Source code for ligo.skymap.postprocess.crossmatch

#
# Copyright (C) 2013-2020  Leo Singer
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see <https://www.gnu.org/licenses/>.
#
"""Catalog cross matching for HEALPix sky maps."""
from collections import namedtuple

import astropy_healpix as ah
from astropy.coordinates import ICRS, SkyCoord, SphericalRepresentation
from astropy import units as u
import healpy as hp
import numpy as np

from .. import distance
from .. import moc

from .cosmology import dVC_dVL_for_DL

__all__ = ('crossmatch', 'CrossmatchResult')


def flood_fill(nside, ipix, m, nest=False):
    """Stack-based flood fill algorithm in HEALPix coordinates.

    Based on <http://en.wikipedia.org/w/index.php?title=Flood_fill&oldid=566525693#Alternative_implementations>.
    """  # noqa: E501
    # Initialize stack with starting pixel index.
    stack = [ipix]
    while stack:
        # Pop last pixel off of the stack.
        ipix = stack.pop()
        # Is this pixel in need of filling?
        if m[ipix]:
            # Fill in this pixel.
            m[ipix] = False
            # Find the pixels neighbors.
            neighbors = hp.get_all_neighbours(nside, ipix, nest=nest)
            # All pixels have up to 8 neighbors. If a pixel has less than 8
            # neighbors, then some entries of the array are set to -1. We
            # have to skip those.
            neighbors = neighbors[neighbors != -1]
            # Push neighboring pixels onto the stack.
            stack.extend(neighbors)


def count_modes(m, nest=False):
    """Count the number of modes in a binary HEALPix image by repeatedly
    applying the flood-fill algorithm.

    WARNING: The input array is clobbered in the process.
    """
    npix = len(m)
    nside = ah.npix_to_nside(npix)
    for nmodes in range(npix):
        nonzeroipix = np.flatnonzero(m)
        if len(nonzeroipix):
            flood_fill(nside, nonzeroipix[0], m, nest=nest)
        else:
            break
    return nmodes


def count_modes_moc(uniq, i):
    n = len(uniq)
    mask = np.concatenate((np.ones(i + 1, dtype=bool),
                           np.zeros(n - i - 1, dtype=bool)))
    sky_map = np.rec.fromarrays((uniq, mask), names=('UNIQ', 'MASK'))
    sky_map = moc.rasterize(sky_map)['MASK']
    return count_modes(sky_map, nest=True)


def cos_angle_distance(theta0, phi0, theta1, phi1):
    """Cosine of angular separation in radians between two points on the
    unit sphere.
    """
    cos_angle_distance = (
        np.cos(phi1 - phi0) * np.sin(theta0) * np.sin(theta1) +
        np.cos(theta0) * np.cos(theta1))
    return np.clip(cos_angle_distance, -1, 1)


def angle_distance(theta0, phi0, theta1, phi1):
    """Angular separation in radians between two points on the unit sphere."""
    return np.arccos(cos_angle_distance(theta0, phi0, theta1, phi1))


# Class to hold return value of find_injection method
CrossmatchResult = namedtuple(
    'CrossmatchResult',
    'searched_area searched_prob offset searched_modes contour_areas '
    'area_probs contour_modes searched_prob_dist contour_dists '
    'searched_vol searched_prob_vol contour_vols probdensity probdensity_vol')
"""Cross match result as returned by
:func:`~ligo.skymap.postprocess.crossmatch.crossmatch`.

Notes
-----
 - All probabilities returned are between 0 and 1.
 - All angles returned are in degrees.
 - All areas returned are in square degrees.
 - All distances are luminosity distances in units of Mpc.
 - All volumes are in units of Mpc³. If :func:`.crossmatch` was run with
   ``cosmology=False``, then all volumes are Euclidean volumes in luminosity
   distance. If :func:`.crossmatch` was run with ``cosmology=True``, then all
   volumes are comoving volumes.

"""
_same_length_as_coordinates = ''' \
Same length as the `coordinates` argument passed to \
:func:`~ligo.skymap.postprocess.crossmatch.crossmatch`.'''
_same_length_as_contours = ''' \
of the probabilities specified by the `contour` argument passed to \
:func:`~ligo.skymap.postprocess.crossmatch.crossmatch`.'''
_same_length_as_areas = ''' \
of the areas specified by the `areas` argument passed to
:func:`~ligo.skymap.postprocess.crossmatch.crossmatch`.'''
CrossmatchResult.searched_area.__doc__ = '''\
Area within the 2D credible region containing each target \
position.''' + _same_length_as_coordinates
CrossmatchResult.searched_prob.__doc__ = '''\
Probability within the 2D credible region containing each target \
position.''' + _same_length_as_coordinates
CrossmatchResult.offset.__doc__ = '''\
Angles on the sky between the target positions and the maximum a posteriori \
position.''' + _same_length_as_coordinates
CrossmatchResult.searched_modes.__doc__ = '''\
Number of disconnected regions within the 2D credible regions \
containing each target position.''' + _same_length_as_coordinates
CrossmatchResult.contour_areas.__doc__ = '''\
Area within the 2D credible regions''' + _same_length_as_contours
CrossmatchResult.area_probs.__doc__ = '''\
Probability within the 2D credible regions''' + _same_length_as_areas
CrossmatchResult.contour_modes.__doc__ = '''\
Number of disconnected regions within the 2D credible \
regions''' + _same_length_as_contours
CrossmatchResult.searched_prob_dist.__doc__ = '''\
Cumulative CDF of distance, marginalized over sky position, at the distance \
of each of the targets.''' + _same_length_as_coordinates
CrossmatchResult.contour_dists.__doc__ = '''\
Distance credible interval, marginalized over sky \
position,''' + _same_length_as_coordinates
CrossmatchResult.searched_vol.__doc__ = '''\
Volume within the 3D credible region containing each target \
position.''' + _same_length_as_coordinates
CrossmatchResult.searched_prob_vol.__doc__ = '''\
Probability within the 3D credible region containing each target \
position.''' + _same_length_as_coordinates
CrossmatchResult.contour_vols.__doc__ = '''\
Volume within the 3D credible regions''' + _same_length_as_contours
CrossmatchResult.probdensity.__doc__ = '''\
2D probability density per steradian at the positions of each of the \
targets.''' + _same_length_as_coordinates
CrossmatchResult.probdensity_vol.__doc__ = '''\
3D probability density per cubic megaparsec at the positions of each of the \
targets.''' + _same_length_as_coordinates


[docs]def crossmatch(sky_map, coordinates=None, contours=(), areas=(), modes=False, cosmology=False): """Cross match a sky map with a catalog of points. Given a sky map and the true right ascension and declination (in radians), find the smallest area in deg^2 that would have to be searched to find the source, the smallest posterior mass, and the angular offset in degrees from the true location to the maximum (mode) of the posterior. Optionally, also compute the areas of and numbers of modes within the smallest contours containing a given total probability. Parameters ---------- sky_map : :class:`astropy.table.Table` A multiresolution sky map, as returned by :func:`ligo.skymap.io.fits.read_sky_map` called with the keyword argument ``moc=True``. coordinates : :class:`astropy.coordinates.SkyCoord`, optional The catalog of target positions to match against. contours : :class:`tuple`, optional Credible levels between 0 and 1. If this argument is present, then calculate the areas and volumes of the 2D and 3D credible regions that contain these probabilities. For example, for ``contours=(0.5, 0.9)``, then areas and volumes of the 50% and 90% credible regions. areas : :class:`tuple`, optional Credible areas in square degrees. If this argument is present, then calculate the probability contained in the 2D credible levels that have these areas. For example, for ``areas=(20, 100)``, then compute the probability within the smallest credible levels of 20 deg² and 100 deg², respectively. modes : :class:`bool`, optional If True, then enable calculation of the number of distinct modes or islands of probability. Note that this option may be computationally expensive. cosmology : :class:`bool`, optional If True, then search space by descending probability density per unit comoving volume. If False, then search space by descending probability per luminosity distance cubed. Returns ------- result : :class:`~ligo.skymap.postprocess.crossmatch.CrossmatchResult` Notes ----- This function is also be used for injection finding; see :doc:`/tool/ligo_skymap_stats`. Examples -------- First, some imports: >>> from astroquery.vizier import VizierClass >>> from astropy.coordinates import SkyCoord >>> from ligo.skymap.io import read_sky_map >>> from ligo.skymap.postprocess import crossmatch Next, retrieve the GLADE catalog using Astroquery and get the coordinates of all its entries: >>> vizier = VizierClass( ... row_limit=-1, columns=['GWGC', '_RAJ2000', '_DEJ2000', 'Dist']) >>> cat, = vizier.get_catalogs('VII/281/glade2') >>> coordinates = SkyCoord(cat['_RAJ2000'], cat['_DEJ2000'], cat['Dist']) Load the multiresolution sky map for S190814bv: >>> url = 'https://gracedb.ligo.org/api/superevents/S190814bv/files/bayestar.multiorder.fits' >>> skymap = read_sky_map(url, moc=True) Perform the cross match: >>> result = crossmatch(skymap, coordinates) Using the cross match results, we can list the galaxies within the 90% credible volume: >>> print(cat[result.searched_prob_vol < 0.9]) _RAJ2000 _DEJ2000 GWGC Dist deg deg Mpc -------------------- -------------------- ---------- -------------------- 9.3396699999999999 -19.9342460000000017 NGC0171 57.56212553960000 20.2009090000000064 -31.1146050000000010 --- 137.16022925600001 8.9144679999999994 -20.1252980000000008 ESO540-003 49.07809291930000 10.6762720000000009 -21.7740819999999999 --- 276.46938505499998 13.5855169999999994 -23.5523850000000010 --- 138.44550704800000 20.6362969999999990 -29.9825149999999958 --- 160.23313164900000 13.1923879999999993 -22.9750179999999986 --- 236.96795954500001 11.7813630000000007 -24.3706470000000017 --- 244.25031189699999 19.1711120000000008 -31.4339490000000019 --- 152.13614001400001 13.6367060000000002 -23.4948789999999974 --- 141.25162979500001 ... ... ... ... 11.3517000000000010 -25.8596999999999966 --- 335.73800000000000 11.2073999999999998 -25.7149000000000001 --- 309.02999999999997 11.1875000000000000 -25.7503999999999991 --- 295.12099999999998 10.8608999999999991 -25.6904000000000003 --- 291.07200000000000 10.6938999999999975 -25.6778300000000002 --- 323.59399999999999 15.4935000000000009 -26.0304999999999964 --- 304.78899999999999 15.2794000000000008 -27.0410999999999966 --- 320.62700000000001 14.8323999999999980 -27.0459999999999994 --- 320.62700000000001 14.5340999999999969 -26.0949000000000026 --- 307.61000000000001 23.1280999999999963 -31.1109199999999966 --- 320.62700000000001 Length = 1479 rows """ # noqa: E501, W291 # Astropy coordinates that are constructed without distance have # a distance field that is unity (dimensionless). if coordinates is None: true_ra = true_dec = true_dist = None else: # Ensure that coordinates are in proper frame and representation coordinates = SkyCoord(coordinates, representation_type=SphericalRepresentation, frame=ICRS) true_ra = coordinates.ra.rad true_dec = coordinates.dec.rad if np.any(coordinates.distance != 1): true_dist = coordinates.distance.to_value(u.Mpc) else: true_dist = None contours = np.asarray(contours) # Sort the pixels by descending posterior probability. sky_map = np.flipud(np.sort(sky_map, order='PROBDENSITY')) # Find the pixel that contains the injection. order, ipix = moc.uniq2nest(sky_map['UNIQ']) max_order = np.max(order) max_nside = ah.level_to_nside(max_order) max_ipix = ipix << np.int64(2 * (max_order - order)) if true_ra is not None: true_theta = 0.5 * np.pi - true_dec true_phi = true_ra true_pix = hp.ang2pix(max_nside, true_theta, true_phi, nest=True) i = np.argsort(max_ipix) true_idx = i[np.digitize(true_pix, max_ipix[i]) - 1] # Find the angular offset between the mode and true locations. mode_theta, mode_phi = hp.pix2ang( ah.level_to_nside(order[0]), ipix[0], nest=True) if true_ra is None: offset = np.nan else: offset = np.rad2deg( angle_distance(true_theta, true_phi, mode_theta, mode_phi)) # Calculate the cumulative area in deg2 and the cumulative probability. dA = moc.uniq2pixarea(sky_map['UNIQ']) dP = sky_map['PROBDENSITY'] * dA prob = np.cumsum(dP) area = np.cumsum(dA) * np.square(180 / np.pi) if true_ra is None: searched_area = searched_prob = probdensity = np.nan else: # Find the smallest area that would have to be searched to find # the true location. searched_area = area[true_idx] # Find the smallest posterior mass that would have to be searched to # find the true location. searched_prob = prob[true_idx] # Find the probability density. probdensity = sky_map['PROBDENSITY'][true_idx] # Find the contours of the given credible levels. contour_idxs = np.digitize(contours, prob) - 1 # For each of the given confidence levels, compute the area of the # smallest region containing that probability. contour_areas = np.interp( contours, prob, area, left=0, right=4*180**2/np.pi).tolist() # For each listed area, find the probability contained within the # smallest credible region of that area. area_probs = np.interp(areas, area, prob, left=0, right=1).tolist() if modes: if true_ra is None: searched_modes = np.nan else: # Count up the number of modes in each of the given contours. searched_modes = count_modes_moc(sky_map['UNIQ'], true_idx) contour_modes = [ count_modes_moc(sky_map['UNIQ'], i) for i in contour_idxs] else: searched_modes = np.nan contour_modes = np.nan # Distance stats now... if 'DISTMU' in sky_map.dtype.names: dP_dA = sky_map['PROBDENSITY'] mu = sky_map['DISTMU'] sigma = sky_map['DISTSIGMA'] norm = sky_map['DISTNORM'] # Set up distance grid. n_r = 1000 distmean, _ = distance.parameters_to_marginal_moments(dP, mu, sigma) max_r = 6 * distmean if true_dist is not None and np.size(true_dist) != 0 \ and np.max(true_dist) > max_r: max_r = np.max(true_dist) d_r = max_r / n_r # Calculate searched_prob_dist and contour_dists. r = d_r * np.arange(1, n_r) P_r = distance.marginal_cdf(r, dP, mu, sigma, norm) if true_dist is None: searched_prob_dist = np.nan else: searched_prob_dist = np.interp(true_dist, r, P_r, left=0, right=1) if len(contours) == 0: contour_dists = [] else: lo, hi = np.interp( np.row_stack(( 0.5 * (1 - contours), 0.5 * (1 + contours) )), P_r, r, left=0, right=np.inf) contour_dists = (hi - lo).tolist() # Calculate volume of each voxel, defined as the region within the # HEALPix pixel and contained within the two centric spherical shells # with radii (r - d_r / 2) and (r + d_r / 2). dV = (np.square(r) + np.square(d_r) / 12) * d_r * dA.reshape(-1, 1) # Calculate probability within each voxel. dP = np.exp( -0.5 * np.square( (r.reshape(1, -1) - mu.reshape(-1, 1)) / sigma.reshape(-1, 1) ) ) * (dP_dA * norm / (sigma * np.sqrt(2 * np.pi))).reshape(-1, 1) * dV dP[np.isnan(dP)] = 0 # Suppress invalid values # Calculate probability density per unit volume. if cosmology: dV *= dVC_dVL_for_DL(r) dP_dV = dP / dV i = np.flipud(np.argsort(dP_dV.ravel())) P_flat = np.cumsum(dP.ravel()[i]) V_flat = np.cumsum(dV.ravel()[i]) contour_vols = np.interp( contours, P_flat, V_flat, left=0, right=np.inf).tolist() P = np.empty_like(P_flat) V = np.empty_like(V_flat) P[i] = P_flat V[i] = V_flat P = P.reshape(dP.shape) V = V.reshape(dV.shape) if true_dist is None: searched_vol = searched_prob_vol = probdensity_vol = np.nan else: i_radec = true_idx i_dist = np.digitize(true_dist, r) - 1 probdensity_vol = dP_dV[i_radec, i_dist] searched_prob_vol = P[i_radec, i_dist] searched_vol = V[i_radec, i_dist] else: searched_vol = searched_prob_vol = searched_prob_dist \ = probdensity_vol = np.nan contour_dists = [np.nan] * len(contours) contour_vols = [np.nan] * len(contours) # Done. return CrossmatchResult( searched_area, searched_prob, offset, searched_modes, contour_areas, area_probs, contour_modes, searched_prob_dist, contour_dists, searched_vol, searched_prob_vol, contour_vols, probdensity, probdensity_vol)