In this work, we propose a deep learning-based method to perform semiparametric regression analysis for spatially dependent data. To be specific, we use a sparsely connected deep neural network with rectified linear unit (ReLU) activation function to estimate the unknown regression function that describes the relationship between response and covariates in the presence of spatial dependence. Under some mild conditions, the estimator is proven to be consistent, and the rate of convergence is determined by three factors: (1) the architecture of neural network class, (2) the smoothness and (intrinsic) dimension of true mean function, and (3) the magnitude of spatial dependence. Our method can handle well large data set owing to the stochastic gradient descent optimization algorithm. Simulation studies on synthetic data are conducted to assess the finite sample performance, the results of which indicate that the proposed method is capable of picking up the intricate relationship between response and covariates. Finally, a real data analysis is provided to demonstrate the validity and effectiveness of the proposed method.

在这项工作中，我们提出了一种基于深度学习的方法来对空间相关数据进行半参数回归分析。具体来说，我们使用具有修正线性单元（ReLU）激活函数的稀疏连接深度神经网络来估计描述响应和协变量在存在空间依赖性的情况下。在一些温和的条件下，估计器被证明是一致的，并且收敛速度由三个因素决定：（1）神经网络类的架构，（2）真实均值函数的平滑度和（内在）维数，以及(3)空间依赖性的大小。由于随机梯度下降优化算法，我们的方法可以很好地处理大数据集。对合成数据进行模拟研究以评估有限样本性能，其结果表明所提出的方法能够识别响应和协变量之间的复杂关系。最后，提供了真实的数据分析来证明所提出方法的有效性和有效性。