A statistical framework for analyzing shape in a time series of random geometric objects

Music City Center 

We introduce a new framework to analyze shape descriptors that capture
the geometric features of an ensemble of point clouds. At the core of
our approach is the point of view that the data arises as sampled
recordings from a metric space-valued stochastic process, possibly of
nonstationary nature, thereby integrating geometric data analysis into
the realm of functional time series analysis. Our framework allows
for natural incorporation of spatial-temporal dynamics, heterogeneous
sampling, and the study of convergence rates. Further, we derive
complete invariants for classes of metric space-valued stochastic
processes in the spirit of Gromov, and relate these invariants to
so-called ball volume processes. Under mild dependence conditions, a
weak invariance principle in $D([0,1]\times [0,\mathscr{R}])$ is
established for sequential empirical versions of the latter, assuming
the probabilistic structure possibly changes over time. Finally, we
use this result to introduce novel test statistics for topological
change, which are distribution-free in the limit under the hypothesis
of stationarity. We explore these test statistics on time series of
single-cell mRNA expression data, using shape descriptors coming from
topological data analysis.

topological data analysis


functional data analysis


persistent homology


locally stationary processes


U-statistics