  Persistent homology is used to track the appearance and disappearance of
features as we move through a nested sequence of topological spaces. Equating
the nested sequence to a filtration and the appearance and disappearance of
features to events, we show that simple event history methods can be used for
the analysis of topological data. We propose a version of the well known
Nelson-Aalen cumulative hazard estimator for the comparison of topological
features of random fields and for testing parametric assumptions. We suggest a
Cox proportional hazards approach for the analysis of embedded metric trees.
The Nelson-Aalen method is illustrated on globally distributed climate data and
on neutral hydrogen distribution in the Milky Way. The Cox method is use to
compare vascular patterns in fundus images of the eyes of healthy and diabetic
retinopathy patients.
