  Persistent homology is a vital tool for topological data analysis. Previous
work has developed some statistical estimators for characteristics of
collections of persistence diagrams. However, tools that provide statistical
inference for observations that are persistence diagrams are limited.
Specifically, there is a need for tests that can assess the strength of
evidence against a claim that two samples arise from the same population or
process. We propose the use of randomization-style null hypothesis significance
tests (NHST) for these situations. The test is based on a loss function that
comprises pairwise distances between the elements of each sample and all the
elements in the other sample. We use this method to analyze a range of
simulated and experimental data. Through these examples we experimentally
explore the power of the p-values. Our results show that the
randomization-style NHST based on pairwise distances can distinguish between
samples from different processes, which suggests that its use for hypothesis
tests upon persistence diagrams is reasonable. We demonstrate its application
on a real dataset of fMRI data of patients with ADHD.
