A Subsampling Based Neural Network for Spatial Data

TL;DR

This paper introduces a localized two-layer deep neural network for spatial data regression, proving its consistency and faster convergence rates, with applications to temperature estimation from satellite images.

Contribution

It proposes a novel neural network model with asymptotic analysis for lattice spatial data, demonstrating improved convergence and generalization over existing methods.

Findings

Proved the consistency of the neural network for bounded and unbounded domains.

Established faster asymptotic convergence rates compared to previous models.

Validated the approach with simulations and real satellite data for temperature estimation.

Abstract

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.