Fr\'echet global regression is extended to the context of bivariate curve stochastic processes with values in a Riemannian manifold. The proposed regression predictor arises as a reformulation of the standard least-squares parametric linear predictor in terms of a weighted Fr\'echet functional mean. Specifically, in our context, in this reformulation, the Euclidean distance is replaced by the integrated quadratic geodesic distance. The regression predictor is then obtained from the weighted Fr\'echet curve mean, lying in the time-varying geodesic submanifold, generated by the regressor process components involved in the time correlation range. The regularized Fr\'echet weights are computed in the time-varying tangent spaces. The uniform weak-consistency of the regression predictor is proved. Model selection is also addressed. A simulation study is undertaken to illustrate the performance of the spherical curve variable selection algorithm proposed in a multivariate framework.

翻译：将 Fréchet 全局回归拓展至取值于黎曼流形的双变量曲线随机过程背景中。提出的回归预测算子通过加权Fréchet泛函均值重新表述标准最小二乘参数线性预测算子。具体而言，在本背景下，该重新表述中用积分二次测地距离替代欧氏距离。随后从加权Fréchet曲线均值中获取回归预测算子，该均值位于由时间相关范围内的回归过程分量生成的时变测地子流形中。正则化Fréchet权重在时变切空间中计算。证明了回归预测算子的均匀弱相合性，并讨论了模型选择问题。通过仿真研究展示了所提出的多元球面曲线变量选择算法的性能。