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.

翻译：本文提出了一种基于深度学习的方法，用于对空间依赖数据进行半参数回归分析。具体而言，我们采用带有线性整流单元激活函数的稀疏连接深度神经网络来估计未知的回归函数，该函数描述了在空间依赖性存在下响应变量与协变量之间的关系。在温和条件下，该估计量被证明具有一致性，且收敛速率由三个因素决定：(1)神经网络类别的架构，(2)真实均值函数的平滑性与(内在)维度，以及(3)空间依赖性的强度。我们的方法得益于随机梯度下降优化算法，能够有效处理大规模数据集。通过合成数据的模拟研究评估了有限样本表现，结果表明所提方法能够捕捉响应变量与协变量之间的复杂关系。最后，通过真实数据分析验证了该方法的有效性和可行性。