In spatial statistics, a common objective is to predict the values of a spatial process at unobserved locations by exploiting spatial dependence. In geostatistics, Kriging provides the best linear unbiased predictor using covariance functions and is often associated with Gaussian processes. However, when considering non-linear prediction for non-Gaussian and categorical data, the Kriging prediction is not necessarily optimal, and the associated variance is often overly optimistic. We propose to use deep neural networks (DNNs) for spatial prediction. Although DNNs are widely used for general classification and prediction, they have not been studied thoroughly for data with spatial dependence. In this work, we propose a novel neural network structure for spatial prediction by adding an embedding layer of spatial coordinates with basis functions. We show in theory that the proposed DeepKriging method has multiple advantages over Kriging and classical DNNs only with spatial coordinates as features. We also provide density prediction for uncertainty quantification without any distributional assumption and apply the method to PM$_{2.5}$ concentrations across the continental United States.

翻译：在空间统计中,一个共同目标是通过利用空间依赖性来预测未观测地点的空间过程值。在地理统计学中,克里金利用共同变量功能提供最佳线性无偏向预测器,而且往往与高斯进程相关。然而,在考虑非高加索和绝对数据的非线性预测时,克里金预测不一定是最佳的,相关差异往往过于乐观。我们提议使用深神经网络进行空间预测。虽然DNN广泛用于一般分类和预测,但并未对具有空间依赖性的数据进行彻底研究。在这项工作中,我们提出一个新的空间预测线性网络结构,在基础功能中增加一个空间坐标嵌入层。我们从理论上表明,拟议的DeepKriging方法具有多种优势,仅比克里金和古典DNNW具有空间坐标特征的多重优势。我们还提出在不作任何分配假设的情况下进行不确定性量化的密度预测,并将这种方法应用于整个美国大陆的PM$2.5}的浓度。