Topological features of multivariate distributions: dependency on the covariance matrix. (English) Zbl 1478.62388
Summary: Topological data analysis provides a new perspective on many problems in the domain of complex systems. Here, we establish the dependency of mean values of functional \(p\)-norms of ‘persistence landscapes’ on a uniform scaling of the underlying multivariate distribution. Furthermore, we demonstrate that average values of \(p\)-norms are decreasing, when the covariance in a system is increasing. To illustrate the complex dependency of these topological features on changes in the variance-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.
MSC:
| 62R40 | Topological data analysis |
| 55N31 | Persistent homology and applications, topological data analysis |
| 62H10 | Multivariate distribution of statistics |
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
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