[论文解读] Some asymptotic results of survival tree and forest models

Yifan Cui, Ruoqing Zhu+2 more|arXiv (Cornell University)|Jul 30, 2017

Statistical Methods and Inference被引用 3

一句话总结

本文提出了一种用于生存树和生存随机森林的偏差校正分裂规则，通过在故障分布估计中考虑右删失的影响，提高了模型的一致性和预测准确性。该方法在低维和高维设置下证明了其一致性，收敛速度仅依赖于与故障相关的变量，并通过模拟实验验证了预测误差的降低。

ABSTRACT

Random survival forest and survival trees are popular models in statistics and machine learning. However, there is a lack of general understanding regarding consistency, splitting rules and influence of the censoring mechanism. In this paper, we investigate the statistical properties of existing methods from several interesting perspectives. First, we show that traditional splitting rules with censored outcomes rely on a biased estimation of the within-node failure distribution. To exactly quantify this bias, we develop a concentration bound of the within-node estimation based on non i.i.d. samples and apply it to the entire forest. Second, we analyze the entanglement between the failure and censoring distributions caused by univariate splits, and show that without correcting the bias at an internal node, survival tree and forest models can still enjoy consistency under suitable conditions. In particular, we demonstrate this property under two cases: a finite-dimensional case where the splitting variables and cutting points are chosen randomly, and a high-dimensional case where the covariates are weakly correlated. Our results can also degenerate into an independent covariate setting, which is commonly used in the random forest literature for high-dimensional sparse models. However, it may not be avoidable that the convergence rate depends on the total number of variables in the failure and censoring distributions. Third, we propose a new splitting rule that compares bias-corrected cumulative hazard functions at each internal node. We show that the rate of consistency of this new model depends only on the number of failure variables, which improves from non-bias-corrected versions. We perform simulation studies to confirm that this can substantially benefit the prediction error.

研究动机与目标

• 为解决生存树与森林模型中一致性、分裂规则及删失效应缺乏理论理解的问题。
• 量化传统分裂规则在依赖非独立同分布删失生存数据时产生的偏差。
• 建立在节点估计中未校正偏差的情况下，生存树与森林仍保持一致的条件。
• 提出一种新的分裂规则，以校正内部节点累积风险估计中的偏差。
• 展示收敛速度仅依赖于与故障相关的变量，而非完整的协变量集合。

提出的方法

• 推导在非独立同分布抽样下，节点内故障分布估计的浓度界，同时考虑删失的影响。
• 分析生存树中单变量分裂导致的故障分布与删失分布之间的纠缠关系。
• 提出一种基于每个内部节点处偏差校正累积风险函数的新分裂规则。
• 在协变量弱相关的情况下，建立新模型在低维与高维设置下的理论一致性。
• 将浓度界应用于整个森林，以量化估计误差的传播。
• 通过模拟研究比较偏差校正与非偏差校正模型之间的预测误差。

实验结果

研究问题

• RQ1 删失偏差如何影响生存树与森林模型的一致性？
• RQ2 当分裂规则产生有偏的节点内故障分布估计时，生存树与森林是否仍能保持一致？
• RQ3 在协变量弱相关且高维设置下，确保一致性的条件是什么？
• RQ4 生存森林模型的收敛速度如何依赖于故障变量与删失变量的数量？
• RQ5 偏差校正分裂规则能否降低预测误差并提升收敛速度？

主要发现

• 传统生存树分裂规则由于删失数据的存在，会在节点内故障分布估计中引入偏差。
• 推导出非独立同分布样本的浓度界，并应用于量化整个森林中的估计误差。
• 即使未进行偏差校正，只要分裂变量与切分点在合适条件下选择，生存树与森林仍能保持一致。
• 所提出的偏差校正分裂规则实现了一致性速率，该速率仅依赖于故障相关变量的数量，而不依赖于协变量总数。
• 模拟研究证实，与非偏差校正版本相比，偏差校正模型显著降低了预测误差。
• 理论结果退化为标准随机森林文献中使用的独立协变量设置，验证了在稀疏高维模型中的一致性。