The accumulative persistence function -- a useful functional summary statistic for topological data analysis

We introduce a new functional summary statistic called the accumulative persistence function (APF) and having several attractive properties: It is a one-dimensional function easier to handle than the two-dimensional functions usually considered in persistence homology; for example, confidence regions are easier to plot and more visually appealing for a one-dimensional function than for a two-dimensional function; often, at least with probability one, there will be a one-to-one correspondence between the APF and the persistent diagram (for each fixed dimension) the APF is a natural way of constructing a monotonic function, and this will ease the proof of e.g. convergence theorems; contrary to the so-called dominant landscape function λ_1 or the silhouette, the APF provides information about topological features without distinguishing between long and short lifetimes. For instance, for application in spatial statistics, short lifetime topological features are relevant. In the talk we focus on extreme rank envelopes, a useful concept to make goodness-of-fit test associated to a confidence region for the APF, while other applications will be briefly discussed, including functional boxplots for APFs, the confidence region for the mean of APFs, and comparing groups of persistence diagrams: the two sample problem, clustering, and supervised classification.