论文标题
形状的随机性和对形状的统计推断通过光滑的Euler特性变换
Randomness of Shapes and Statistical Inference on Shapes via the Smooth Euler Characteristic Transform
论文作者
论文摘要
在本文中，我们建立了数学基础，用于建模形状的随机性并使用光滑的Euler特征转换对形状进行统计推断。基于这些基础，我们提出了两种基于卡方的基于统计的算法，用于在随机形状上测试假设。提出了模拟研究以验证我们的数学推导，并将我们的算法与最先进的方法进行比较，以证明我们提出的框架的实用性。作为实际应用，我们分析了来自四个灵长类动物属的下颌磨牙的数据集，并表明我们的算法有能力检测显着的形状差异，从而概括了各个下属的已知形态变化。总而言之，我们的讨论桥接了以下领域：代数和计算拓扑，概率理论和随机过程，Sobolev空间和功能分析，功能数据方差分析以及几何形态图。
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 data set 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.
