提出一种在流形上预测时间相关曲线的新方法，能更好处理地球磁场数据。
Local Fréchet functional regression in manifolds from time-correlated bivariate curve data
- 基于加权弗雷歇均值，在流形上构建局部线性预测模型。
- 理论证明该方法渐近最优，模拟与真实数据表现优于传统方法。
- 适合需要高精度动态轨迹预测的研究者，如地磁或生物运动分析。
在较弱条件下，推导出响应变量与自变量均取值于可分希尔伯特空间时的最小二乘局部线性弗雷歇曲线预测器。建立了在向量函数的外部L2空间中实现局部线性弗雷歇功能预测器的条件，该空间取值于紧致黎曼流形的时间变化切空间。随后提出了基于加权弗雷歇均值的内在局部线性弗雷歇曲线预测器，并证明其渐近最优性。通过模拟和真实数据应用，分析了两种预测器经验版本在有限样本下的表现，与测地线Nadaraya-Watson型曲线预测器进行对比。真实数据应用中，利用NASA MAGSAT卫星观测的地理纬度和经度随时间的变化，预测地球磁场的时间变化球坐标。
原文摘要 · Abstract (English)
Under mild conditions, a least-squares local linear Fréchet curve predictor is derived for a response and a regressor evaluated in a separable Hilbert space. The conditions that allow the implementation of the local linear Fréchet functional predictor in the ambient L2-space of vector functions, with values in the time-varying tangent space of a compact Riemannian manifold, are established. An intrinsic local linear Fréchet curve predictor on such a manifold is then proposed, based on a weighted Fréchet mean approach. Its asymptotic optimality is proved. Simulations and a real-data application are considered to analyze the finite-sample performance of the empirical versions of both predictors, compared with a geodesic Nadaraya-Watson-type curve predictor. In the real-data application, the functional prediction of the time-varying spherical coordinates of the Earth's magnetic field is addressed usingobservations through time of the geocentric latitude and longitude of the NASA MAGSAT spacecraft.
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