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:我们提出了一个新框架,用于分析捕获点云集合的几何特征的形状描述符。在我们的方法核心是使用度量空间-值随机过程的采样记录来解释数据的产生,这可能是非平稳的。因此,将几何数据分析集成到函数时间序列分析中。我们聚焦于由拓扑数据分析产生的描述符。我们的框架允许自然地综合空间-时间动态、异质采样和收敛速率的研究。此外,我们导出了类Gromov统计空间-值随机过程的完备不变量,并将这些不变量与所谓的球体积过程联系起来。在轻微的依赖条件下,我们假设可能随时间改变概率结构的后续经验版本的弱不变性原理在$D([0,1]\times[0,\mathscr{R}])$中得到证明。最后,我们利用此结果引入了用于拓扑变化的新型检验统计量,这些统计量在平稳假设下在极限情况下是分布自由的。