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A Subsampling Based Neural Network for Spatial Data

Debjoy Thakur

TL;DR

A consistent localized two-layer deep neural network-based regression for spatial data using asymptotic analysis of deep neural networks for lattice data and proves the consistency of this deep neural network for bounded and unbounded spatial domains under a fixed sampling design of mixed-increasing spatial regions.

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.

A Subsampling Based Neural Network for Spatial Data

TL;DR

A consistent localized two-layer deep neural network-based regression for spatial data using asymptotic analysis of deep neural networks for lattice data and proves the consistency of this deep neural network for bounded and unbounded spatial domains under a fixed sampling design of mixed-increasing spatial regions.

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.

Paper Structure

This paper contains 15 sections, 4 theorems, 31 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Localized DNN
  4. Spatial Sampling Regions
  5. Spatial DNN
  6. Asymptotic Analysis
  7. Asymptotic Convergence
  8. Uncertainity Quantification
  9. Results and Discussion
  10. Consistency for Unbounded Spatial Domain: Simulation Study
  11. Consistency for Bounded Spatial Domain: Satellite Image
  12. Conclusion
  13. Proof of Theorem \ref{['thm:existance']}
  14. Proof of Theorem \ref{['thm:convergence']}
  15. Proof of Theorem \ref{['thm:conv_rate']}

Key Result

Theorem 1

[Existence] Let $(\Omega, \mathcal{A}, P)$ be a complete probability space and let $(\mathcal{F}, \rho)$ be a pseudo-metric. Let $\{\mathcal{F}_{r,n}\}$ be a sequence of compact subsets of $\mathcal{F}$. Let $\mathbb{L}_{n}:\Omega \times \mathcal{F}_{r,n} \to \Bar{\mathbb{R}}$ be $\mathcal{A} \otime

Figures (6)

  • Figure 1: Flowchart of Localized DNN Estimator
  • Figure 2: 95$\%$ CI of localized 2-DNN functional.
  • Figure 3: Asymptotic convergence of KL divergence with increasing $\delta_n$.
  • Figure 4: Variation of KL Divergence with $\kappa$ values.
  • Figure 5: Satellite images with pixel spacing, $\eta_n = 0.25$ and the dotted points are indicating monthly (January) average of 1982 of SKT of cities in USA.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 2.1
  • Proof 2.1
  • proof
  • proof
  • ...and 1 more