Randomness of Shapes and Statistical Inference on Shapes via the Smooth Euler Characteristic Transform

In this article, we establish the mathematical foundations for modeling the randomness of shapes and conducting statistical inference on shapes using the smooth Euler characteristic transform. Based on these foundations, we propose two Chi-squared statistic-based algorithms for testing hypotheses on random shapes. Simulation studies are presented to validate our mathematical derivations and to compare our algorithms with state-of-the-art methods to demonstrate the utility of our proposed framework. As real applications, we analyze a dataset of mandibular molars from four genera of primates and show that our algorithms have the power to detect significant shape differences that recapitulate known morphological variation across suborders. Altogether, our discussions bridge the following fields: algebraic and computational topology, probability theory and stochastic processes, Sobolev spaces and functional analysis, analysis of variance for functional data, and geometric morphometrics. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.本文为基于光滑欧拉特征变换（smooth Euler characteristic transform）对形状随机性进行建模并开展形状统计推断建立了数学基础。基于上述理论基础，我们提出两种基于卡方统计量（Chi-squared statistic）的算法，用于随机形状的假设检验。文中通过仿真实验验证了数学推导的正确性，并将所提算法与当前顶尖方法进行对比，以展示所提出框架的实用价值。作为实际应用案例，我们分析了来自4个灵长类属的下颌臼齿数据集，结果表明所提算法能够检测出显著的形状差异，且该差异可复现已知的跨亚目形态学变异。综上，本文的研究横跨以下领域：代数与计算拓扑学、概率论与随机过程、索伯列夫空间与泛函分析、功能数据方差分析以及几何形态测量学（geometric morphometrics）。本文的在线补充材料包含了可用于复现本研究的标准化材料说明。