# Calculating errors

Hi All,

Sorry if this question is a bit simple but I want to check that my calculation of comp star errors is sound. SeqPlot provides the V magnitude and B-V color index of potential comp stars. The error of each is also reported, i.e.:

V = 5.000, error = 0.010 and B-V = 0.500, error = 0.015

So the B magnitude is V + (B-V) = 5.500, simple enough. But errors should be added in quadrature:

B error = SQRT(0.010*0.010 + 0.015*0.015) = 0.018

Is this correct? Cheers,

Mark

Hi Jim,

Thanks for your feedback. I understand what you are saying, but you used the value of (B-V) instead of it's error. So in the example:

Berr = sqrt(0.015^2-0.01^2) = 0.011.....

This was my first thought for calculating Berr, however SeqPlot occassionally reports (B-V)err = Verr, which implies zero error in B. Even worse, sometimes (B-V)err is less than Verr implying Berr is an imaginary number.

I haven't made an exhaustive study of this, but my feeling is that the General Catalogue of Photometric Data (source 20 in SeqPlot) is most likely to give these low (B-V)err values. Maybe the GCPD used different sources for their V and (B-V) data?

So I'll use your suggestion when the result is physically reasonable. For situations when (B-V)err is less than or equal to Verr I'll assume Berr = Verr.

Thanks again. Cheers,

Mark

Mark's calculation is correct: errors add in quadrature. This has always been an interesting issue with traditional reporting (V and colors) vs. modern reporting (individual bandpasses). Most astronomers prefer the traditional method as it give the colors of objects, more astrophysically relevant information. This means that converting such measures into individual bandpasses (B,V,Rc, etc.) will end up with higher uncertainty. The logical solution is to calculate individual bandpasses *and* colors. We may do that with the final version of APASS.

Arne

I think Mark is needs to go backward Arne. He has (B-V)err and Verr. He needs to calculate what was added in quadrature to Verr to produce (B-V)err. That is what is the value of Berr.

Jim

Hi Jim,

It depends on how (B-V) was calculated. If the original researcher solved for V and (B-V), then the appropriate error for B is sqrt (Verr**2 + BVerr**2). If the original researcher solved for V and B, and the program reported (B-V), then your technique is correct. It is easy for error propagation to get out of hand.

Arne

Arne,

Is the choice to calculate the Berr from the quadrature sum of the Verr and BVerr based upon the fact that we don't know whether the BVerr is a quadrature calculation using Berr and Verr, or is the standard deviation empirically determined from the spread of the B-V values themselves? Sometimes the color index error values tell us they aren't quadrature sum because they are less than the corresponding Verr. I have seen this a number of times in sequence tables. When that isn't the case, and you don't know how the BVerr was determined, is the rule that you make the conservative assumption that the Verr and the Color index error values are not correlated rather than assuming the Verr is perfectly correlated to a V error used in a quadrature calculation of the color index error?

Brad Walter, WBY

May I be permitted to throw a spanner into the works here? Just recently, I've been reducing some BVRI mags to the standard system for an amateur astronomer friend here in the U.K. For each star & bandpass he has used about 6 different comparison stars (lucky he had that many comps on his images I guess) to produce 6 mag values for the variable. How should one go about determining the error in the variable's mag? One would of course noramlly determine the mean & then the standard error (using the 'n-1' version of the standard deviation); but then there are the errors in the comp star mags. Is this a case of as Arne says; errors getting out of hand?

Keith Robinson RKO

Hi Brad,

That is correct - unless you know how (B-V) was determined, you should assume that (B-V) and V errors are uncorrelated and their values should be added in quadrature.

Arne

Hi Keith,

Error calculation is in general complex. For your case, you need the photometric errors for the target and the 6 comp stars, plus the calibration errors for the 6 comp stars, plus any systematic errors that might be present (such as lack of transformation or the transformation error itself). The quick-and-dirty method is to calculate (v-cx) + Cx for all 6 comp stars (where lowercase are instrumental magnitudes and uppercase is the standardized/calibration magnitude), and then form mean and standard deviation for the set of 6 estimates. This calculation incorporates the calibration error for the comparison stars and the error for the target-comp difference (the quadrature sum of the photometric errors of the two stars), so includes nearly all of the uncertainty terms. I consider it "close enough". Note that this is not robust; you really only have one measure of the target star that is shared between the 6 calculations, so it will underestimate the true uncertainty. To get a true estimate of the target uncertainty would require multiple images, and that is not always possible if the star is varying rapidly.

If you form your ensemble in other ways, such as flux weighted, then you should probably calculate the uncertainty for each part of the calculation and add them in quadrature.

Arne

Phew!!!

Thanks Arne

Keith

Hi All,

Thanks everyone for your responses. It seemed such a simple question at the time.

I'm looking forward to the next installment of the CHOICE "Uncertainty about Uncertainty" course. Anyone know when this is likely to be run again? Or if course notes are available?

Arne, thanks for the description of error calculation for ensemble photometry. I'll try to impliment it in my analysis spreadsheets. Cheers,

Mark

Uncertainty estimation is one of the most complicated areas of any research project. As I read more and more of the referreed papers it has become obvious that often most of the effort goes into the error estimation and elimination of systematic errors. It also has become clear that there is a lot of art and assumption involved in error analysis.

The most dog eared book on my shelf is __Data Reduction and Error Analysis for the Physical Sciences__, aka Bevington and Robinson.

Hello Mark,

For any updates to CHOICE courses, you can check this thread: http://www.aavso.org/choice-astronomy-online-courses

According to that thread, we have not planned the next Uncertainty about Uncertainty course yet. If it is offered again, an update will be posted in that thread, so please keep an eye on it!

Mark

I *think*:

((B-V)err)^2 = Berr^2 + Verr^2

Solving for Berr

Berr^2 = ((B-V)err)^2 - Verr^2

So Berr = sqrt(0.5^2-0.01^2) = 0.48989.....

Jim Jones