Bottleneck Profiles and Discrete Prokhorov Metrics for Persistence Diagrams

In topological data analysis (TDA), persistence diagrams have been a succesful tool. To compare them, Wasserstein and Bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in a systematic way by introducing bottleneck profiles. This leads to a notion of discrete Prokhorov metrics for persistence diagrams as a generalization of the Bottleneck distance. They satisfy a stability result and bounds with respect to Wasserstein metrics. We provide algorithms to compute the newly introduced quantities and end with an discussion about experiments.Comment: 31 pages, 12 figures; improved exposition, fixed various inaccuracies, added another experimen

Topology-Driven Goodness-of-Fit Tests in Arbitrary Dimensions

This paper adopts a tool from computational topology, the Euler characteristic curve (ECC) of a sample, to perform one- and two-sample goodness of fit tests, we call TopoTests. The presented tests work for samples in arbitrary dimension, having comparable power to the state of the art tests in the one dimensional case. It is demonstrated that the type I error of TopoTests can be controlled and their type II error vanishes exponentially with increasing sample size. Extensive numerical simulations of TopoTests are conducted to demonstrate their power.Comment: 28 pages, 13 figure

OxCOVID19 Database, a multimodal data repository for better understanding the global impact of COVID-19

Oxford COVID-19 Database (OxCOVID19 Database) is a comprehensive source of information related to the COVID-19 pandemic. This relational database contains time-series data on epidemiology, government responses, mobility, weather and more across time and space for all countries at the national level, and for more than 50 countries at the regional level. It is curated from a variety of (wherever available) official sources. Its purpose is to facilitate the analysis of the spread of SARS-CoV-2 virus and to assess the effects of non-pharmaceutical interventions to reduce the impact of the pandemic. Our database is a freely available, daily updated tool that provides unified and granular information across geographical regions