ 2 months on a cluster of 10 PCs with 6 CPU cores each, but I can't confirm the information here is true). LS and GLS periodograms can both use trigonometric recursions or use FFT to significantly speed up the calculations in comparison to the basic implementation of periodogram. More precisely in the case of data I've been testing it on recursive periodograms were ~30 times faster than the basic ones, and for FFT the factor exceeded 100.

There is also the possibility of running periodograms on GPUs/TPUs, but it's a relatively new thing, and I'm almost sure currently only the basic LS periodogram was implemented in that way. For the more advanced ones limited static memory is the biggest issue making the implementation much more difficult, than it looks.

TLDR; Once you have a list of variable star candidates usage of a phase-folding periodogram will almost always be a better choice. Trigonometric ones are better suited for finding variable candidates from large databases.

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