Model-based Fréchet regression in (quotient) metric spaces with a focus on elastic curves
提出一种在度量空间中基于模型的Fréchet回归方法，并应用于商度量空间，处理弹性曲线数据（如手写字母、运动路径），通过模拟和MRI海马轮廓数据验证，用于分析年龄、阿尔茨海默病和性别对海马形状的影响。
We introduce model-based Fréchet regression in metric spaces. Instead of starting from point-wise conditional Fréchet means, our approach is defined as a constrained minimization problem over a model class of functions. The approach is then applied to develop a general framework of regression for quotient metric spaces with distances induced by isometric group actions. Such spaces arise naturally in applications where objects are considered equivalent up to transformations. We first establish general existence and consistency results for model-based Fréchet regression, with our quotient space regression model as a special case. As an important example we consider regression for elastic curves in the square-root velocity framework. This addresses data such as handwritten letters, movement paths, or outlines of objects, where only the image but not the parametrization of the curves is of interest. To handle sparsely or irregularly sampled curves, we model smooth conditional mean curves using splines. We validate our approach through simulations and an application to hippocampal outlines extracted from Magnetic Resonance Imaging scans. Here we model how the shape of the irregularly sampled hippocampus is related to age, Alzheimer’s disease and sex, to disentangle the shrinking effects of Alzheimer’s from normal aging.