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 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：Fr\'echet全局回归被推广到取值为黎曼流形的双变量曲线随机过程的情境中。所提出的回归预测器源于标准最小二乘参数线性预测器在加权Fr\'echet泛函均值框架下的重构。具体而言，在我们的设定中，该重构将欧几里得距离替换为积分二次测地距离。随后，回归预测器通过加权Fr\'echet曲线均值获得，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。正则化的Fr\'echet权重在时变切空间中计算。回归预测器的弱一致性得到证明。同时讨论了模型选择问题。通过模拟研究，验证了本文提出的多元框架下球面曲线变量选择算法的性能。