Semiparametric Regression for Spatial Data via Deep Learning

TL;DR

This paper introduces a deep learning approach for semiparametric regression on spatial data, demonstrating consistency, convergence, and effectiveness through simulations and real data analysis.

Contribution

It develops a novel neural network-based method for spatial semiparametric regression, with theoretical guarantees and practical validation.

Findings

Method accurately captures complex spatial relationships.

Estimator is proven consistent with known convergence rates.

Effective on large datasets with real-world spatial data.

Abstract

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.