  How can we tell complex point clouds with different small scale
characteristics apart, while disregarding global features? Can we find a
suitable transformation of such data in a way that allows to discriminate
between differences in this sense with statistical guarantees? In this paper,
we consider the analysis and classification of complex point clouds as they are
obtained, e.g., via single molecule localization microscopy. We focus on the
task of identifying differences between noisy point clouds based on small scale
characteristics, while disregarding large scale information such as overall
size. We propose an approach based on a transformation of the data via the
so-called Distance-to-Measure (DTM) function, a transformation which is based
on the average of nearest neighbor distances. For each data set, we estimate
the probability density of average local distances of all data points and use
the estimated densities for classification. While the applicability is
immediate and the practical performance of the proposed methodology is very
good, the theoretical study of the density estimators is quite challenging, as
they are based on i.i.d. observations that have been obtained via a complicated
transformation. In fact, the transformed data are stochastically dependent in a
non-local way that is not captured by commonly considered dependence measures.
Nonetheless, we show that the asymptotic behaviour of the density estimator is
driven by a kernel density estimator of certain i.i.d. random variables by
using theoretical properties of U-statistics, which allows to handle the
dependencies via a Hoeffding decomposition. We show via a numerical study and
in an application to simulated single molecule localization microscopy data of
chromatin fibers that unsupervised classification tasks based on estimated
DTM-densities achieve excellent separation results.
