We propose regression models for curve-valued responses in two or more dimensions, where only the image but not the parametrization of the curves is of interest. Examples of such data are handwritten letters, movement paths or outlines of objects. In the square-root-velocity framework, a parametrization invariant distance for curves is obtained as the quotient space metric with respect to the action of re-parametrization, which is by isometries. With this special case in mind, we discuss the generalization of 'linear' regression to quotient metric spaces more generally, before illustrating the usefulness of our approach for curves modulo re-parametrization. We address the issue of sparsely or irregularly sampled curves by using splines for modeling smooth conditional mean curves. We test this model in simulations and apply it to human hippocampal outlines, obtained 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.

翻译：我们针对二维或更高维曲线值响应提出了回归模型，此类数据中仅关注曲线的图像而非参数化形式。这类数据的典型示例包括手写字母、运动轨迹或物体轮廓。在平方根速度框架中，曲线的参数化不变距离被定义为重参数化（等距变换群作用）下的商空间度量。围绕这一特例，本文首先讨论将"线性"回归推广至一般商度量空间，随后通过重参数化模曲线实例展示本方法的实用性。针对稀疏或不规则采样曲线问题，我们采用样条函数建模平滑条件均值曲线。通过数值模拟验证模型性能后，我们将该方法应用于磁共振成像扫描获得的人脑海马体轮廓数据，用于建模不规则采样海马体形状与年龄、阿尔茨海默病及性别的关联关系。