Source code for ligo.skymap.postprocess.ellipse

#
# Copyright (C) 2013-2020  Leo Singer
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import astropy_healpix as ah
from astropy import units as u
from astropy.wcs import WCS
import healpy as hp
import numpy as np

from .. import moc
from ..extern.numpy.quantile import quantile

__all__ = ('find_ellipse',)


[docs]def find_ellipse(prob, cl=90, projection='ARC', nest=False): """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 ---------- prob : np.ndarray, astropy.table.Table The HEALPix probability map, either as a full rank explicit array or as a multi-order map. cl : float The desired credible level (default: 90). projection : str, optional The WCS projection (default: 'ARC', or zenithal equidistant). For a list of possible values, see the Astropy documentation [3]_. nest : bool HEALPix pixel ordering (default: False, or ring ordering). Returns ------- ra : float The ellipse center right ascension in degrees. dec : float The ellipse center right ascension in degrees. a : float The lenth of the semimajor axis in degrees. b : float The length of the semiminor axis in degrees. pa : float The orientation of the ellipse axis on the plane of the sky in degrees. area : float 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] http://aladin.u-strasbg.fr/java/AladinScriptManual.gml#draw .. [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.utils.data import download_file >>> 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' # doctest: +SKIP >>> filename = download_file(url, cache=True, show_progress=False) # doctest: +SKIP >>> _, healpix_hdu = fits.open(filename) # doctest: +SKIP >>> prob = hp.read_map(healpix_hdu, verbose=False) # doctest: +SKIP Then, we calculate ellipse and write it to a DS9 region file. >>> ra, dec, a, b, pa, area = find_ellipse(prob) # doctest: +SKIP >>> print(*np.around([ra, dec, a, b, pa, area], 5)) # doctest: +SKIP 195.03732 -19.29358 8.66545 1.1793 63.61698 32.07665 >>> s = 'fk5;ellipse({},{},{},{},{})'.format(ra, dec, a, b, pa) # doctest: +SKIP >>> open('ds9.reg', 'w').write(s) # doctest: +SKIP Then, we reproject a small patch of the HEALPix map, and save it to a file. >>> wcs = WCS() # doctest: +SKIP >>> wcs.wcs.ctype = ['RA---ARC', 'DEC--ARC'] # doctest: +SKIP >>> wcs.wcs.crval = [ra, dec] # doctest: +SKIP >>> wcs.wcs.crpix = [128, 128] # doctest: +SKIP >>> wcs.wcs.cdelt = [-0.1, 0.1] # doctest: +SKIP >>> img, _ = reproject_from_healpix(healpix_hdu, wcs, [256, 256]) # doctest: +SKIP >>> img_hdu = fits.ImageHDU(img, wcs.to_header()) # doctest: +SKIP >>> img_hdu.writeto('skymap.fits') # doctest: +SKIP 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: .. code-block:: sh $ 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. 1. Open 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: 1. Open 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 # doctest: +SKIP >>> skymap_moc = read_sky_map(healpix_hdu, moc=True) # doctest: +SKIP >>> ellipse = find_ellipse(skymap_moc) # doctest: +SKIP >>> print(*np.around(ellipse, 5)) # doctest: +SKIP 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 = ah.nside_to_npix(nside) ... theta, phi = hp.pix2ang(nside, np.arange(npix)) ... theta_max = np.deg2rad(radius) ... 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 = find_ellipse(prob) >>> dec, a, b, area # doctest: +FLOAT_CMP (89.90862520480792, 0.8703361458208101, 0.8703357768874356, 2.3788811576269793) >>> prob = make_uniform_in_sin_theta(10) >>> ra, dec, a, b, pa, area = find_ellipse(prob) >>> dec, a, b, area # doctest: +FLOAT_CMP (89.90827657529562, 9.024846562072119, 9.024842703023802, 255.11972196535515) >>> prob = make_uniform_in_sin_theta(120) >>> ra, dec, a, b, pa, area = find_ellipse(prob) >>> dec, a, b, area # doctest: +FLOAT_CMP (90.0, 107.9745037610576, 107.97450376105758, 26988.70467497216) **Example 4** These are approximately Gaussian distributions. >>> from scipy import stats >>> def make_gaussian(mean, cov, nside=512): ... npix = ah.nside_to_npix(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], ... np.square(np.deg2rad(1))) ... >>> find_ellipse(prob) # doctest: +FLOAT_CMP (45.0, 0.0, 2.1424077148886744, 2.1420790721225518, 90.0, 14.467701995920123) 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], ... np.diag(np.square(np.deg2rad([1, 1, 10])))) ... >>> find_ellipse(prob) # doctest: +FLOAT_CMP (44.99999999999999, 0.0, 13.58768882719899, 2.0829846178241853, 90.0, 88.57796576937031) This one is centered at RA=0°, Dec=0°, and is elongated in the east-west direction. >>> prob = make_gaussian( ... [1, 0, 0], ... np.diag(np.square(np.deg2rad([1, 10, 1])))) ... >>> find_ellipse(prob) # doctest: +FLOAT_CMP (0.0, 0.0, 13.583918022027149, 2.0823769912401433, 0.0, 88.54622940628761) 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]]) ... >>> find_ellipse(prob) # doctest: +FLOAT_CMP (0.0, 0.0, 64.7713312709293, 33.50754131182681, 80.78231196786838, 6372.344658663038) 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]]) ... >>> find_ellipse(prob) # doctest: +FLOAT_CMP (0.0, 0.0, 64.77133127093047, 33.50754131182745, 99.21768803213159, 6372.344658663096) 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]]) ... >>> find_ellipse(prob) # doctest: +FLOAT_CMP (0.0, 0.0, 64.7756448603915, 33.509863018519894, 170.78252287327365, 6372.425731592412) 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]]) ... >>> find_ellipse(prob) # doctest: +FLOAT_CMP (0.0, 0.0, 64.77564486039148, 33.50986301851987, 9.217477126726322, 6372.42573159241) """ # noqa: E501 try: prob['UNIQ'] except (IndexError, KeyError, ValueError): npix = len(prob) nside = ah.npix_to_nside(npix) ipix = range(npix) area = ah.nside_to_pixel_area(nside).to_value(u.deg**2) else: order, ipix = moc.uniq2nest(prob['UNIQ']) nside = 1 << order.astype(int) ipix = ipix.astype(int) area = ah.nside_to_pixel_area(nside).to_value(u.sr) prob = prob['PROBDENSITY'] * area area *= np.square(180 / np.pi) nest = True # Find median a posteriori sky position. xyz0 = [quantile(x, 0.5, weights=prob) for x in hp.pix2vec(nside, ipix, nest=nest)] (ra,), (dec,) = hp.vec2ang(np.asarray(xyz0), lonlat=True) # Construct WCS with the specified projection # and centered on mean direction. w = WCS() w.wcs.crval = [ra, dec] w.wcs.ctype = ['RA---' + projection, 'DEC--' + projection] # Transform HEALPix to zenithal equidistant coordinates. xy = w.wcs_world2pix( np.transpose( hp.pix2ang( nside, ipix, nest=nest, lonlat=True)), 1) # Keep only values that were inside the projection. keep = np.logical_and.reduce(np.isfinite(xy), axis=1) xy = xy[keep] prob = prob[keep] if not np.isscalar(area): area = area[keep] # Find covariance matrix, performing three rounds of sigma-clipping # to reject outliers. keep = np.ones(len(xy), dtype=bool) for _ in range(3): c = np.cov(xy[keep], aweights=prob[keep], rowvar=False) nsigmas = np.sqrt(np.sum(xy.T * np.linalg.solve(c, xy.T), axis=0)) keep &= (nsigmas < 3) # Find the number of sigma that enclose the cl% credible level. i = np.argsort(nsigmas) nsigmas = nsigmas[i] cls = np.cumsum(prob[i]) if np.isscalar(area): careas = np.arange(1, len(i) + 1) * area else: careas = np.cumsum(area[i]) nsigma = np.interp(1e-2 * cl, cls, nsigmas) area = np.interp(1e-2 * cl, cls, careas) # If the credible level is not within the projection, # then stop here and return all nans. if 1e-2 * cl > cls[-1]: return np.nan, np.nan, np.nan, np.nan, np.nan # Find the eigendecomposition of the covariance matrix. w, v = np.linalg.eigh(c) # Find the semi-minor and semi-major axes. b, a = nsigma * np.sqrt(w) # Find the position angle. pa = np.rad2deg(np.arctan2(*v[0])) # An ellipse is symmetric under rotations of 180°. # Return the smallest possible positive position angle. pa %= 180 # Done! return ra, dec, a, b, pa, area