Statistical Topology and the Random Interstellar Medium

Topological data analysis is used for the analysis of data Z(s) indexed by a location s in some space, for example an image or a random field. A key element is the notion of persistent homology: how features change as we filter in some way.  Often the filtration is based on level, t say, and features are components, holes, loops and so on that are apparent in level sets made up of the locations of all points with values Z(s)>t.

Equating level with time (or log time) and the appearance and disappearance of features as births and deaths, then an alternative interpretation of persistent homology is as an event history process.  We explore the use of familiar methods in survival and event history analysis in the analysis of hydrogen concentration in the interstellar medium. We use topological event history methods to investigate the small-scale variation and local spatial characteristics  in three regions of the southern sky. We demonstrate that there are circumstances where topological methods can identify differences in distributions when conventional marginal or correlation analyses may not.