随机森林的MDA：不一致性，以及通过Sobol-MDA的实用解决方案
MDA for random forests: inconsistency, and a practical solution via the Sobol-MDA
摘要：
变量重要性度量是分析随机森林黑箱机制的主要工具。虽然平均减少精度（MDA）被广泛接受为最有效的随机森林变量的重要性措施，很少有人知道它的统计特性。事实上，MDA的定义在主要的随机森林软件中各不相同。在本文中，我们的目标是严格分析主要MDA实现的行为。因此，我们数学形式化的各种实现MDA算法，然后建立他们的限制时，样本量增加。这种渐近分析表明，这些MDA版本不同的重要性措施，因为它们收敛到不同的数量。更重要的是，我们将这些限制分解为三个部分：前两项与Sobol指数相关，Sobol指数是协变量对响应方差的贡献的定义明确的度量，广泛用于敏感性分析领域，而第三项则相反，其值随着协变量内的依赖性而增加。因此，我们从理论上证明，MDA不针对正确的数量来检测有影响力的协变量在一个依赖的设置，一个事实，已经注意到实验。为了解决这个问题，我们为随机森林定义了一个新的重要性度量，Sobol-MDA，它修复了原始MDA的缺陷，并一致地估计了在没有给定协变量的情况下重新训练的森林的准确性下降，但具有有效的计算成本。Sobol-MDA在变量选择的模拟和真实数据上的经验表现优于其竞争对手。在R和C++中的开源实现可以在线获得。
英文摘要：
Variable importance measures are the main tools to analyze the black-box mechanisms of random forests. Although the mean decrease accuracy (MDA) is widely accepted as the most efficient variable importance measure for random forests, little is known about its statistical properties. In fact, the definition of MDA varies across the main random forest software. In this article, our objective is to rigorously analyze the behavior of the main MDA implementations. Consequently, we mathematically formalize the various implemented MDA algorithms, and then establish their limits when the sample size increases. This asymptotic analysis reveals that these MDA versions differ as importance measures, since they converge towards different quantities. More importantly, we break down these limits into three components: the first two terms are related to Sobol indices, which are well-defined measures of a covariate contribution to the response variance, widely used in the sensitivity analysis field, as opposed to the third term, whose value increases with dependence within covariates. Thus, we theoretically demonstrate that the MDA does not target the right quantity to detect influential covariates in a dependent setting, a fact that has already been noticed experimentally. To address this issue, we define a new importance measure for random forests, the Sobol-MDA, which fixes the flaws of the original MDA, and consistently estimates the accuracy decrease of the forest retrained without a given covariate, but with an efficient computational cost. The Sobol-MDA empirically outperforms its competitors on both simulated and real data for variable selection. An open source implementation in R and C++ is available online.
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