Model-based Fréchet Regression in (quotient) Metric Spaces with a Focus on Elastic Curves

We introduce model-based Fr & eacute;chet regression in metric spaces. Instead of starting from point-wise conditional Fr & eacute;chet 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 Fr & eacute;chet 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.