Is there a widely accepted software package for determining the Time of Minimum and the standard error of the ToM from unevenly spaced time series observations?
I have written a python script to apply the Kwee and van Woerden (1956) method, but would like to be able to reference a well known and accepted software instead of “John’s homebrew script”!
Some of my questions are:
Is it still accepted to perform a linear interpolation between the unevenly spaced observations to create an evenly spaced sequence?
If I have an even number of observations, should I interpolate to an odd number by adding an evenly spaced point, or by dropping one point. If N is even, should I use N+1 or N-1 since KvW requires an odd number of evenly spaced observations.
Is the mid-exposure time used for the calculations, or should the time of the observation be flux weighted since the light curve gradients are high near the ToM?
How is the uncertainty of the individual observatons utilized in the calculation? KvW were silent on this. As the variable becomes faint, the uncertainty normally increases.
Is there an accepted software that makes the “right” choices in performing the ToM analysis, and importantly the standard error of ToM?
Thanks in advance for your replies,
John
One last bonus question: How should one determine the ToM from survey data where the observations are widely spaced with respect to the period?
I’ve calculated a lot of times of minimum of eclipsing binaries, sometimes using KVW, but mostly polynomial fitting, but I can’t answer your questions directly. These days I find an AI search mostly very helpful in these sorts of complex questions (disclaimer: AI can make errors). For example, I fed your first question into Google AI, making clear in the question that the subject matter was eclipsing binary stars. I received a fairly detailed response.
I then fed your question re the acceptability or otherwise of a linear interpolation and received a reply in the affirmative, but with the qualification that modern methods have moved away from KVW. At some point there was a comment that KVW underestimates the error of the determination.
I didn’t keep going with the rest of your questions.
Concerning your last question re ToM from sparse data, I wrote an article on a laborious method for sparse TESS data in the April 2024 Newsletter of Variable Stars South. I’ll attempt to attach a MS Word document of the paper.
As you might imagine, I’ve been down the AI path and have reached my own conclusions but I wanted thoughtful answers from people like yourself with practical experience.
I am definitely interested in the paper you provided.
You didn’t mention a S/W. If you don’t mind I would be interested in that.
I’ve used Peranso for both KVW and polynomial fitting and also VStar for polynomial fitting. Peranso has an unusual method of determining the error estimate of the polynomial fit ToM, taking the RMS value then somehow transposing it to the x (time) axis. I thought I had a copy of the method (sent to me on request by Tonny Vanmunster), but I can’t find it.
When I use VStar for the polynomial fit (usually 6th or 7th degree) I simply quote the RMS value, which of course is not actually the error of the ToM estimate itself.
I took a quick glance at the paper you attached. I’m a bit confused on what you are calling sparse data. In figure 6 you show the region you selected to be used to determine the time-of-minimum (ToM). I would not have selected such a wide region around the minimum you were analyzing. I would have selected for that star something around the +13.5 level or higher. Why you ask: because the outer regions have most likely very little to contribute to the actual ToM in a physical sense. There may be other physical processes like the O’Connel effect, star spots, etc. going on so far from the actual eclipse minimum.
I have some fortran from 1989 (not written by me) that indicates Kwee & van Woerden (KvW) computes a mean error of the differences used in the calculation of the ToM. I presume Paranso KvW does the same–I haven’t used Paranse a lot nor in a long time so not sure.
TESS produces very precise, low noise measurements generally so that begs the question of what is an appropriate error estimate to use for a determination of a ToM of an eclipser. The RMS error of the residuals of the fitting function maybe? A quick look at Deeg (2020) at least to me seems to be a lot of hand waving with tests done with “added artificial noise” for example. The “Identification of the Problem” section is a little light on proof there is a problem IMHO! The NaN in that paper to me screams very high precision, very few points so it is not too surprising you might get NaN in a mean-error computation like in KvW. Is a NaN a problem or does it mean you should be very happy that at least numerically you have basically no error in the estimate of the ToM?
The original Kwee & van Woerden method is simple and it works.
The next release of VStar will have a KvW (classic and Deeg 2020 variant in one) plug-in.
This old forum topic talked about an alternative (Brat+) method:
Your article also refers to O-C, which I also plan to have in the next release (as for KvW, has been on the list for awhile).
Development snapshot releases will appear before the official release.
The current release also added a piecewise linear model creator plug-in that converts a binned mean series to a piecewise linear function. As with polynomial models, extrema can be determined from this as well. The current release also fixed a problem in the generated loess model creator plug-in as well.
Thanks for your comments David, and thanks also (thanks are really not enough) for your huge contribution to the astronomical community with your creation of VStar.
Thanks for your comments Jim. Discussion of the ‘subjective’ elements in using polynomial fit for ToM determination could go on forever. I’ve done substantial testing on the very precise TESS data as well as the far less precise data of my own from EBs. Selection of the range of observations to which the polynomial fit is applied is one of these. The other is the degree of the polynomial. I agree that it would be OK to use a shorter segment of the light curve in Fig. 6 of my article, and that it because the example uses the very precise TESS data. Not so for my own data, and I simply use a substantial part of the eclipse for any ToM determination, TESS or other, with a high order polynomial. Leaving the sample constant, I often determine the ToM with several different polynomial degrees (for example, 4 to 8). The results converge and I make the assumption, for which I cannot give you a mathematical proof, that the true mathematical result is close to the values near the convergence.
As I said, the discussion could go on forever about polynomial fits, KVW and other methods. What matters is the result. An advantage of TESS is that each sector covers close to a months worth of data, and with the precise results (from the amazingly small lenses) it is easy to construct O-C diagrams from even just one sector. Those diagrams show show how precise the O-C values are.
One final comment. The article I attached to my post above mentions the time conversion utility previously hosted by the State University of Ohio, but not any more. Converting BJD_TDB (TESS data) to HJD_UTC (for O-C diagrams) thus becomes a problem. For those who speak Python, there is a utility in Astropy which does the heavy lifting. I don’t speak Python, but my grandson does, and he wrote a routine for me which does the job.
Sure there is a large amount of literature, in astronomy, and mathematics on determination of extrema for sure. We should all try to use standard tools and methods, not roll our own, for hopefully obvious reasons. Physical models are best, of course. I’m sure you know that polynomial fits can do strange things in the face of noise as can Fourier fits. Sir Ocam rules and I still very much like the Kwee & van Woerden method and am not concerned at all about the tiny error estimates in the face of high-precision photometry from TESS.
Unfortunately the link in the old forum thread no longer works. Anyone still have a copy of the pdf to share? I’ll google when done here. The old AAVSO forum thread is mostly talking about “I really like the Brat+” method and I code it up in my fav language! Sorry to all you who love to write code… you have my sympathy!
In my old fortran the header has this in it…
C AN EXCELLENT WORKING DESCRIPTION IS FOUND
C IN HENDEN, A.A., AND KATICHUCK, R. H. (1982)) "ASTRONOMICAL
C PHOTOMETRY, VAN NOSTRAND REINHOLD, NEW YORK, PP.266-268,
C BUT THEIR EXAMPLE CONTAINS NUMERICAL ERRORS.
In my book what at first glance looks like Kwee & van Woerden references Hertsprung, E 1928. BAN, Vol. 4, 179 not Kwee & van Woerden.
I’m confused now…
Google AI suggests SILICUPS from Moravian Instruments which is still downloadable. Anyone using or trying that on the TESS data?
I did a google AI search using “general mathematical method for determination of a time of minimum in data with error estimate for the time”. The AI is clearly biased from my previous searches!
Here is the AI response: " The Kwee-van Woerden (KvW) method, a non-parametric approach, determines the time of minimum by identifying the symmetry axis of a light curve using a parabola fit to the symmetry function (S(T_0)) and a grid-based calculation of errors [1956]. Alternatively, modern techniques, often implemented in tools like SILICUPS, use Non-Linear Least Squares fitting of polynomial templates, using the covariance matrix or Monte Carlo bootstrapping for error estimation. [1, 2]"
I leave that as a jumping off point… I’m movin’ on…
One of the softwares my students and I use is Peranso. It can pull in a number of different data sets, including TESS. We also use a range of other methods to do max and min. The main issue with Peranso is the cost. There is the upfront cost and then a yearly fee to stay with updated software. The software also does period searches with at least 14 differnt methods, which is nice. For my situation I have university funding to cover the cost, but I can understand people wanting a cheaper software.
A dataset containing a minima (could be a maximum or any extrema) if viewed from a statistical sampling and information theory point-of-view contain the same information.
Sure some methods may be more “optimized” for unequally spaced data sampling. Some only use equally spaced data.
What if the minimum is flat bottomed but with a slight non-zero slope? What if there is a gas stream changing the shape of the flat bottomed minimum slightly from cycle to cycle? Where exactly is this mythical minimum anyway? Geometrically a flat-bottomed minimum contains no information where the minimum is, but the rise and decline data does.
Here is a paper on using wavelet analysis and a noise process to determine Tc (ToM) for transits to start a conversation.
Western fire smoke killing any observing for me after the rain and cloud period. No data with extrema in it for me to look at!
In the case of eclipsing binaries, I use the online Time of Minimum calculator at VarAstro - portal for publication and sharing photometric observations of variable stars and exoplanetary transits, the web site of the Czech Astronomical Society. After uploading an AAVSO dataset and an accompanying chart, the program has a button that fits a 6-parameter function to the data to find the time of minimum. I also wrote a Python program that accomplishes the same thing, but I now prefer using the VarAstro calculator because it is quick, accurate, and modern.
with tests done with “added artificial noise” for example. The “Identification of the Problem” section is a little light on proof there is a problem IMHO! The NaN in that paper to me screams very high precision, very few points so it is not too surprising you might get NaN in a mean-error computation like in KvW. Is a NaN a problem or does it mean you should be very happy that at least numerically you have basically no error in the estimate of the ToM?