Combining estimated errors (galaxy properties, observational)

[Jean Tate]

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?

 

[mfb]

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?

 

[Jean Tate]

Thanks mfb!

You can always subtract those values with Gaussian error propagation, assuming uncorrelated (!) Gaussian uncertainties.

Yes, as an initial assumption, I think it's safe to assume uncorrelated uncertainties (I'm not 100% sure I know what this means, could you clarify please?)

What is "Gaussian error propagation"? And how does it work, in this case?

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.

If you assume no band-specific asymmetries, the two observations are then of the 'same galaxy'. Simplest to pick the mid-point as the 'true', or 'most likely' (or some such) PA: -0.35°. If you wanted to quote an estimated error, what would it be (given the above assumption of "uncorrelated Gaussian uncertainty")?

What about the axis ratio estimates?

And the r_e ones?

10° uncertainty of a position measurement? Really?

It's not a position measurement, but a position angle measurement.

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).

 

[mfb]

Yes, as an initial assumption, I think it's safe to assume uncorrelated uncertainties (I'm not 100% sure I know what this means, could you clarify please?)

Correlated uncertainties would mean that the error of the individual measurements depend on each other - like the calibration of the whole instrument, which might be the same for both measurements.

What is "Gaussian error propagation"? And how does it work, in this case?

$\sigma_{a-b}=\sqrt{\sigma_a^2+\sigma_b^2}$

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.

What about the axis ratio estimates?

And the r_e ones?

It does not matter which parameter you consider, the formulas are general enough.

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).

Ah, they are oriented like that. But shouldn't that give a range of (-180° to 180°)? In addition, different parameters or a two-dimensional analysis might be better in this case.

 

[Jean Tate]

Thanks again mfb.

Correlated uncertainties would mean that the error of the individual measurements depend on each other - like the calibration of the whole instrument, which might be the same for both measurements.

Yes, that's pretty much what I understood the term to mean.

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!

$\sigma_{a-b}=\sqrt{\sigma_a^2+\sigma_b^2}$

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.

Again, thanks.

This is the same as "adding the errors in quadrature", isn't it?

It does not matter which parameter you consider, the formulas are general enough.

I realize that I am not expressing myself well, apologies (I'm doing all this on my own, as a pure amateur).

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).

Ah, they are oriented like that. But shouldn't that give a range of (-180° to 180°)?

As I understand it, in double/binary star work the PA can indeed range from -180° to 180°, because the brighter star is chosen as the reference star/point. However, for galaxies the angles are undirected; rotate 180° and you have the same PA.

In addition, different parameters or a two-dimensional analysis might be better in this case.

And there are many such, especially where there is clear structure such as bars or arms.

However, at its simplest, a (spiral) galaxy is assumed to be a circular, essentially zero-thickness 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 face-on (shape is a circle, ab=1) -> angle of inclination (i) 90°. Of course, spiral galaxies are not spherical cows!

 

[mfb]

This is the same as "adding the errors in quadrature", isn't it?

Right

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?

Right, you get asymmetric uncertainties in those regions. Other parameters might be better to study, like the "length" and "width", as they are not close to their boundary (0). The ratio smallerone/largerone can be calculated afterwards.