Statistical analysis is increasingly confronted with complex data from metric spaces. Petersen and M\"uller (2019) established a general paradigm of Fr\'echet regression with complex metric space valued responses and Euclidean predictors. However, the local approach therein involves nonparametric kernel smoothing and suffers from the curse of dimensionality. To address this issue, we in this paper propose a novel random forest weighted local Fr\'echet regression paradigm. The main mechanism of our approach relies on a locally adaptive kernel generated by random forests. Our first method utilizes these weights as the local average to solve the conditional Fr\'echet mean, while the second method performs local linear Fr\'echet regression, both significantly improving existing Fr\'echet regression methods. Based on the theory of infinite order U-processes and infinite order Mmn -estimator, we establish the consistency, rate of convergence, and asymptotic normality for our local constant estimator, which covers the current large sample theory of random forests with Euclidean responses as a special case. Numerical studies show the superiority of our methods with several commonly encountered types of responses such as distribution functions, symmetric positive-definite matrices, and sphere data. The practical merits of our proposals are also demonstrated through the application to human mortality distribution data.
翻译:Petersen 和 M\'uller (2019年) 建立了Fr\'echet 回归的总范式, 并配有复杂的衡量空间价值的响应和 Euclidean 预测器。 但是, 当地的方法涉及非对等内核平滑, 并受到维度诅咒的影响。 为了解决这个问题, 我们本文提出一个新的随机森林加权本地Fr\'echet 回归范式。 我们方法的主要机制依赖于随机森林产生的本地适应性内核。 我们的第一种方法利用这些权重作为当地平均值来解决条件的Fr\'echet mean, 而第二种方法则进行局部线性Fr\'echet回归, 两者都大大改进了现有的Fr\'echet回归法。 根据无限顺序U- proaches和无限顺序 Mmn-simantator的理论, 我们为本地恒定的估算器设定了一致性、趋同率、 随机森林和Euclideidean 反应作为特例的样本理论。 Numericalal 研究也展示了我们实际数据分布方式的优势。