AI 中文总结
该研究提出函数分布随机森林方法，实现函数响应条件分布的非参数估计，在模拟和NHANES数据应用中均表现优于基线方法，可捕捉分布变化与协变量关联。
AI 中文摘要
许多函数数据分析会将随机函数简化为标量摘要或条件均值曲线，当我们希望理解协变量如何影响整个函数响应的分布（包括其形状、时间或变异性）时，这种做法存在局限。我们研究函数结果的条件律估计问题，证明这些对象可在实用的非参数框架中进行估计与评估。为此，我们引入函数分布随机森林（functional distributional random forests），该方法通过训练随机森林以最小化决策树叶节点内基于核的最大均值差异，将每个条件律估计为依赖协变量的采样函数分布。这支持对条件分布的任意泛函进行推断，同时保持预测样本与真实曲线的关联。我们考虑定义在函数空间上的多种核，包括Sobolev核和算子诱导核；还提供了估计量的一致性条件，并开发评分规则以将其与基线估计量进行比较。在模拟实验中，我们的方法能恢复基线方法遗漏的分布变化；在NHANES加速度计数据的应用中，它识别出协变量相关的中位活动曲线及预测离散度的有趣变化。
英文摘要
Many functional data analyses reduce random functions to scalar summaries or conditional mean curves. This is limiting when we wish to understand how covariates affect the distribution of entire functional responses, including their shape, timing, or variability. We study the problem of estimating conditional laws of functional outcomes and show that these objects can be estimated and evaluated in a practical nonparametric framework. To do this, we introduce functional distributional random forests, which estimate each conditional law as a covariate-dependent distribution over sampled functions by training a random forest to minimize a kernel-based maximum mean discrepancy within the leaf nodes of the decision tree. This supports inference on arbitrary functionals of the conditional distribution while keeping predictive samples tied to realistic curves. We consider a variety of kernels defined on function spaces, including Sobolev and operator-induced kernels. We also provide conditions for consistency of our estimator and develop scoring rules for comparing it to baseline estimators. In simulations, our method recovers distributional changes that are missed by baseline methods. In an application to NHANES accelerometer data, it identifies interesting covariate-associated changes in both median activity profiles and predictive dispersion.
