Sequence-Specific Photometric Transformation Using Ensemble-Based Linear Regression

Science Smart Telescope Photometry
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I have only recently begun following this forum, and I find the ideas shared here very interesting.

Dr. Brian Kloppenborg suggested that I share this approach here with the Smart Telescope Working Group.

I also own a Celestron Origin telescope, which I primarily use for live online astronomy presentations. Recently, however, I have started experimenting with photometric observations using the Origin, mainly targeting RR Lyrae and δ Cephei type variables.

In parallel with this, I continue to perform photometric measurements using Newtonian telescopes mounted on equatorial mounts, equipped with a monochrome CMOS camera (ZWO ASI1600MM Pro), ASIAIR control, and non-standard (ZWO New LRGB) filters. During the reduction of these observations, I developed a method to convert instrumental magnitudes into standard magnitudes.

The essence of the method is the following:

For each image series, I perform ensemble differential photometry on all stars within the field of view that have known standard magnitudes listed on AAVSO charts, and I determine their instrumental magnitudes.

For the non-variable stars, the instrumental magnitudes obtained from the ensemble photometry are averaged over the entire duration of the observing sequence. Since these stars are constant in time, averaging reduces measurement noise, while the standard deviation provides a useful characterization of the internal stability of the observing sequence and the observing conditions.

Each star thus contributes a single data point consisting of its averaged instrumental magnitude and its corresponding standard magnitude. These data points form the basis of a linear regression. From this regression, I determine the transformation equation valid for the given observing sequence, which I then directly use to convert the variable star’s instrumental magnitude into a standard magnitude.

The primary goal of this approach is to determine the standard magnitude of the observed variable star within a given observing sequence with the highest possible internal consistency.

The correlation coefficient obtained from the linear regression and the residual scatter provide immediate, quantitative feedback on the internal precision of the observing sequence and indicate the level of scientific use for which the results may be suitable.

In this approach, I do not treat the transformation as a constant system parameter, but rather as a local fitting problem: for each observing sequence, I determine an empirical relationship fitted to the standard stars present in the field of view.

Conceptually, this sequence-level fitting may be related to the nightly coefficient strategies currently discussed within the STWG, although implemented here at the level of a single observed field.

I would be interested to learn whether similar sequence-specific transformation strategies are being considered within the current STWG workflow.

If this is of interest, I would be happy to return with a more detailed and mathematically

Best regards,

Vilmos Tekler

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The only potential problem I see with this approach is that the errors of the magnitudes and colour indices of AAVSO comparison stars are often larger than the corresponding errors in AAVSO photometric standard stars.

Therefore, the transformation coefficients derived from comparison stars may not be as accurate as those calculated from photometric standards. If this is the case, and particularly since you are using non-scientific filters for photometry for which TCs may for some filters be very different from those of scientific filters, it would be advantageous to get the ‘best’ transformation coefficients possible.

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Hi Vilmos,

Thanks for bringing this topic up.

If I’m not mistaken, this approach was also proposed by Bruce L. Gary (CCD Transformation Equations). I tried it for last couple of years for observing short period variables (e.g. Lyrids).

The main outcome for me was: the numbers of comparison stars in the sequence is crucial.

My setup is rather modest (150 mm Newtonian, ZWO ASI585MM Pro + filter wheel), as well as viewing conditions (backyard at Seattle suburbia). So far the only target I’m satisfied is SZ Lyncis, which has a dozen of comparison stars inside 60 arcsec FOV. I was able to achieve uncertainty about 30 mmag for V, 50 mmag for B and Rc.

Measurements of stars with lesser number of comparison stars produce very noisy data.

It would be very nice if you publish the details of your approach. My algorithm (quite straightforward) is here: vsopy-pkg/src/vsopy/phot/transform.py at dev · dmitrymu/vsopy-pkg · GitHub .

Best regards,
Dmitry (MDMD)

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Vilmos:
Thank you for experimenting with this and for sharing what you’ve found.

You’ve done a couple of things that I find interesting:

  • You are averaging instrumental magnitudes. It’s never occurred to me to do that, mostly because I’m so used to miserable sky conditions that I expect transient cloud cover all the time. But the technique that you’re using is commonly used by research-class observatories with well-understood sky conditions for exactly the reasons that you’ve noted. It has to be used with caution, though, if you have a long time series where airmass/extinction is changing noticeably throughout the sequence.
  • By limiting your regression fitting to a single field in the sky, you eliminate some of the field-to-field issues that happen when you combine all data across a single night. When the STWG does a linear fit across an entire session’s worth of observations, we are doing it to extract two sets of information: the correction coefficients for our color/vignetting/airmass correction algorithms, and a set of zero points for our images. We are rarely using existing AAVSO photometric sequences, since we get “better” results using APASS or ATLAS Refcat-2 catalog data (either of which is more homogeneous than the AAVSO sequences). Right now we find ourselves trying to get “more stars, more stars” into those regressions (looking for 10,000 to 100,000 points from each session) in order to average out errors.

Please continue doing what you’re doing – and sharing your results with us!

– Mark

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Dear Roy and Dmitry,

First of all, thank you both for your thoughtful comments and for taking the time to respond. I apologize for my delayed reply: I was travelling abroad for the past couple of weeks and unfortunately did not receive any notification that new comments had been posted.

Roy raises an important point regarding the accuracy of AAVSO comparison stars. It is indeed true that the magnitude and colour index uncertainties of comparison stars are often larger than those of dedicated photometric standard stars, and classical transformation procedures therefore aim to determine stable, instrument-specific transformation coefficients using such standards.

The goal of the approach I described, however, is somewhat different. The method does not attempt to determine universal transformation coefficients for a given instrument system. Instead, it aims to determine an empirical relationship between instrumental and standard magnitudes within a specific observing sequence, using the stars present in the observed field.

In this framework, the transformation is determined separately for each observing sequence using all stars in the field with known standard magnitudes. When several comparison stars are available, the random errors of the catalog magnitudes partly average out statistically; approximately, the uncertainty decreases by a factor of √N, where N is the number of comparison stars used.

An important aspect of the method is that the accuracy of the transformation is not assumed a priori. The residual scatter obtained from the linear regression provides a direct empirical estimate of how accurately the transformation can be applied in that particular observing sequence. In this way, the effects of catalog uncertainties, instrumental characteristics, and observing conditions all appear together in the regression, and the residual scatter immediately indicates the practical reliability of the transformation.

Dmitry’s experience illustrates this practical aspect very well. The stability of the method indeed depends strongly on the number of suitable comparison stars available in the field. When a sufficient number of stars can be used, the regression becomes stable and the residual scatter provides a meaningful measure of the internal precision of the measurement. With fewer comparison stars, the noise naturally increases.

For clarity, the practical steps of the method are roughly the following:

  1. Acquire an image sequence of the target field.
  2. Perform ensemble differential photometry for all stars with known standard magnitudes listed on the AAVSO charts.
  3. For non-variable stars, average the instrumental magnitudes over the entire observing sequence.
  4. Construct pairs of (instrumental magnitude, catalog standard magnitude).
  5. Fit a linear regression to these data points.
  6. Evaluate the quality of the fit using the correlation coefficient and the residual scatter.
  7. Apply the resulting transformation to the instrumental magnitude of the variable star.

The purpose of the method is therefore not to determine universal transformation coefficients, but to obtain the most internally consistent estimate of the standard magnitude within the given observing sequence.

Thank you also, Dmitry, for mentioning Bruce L. Gary’s work. There are certainly conceptual similarities with some of those approaches. If there is interest, I would be happy to describe the method and its practical implementation in more detail.

Best regards,
Vilmos Tekler

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Dear Mark,

Thank you very much for your thoughtful comments and for the encouraging words.

Regarding the averaging of instrumental magnitudes, I completely agree that this must be used with caution if the airmass changes significantly during the observing sequence. In the reduction workflow I currently use, each individual measurement is first corrected for the airmass calculated for that specific moment, and the averaging is performed on these corrected values. This helps to reduce the influence of changing extinction during the sequence.

In practice, the regression is usually performed using the stars within a single field of view, and the main goal is to obtain the most internally consistent estimate of the standard magnitude for the variable star within that particular observing sequence. In this sense, the approach can be regarded as an empirical calibration valid for a specific observing sequence rather than an attempt to determine global transformation coefficients.

For a given variable star I normally use a predefined ensemble of comparison stars, typically taken from the AAVSO sequence for that field. This ensures that the same photometric reference set is used consistently across different observing sessions.

I should also mention that so far I have mainly applied this method to observations obtained with a monochrome camera and photometric filters. With the Celestron Origin I am still in the early stages of experimenting with variable star observations, so in that case I am currently exploring what approaches may work best.

I also found your description of the STWG session-level fits very interesting. I would be very interested to learn more about how these regressions are implemented in practice and which parameters are typically fitted simultaneously in your workflow.

Thank you again for your comments and for the encouragement.

Best regards,
Vilmos Tekler

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Dear Vilmos,

You said

each individual measurement is first corrected for the airmass calculated for that specific moment, and the averaging is performed on these corrected values.

It would be very interesting to see this method in more formal presentation.

IMHO, if we apply regression to the all measurements during a night, we don’t need to differ much from the classical approach with extinction correction followed by color transformation. We may just calculate al these coefficients at once using multilinear regression. This approach is described, for example, in “Photoelectric photometry - an approach to data reduction”, Harris, Fitzgerald, Reed, PASP, v93, 1981 (1981PASP...93..507H Page 507).

I applied a prototype implementation of this method to SZ Lyn data I have in hands. It looks like the results don’t differ much in accuracy and precision form the results of per-point transformation by Gary’s method.

Best regards,
Dmitry (MDMD)

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Dear Dmitry,

Thank you for your comment and for pointing me to the Harris, Fitzgerald & Reed (1981) paper.

In my approach, I do not derive transformation parameters for an entire night. Instead, I build the regression model directly from the comparison stars in the same field as the variable, for each observing series — and in some cases even for each individual measurement point.

This way, atmospheric effects are handled locally and implicitly, rather than through a global nightly solution.

If you are interested, I describe my workflow in more detail below.

Detailed explanation:

The regression approach described by Harris et al. represents a classical nightly solution, where standard stars observed during a night are used to determine extinction, zero point, and color transformation terms.

In contrast, my approach is based on local ensembles: for each observing series, I use the comparison stars located in the same field of view as the variable (typically from AAVSO sequences) to construct the regression model. Therefore, the model is built from exactly the same stars that are used to derive the variable’s magnitude.

The underlying assumption is that all effects affecting both the variable and the comparison stars — such as airmass, transparency variations, or even passing clouds — are implicitly included in the regression. The quality of the solution is evaluated using the residual scatter and regression coefficients.

The exact implementation depends on the nature of the observing series.

For short series (typically 10–20 minutes), where a single representative magnitude is required (e.g., Mira variables or recurrent novae), I average the measurements after airmass correction, and derive the regression parameters from these averaged values. These parameters are then applied to the variable.

For longer time-series (spanning several hours), I construct the regression model separately for each measurement point (again after airmass correction), and apply the resulting parameters to the variable at the same time. This provides a time-dependent, dynamic correction.

In practice, I choose between these approaches based on the behavior of the comparison stars. If their measured magnitudes remain stable and follow a flat trend, averaging is sufficient. However, if I observe increased scatter or time-dependent variations — for example due to changing atmospheric conditions such as humidity or thin clouds — I switch to the point-by-point regression approach. This allows short-timescale variations to be properly handled without biasing the variable’s light curve.

Although this approach differs from the classical nightly solution, under stable atmospheric conditions the two methods are expected to yield similar results — as you also noted in your SZ Lyn observations.

Best regards,
Vilmos Tekler (TVIB)

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Vilmos,

I believe I understand your method, and clearly you apply it diligently.

However, it rests on transformations determined from the comparison stars in the same field as the variable star, and I therefore come back to my original point in an earlier post: the accuracy of magnitudes and colour indices of comparison stars is inevitably not as good as the accuracy of photometric standards. Therefore your transformations are likely to be of lesser accuracy than those determined from such standards.

I may have missed it, but have you tested this point by comparing the accuracies of the two different methods using standard stars (instead of variables) as targets?

1 Like
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I agree with Roy that the basic idea here is correct. But the method assumes that the sequences in each are actually quite good, whereas as has been mentioned, the AAVSO sequences often have substantial errors both internal and systematic. In comparing star-by-star where there is overlap with Landolt and other high-weight sources, APASS DR9 can be off by 0.1 - 0.2 mag in either direction.
As an interim, you want to get color terms via standards, but use the local field comp stars to set the zero-point. That value may well be off, but until better data show up, that’s what you have to work with short of doing new field calibrations yourself on stable nights (which I would encourage, by the way).

\Brian

1 Like
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Dear Dmitry, Roy, and Brian,

Thank you all for your comments — you have raised very important and insightful points.

I fully agree that the magnitudes and colour indices of photometric standard stars are generally more accurate than those of comparison stars, and that AAVSO sequences — particularly those based on APASS data — may contain significant internal and systematic errors, sometimes at the level of 0.1–0.2 mag. From the perspective of absolute photometric accuracy, this is clearly a limiting factor.

At the same time, I believe that the evaluation of the method depends strongly on the scientific goal. In many cases of variable star research, the primary objective is not the highest possible absolute photometric accuracy, but rather the precise characterization of variability — in particular the shape of the light curve and its time behaviour.

In this context, the relative importance of different error sources may change. While transformations based on standard stars aim to maximize absolute calibration accuracy, the local, differential approach has the advantage that the variable and the comparison stars are observed under identical conditions. As a result, atmospheric and other time-dependent effects are incorporated in the same way into both, and are therefore implicitly accounted for in the regression.

It is also worth noting that systematic offsets in the magnitudes of comparison stars primarily affect the absolute scale, but do not necessarily degrade the internal consistency of the light curve. This is especially relevant when the goal is to track relative variations as accurately as possible.

Interestingly, Dmitry reported similar results for SZ Lyn, where the classical nightly regression and the local method did not show a significant difference in accuracy and precision. This may suggest that, under practical observing conditions, the dominant source of error is not necessarily the catalog accuracy alone, but rather atmospheric and observational effects.

Brian’s suggestion of a hybrid approach is very valuable, and I fully agree that deriving colour transformation terms from standard stars can be important. At the same time, determining the zero-point from local comparison stars often provides a practical and effective compromise.

Roy’s suggestion to directly compare the methods using standard stars as targets is also very important. I have not yet performed such a test, but I agree that it would be highly useful and I plan to carry it out in the future.

Overall, I see these approaches not as alternatives, but as methods optimized for different observational goals: one prioritizes absolute photometric accuracy, while the other emphasizes consistent and stable tracking of variability.

Best regards,
Vilmos Tekler (TVIB)

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If you do any real-world testing… one or two 'tests" aren’t enough. The olde rule-of-thumb is 200 tests or more to have any statistical significance.

Dmitry… This one is for you! :rofl: You typed above:

> I applied a prototype implementation of this method to SZ Lyn data I have in hands. It looks like the results don’t differ much in accuracy and precision form the results of per-point transformation by Gary’s method.

Do you have some real statistical-type values that quantify “don’t differ much…” for your single comparison of processing? :open_mouth:

I too would like to see a written presentation of the original proposal of the Vilmos method compared to the standard application of transformations that most of us do. There are many noise sources that combine to give you that single one sigma error estimate for a photometric point you report, the error budget.

Going back to sleep now… :sleeping_face:

Jim (DEY)

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Vilmos:

I think the claim about accurate relative tracking assumes that all the observers involved continue to use the same set of comparison stars.

Tom