The aim of this paper is to develop a general regression framework for the analysis of manifold-valued response in a Riemannian symmetric space (RSS) and its association with multiple covariates of interest, such as age or gender, in Euclidean space. Such RSS-valued data arises frequently in medical imaging, surface modeling, and computer vision, among many others. We develop an intrinsic regression model solely based on an intrinsic conditional moment assumption, avoiding specifying any parametric distribution in RSS. We propose various link functions to map from the Euclidean space of multiple covariates to the RSS of responses. We develop a two-stage procedure to calculate the parameter estimates and determine their asymptotic distributions. We construct the Wald and geodesic test statistics to test hypotheses of unknown parameters. We systematically investigate the geometric invariant property of these estimates and test statistics. Simulation studies and a real data analysis are used to evaluate the finite sample properties of our methods.

中文翻译：

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黎曼对称空间上的回归模型。

本文的目的是开发一个通用回归框架，用于分析黎曼对称空间（RSS）中的流形值响应以及其与欧氏空间中感兴趣的多个协变量（例如年龄或性别）的关联。此类RSS值数据经常出现在医学成像，表面建模和计算机视觉等方面。我们仅基于内在条件矩假设开发内在回归模型，避免在RSS中指定任何参数分布。我们提出了各种链接函数，以从多个协变量的欧几里得空间映射到响应的RSS。我们开发了一个两阶段程序来计算参数估计并确定其渐近分布。我们构造Wald和测地线检验统计量来检验未知参数的假设。我们系统地研究了这些估计值和检验统计量的几何不变性。仿真研究和真实数据分析用于评估我们方法的有限样本属性。