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.

拓扑数据分析为复杂系统领域的许多问题提供了新的视角。在这里，我们建立了函数平均值的依赖关系$磷$-基础多元分布的统一标度上的“持久性景观”规范。此外，我们证明了平均值$磷$- 当系统中的协方差增加时，范数减少。为了说明这些拓扑特征对方差-协方差矩阵变化的复杂依赖性，我们利用具有已知统计特性的双变量分布进行数值实验。我们的结果有助于解释令人费解的行为$磷$- 从 COVID-19 大流行引起的全球金融危机开始阶段欧洲和美国市场主要股票指数的每日对数回报得出的规范。