We introduce a new framework to analyze shape descriptors that capture the geometric features of an ensemble of point clouds. At the core of our approach is the point of view that the data arises as sampled recordings from a metric space-valued stochastic process, possibly of nonstationary nature, thereby integrating geometric data analysis into the realm of functional time series analysis. We focus on the descriptors coming from topological data analysis. Our framework allows for natural incorporation of spatial-temporal dynamics, heterogeneous sampling, and the study of convergence rates. Further, we derive complete invariants for classes of metric space-valued stochastic processes in the spirit of Gromov, and relate these invariants to so-called ball volume processes. Under mild dependence conditions, a weak invariance principle in $D([0,1]\times [0,\mathscr{R}])$ is established for sequential empirical versions of the latter, assuming the probabilistic structure possibly changes over time. Finally, we use this result to introduce novel test statistics for topological change, which are distribution free in the limit under the hypothesis of stationarity.

翻译：我们提出了一种新的框架，用于分析能够捕捉点云集合几何特征的形状描述符。该方法的核心视角基于数据源自度量空间值随机过程的采样记录（可能具有非平稳性质），从而将几何数据分析融入函数型时间序列分析领域。我们重点关注拓扑数据分析中的描述符。该框架能够自然地纳入时空动力学、异质性采样以及收敛速度研究。此外，我们遵循格罗莫夫思想，推导了度量空间值随机过程类的完备不变量，并将这些不变量与所谓的球体积过程相关联。在温和的依赖性条件下，对于这些过程的序列经验版本，建立了在 $D([0,1]\times [0,\mathscr{R}])$ 空间中的弱不变原理，同时假设概率结构可能随时间变化。最后，我们利用该结果引入新的拓扑变化检验统计量，该统计量在平稳性假设下具有渐近分布无关性。