[Aavso-photometry] Standard deviation calculation

[Brad Walter]

OK, makes sense. 

Thanks. 
-----Original Message-----
From: arne [mailto:arne at aavso.org] 
Sent: Saturday, December 15, 2007 8:14 AM
To: aavso-photometry at mira.aavso.org
Cc: Brad Walter
Subject: Standard deviation calculation

Brad Walter wrote:
> 1. I don't understand why the STDEV of K-C is added in quadrature to 
> the Poisson noise of the target star. When you measure the target star 
> in differential photometry you are actually measuring K-T. When you 
> measure K-C you are including all of the stochastic error elements 
> that are in the K-T measurements if C and T are the same magnitude.
> 2. How do you correct the error estimate for differences in magnitudes 
> between C and T? It seems to me you have to do a calculation something 
> like the following:
> STDEV(k-t) = SQRT((STDEV(k-c)^2 - SE(k)^2)*SE(t)^2/SE(c)^2 + SE(k)^2)
> 
> Where SE(x) = -2.5*Log(1+1/SNR(x))
> 
------------------------
> Usually you work with three stars for time series: the target (V), the 
> comparison star (C), and the check star (K).  Usually you obtain 
> measures for all three stars on every image.  If you form the 
> difference between the check and the comparison star (K-C), this 
> represents the major part of the error in estimating the target, since 
> typically these two stars are similar in magnitude to the target and 
> their difference roughly matches the error of a constant star with the 
> target's brightness.
> 
> So the usual way of handling a time series is to calculate the mean 
> and standard deviation of (K-C), and report the standard deviation of 
> (K-C), added in quadrature to the Poisson noise of the target, as the 
> error for the target star.
> 
> If your sky conditions are varying, you may want to break up the 
> calculation of the standard deviation of (K-C) into smaller time 
> intervals, and modify the reported error for each of those intervals.
> Arne
> 
I walk a fine line between giving a full complex description of the error
calculation process, and giving something that gets most of the way there so
that the average report can be improved beyond a pure reporting of Poisson
error.

The full error equation has been given by Michael Newberry on this list if I
remember correctly, and can also be found in Steve Howell's book.
Remember that you need to work in flux space and not magnitude space for a
careful calculation.

What Brad is saying is that you are working with (at least) three stars in
this process: target (T), comparison (C) and check (K).  Each of these stars
will be of different brightness than the others, and therefore will have
different Poisson error.  I think that what needs to be remembered is that,
for most amateur situations, Poisson error is actually nearly negligible in
comparison with the other error sources.  That is why the
(K-C) estimate is suggested in reporting errors to the AAVSO, as it includes
implicitly many of the possible error souces and is simple to calculate.  I
included the Poisson error of the target in quadrature to guard against
those cases where the comparison/check are much brighter than the target
(such as estimating the brightness of a cataclysmic variable in quiescence).
If the target/check are fainter, then including the target Poisson error
will be a very minor overestimate.  So the quadrature summation presented in
my posting above is an approximation and should have been described that
way.

There is no simple mechanism that gives you a perfect estimate of the error.
The (K-C) process gets you most of the way there and is one that we might
get most observers to use (rather than the very optimistic Poisson error
estimate that many use now).  If you want to try something more accurate,
then by all means look at the real error equation and see what results you
obtain.
Arne