# Uncertainty Ellipses (ligo.skymap.postprocess.ellipse)¶

ligo.skymap.postprocess.ellipse.find_ellipse(prob, cl=90, projection='ARC', nest=False)[source]

For a HEALPix map, find an ellipse that contains a given probability.

The orientation is defined as the angle of the semimajor axis counterclockwise from west on the plane of the sky. If you think of the semimajor distance as the width of the ellipse, then the orientation is the clockwise rotation relative to the image x-axis. Equivalently, the orientation is the position angle of the semi-minor axis.

These conventions match the definitions used in DS9 region files [1] and Aladin drawing commands [2].

Parameters
probnp.ndarray, astropy.table.Table

The HEALPix probability map, either as a full rank explicit array or as a multi-order map.

clfloat

The desired credible level (default: 90).

projectionstr, optional

The WCS projection (default: ‘ARC’, or zenithal equidistant). For a list of possible values, see the Astropy documentation [3].

nestbool

HEALPix pixel ordering (default: False, or ring ordering).

Returns
rafloat

The ellipse center right ascension in degrees.

decfloat

The ellipse center right ascension in degrees.

afloat

The lenth of the semimajor axis in degrees.

bfloat

The length of the semiminor axis in degrees.

pafloat

The orientation of the ellipse axis on the plane of the sky in degrees.

areafloat

The area of the ellipse in square degrees.

Notes

The center of the ellipse is the median a posteriori sky position. The length and orientation of the semi-major and semi-minor axes are measured as follows:

1. The sky map is transformed to a WCS projection that may be specified by the caller. The default projection is ARC (zenithal equidistant), in which radial distances are proportional to the physical angular separation from the center point.

2. A 1-sigma ellipse is estimated by calculating the covariance matrix in the projected image plane using three rounds of sigma clipping to reject distant outlier points.

3. The 1-sigma ellipse is inflated until it encloses an integrated probability of cl (default: 90%).

The function returns a tuple of the right ascension, declination, semi-major distance, semi-minor distance, and orientation angle, all in degrees.

References

1

http://ds9.si.edu/doc/ref/region.html

2

3

http://docs.astropy.org/en/stable/wcs/index.html#supported-projections

Examples

Example 1

First, we need some imports.

>>> from astropy.io import fits
>>> from astropy.wcs import WCS
>>> import healpy as hp
>>> from reproject import reproject_from_healpix
>>> import subprocess


Next, we download the BAYESTAR sky map for GW170817 from the LIGO Document Control Center.

>>> url = 'https://dcc.ligo.org/public/0146/G1701985/001/bayestar.fits.gz'
>>> _, healpix_hdu = fits.open(filename)


Then, we calculate ellipse and write it to a DS9 region file.

>>> ra, dec, a, b, pa, area = find_ellipse(prob)
>>> print(*np.around([ra, dec, a, b, pa, area], 5))
195.03732 -19.29358 8.66545 1.1793 63.61698 32.07665
>>> s = 'fk5;ellipse({},{},{},{},{})'.format(ra, dec, a, b, pa)
>>> open('ds9.reg', 'w').write(s)


Then, we reproject a small patch of the HEALPix map, and save it to a file.

>>> wcs = WCS()
>>> wcs.wcs.ctype = ['RA---ARC', 'DEC--ARC']
>>> wcs.wcs.crval = [ra, dec]
>>> wcs.wcs.crpix = [128, 128]
>>> wcs.wcs.cdelt = [-0.1, 0.1]
>>> img, _ = reproject_from_healpix(healpix_hdu, wcs, [256, 256])
>>> img_hdu.writeto('skymap.fits')


Now open the image and region file in DS9. You should find that the ellipse encloses the probability hot spot. You can load the sky map and region file from the command line:

\$ ds9 skymap.fits -region ds9.reg


Or you can do this manually:

1. Open DS9.

2. Open the sky map: select “File->Open…” and choose skymap.fits from the dialog box.

3. Open the region file: select “Regions->Load Regions…” and choose ds9.reg from the dialog box.

Now open the image and region file in Aladin.

2. Open the sky map: select “File->Load Local File…” and choose skymap.fits from the dialog box.

3. Open the sky map: select “File->Load Local File…” and choose ds9.reg from the dialog box.

You can also compare the original HEALPix file with the ellipse in Aladin:

2. Open the HEALPix file by pasting the URL from the top of this example in the Command field at the top of the window and hitting return, or by selecting “File->Load Direct URL…”, pasting the URL, and clicking “Submit.”

3. Open the sky map: select “File->Load Local File…” and choose ds9.reg from the dialog box.

Example 2

This example shows that we get approximately the same answer for GW171087 if we read it in as a multi-order map.

>>> from ..io import read_sky_map
>>> ellipse = find_ellipse(skymap_moc)
>>> print(*np.around(ellipse, 5))
195.03709 -19.27589 8.67611 1.18167 63.60454 32.08015


Example 3

I’m not showing the ra or pa output from the examples below because the right ascension is arbitary when dec=90° and the position angle is arbitrary when a=b; their arbitrary values may vary depending on your math library. Also, I add 0.0 to the outputs because on some platforms you tend to get values of dec or pa that get rounded to -0.0, which is within numerical precision but would break the doctests (see https://stackoverflow.com/questions/11010683).

This is an example sky map that is uniform in sin(theta) out to a given radius in degrees. The 90% credible radius should be 0.9 * radius. (There will be deviations for small radius due to finite resolution.)

>>> def make_uniform_in_sin_theta(radius, nside=512):
...     npix = hp.nside2npix(nside)
...     theta, phi = hp.pix2ang(nside, np.arange(npix))
...     prob = np.where(theta <= theta_max, 1 / np.sin(theta), 0)
...     return prob / prob.sum()
...

>>> prob = make_uniform_in_sin_theta(1)
>>> ra, dec, a, b, pa, area = np.around(find_ellipse(prob), 5) + 0
>>> print(dec, a, b, area)
89.90863 0.87034 0.87034 2.37888

>>> prob = make_uniform_in_sin_theta(10)
>>> ra, dec, a, b, pa, area = np.around(find_ellipse(prob), 5) + 0
>>> print(dec, a, b, area)
89.90828 9.02485 9.02484 255.11972

>>> prob = make_uniform_in_sin_theta(120)
>>> ra, dec, a, b, pa, area = np.around(find_ellipse(prob), 5) + 0
>>> print(dec, a, b, area)
90.0 107.9745 107.9745 26988.70467


Example 4

These are approximately Gaussian distributions.

>>> from scipy import stats
>>> def make_gaussian(mean, cov, nside=512):
...     npix = hp.nside2npix(nside)
...     xyz = np.transpose(hp.pix2vec(nside, np.arange(npix)))
...     dist = stats.multivariate_normal(mean, cov)
...     prob = dist.pdf(xyz)
...     return prob / prob.sum()
...


This one is centered at RA=45°, Dec=0° and has a standard deviation of ~1°.

>>> prob = make_gaussian(
...     [1/np.sqrt(2), 1/np.sqrt(2), 0],
...
>>> ra, dec, a, b, pa, area = np.around(find_ellipse(prob), 5) + 0
>>> print(ra, dec, a, b, area)
45.0 0.0 2.14241 2.14208 14.4677


This one is centered at RA=45°, Dec=0°, and is elongated in the north-south direction.

>>> prob = make_gaussian(
...     [1/np.sqrt(2), 1/np.sqrt(2), 0],
...
>>> ra, dec, a, b, pa, area = np.around(find_ellipse(prob), 5) + 0
>>> print(ra, dec, a, b, pa, area)
45.0 0.0 13.58769 2.08298 90.0 88.57797


This one is centered at RA=0°, Dec=0°, and is elongated in the east-west direction.

>>> prob = make_gaussian(
...     [1, 0, 0],
...
>>> ra, dec, a, b, pa, area = np.around(find_ellipse(prob), 5) + 0
>>> print(dec, a, b, pa, area)
0.0 13.58392 2.08238 0.0 88.54623


This one is centered at RA=0°, Dec=0°, and has its long axis tilted about 10° to the west of north.

>>> prob = make_gaussian(
...     [1, 0, 0],
...     [[0.1, 0, 0],
...      [0, 0.1, -0.15],
...      [0, -0.15, 1]])
...
>>> ra, dec, a, b, pa, area = np.around(find_ellipse(prob), 5) + 0
>>> print(dec, a, b, pa, area)
0.0 64.77133 33.50754 80.78231 6372.34466


This one is centered at RA=0°, Dec=0°, and has its long axis tilted about 10° to the east of north.

>>> prob = make_gaussian(
...     [1, 0, 0],
...     [[0.1, 0, 0],
...      [0, 0.1, 0.15],
...      [0, 0.15, 1]])
...
>>> ra, dec, a, b, pa, area = np.around(find_ellipse(prob), 5) + 0
>>> print(dec, a, b, pa, area)
0.0 64.77133 33.50754 99.21769 6372.34466


This one is centered at RA=0°, Dec=0°, and has its long axis tilted about 80° to the east of north.

>>> prob = make_gaussian(
...     [1, 0, 0],
...     [[0.1, 0, 0],
...      [0, 1, 0.15],
...      [0, 0.15, 0.1]])
...
>>> ra, dec, a, b, pa, area = np.around(find_ellipse(prob), 5) + 0
>>> print(dec, a, b, pa, area)
0.0 64.77564 33.50986 170.78252 6372.42573


This one is centered at RA=0°, Dec=0°, and has its long axis tilted about 80° to the west of north.

>>> prob = make_gaussian(
...     [1, 0, 0],
...     [[0.1, 0, 0],
...      [0, 1, -0.15],
...      [0, -0.15, 0.1]])
...
>>> ra, dec, a, b, pa, area = np.around(find_ellipse(prob), 5) + 0
>>> print(dec, a, b, pa, area)
0.0 64.77564 33.50986 9.21748 6372.42573