The accumulated persistence function, a new useful functional summary statistic for topological data analysis, with a view to brain artery trees and spatial point process applications

TL;DR

This paper introduces the accumulated persistence function, a new one-dimensional functional summary statistic for topological data analysis that retains full information of persistence diagrams, demonstrated on brain artery trees and spatial point processes.

Contribution

It proposes a novel one-dimensional functional summary statistic that captures all information of persistence diagrams, simplifying statistical analysis in topological data analysis.

Findings

The accumulated persistence function effectively summarizes topological features.

It is applicable to brain artery trees and spatial point processes.

The method retains full information of persistence diagrams.

Abstract

We start with a simple introduction to topological data analysis where the most popular tool is called a persistent diagram. Briefly, a persistent diagram is a multiset of points in the plane describing the persistence of topological features of a compact set when a scale parameter varies. Since statistical methods are difficult to apply directly on persistence diagrams, various alternative functional summary statistics have been suggested, but either they do not contain the full information of the persistence diagram or they are two-dimensional functions. We suggest a new functional summary statistic that is one-dimensional and hence easier to handle, and which under mild conditions contains the full information of the persistence diagram. Its usefulness is illustrated in statistical settings concerned with point clouds and brain artery trees. The appendix includes additional methods and examples, together with technical details. The R-code used for all examples is available at http://people.math.aau.dk/~christophe/Rcode.zip.