#
# Copyright (C) 2013-2023 Leo Singer
#
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# the Free Software Foundation, either version 3 of the License, or
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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, np.ndarray
The desired credible level or levels (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, np.ndarray
The lenth of the semimajor axis in degrees.
b : float, np.ndarray
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, np.ndarray
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.
If no ellipse can be found that contains integrated probability greater
than or equal to the desired credible level ``cl``, then the return values
``a``, ``b``, and ``area`` will be set to nan.
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)
***Example 5***
You can ask for other credible levels:
>>> find_ellipse(prob, cl=50) # doctest: +FLOAT_CMP
(0.0, 0.0, 37.054207653285076, 19.168955020015982, 9.217477126726322, 2182.5580135410632)
Or even for multiple credible levels:
>>> find_ellipse(prob, cl=[50, 90]) # doctest: +FLOAT_CMP
(0.0, 0.0, array([37.05420765, 64.77564486]), array([19.16895502, 33.50986302]), 9.217477126726322, array([2182.55801354, 6372.42573159]))
""" # 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 the specified projection.
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])
# np.multiply rather than * to automatically convert to ndarray if needed
cl = np.multiply(cl, 1e-2)
nsigma = np.interp(cl, cls, nsigmas, right=np.nan)
area = np.interp(cl, cls, careas, right=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 * root_w for root_w in 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