cKDTree.count_neighbors(self, other, r, p=2., weights=None, cumulative=True)

Count how many nearby pairs can be formed. (pair-counting)

Count the number of pairs (x1,x2) can be formed, with x1 drawn from self and x2 drawn from other, and where distance(x1, x2, p) <= r.

Data points on self and other are optionally weighted by the weights argument. (See below)

The algorithm we implement here is based on [R349]. See notes for further discussion.


other : cKDTree instance

The other tree to draw points from, can be the same tree as self.

r : float or one-dimensional array of floats

The radius to produce a count for. Multiple radii are searched with a single tree traversal. If the count is non-cumulative(cumulative=False), r defines the edges of the bins, and must be non-decreasing.

p : float, optional

1<=p<=infinity. Which Minkowski p-norm to use. Default 2.0.

weights : tuple, array_like, or None, optional

If None, the pair-counting is unweighted. If given as a tuple, weights[0] is the weights of points in self, and weights[1] is the weights of points in other; either can be None to indicate the points are unweighted. If given as an array_like, weights is the weights of points in self and other. For this to make sense, self and other must be the same tree. If self and other are two different trees, a ValueError is raised. Default: None

cumulative : bool, optional

Whether the returned counts are cumulative. When cumulative is set to False the algorithm is optimized to work with a large number of bins (>10) specified by r. When cumulative is set to True, the algorithm is optimized to work with a small number of r. Default: True


result : scalar or 1-D array

The number of pairs. For unweighted counts, the result is integer. For weighted counts, the result is float. If cumulative is False, result[i] contains the counts with (-inf if i == 0 else r[i-1]) < R <= r[i]


Pair-counting is the basic operation used to calculate the two point correlation functions from a data set composed of position of objects.

Two point correlation function measures the clustering of objects and is widely used in cosmology to quantify the large scale structure in our Universe, but it may be useful for data analysis in other fields where self-similar assembly of objects also occur.

The Landy-Szalay estimator for the two point correlation function of D measures the clustering signal in D. [R350]

For example, given the position of two sets of objects,

  • objects D (data) contains the clustering signal, and
  • objects R (random) that contains no signal,
\[\begin{split}\xi(r) = \frac{<D, D> - 2 f <D, R> + f^2<R, R>}{f^2<R, R>},\end{split}\]

where the brackets represents counting pairs between two data sets in a finite bin around r (distance), corresponding to setting cumulative=False, and f = float(len(D)) / float(len(R)) is the ratio between number of objects from data and random.

The algorithm implemented here is loosely based on the dual-tree algorithm described in [R349]. We switch between two different pair-cumulation scheme depending on the setting of cumulative. The computing time of the method we use when for cumulative == False does not scale with the total number of bins. The algorithm for cumulative == True scales linearly with the number of bins, though it is slightly faster when only 1 or 2 bins are used. [R353].

As an extension to the naive pair-counting, weighted pair-counting counts the product of weights instead of number of pairs. Weighted pair-counting is used to estimate marked correlation functions ([R351], section 2.2), or to properly calculate the average of data per distance bin (e.g. [R352], section 2.1 on redshift).

[R349](1, 2) Gray and Moore, “N-body problems in statistical learning”, Mining the sky, 2000,
[R350]Landy and Szalay, “Bias and variance of angular correlation functions”, The Astrophysical Journal, 1993,
[R351]Sheth, Connolly and Skibba, “Marked correlations in galaxy formation models”, Arxiv e-print, 2005,
[R352]Hawkins, et al., “The 2dF Galaxy Redshift Survey: correlation functions, peculiar velocities and the matter density of the Universe”, Monthly Notices of the Royal Astronomical Society, 2002,