There has been increasing interest in statistical analysis of data lying in manifolds. This paper generalizes a smoothing spline fitting method to Riemannian manifold data based on the technique of unrolling, unwrapping and wrapping originally proposed by Jupp and Kent for spherical data. In particular, we develop such a fitting procedure for shapes of configurations in general m‐dimensional Euclidean space, extending our previous work for two‐dimensional shapes. We show that parallel transport along a geodesic on Kendall shape space is linked to the solution of a homogeneous first‐order differential equation, some of whose coefficients are implicitly defined functions. This finding enables us to approximate the procedure of unrolling and unwrapping by simultaneously solving such equations numerically, and so to find numerical solutions for smoothing splines fitted to higher dimensional shape data. This fitting method is applied to the analysis of some dynamic 3D peptide data.

中文翻译：

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黎曼流形上的花键平滑处理，并应用于3D形状空间

人们对流形数据的统计分析越来越感兴趣。本文基于Jupp和Kent最初提出的针对球面数据的展开，展开和包装技术，将一种平滑样条拟合方法推广到黎曼流形数据。特别是，我们开发这样的拟合程序在一般配置的形状米维欧几里得空间，扩展了我们先前对二维形状的工作。我们表明，沿Kendall形状空间上的测地线的平行传输与齐次一阶微分方程的解相关，其中一些系数是隐式定义的函数。这一发现使我们能够通过同时数值求解此类方程来近似展开和展开的过程，从而找到用于拟合高维形状数据的花键平滑的数值解。这种拟合方法适用于一些动态3D肽数据的分析。