The application of deep neural networks in geospatial data has become a trending research problem in the present day. A significant amount of statistical research has already been introduced, such as generalized least square optimization by incorporating spatial variance-covariance matrix, considering basis functions in the input nodes of the neural networks, and so on. However, for lattice data, there is no available literature about the utilization of asymptotic analysis of neural networks in regression for spatial data. This article proposes a consistent localized two-layer deep neural network-based regression for spatial data. We have proved the consistency of this deep neural network for bounded and unbounded spatial domains under a fixed sampling design of mixed-increasing spatial regions. We have proved that its asymptotic convergence rate is faster than that of \cite{zhan2024neural}'s neural network and an improved generalization of \cite{shen2023asymptotic}'s neural network structure. We empirically observe the rate of convergence of discrepancy measures between the empirical probability distribution of observed and predicted data, which will become faster for a less smooth spatial surface. We have applied our asymptotic analysis of deep neural networks to the estimation of the monthly average temperature of major cities in the USA from its satellite image. This application is an effective showcase of non-linear spatial regression. We demonstrate our methodology with simulated lattice data in various scenarios.

翻译：深度神经网络在地理空间数据中的应用已成为当前的研究热点。已有大量统计学研究被引入，例如通过纳入空间方差-协方差矩阵的广义最小二乘优化、在神经网络输入节点中考虑基函数等。然而，对于格点数据，目前尚无关于神经网络在空间数据回归中的渐近分析应用的文献。本文提出了一种基于局部化双层深度神经网络的空间数据一致性回归方法。我们证明了该深度神经网络在混合增长空间区域的固定抽样设计下，对于有界和无界空间域均具有一致性。我们证明了其渐近收敛速度优于\cite{zhan2024neural}的神经网络，并且是对\cite{shen2023asymptotic}神经网络结构的改进推广。我们通过实验观测了观测数据与预测数据的经验概率分布之间差异度量的收敛速度，发现对于光滑度较低的空间曲面，该收敛速度会更快。我们将深度神经网络的渐近分析应用于美国主要城市月平均温度的卫星图像估计。该应用有效展示了非线性空间回归的实际效果。我们通过多种场景下的模拟格点数据验证了所提方法的有效性。