Local Fr\'echet regression is a nonparametric regression method for metric space valued responses and Euclidean predictors, which can be utilized to obtain estimates of smooth trajectories taking values in general metric spaces from noisy metric space valued random objects. We derive uniform rates of convergence, which so far have eluded theoretical analysis of this method, for both fixed and random target trajectories, where we utilize tools from empirical processes. These results are shown to be widely applicable in metric space valued data analysis. In addition to simulations, we provide two pertinent examples where these results are important: The consistent estimation of the location of properly defined extrema in metric space valued trajectories, which we illustrate with the problem of locating the age of minimum brain connectivity as obtained from fMRI data; Time warping for metric space valued trajectories, illustrated with yearly age-at-death distributions for different countries.

翻译：局部Fr\'echet回归是一种非参数回归法,用于测量空间有价值反应和Euclidean预测器,可用于从噪音的光度空间有价值随机物体中测得一般光度空间数值的光滑轨迹估计值。我们得出统一的趋同率,迄今为止,在固定和随机目标轨道上,我们利用经验过程的工具对这种方法进行理论分析。这些结果被证明广泛适用于空间有价值数据定性分析。除了模拟之外,我们提供了两个重要相关的例子:一致估计在有价值空间有价值空间的光度空间上正确定义的极限体的位置,我们从FMRI数据中说明了确定最小脑连接年龄的问题;用每年不同国家的死亡年龄分布图解,对具有生命价值的基准空间轨迹进行时间扭曲。