Statistical Learning of High-Dimensional Spatial Dependence Structures

该项目处理空间和时空统计的一个重要的、基本的问题——对潜在的空间依赖结构的全面估计。对于这些模型，迄今为止的重点是在条件手段中显示依赖性的过程。也就是说，随机过程在特定测量点的实现的均值取决于相邻的观测值。这一发现可以追溯到托布勒地理第一定律。周围的观测值是根据它们的地理邻近度来定义的，尽管这并不一定导致随机变量观测值的依赖性，即协方差。地统计模型和空间计量模型都假定了一定的空间依赖性结构，然而，这种结构通常是未知的。因此，在这个项目中，将开发新的统计方法，允许对空间依赖结构进行完整的估计。为此，将使用机器/统计学习方法。除了在条件均值中具有（自回归）依赖性的经典模型外，还将研究在条件方差中具有依赖性的模型。这些模式就是所谓的空间ARCH过程——类似于Robert F. Engle（1982）的时间ARCH模式。最后，将使用各种应用示例来演示如何解释估计的参数。在这里，重点将放在环境中的自然过程上，比如空气污染或颗粒物。利用可免费获得的传感器数据，结果可用于，例如，获得城市地区细尘污染的当地预测，然后可用于关于空气质量的最佳路线。

The project deals with an important, fundamental problem of spatial and spatiotemporal statistics – the full estimation of the underlying spatial dependence structure. For these models, the focus has so far been on processes showing a dependence in the conditional means. That is, the mean of a realization of the random process at a particular measurement point depends on the adjacent observations. This finding goes back to the Tobler’s first law of Geography. The surrounding observations are defined on the basis of their geographical proximity, although this does not necessarily lead to a dependence of the observations of the random variables, i.e. the covariances.Both geostatistical and spatial econometric models assume a certain structure of the spatial dependence, which, however, is typically unknown. In this project, therefore, new statistical methods will be developed that allow the complete estimation of the spatial dependence structure. For this purpose, machine/statistical learning methods will be used.Besides classical models with (autoregressive) dependencies in the conditional means, models with dependencies in conditional variances will be investigated. These models are so-called spatial ARCH processes - analogous to the temporal ARCH model of Robert F. Engle (1982).Finally, various application examples will be used to demonstrate how the estimated parameters can be interpreted. Here, the focus will be on natural processes in the environment, such as air pollution or particulate matter. Using freely available sensor data, the results can be used, for example, to obtain local predictions of fine dust pollution in an urban area, which can then be used for optimal routing with respect to air quality.