Flexible regression methods for curve and shape data

曲线和形状数据的灵活回归方法

• 批准号: 431707411
• 负责人: Professorin Dr. Sonja Greven
• 金额: --
• 依托单位: Lehrstuhl für Statistik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2020
• 资助国家: 德国
• 起止时间: 2019-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/431707411?language=en
• 关键词: Flexible; regression; methods; curve; shape

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.

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的曲线。例如运动模式和骨骼轮廓。在某些设置中，这些可以被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，该等价类解释了沿着曲线的平移、旋转、缩放和重新参数化的不变性。这在所得的商空间（形状空间）上诱导出非欧几里德几何。这个项目的目标是推进功能形状分析领域的理论和真实的数据分析问题的有用的适用方法。特别是，一个通用的和灵活的回归框架曲线和形状在两个从函数数据的加性模型到多变量函数数据和函数形状数据的加性模型，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重新参数化;根据特定数据场景中的需要，内在地和模块化地考虑关于重新参数化、平移、旋转和/或缩放的不变性的不同组合;允许曲线和形状被不规则地或稀疏地采样，或者作为形状的集合被观察;并且包括各种加性协变量效应类型，包括线性、非线性和随机效应。此外，将开发曲线上标量和形状回归的适当效应。虽然建立在可解释的线性和添加剂预测，框架将符合从各自的不变性所产生的空间的内在几何形状。所有开发都将在开源软件R中实现，并应用于合作项目。总体而言，开发的框架将大大扩展曲线和形状分析回归模型的可用性和灵活性。