r"""
Module containing the `~halotools.mock_observables.alignments.ii_plus_projected` function used to
calculate the projected intrinsic ellipticity-ellipticity (II) correlation
"""
from __future__ import absolute_import, division, print_function, unicode_literals
import numpy as np
from math import pi
from .alignment_helpers import process_projected_alignment_args
from ..mock_observables_helpers import (enforce_sample_has_correct_shape,
get_separation_bins_array, get_line_of_sight_bins_array, get_period, get_num_threads)
from ..pair_counters.mesh_helpers import _enforce_maximum_search_length
from ..pair_counters import positional_marked_npairs_xy_z, marked_npairs_xy_z
__all__ = ['ii_plus_projected']
__author__ = ['Duncan Campbell']
np.seterr(divide='ignore', invalid='ignore') # ignore divide by zero in e.g. DD/RR
[docs]
def ii_plus_projected(sample1, orientations1, ellipticities1, sample2, orientations2, ellipticities2,
rp_bins, pi_max, randoms1=None, randoms2=None, weights1=None, weights2=None,
ran_weights1=None, ran_weights2=None, estimator='Natural',
period=None, num_threads=1, approx_cell1_size=None, approx_cell2_size=None):
r"""
Calculate the projected intrinsic ellipticity-ellipticity correlation function (II),
:math:`w_{++}(r_p)`, where :math:`r_p` is the separation perpendicular to the line-of-sight (LOS)
between two galaxies. See the 'Notes' section for details of this calculation.
The first two dimensions define the plane for perpendicular distances. The third
dimension is used for parallel distances, i.e. x,y positions are on the plane of the
sky, and z is the redshift coordinate. This is the 'distant observer' approximation.
Note in particular that the `~halotools.mock_observables.alignments.ii_plus_projected` function does not
accept angular coordinates for the input ``sample1`` or ``sample2``.
Parameters
----------
sample1 : array_like
Npts1 x 3 numpy array containing 3-D positions of points with associated
orientations and ellipticities.
See the :ref:`mock_obs_pos_formatting` documentation page, or the
Examples section below, for instructions on how to transform
your coordinate position arrays into the format accepted by the ``sample1`` and ``sample2`` arguments.
Length units are comoving and assumed to be in Mpc/h, here and throughout Halotools.
orientations1 : array_like
Npts1 x 2 numpy array containing projected orientation vectors for each point in ``sample1``.
these will be normalized if not already.
ellipticities1: array_like
Npts1 x 1 numpy array containing ellipticities for each point in ``sample1``.
sample2 : array_like, optional
Npts2 x 3 array containing 3-D positions of points with associated
orientations and ellipticities.
orientations12 : array_like
Npts1 x 2 numpy array containing projected orientation vectors for each point in ``sample2``.
these will be normalized if not already.
ellipticities2: array_like
Npts1 x 1 numpy array containing ellipticities for each point in ``sample2``.
rp_bins : array_like
array of boundaries defining the radial bins perpendicular to the LOS in which
pairs are counted.
Length units are comoving and assumed to be in Mpc/h, here and throughout Halotools.
pi_max : float
maximum LOS distance defining the projection integral length-scale in the z-dimension.
Length units are comoving and assumed to be in Mpc/h, here and throughout Halotools.
randoms1 : array_like, optional
Nran1 x 3 array containing 3-D positions of randomly distributed points corresponding to ``sample1``.
If no randoms are provided (the default option), the
calculation can proceed using analytical randoms
(only valid for periodic boundary conditions).
randoms2 : array_like, optional
Nran2 x 3 array containing 3-D positions of randomly distributed points corresponding to ``sample2``.
If no randoms are provided (the default option), the
calculation can proceed using analytical randoms
(only valid for periodic boundary conditions).
weights1 : array_like, optional
Npts1 array of weghts. If this parameter is not specified, it is set to numpy.ones(Npts1).
weights2 : array_like, optional
Npts2 array of weghts. If this parameter is not specified, it is set to numpy.ones(Npts2).
ran_weights1 : array_like, optional
Npran1 array of weghts. If this parameter is not specified, it is set to numpy.ones(Nran1).
ran_weights2 : array_like, optional
Nran2 array of weghts. If this parameter is not specified, it is set to numpy.ones(Nran2).
estimator : string, optional
string indicating which estimator to use
period : array_like, optional
Length-3 sequence defining the periodic boundary conditions
in each dimension. If you instead provide a single scalar, Lbox,
period is assumed to be the same in all Cartesian directions.
If set to None (the default option), PBCs are set to infinity,
in which case ``randoms`` must be provided.
Length units are comoving and assumed to be in Mpc/h, here and throughout Halotools.
num_threads : int, optional
Number of threads to use in calculation, where parallelization is performed
using the python ``multiprocessing`` module. Default is 1 for a purely serial
calculation, in which case a multiprocessing Pool object will
never be instantiated. A string 'max' may be used to indicate that
the pair counters should use all available cores on the machine.
approx_cell1_size : array_like, optional
Length-3 array serving as a guess for the optimal manner by how points
will be apportioned into subvolumes of the simulation box.
The optimum choice unavoidably depends on the specs of your machine.
Default choice is to use Lbox/10 in each dimension,
which will return reasonable result performance for most use-cases.
Performance can vary sensitively with this parameter, so it is highly
recommended that you experiment with this parameter when carrying out
performance-critical calculations.
approx_cell2_size : array_like, optional
Analogous to ``approx_cell1_size``, but for sample2. See comments for
``approx_cell1_size`` for details.
Returns
-------
correlation_function : numpy.array
*len(rp_bins)-1* length array containing the correlation function :math:`w_{++}(r_p)`
computed in each of the bins defined by input ``rp_bins``.
Notes
-----
The projected II-correlation function is calculated as:
.. math::
w_{++}(r_p) = 2 \int_0^{\pi_{\rm max}} \xi_{++}(r_p, \pi) \mathrm{d}\pi
If the Natural estimator is indicated, the projected II-correlation function is calculated as:
.. math::
\xi_{++}(r_p, \pi) = \frac{S_{+}S_{+}}{R_sR_s}
where
.. math::
S_{+}S_{+} = \sum_{i \neq j} w_jw_i e_{+}(j|i)e_{+}(i|j)
:math:`w_j` and :math:`w_j` are weights. Weights are set to 1 for all galaxies by default.
The alingment of the :math:`j`-th galaxy relative to the direction to the :math:`i`-th galaxy is given by:
.. math::
e_{+}(j|i) = e_j\cos(2\phi)
where :math:`e_j` is the ellipticity of the :math:`j`-th galaxy. :math:`\phi` is the angle between the
orientation vector, :math:`\vec{o}_j`, and the projected direction between the :math:`j`-th
and :math:`i`-th galaxy, :math:`\vec{r}_{p i,j}`.
.. math::
\cos(\phi) = \vec{o}_j \cdot \vec{r}_{p i,j}
:math:`R_sR_s` are random pair counts,
Examples
--------
For demonstration purposes we create a randomly distributed set of points within a
periodic cube of Lbox = 250 Mpc/h.
>>> Npts = 1000
>>> Lbox = 250
>>> x = np.random.uniform(0, Lbox, Npts)
>>> y = np.random.uniform(0, Lbox, Npts)
>>> z = np.random.uniform(0, Lbox, Npts)
We transform our *x, y, z* points into the array shape used by the pair-counter by
taking the transpose of the result of `numpy.vstack`. This boilerplate transformation
is used throughout the `~halotools.mock_observables` sub-package:
>>> sample1 = np.vstack((x,y,z)).T
Alternatively, you may use the `~halotools.mock_observables.return_xyz_formatted_array`
convenience function for this same purpose, which provides additional wrapper
behavior around `numpy.vstack` such as placing points into redshift-space.
We then create a set of random orientation vectors and ellipticities for each point
>>> random_orientations = np.random.random((Npts,2))
>>> random_ellipticities = np.random.random(Npts)
We can the calculate the projected auto-GI correlation between these points:
>>> rp_bins = np.logspace(-1,1,10)
>>> pi_max = 0.25
>>> w = ii_plus_projected(sample1, random_orientations, random_ellipticities, sample1, random_orientations, random_ellipticities, rp_bins, pi_max, period=Lbox)
"""
# process arguments
alignment_args = (sample1, orientations1, ellipticities1, weights1,
sample2, orientations2, ellipticities2, weights2,
randoms1, ran_weights1, randoms2, ran_weights2)
sample1, orientations1, ellipticities1, weights1, sample2,\
orientations2, ellipticities2, weights2, randoms1, ran_weights1,\
randoms2, ran_weights2 = process_projected_alignment_args(*alignment_args)
function_args = (sample1, rp_bins, pi_max, sample2, randoms1, randoms2,
period, num_threads, approx_cell1_size, approx_cell2_size)
sample1, rp_bins, pi_bins, sample2, randoms1, randoms2,\
period, num_threads, PBCs, no_randoms = _ii_plus_projected_process_args(*function_args)
# How many points are there (for normalization purposes)?
N1 = len(sample1)
N2 = len(sample2)
if no_randoms: # set random density the the same as the sampels
NR1 = N1
NR2 = N2
else:
NR1 = len(randoms1)
NR2 = len(randoms2)
#define merk vectors to use in pair counting
# sample 1
marks1 = np.ones((N1, 3))
marks1[:, 0] = ellipticities1 * weights1
marks1[:, 1] = orientations1[:, 0]
marks1[:, 2] = orientations1[:, 1]
# sample 2
marks2 = np.ones((N2, 3))
marks2[:, 0] = ellipticities2 * weights2
marks2[:, 1] = orientations2[:, 0]
marks2[:, 2] = orientations2[:, 1]
# randoms 1
ran_marks1 = np.ones((NR1, 3))
ran_marks1[:, 0] = ran_weights1
ran_marks1[:, 1] = 0 # dummy
ran_marks1[:, 2] = 0 # dummy
# randoms 2
ran_marks2 = np.ones((NR2, 3))
ran_marks2[:, 0] = ran_weights2
ran_marks2[:, 1] = 0 # dummy
ran_marks2[:, 2] = 0 # dummy
# define pi bins
pi_bins = np.array([0.0, pi_max])
do_SS, do_RR = II_estimator_requirements(estimator)
# count marked pairs
if do_SS:
SS = marked_pair_counts(sample1, sample2, marks1, marks2,
rp_bins, pi_bins, period, num_threads,
approx_cell1_size, approx_cell2_size)
else:
SS = None
# count random pairs
if do_RR:
RR = random_counts(randoms1, randoms2, ran_weights1, ran_weights2,
rp_bins, pi_bins, N1, N2, no_randoms, period, PBCs,
num_threads, approx_cell1_size, approx_cell2_size)
else:
RR = None
result = II_estimator(SS, RR, N1, N2, NR1, NR2, estimator)
return result*2.0*pi_max # factor of 2pi_max accounts for integration
def II_estimator(SS, RR, N1, N2, NR1, NR2, estimator='Natural'):
r"""
apply the supplied GI estimator to calculate the correlation function.
"""
if estimator == 'Natural':
factor = (NR1*NR2)/(N1*N2)
return factor*(SS/RR)
else:
msg = ('The estimator provided is not supported.')
raise ValueError(msg)
def II_estimator_requirements(estimator):
r"""
Return the requirments for the supplied GI estimator.
"""
do_RR = False
do_SS = False
if estimator == 'Natural':
do_SS = True
do_RR = True
return do_SS, do_RR
else:
msg = ('The estimator provided is not supported.')
raise ValueError(msg)
def marked_pair_counts(sample1, sample2, weights1, weights2, rp_bins, pi_bins, period,
num_threads, approx_cell1_size, approx_cell2_size):
r"""
Count marked pairs.
"""
weight_func_id = 5
SS = positional_marked_npairs_xy_z(sample1, sample2, rp_bins, pi_bins, period=period,
weights1=weights1, weights2=weights2, weight_func_id=weight_func_id,
num_threads=num_threads, approx_cell1_size=approx_cell1_size,
approx_cell2_size=approx_cell1_size)[0]
SS = np.diff(np.diff(SS, axis=0), axis=1)
SS = SS.flatten()
return SS
def random_counts(randoms1, randoms2, ran_weights1, ran_weights2, rp_bins, pi_bins,
N1, N2, no_randoms, period,
PBCs, num_threads, approx_cell1_size, approx_cell2_size):
r"""
Count random pairs.
"""
if no_randoms is False:
RR = marked_npairs_xy_z(randoms1, randoms2, rp_bins, pi_bins,
period=period, num_threads=num_threads, weight_func_id=1,
weights1=ran_weights1, weights2=ran_weights2,
approx_cell1_size=approx_cell1_size,
approx_cell2_size=approx_cell2_size)
RR = np.diff(np.diff(RR, axis=0), axis=1)
RR = RR.flatten()
return RR
else:
# set 'number' or randoms
# setting Nran to Ndata makes normalization simple
NR1 = N1
NR2 = N2
# do volume calculations
v = cylinder_volume(rp_bins, 2.0*pi_bins)
dv = np.diff(np.diff(v, axis=0), axis=1)
global_volume = period.prod()
# calculate the random-random pairs.
rhor = (NR1*NR2)/global_volume
RR = (dv*rhor)
return RR.flatten()
def cylinder_volume(R, h):
r"""
Calculate the volume of a cylinder(s), used for the analytical randoms.
"""
return pi*np.outer(R**2.0, h)
def _ii_plus_projected_process_args(sample1, rp_bins, pi_max, sample2, randoms1, randoms2,
period, num_threads, approx_cell1_size, approx_cell2_size):
r"""
Private method to do bounds-checking on the arguments passed to
`~halotools.mock_observables.alignments.alignments.ii_plus_projected`.
"""
sample1 = enforce_sample_has_correct_shape(sample1)
if randoms1 is not None:
randoms1 = np.atleast_1d(randoms1)
no_randoms1 = False
else: no_randoms1 = True
if randoms2 is not None:
randoms2 = np.atleast_1d(randoms2)
no_randoms2 = False
else: no_randoms2 = True
#if one of the randoms is missing, raise an error
no_randoms = True
if no_randoms1:
if no_randoms2 is False:
msg = "if one set of randoms is provided, both randoms must be provided.\n"
raise ValueError(msg)
elif no_randoms2:
if no_randoms1 is False:
msg = "if one set of randoms is provided, both randoms must be provided.\n"
raise ValueError(msg)
else:
no_randoms = False
pi_max = float(pi_max)
pi_bins = np.array([0.0, pi_max])
rp_bins = get_separation_bins_array(rp_bins)
rp_max = np.amax(rp_bins)
period, PBCs = get_period(period)
_enforce_maximum_search_length([rp_max, rp_max, pi_max], period)
if (randoms1 is None) & (PBCs is False):
msg = "If no PBCs are specified, both randoms must be provided.\n"
raise ValueError(msg)
num_threads = get_num_threads(num_threads)
return sample1, rp_bins, pi_bins, sample2, randoms1, randoms2, period, num_threads, PBCs, no_randoms