r""" Module containing the `~halotools.mock_observables.npairs_s_mu` function
used to count pairs as a function of separation.
"""
from __future__ import (absolute_import, division, print_function, unicode_literals)
import numpy as np
import multiprocessing
from functools import partial
from .rectangular_mesh import RectangularDoubleMesh
from .mesh_helpers import _set_approximate_cell_sizes, _cell1_parallelization_indices
from .cpairs import npairs_s_mu_engine
from .npairs_3d import _npairs_3d_process_args
from ...utils.array_utils import array_is_monotonic
__author__ = ('Andrew Hearin', 'Duncan Campbell')
__all__ = ('npairs_s_mu', )
[docs]
def npairs_s_mu(sample1, sample2, s_bins, mu_bins, period=None,
num_threads=1, approx_cell1_size=None, approx_cell2_size=None):
r"""
Function counts the number of pairs of points separated by less than
radial separation, :math:`s`, given by ``s_bins`` and
angular distance, :math:`\mu\equiv\cos(\theta_{\rm los})`, given by ``mu_bins``,
where :math:`\theta_{\rm los}` is the angle between :math:`\vec{s}` and
the line-of-sight (LOS).
The first two dimensions (x, y) define the plane for perpendicular distances.
The third dimension (z) defines the LOS. i.e. x,y positions are on
the plane of the sky, and z is the radial distance coordinate. This is the 'distant
observer' approximation.
A common variation of pair-counting calculations is to count pairs with
separations *between* two different distances, e.g. [s1 ,s2] and [mu1, mu2].
You can retrieve this information from `~halotools.mock_observables.npairs_s_mu`
by taking `numpy.diff` of the returned array along each axis.
See Notes section for further clarification.
Parameters
----------
sample1 : array_like
Numpy array of shape (Npts1, 3) containing 3-D positions of points.
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.
sample2 : array_like
Numpy array of shape (Npts2, 3) containing 3-D positions of points.
Should be identical to sample1 for cases of auto-sample pair counts.
Length units are comoving and assumed to be in Mpc/h, here and throughout Halotools.
s_bins : array_like
numpy array of shape (num_s_bin_edges, ) storing the :math:`s`
boundaries defining the bins in which pairs are counted.
mu_bins : array_like
numpy array of shape (num_mu_bin_edges, ) storing the
:math:`\cos(\theta_{\rm LOS})` boundaries defining the bins in
which pairs are counted. All values must be between [0,1].
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.
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
-------
num_pairs : array of shape (num_s_bin_edges, num_mu_bin_edges) storing the
number of pairs separated by less than (s, mu)
Notes
-----
Let :math:`\vec{s}` be the radial vector connnecting two points.
The magnitude, :math:`s`, is:
.. math::
s = \sqrt{r_{\parallel}^2+r_{\perp}^2},
where :math:`r_{\parallel}` is the separation parallel to the LOS
and :math:`r_{\perp}` is the separation perpednicular to the LOS. :math:`\mu` is
the cosine of the angle, :math:`\theta_{\rm LOS}`, between the LOS
and :math:`\vec{s}`:
.. math::
\mu = \cos(\theta_{\rm LOS}) \equiv r_{\parallel}/s.
Along the first dimension of ``num_pairs``, :math:`s` increases.
Along the second dimension, :math:`\mu` decreases,
i.e. :math:`\theta_{\rm LOS}` increases.
If sample1 == sample2 that the `~halotools.mock_observables.npairs_s_mu` function
double-counts pairs. If your science application requires sample1==sample2 inputs
and also pairs to not be double-counted, simply divide the final counts by 2.
One final point of clarification concerning double-counting may be in order.
Suppose sample1==sample2 and s_bins[0]==0. Then the returned value for this bin
will be len(sample1), since each sample1 point has distance 0 from itself.
Examples
--------
For demonstration purposes we create randomly distributed sets of points within a
periodic unit cube.
>>> Npts1, Npts2, Lbox = 1000, 1000, 200.
>>> period = [Lbox, Lbox, Lbox]
>>> s_bins = np.logspace(-1, 1.25, 15)
>>> mu_bins = np.linspace(0, 1)
>>> x1 = np.random.uniform(0, Lbox, Npts1)
>>> y1 = np.random.uniform(0, Lbox, Npts1)
>>> z1 = np.random.uniform(0, Lbox, Npts1)
>>> x2 = np.random.uniform(0, Lbox, Npts2)
>>> y2 = np.random.uniform(0, Lbox, Npts2)
>>> z2 = np.random.uniform(0, Lbox, Npts2)
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([x1, y1, z1]).T
>>> sample2 = np.vstack([x2, y2, z2]).T
>>> from halotools.mock_observables.pair_counters import npairs_s_mu
>>> result = npairs_s_mu(sample1, sample2, s_bins, mu_bins, period=period)
"""
# Process the inputs with the helper function
result = _npairs_3d_process_args(sample1, sample2, s_bins, period,
num_threads, approx_cell1_size, approx_cell2_size)
x1in, y1in, z1in, x2in, y2in, z2in = result[0:6]
s_bins, period, num_threads, PBCs, approx_cell1_size, approx_cell2_size = result[6:]
xperiod, yperiod, zperiod = period
rmax = np.max(s_bins)
# process mu_bins parameter separately
mu_bins = np.atleast_1d(mu_bins)
try:
assert mu_bins.ndim == 1
assert len(mu_bins) > 1
if len(mu_bins) > 2:
assert array_is_monotonic(mu_bins, strict=True) == 1
except AssertionError:
msg = ("\n Input `mu_bins` must be a monotonically increasing \n"
"1D array with at least two entries")
raise ValueError(msg)
# convert to mu=sin(theta_los) binning used by the cython engine.
mu_bins_prime = np.sin(np.arccos(mu_bins))
mu_bins_prime = np.sort(mu_bins_prime)
# increasing mu_prime now corresponds to increasing theta_LOS
search_xlength, search_ylength, search_zlength = rmax, rmax, rmax
# Compute the estimates for the cell sizes
approx_cell1_size, approx_cell2_size = (
_set_approximate_cell_sizes(approx_cell1_size, approx_cell2_size, period)
)
approx_x1cell_size, approx_y1cell_size, approx_z1cell_size = approx_cell1_size
approx_x2cell_size, approx_y2cell_size, approx_z2cell_size = approx_cell2_size
# Build the rectangular mesh
double_mesh = RectangularDoubleMesh(x1in, y1in, z1in, x2in, y2in, z2in,
approx_x1cell_size, approx_y1cell_size, approx_z1cell_size,
approx_x2cell_size, approx_y2cell_size, approx_z2cell_size,
search_xlength, search_ylength, search_zlength, xperiod, yperiod, zperiod, PBCs)
# Create a function object that has a single argument, for parallelization purposes
engine = partial(npairs_s_mu_engine,
double_mesh, x1in, y1in, z1in, x2in, y2in, z2in, s_bins, mu_bins_prime)
# Calculate the cell1 indices that will be looped over by the engine
num_threads, cell1_tuples = _cell1_parallelization_indices(
double_mesh.mesh1.ncells, num_threads)
if num_threads > 1:
pool = multiprocessing.Pool(num_threads)
result = pool.map(engine, cell1_tuples)
counts = np.sum(np.array(result), axis=0)
pool.close()
else:
counts = engine(cell1_tuples[0])
return np.array(counts)