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。该方法的核心在于将数据视为度量空间值随机过程（可能具有非平稳性）的采样记录，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，该框架能够自然地整合时空动态特性、异质性采样以及收敛速度研究。进一步地，我们以Gromov风格推导出度量空间值随机过程类的完全不变量，并将这些不变量与所谓的球体体积过程相关联。在温和依赖条件下，针对后者的序贯经验版本，我们建立了$D([0,1]\times [0,\mathscr{R}])$空间中的弱不变原理，此时概率结构可能随时间变化。最后，利用这一结果，我们引入了用于检测拓扑变化的新型检验统计量，该统计量在平稳性假设下的极限分布中具有无分布特性。