Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics
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The present contribution investigates multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. Existing limit theorems for topological statistics prove difficult to use in practice for the construction of confidence intervals, motivating the use of the bootstrap in this capacity. However, the standard nonparametric bootstrap does not provide for asymptotically valid confidence intervals in some situations. The smoothed bootstrap, instead, is shown to give consistent estimation where the standard bootstrap fails. The present work relates to other general results in the area of stabilizing statistics, including central limit theorems for functionals of Poisson and Binomial processes in the critical regime. Specific statistics considered include the persistent Betti numbers of Ĉech and Vietoris-Rips complexes over point sets in R^d, along with Euler characteristics, and minimum spanning trees. We further define a new type of B-bounded persistent homology, and investigate its fundamental properties. A simulation study is provided to assess the performance of the bootstrap for finite sample sizes. Data application is made to a cosmic web dataset from the Sloan Digital Sky Survey (SDSS).
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