Nonparametric Estimation for High-Dimensional Space Models Based on a Deep Neural Network

With high dimensionality and dependence in spatial data, traditional parametric methods suffer from the curse of dimensionality problem. The theoretical properties of deep neural network estimation methods for high-dimensional spatial models with dependence and heterogeneity have been investigated only in a few studies. In this paper, we propose a deep neural network with a ReLU activation function to estimate unknown trend components, considering both spatial dependence and heterogeneity. We prove the compatibility of the estimated components under spatial dependence conditions and provide an upper bound for the mean squared error (MSE). Simulations and empirical studies demonstrate that the convergence speed of neural network methods is significantly better than that of local linear methods.