Flexible regression methods for curve and shape data

使用现代成像和跟踪设备，研究人员在广泛的领域收集越来越多的数据，其中每个观察对应于一个二维或更高维度的曲线。例如运动模式和骨骼轮廓。在某些情况下，可以将这些数据视为多变量函数数据。在其他情况下，函数形状主要是感兴趣的，即曲线的等价类，说明沿曲线的平移，旋转，缩放和重新参数化的不变性。这在得到的商空间（形状空间）上推导出非欧几里德几何。该项目的目标是在理论和实际数据分析问题的有效应用方法方面推进功能形状分析领域。特别是，将开发和实现两个（或可能更高）维度的曲线和形状的通用和灵活的回归框架。从功能数据的可加模型依次推广到多变量功能数据和功能形状数据的可加模型，该框架将在以下方面提供前所未有的灵活性：它将允许建模曲线和形状响应模再参数化；根据特定数据场景的需要，从本质上和模块化地考虑有关重新参数化、平移、旋转和/或缩放的不变性的不同组合；允许曲线和形状不规则或稀疏采样，或作为形状的集合观察；并包括各种可加的协变量效应类型，包括线性、非线性和随机效应。此外，对曲线上的标量和形状回归的适当影响将被开发。虽然建立在可解释的线性和可加性预测因子上，但该框架将符合由各自的不变性产生的空间的固有几何形状。所有的开发都将在开源软件R中实现，并应用于协作项目。总的来说，开发的框架将因此大大扩展曲线和形状分析的回归模型的可用性和灵活性。

Using modern imaging and tracking devices, researchers in a wide range of areas collect more and more data, where each observation corresponds to a two- or higher-dimensional curve. Examples are movement patterns and bone outlines. In some settings, these can be viewed as multivariate functional data. In others, the functional shape is primarily of interest, i.e. the equivalence class of the curve accounting for invariance to translation, rotation, scaling and re-parameterization along the curve. This induces a non-Euclidean geometry on the resulting quotient spaces (shape spaces). The goal of this project is to advance the field of functional shape analysis both in terms of the theory and in terms of usefully applicable methods for real data analysis problems. In particular, a general and flexible regression framework for curves and shapes in two (or potentially higher) dimensions will be developed and implemented.Successively generalizing from additive models for functional data to those for multivariate functional data and for functional shape data, this framework will offer unprecedented flexibility in the following respects: it will allow for modeling curve and shape responses modulo re-parameterization; intrinsically and modularly account for different combinations of invariances with respect to re-parameterization, translation, rotation and/or scaling according to the needs in a specific data scenario; allow for curves and shapes to be irregularly or sparsely sampled, or observed as an ensemble of shapes; and include various additive covariate effect types including linear, non-linear and random effects. Additionally, appropriate effects for scalar-on-curve and -shape regression will be developed. While building on interpretable linear and additive predictors, the framework will conform to the intrinsic geometries of the spaces arising from the respective invariances. All developments will be implemented in the open-source software R and applied in collaborative projects. Overall, the developed framework will thus greatly extend the availability and flexibility of regression models for curve and shape analysis.