Fix centroid and gaussian width analysis functions, add distribution-based implementation

[rosteen]

We've had some reports that the uncertainties reported by centroid, gaussian_sigma_width, and gaussian_fwhm are suspect/wrong. This PR will update to using astropy.uncertainty Monte Carlo distributions to estimate the value and uncertainty for these quantities instead of the suspect analytic solutions, and give the option to use an updated analytic solution for centroid that @eteq re-derived. Opening this now to get eyes on my implementation, but I still need to update the gaussian measures.

[eteq]

For backstory, a picture of the derivation for the analytic solution is below.

I cross-checked this solution and the implementation in this PR in a notebook here:
https://gist.github.com/eteq/37017d89ce3017e387de1e72edb40440

@rosteen, maybe the thing to do is to update this PR to include tests based on that notebook? That is, a test where you generate a fake spectrum with known uncertainty a bunch of times, compute the centroid, and then compare the stddev of those simulations to the uncertanty derived here? As long as you set a random number generator seed for that and a sufficient tolerance it shouldn't lead to mysterious failures due to bad luck of the draw.

[eteq]

Oh, and given the growing complexity, maybe we want to change this PR to just the centroid, finish it up, and then do a separate PR for the others?

[rosteen]

Oh, and given the growing complexity, maybe we want to change this PR to just the centroid, finish it up, and then do a separate PR for the others?

I'm going to try to get the gaussian width measures in here too, I'd rather not be inconsistent between those and the centroiding.

A nice side effect of this is that I think I figured out the generalization of the centroiding math to multiple dimensions, so we aren't restricted to 1D or 2D like we were in the previous implementation of centroid.

[codecov]

Codecov Report

Attention: Patch coverage is 94.59459% with 4 lines in your changes missing coverage. Please review.

Project coverage is 70.72%. Comparing base (0613ba1) to head (55c497e).
Report is 72 commits behind head on main.

Files with missing lines	Patch %	Lines
specutils/analysis/location.py	90.32%	3 Missing ⚠️
specutils/analysis/width.py	97.67%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #1057      +/-   ##
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+ Coverage   70.35%   70.72%   +0.36%     
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  Files          64       64              
  Lines        4406     4485      +79     
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+ Hits         3100     3172      +72     
- Misses       1306     1313       +7     

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[rosteen]

@eteq this is now ready for review. I ended up leaving the default as the analytic solution, since we're fairly certain it's correct now, and leaving the distribution-based solution as an option - I figured this is less disruptive/doesn't change a default that users might not notice. I noticed and fixed a couple other things along the way while adding/fixing tests, for example the sigma width uncertainty was previously just adding sigma.unit to the fractional uncertainty rather than correctly multiplying by sigma itself.

The test failure from the latest run was an unrelated file download error.

[camipacifici]

From the specviz side: left is old, right is new.
I'd say good job!! :)