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Combining estimated errors (galaxy properties, observational) 
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#1
Feb1513, 08:53 AM

P: 6

Not sure if this is the right place to put this (sorry, first post here)
Suppose I determine that the position angle (PA) of a galaxy, in one band  observed by SDSS  is 85.4 ± 8.9°, and is 84.7 ± 10.9° in another. Assuming  for now  that the errors have a Gaussian distribution, and that the "±" numbers are 1σ, how do I go about determining if the two (band) PAs are the same "within 1σ"? (Actually it's more like the binary "are the data consistent with the hypothesis that ...?") PA is nice an linear, and  modulo something subtle and potentially interesting (either an SDSS systematic or weak gravitational lensing, say)  the PAs will be distributed evenly over the interval (90, 90), a distribution which wraps around (i.e. 90 = 90). Suppose I determine that the axis ratio of a galaxy ("ab"), in one band is 0.73±0.05, and 0.82±0.04 in another. And I want to ask a similar question. In this case, ab isn't distributed evenly over (0, 1)  at least I don't think it is  and certainly doesn't 'wrap around'. Does that make the calculations needed to answer the question different? (again, assume no systematics). Next: effective radius (r_{e}), 11.6±0.4 and 9.2±0.4 say (unit? pixels, but it doesn't matter, does it?). In this case, the question becomes a lot more complicated, does it not? I mean, r_{e} carries with it the value of n, the Sérsic profile index (or some other model), which is not  necessarily  the same for both bands. And the distribution is far from linear, isn't it? What is needed to do the calculations in cases like this? 


#2
Feb1513, 03:19 PM

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P: 11,928

You can always subtract those values with Gaussian error propagation, assuming uncorrelated (!) Gaussian uncertainties. The distributions do not matter, and wrapping around is not an issue  a difference compatible with 180° is the same as a difference compatible with 0° then.
10° uncertainty of a position measurement? Really? 


#3
Feb1513, 06:35 PM

P: 6

Thanks mfb!
What is "Gaussian error propagation"? And how does it work, in this case? What about the axis ratio estimates? And the r_{e} ones? In the case of a galaxy as imaged by SDSS, it's the angle the estimated major axis makes with North, with angles towards the East being positive. As ab approaches 1 (which is a circle), estimates of the PA become increasingly uncertain, cet. par.; at 1, PA is undefined. The PA examples I chose are real  they refer to estimates of the PA of a galaxy imaged by SDSS (in DR7)  but are not related to the ab and r_{e} examples (they're just numbers, for the purposes of my questions). 


#4
Feb1613, 11:38 AM

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Combining estimated errors (galaxy properties, observational)
If you want to estimate the true value (of some parameter) for a galaxy, given two different measurements, the best way to estimate this is a weighted average, with the inverse variance (=squared standard deviation) as weight of the individual measurements. 


#5
Feb1813, 09:55 AM

P: 6

Thanks again mfb.
In observational astronomy it can sometimes be very difficult to show  empirically  that the correlation between uncertainties is as close to zero as never mind! This is the same as "adding the errors in quadrature", isn't it? Take the axis ratio ("ab"): it cannot be > 1, nor < 0. An analysis of the photometry of a galaxy (from SDSS, say) which fits an ellipse can produce an estimate of ab. It can also produce an estimate of the error (uncertainty) of this estimate. Earlier I said 'assume that the errors have a Gaussian distribution'; I now realize that, for ab, they cannot have such a distribution (at least, not close to 1 or 0). For example, 0.95 ± 0.1 (σ) leads to a meaningless interpretation, if σ comes from a Gaussian: ab cannot be 1.05! But what does the error distribution look like; what can it look like? Is there, for example, a realistic (and analytically tractable) transformation one can do that makes the error distribution approximately Gaussian and is also free of nonsense implications? (I'll leave further consideration of r_{e} for later). However, at its simplest, a (spiral) galaxy is assumed to be a circular, essentially zerothickness disk, at the center of which is a spherical bulge; the observed PA can then be interpreted in terms of the angle of inclination; fully faceon (shape is a circle, ab=1) > angle of inclination (i) 90°. Of course, spiral galaxies are not spherical cows! 


#6
Feb1813, 03:05 PM

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