Modelling persistence diagrams with planar point processes, and revealing topology with bagplots

Abstract

We introduce a new model for planar point processes, with the aim of capturing the structure of point interaction and spread in persistence diagrams. Persistence diagrams themselves are a key tool of topological data analysis (TDA), crucial for the delineation and estimation of global topological structure in large data sets. To a large extent, the statistical analysis of persistence diagrams has been hindered by difficulties in providing replications, a problem that was addressed in an earlier paper, which introduced a procedure called replicating statistical topology (RST). Here we significantly improve on the power of RST via the introduction of a more realistic class of models for the persistence diagrams. In addition, we introduce to TDA the idea of bagplotting, a powerful technique from non-parametric statistics well adapted for differentiating between topologically significant points, and noise, in persistence diagrams. Outside the setting of TDA, our model provides a setting for fashioning point processes, in any dimension, in which both local interactions between the points, along with global restraints on the overall, global, shape of the point cloud, are important and perhaps competing.