  Persistent homology, while ostensibly measuring changes in topology, captures
multiscale geometrical information. It is a natural tool for the analysis of
point patterns. In this paper we explore the statistical power of the
(persistent homology) rank functions. For a point pattern $X$ we construct a
filtration of spaces by taking the union of balls of radius $a$ centered on
points in $X$, $X_a = \cup_{x\in X}B(x,a)$. The rank function
${\beta}_k(X):{\{(a,b)\in \mathbb{R}^2: a\leq b\}} \to \mathbb{R}$ is then
defined by ${\beta}_k(X)(a,b) = rank ( \iota_*:H_k(X_a) \to H_k(X_b))$ where
$\iota_*$ is the induced map on homology from the inclusion map on spaces. We
consider the rank functions as lying in a Hilbert space and show that under
reasonable conditions the rank functions from multiple simulations or
experiments will lie in an affine subspace. This enables us to perform
functional principal component analysis which we apply to experimental data
from colloids at different effective temperatures and of sphere packings with
different volume fractions. We also investigate the potential of rank functions
in providing a test of complete spatial randomness of 2D point patterns using
the distances to an empirically computed mean rank function of binomial point
patterns in the unit square.
