# underdensity_prob_func¶

halotools.mock_observables.underdensity_prob_func(sample1, rbins, n_ran=None, random_sphere_centers=None, period=None, sample_volume=None, u=0.2, num_threads=1, approx_cell1_size=None, approx_cellran_size=None, seed=None)[source]

Calculate the underdensity probability function (UPF), $$P_U(r)$$.

$$P_U(r)$$ is defined as the probability that a randomly placed sphere of size $$r$$ encompases a volume with less than a specified number density.

See the Formatting your xyz coordinates for Mock Observables calculations documentation page for instructions on how to transform your coordinate position arrays into the format accepted by the sample1 argument.

Parameters:
sample1array_like

Npts1 x 3 numpy array 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.

rbinsfloat

size of spheres to search for neighbors Length units are comoving and assumed to be in Mpc/h, here and throughout Halotools.

n_ranint, optional

integer number of randoms to use to search for voids. If n_ran is not passed, you must pass random_sphere_centers.

random_sphere_centersarray_like, optional

Npts x 3 array of randomly selected positions to drop down spheres to use to measure the void_prob_func. If random_sphere_centers is not passed, n_ran must be passed.

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. If set to None, PBCs are set to infinity, in which case sample_volume must be specified so that the global mean density can be estimated. In this case, it is still necessary to drop down randomly placed spheres in order to compute the UPF. To do so, the spheres will be dropped inside a cubical box whose sides are defined by the smallest/largest coordinate distance of the input sample1. Length units are comoving and assumed to be in Mpc/h, here and throughout Halotools.

sample_volumefloat, optional

If period is set to None, you must specify the effective volume of the sample. Length units are comoving and assumed to be in Mpc/h, here and throughout Halotools.

ufloat, optional

density threshold in units of the mean object density

number of ‘threads’ to use in the pair counting. if set to ‘max’, use all available cores. num_threads=0 is the default.

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 max(rbins) 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_cellran_sizearray_like, optional

Analogous to approx_cell1_size, but for used for randoms. See comments for approx_cell1_size for details.

seedint, optional

Random number seed used to randomly lay down spheres, if applicable. Default is None, in which case results will be stochastic.

Returns:
upfnumpy.array

len(rbins) length array containing the underdensity probability function $$P_U(r)$$ computed for each $$r$$ defined by input rbins.

Notes

This function requires the calculation of the number of pairs per randomly placed sphere, and thus storage of an array of shape(n_ran,len(rbins)). This can be a memory intensive process as this array becomes large.

Examples

For demonstration purposes we create a randomly distributed set of points within a periodic unit cube.

>>> Npts = 10000
>>> Lbox = 1.0
>>> period = np.array([Lbox,Lbox,Lbox])

>>> x = np.random.random(Npts)
>>> y = np.random.random(Npts)
>>> z = np.random.random(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 mock_observables sub-package:

>>> coords = np.vstack((x,y,z)).T

>>> rbins = np.logspace(-2,-1,20)
>>> n_ran = 1000
>>> upf = underdensity_prob_func(coords, rbins, n_ran=n_ran, period=period, u=0.2)