递归弯曲环面上的投影正态模型

中文摘要 AI 辅助

我们为定义在递归嵌入的弯曲环面 $\mathbb{T}^n\hookrightarrow\mathbb{R}^{n+1}$ 上的角度数据开发了一个计算上可行的统计框架。由于递归结构，我们通过推广弯曲二维环面的过程构建了一个精确的均匀采样器。我们将投影正态分布推广到弯曲 $n$-环面，并为这些分布发展了推断理论。为此，我们利用最大对数似然估计和基于能量的 Fréchet 估计来处理投影正态分布。这是通过建立各向同性投影正态分布的可识别性，并应用弯曲环面的极大似然理论和内在统计来实现的。我们在数值上验证了几何、采样器、密度变换和能量求解器。最后，我们使用真实数据证明了所提出模型的适用性。

英文摘要

We develop a computationally tractable statistical framework for angular data supported on a recursively embedded curved torus $\mathbb{T}^n\hookrightarrow\mathbb{R}^{n+1}$. Due to the recursive structure we build an exact uniform sampler by generalizing the process from the curved 2-torus. We generalize projected normal distributions to the curved $n$-torus and we develop inference theory for these distributions. We utilize both maximum-log-likelihood estimates and energy-based Fréchet estimates for the projected normal distribution for this purpose. This is done by establishing identifiability for isotropic projected normal distributions and by applying theory of maximum likelihood and intrinsic statistics for the curved tori. We numerically verify the geometry, sampler, density transformation, and energy solver. Lastly, we demonstrate the applicability of the proposed model using real data.

发表机构

• Hausdorff Research Institute for Mathematics, University of Bonn（波恩大学豪斯多夫数学研究所）
• Umeå University（于默奥大学）

机构由 AI 辅助整理，请以论文原文为准。