Look, I am figuring out the period of a DCEP(B) variable star (AS Cas) with the aid of the DC DFT algorithm, and just want to know how can I estimate the error of the calculations that this algorithm performs.
For polynomial fit, error estimates are available, or at least goodness-of fit metrics are, whereas error estimates for DCDFT aren't made available in VStar at this point.
This topic has come up before. I'd appreciate guidance from VStar users regarding the best approach to providing DCDFT error.
For example, is an error estimate of a Fourier model (created via DCDFT or the Fourier model plugin) appropriate, or is something else required?
Hey again, David.
I really appreciate your support. Look, the question you're asking is very good, and sincerely I do not know exactly how to answer it. I suppose that an error estimate of a DC DFT and the Fourier model plugin would do, but maybe I am mistaken.
Section 7.10 of Grant Foster's excellent book Analyzing Light Curves A Practical Guide gives several methods of estimating the uncertainty of a DCDFT frequency estimate. I just checked and this book is once again available on-line from Lulu Publishing.
I strongly recommend this book to anyone doing light curve analysis with VStar. This is the book I use as the text for the AAVSO CHOICE course Analyzing Data with VStar *.
Grant outlines a few methods of estimating the uncertainty of the estimated frequency of a signal, but recommends the frequency peak FWHM method. This is the one to use since the theoretical methods depend on too many assumptions that are essentially never completely true and often not close to being true.
If you send me your e-mail address via the contact tab on my profile I will send you information discussed in sections 7.9 and 7.10 of the book. Section 7.9 contains information about selecting sampling rates for periodograms (actually frequency spectra). Note: when doing DCDFT analysis of a light curve it is better to work with frequencies rather than periods because spectral features (harmonics, aliases, etc.) generally linear combinations of frequencies not periods.
* "data" rather than "light curves" is used in the course name because VStar is useful for analyzing many kinds of data, not just light curves.
Brad Walter, WBY
Hi Brad, Enrique
I have captured the uncertainty estimate methods in these two issues:
I've implemented standard error of the frequency & semi-amplitude and I'm currently testing these.
Then I'll move on to FWHM.
From issue 255, minimally adding to what Brad has said:
Foster's methods fall into two categories (page 153 of his Analyzing Light Curves book):
- standard error of the frequency and standard error of the semi-amplitude based upon Fourier model residuals
- FWHM, the period range near the peak over which the periodogram power is at least half its peak value.
Foster also says (page 154 of Analyzing Light Curves) that: In most cases the possible range estimated by the theoretical formula is too liberal while the range estimated by FWHM is too conservative.
So, implementing standard error and FWHM seems reasonable, in the same way that for polynomial fit we have AIC, BIC, RMS.
I'm verifying the Java implementation in VStar against Grant's R code as well.
While the discussion is on error stats, one that would be advantageous to add, if possible, would be the error of the estimate of a time of minimum or maximum from a polynomial fit.
RMS, AIC and BIC are available, but none of these actually represent an error estiamte for the time itself.
Thanks David. I'll see what I can find.
The only method I have a description of at this point is used in Peranso. I'll look for it and send it to you offline. It includes a diagram which can't be posted here.
Two papers I have found are https://www.astro.sk/caosp/Eedition/FullTexts/vol43no3/pp382-387.pdf and https://oejv.physics.muni.cz/issues/oejv_0215.pdf . I have used least squares fitting of parabolas and 4th order polynomials to minima and then used Goal Seek in Excel to determine the Δt that corresponds to moving up the model curve for minima (or down the model curve for well-defined maxima) by the RMS of residuals of the data around the minimum (or maximum) from the model. When I say around I mean between the inflection points where the 2nd derivative of the model changes sign e.g. from positive to negative for minima and negative to positive for maxima.
It is very important to detrend the data before fitting the model.
Brad Walter, WBY
Below is a larger list of papers and references on methods of determining the times of minima and maxima and estimating errors for the times. Kwee-van Woerden seems to be the most widely used method and a version of it is used by Peranso. However, at least the original KvW significantly underestimates the error of the estimated time and has some assumptions about gaps and symmetry. The list includes an interesting article on a modification of KvW that results in more realistic error estimates. Some of the articles discuss alternatives to KvW that are being used.
https://articles.adsabs.harvard.edu/full/1956BAN....12..327K Original paper by Kwee & van Woerden
https://www.astro.sk/caosp/Eedition/FullTexts/vol43no3/pp382-387.pdf - Kwee-van Woerden Method to Use or Not to Use?
https://oejv.physics.muni.cz/issues/oejv_0215.pdf - Minima and Maxima Timings of Several Variable Stars.
https://arxiv.org/ftp/arxiv/papers/2011/2011.09231.pdf - paper Modified KvW Method w Reliable error estimates
https://www.mirametrics.com/brief_kwee.php - Mirametrics (Michael Newberry) summary of KvW method
https://arxiv.org/abs/2011.09231 - arXiv entry for Modified KvW Method w Reliable error estimates
Research Gate page for Two Methods for Light Curve Extrema determination
Thanks Brad. I recall some discussion several years ago at a Variable Stars South symposium about a modified KvW method that reduces the error underestimation. This article makes mention of it as well:
I'll read the papers you cite with interest.
The current extrema finder in VStar uses a derivative method but with a small fixed sized step around the inflection point rather than a dynamically determined Δt such as via Excel's Goal Seek. Thanks for these paper and the ones in your subsequent post.
Do your really mean inflection point in your description of the current extrema finder in VStar in your post? I normally think of an inflection point as in the following definition:
“In calculus, the inflection point is where a graph's concavity changes from either up to down or down to up. This change may be slow or dramatic but it is regarded as the point where the slope starts to change”
So these are points where the second derivative changes sign.
Agreed, my use of "inflection point" is not correct here and I agree with what you say.
The extrema (minimum/maximum) finder looks for where the first derivative goes to zero, i.e. changes between positive and negative.
The sign of the second derivative is used to determine concavity up/down, i.e. type of extremum.
For complicated light curves I sometimes use the inflection points as the boundaries of the section of the curve to which I fit polynomials fits estimate time of minimum (or maximum).
Hi Brad, Roy
I haven't got to the issue of ToM error determination for the latest release (2.22.0 yesterday) but have captured the issue here:
and thinking about approaches.
Thanks for your input so far.