Topological Features of Multivariate Distributions: Dependency on the Covariance Matrix

Abstract

Topological data analysis provides a new perspective on many problems in the domain of complex systems. Here, we establish the dependency of the mean value of functional $p$-norms of 'persistence landscapes' on a uniform scaling of the underlying multivariate distribution. Furthermore, we demonstrate that the average value of $p$-norms is decreasing, when the covariance in a system is increasing. To illustrate the complex dependency of these topological features on changes of the covariance matrix, we conduct numerical experiments utilizing bi-variate distributions with known statistical properties. Our results help to explain the puzzling behavior of p-norms derived from daily log-returns of major equity indices on European and US markets at the inception phase of the global financial meltdown caused by the COVID-19 pandemic.