halotools.mock_observables.pair_counters.npairs_s_mu(sample1, sample2, s_bins, mu_bins, period=None, num_threads=1, approx_cell1_size=None, approx_cell2_size=None)[source]

Function counts the number of pairs of points separated by less than radial separation, \(s\), given by s_bins and angular distance, \(\mu\equiv\cos(\theta_{\rm los})\), given by mu_bins, where \(\theta_{\rm los}\) is the angle between \(\vec{s}\) 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 npairs_s_mu by taking numpy.diff of the returned array along each axis.

See Notes section for further clarification.


Numpy array of shape (Npts1, 3) containing 3-D positions of points. See the Formatting your xyz coordinates for Mock Observables calculations 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.


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.


numpy array of shape (num_s_bin_edges, ) storing the \(s\) boundaries defining the bins in which pairs are counted.


numpy array of shape (num_mu_bin_edges, ) storing the \(\cos(\theta_{\rm LOS})\) boundaries defining the bins in which pairs are counted. All values must be between [0,1].

periodarray_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_threadsint, 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_sizearray_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_sizearray_like, optional

Analogous to approx_cell1_size, but for sample2. See comments for approx_cell1_size for details.

num_pairsarray of shape (num_s_bin_edges, num_mu_bin_edges) storing the

number of pairs separated by less than (s, mu)


Let \(\vec{s}\) be the radial vector connnecting two points. The magnitude, \(s\), is:

\[s = \sqrt{r_{\parallel}^2+r_{\perp}^2},\]

where \(r_{\parallel}\) is the separation parallel to the LOS and \(r_{\perp}\) is the separation perpednicular to the LOS. \(\mu\) is the cosine of the angle, \(\theta_{\rm LOS}\), between the LOS and \(\vec{s}\):

\[\mu = \cos(\theta_{\rm LOS}) \equiv r_{\parallel}/s.\]

Along the first dimension of num_pairs, \(s\) increases. Along the second dimension, \(\mu\) decreases, i.e. \(\theta_{\rm LOS}\) increases.

If sample1 == sample2 that the 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.


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 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)