Random Forests for Spatially Dependent Data
Random Forests for Spatially Dependent Data
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DOI：
10.1080/01621459.2021.1950003
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发表时间：
2021-08
影响因子：
3.7
通讯作者：
Arkajyoti Saha;Sumanta Basu;A. Datta
中科院分区：
文献类型：
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作者：
Arkajyoti Saha;Sumanta Basu;A. Datta
空间线性混合模型由线性协变量效应和高斯过程（GP）分布空间随机效应组成，被广泛用于地理空间数据分析。我们考虑协变量效应是非线性的情况。随机森林是估计非线性函数的常用方法，但将其应用于空间数据时往往忽略了空间相关性。我们表明，这对射频性能有不利影响。我们提出了一种新的、原理良好的RF扩展RF- gls，用于估计空间混合模型中的非线性协变量效应，其中空间相关性使用GP建模。RF-GLS以与广义最小二乘（GLS）相同的方式从根本上扩展了普通最小二乘（OLS），以适应线性模型中的依赖性。RF成为RF- gls的一个特例，在具有空间相关数据的大量数值实验中，RF- gls在估计和预测方面的性能都大大优于RF- gls。RF-GLS可用于其他类型的相关数据（如时间序列）的函数估计。我们证明了RF-GLS在β-混合相关误差过程中的一致性，其中包括流行的空间matsamrn GP。作为一个副产品，据我们所知，我们还建立了依赖下RF的第一个一致性结果。我们建立了独立重要性的结果，包括数据驱动函数类的GLS优化器的一般一致性结果，以及在较弱假设下β-混合依赖下的大数统一定律。这些新工具可以潜在地用于其他gls风格的估计器在非参数回归中与相关数据的渐近分析。
Abstract Spatial linear mixed-models, consisting of a linear covariate effect and a Gaussian process (GP) distributed spatial random effect, are widely used for analyses of geospatial data. We consider the setting where the covariate effect is nonlinear. Random forests (RF) are popular for estimating nonlinear functions but applications of RF for spatial data have often ignored the spatial correlation. We show that this impacts the performance of RF adversely. We propose RF-GLS, a novel and well-principled extension of RF, for estimating nonlinear covariate effects in spatial mixed models where the spatial correlation is modeled using GP. RF-GLS extends RF in the same way generalized least squares (GLS) fundamentally extends ordinary least squares (OLS) to accommodate for dependence in linear models. RF becomes a special case of RF-GLS, and is substantially outperformed by RF-GLS for both estimation and prediction across extensive numerical experiments with spatially correlated data. RF-GLS can be used for functional estimation in other types of dependent data like time series. We prove consistency of RF-GLS for β-mixing dependent error processes that include the popular spatial Matérn GP. As a byproduct, we also establish, to our knowledge, the first consistency result for RF under dependence. We establish results of independent importance, including a general consistency result of GLS optimizers of data-driven function classes, and a uniform law of large number under β-mixing dependence with weaker assumptions. These new tools can be potentially useful for asymptotic analysis of other GLS-style estimators in nonparametric regression with dependent data.