论文标题
3D功能数据的曲率和扭转估计：在Frenet Serret框架下建立平均形状的几何方法
Curvature and Torsion estimation of 3D functional data: A geometric approach to build the mean shape under the Frenet Serret framework
论文作者
论文摘要
曲线的分析常规使用了使用功能数据分析中的工具来处理。但是，由于其固有的几何特征很难使用依赖线性近似的经典方法来捕获，因此其扩展到多维曲线构成了新的挑战。我们开发出反映曲线形状变化的平均值的替代表征。基于曲线通过FRENET-SERRET普通微分方程的几何表示，我们引入了平均曲率和平均扭转的新定义，以及通过平均向量场概念的平均形状。多维曲线的平均值的新公式使我们能够将形状特征的参数集成到统一的功能数据建模框架中。我们在惩罚回归中提出功能参数的估计问题，并开发有效的算法。我们通过模拟数据和实际数据示例证明了我们的方法。
The analysis of curves has been routinely dealt with using tools from functional data analysis. However its extension to multi-dimensional curves poses a new challenge due to its inherent geometric features that are difficult to capture with the classical approaches that rely on linear approximations. We develop an alternative characterization of a mean that reflects shape variation of the curves. Based on a geometric representation of the curves through the Frenet-Serret ordinary differential equations, we introduce a new definition of mean curvature and mean torsion, as well as mean shape through the notion of mean vector field. This new formulation of the mean for multi-dimensional curves allows us to integrate the parameters for the shape features into the unified functional data modelling framework. We formulate the estimation problem of the functional parameters in a penalized regression and develop an efficient algorithm. We demonstrate our approach with both simulated data and real data examples.
