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We start with a simple introduction to topological data analysis where themost popular tool is called a persistent diagram. Briefly, a persistent diagramis a multiset of points in the plane describing the persistence of topologicalfeatures of a compact set when a scale parameter varies. Since statisticalmethods are difficult to apply directly on persistence diagrams, variousalternative functional summary statistics have been suggested, but either theydo not contain the full information of the persistence diagram or they aretwo-dimensional functions. We suggest a new functional summary statistic thatis one-dimensional and hence easier to handle, and which under mild conditionscontains the full information of the persistence diagram. Its usefulness isillustrated in statistical settings concerned with point clouds and brainartery trees. The appendix includes additional methods and examples, togetherwith technical details. The R-code used for all examples is available athttp://people.math.aau.dk/~christophe/Rcode.zip.

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