论文标题
随机加权的局部加权局部弗雷奇回归与随机物体
Random Forest Weighted Local Fréchet Regression with Random Objects
论文作者
论文摘要
统计分析越来越多地面临来自度量空间的复杂数据。 Petersen和Müller（2019）建立了Fréchet回归的一般范式，具有复杂的度量空间值响应和欧几里得预测因子。但是，其中的局部方法涉及非参数核平滑，并受到维度的诅咒。为了解决这个问题，我们在本文中提出了一种新型的随机森林加权当地的弗雷切特回归范式。我们方法的主要机制取决于随机森林产生的局部自适应内核。我们的第一种方法将这些权重用作局部平均水平来解决条件的Fréchet平均值，而第二种方法则执行局部线性fréchet回归，都显着改善了现有的Fréchet回归方法。基于无限顺序u过程和无限顺序$ m_ {m_n} $ - 估算器的理论，我们为局部恒定估计器建立了一致性，收敛速度和渐近正态性，这涵盖了当前的大型随机样本理论，其随机森林具有欧几里得响应的特殊情况。数值研究表明，我们方法具有几种常见的响应类型，例如分布函数，对称的正定矩阵和球体数据。我们的提案的实际优点也通过应用于纽约出租车数据和人类死亡率数据的应用来证明。
Statistical analysis is increasingly confronted with complex data from metric spaces. Petersen and Müller (2019) established a general paradigm of Fréchet 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 Fréchet regression paradigm. The main mechanism of our approach relies on a locally adaptive kernel generated by random forests. Our first method uses these weights as the local average to solve the conditional Fréchet mean, while the second method performs local linear Fréchet regression, both significantly improving existing Fréchet regression methods. Based on the theory of infinite order U-processes and infinite order $M_{m_n}$-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 New York taxi data and human mortality data.
