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

Author

Listed:
  • Hongxia Wang

    (School of Statistics and Data Science, Nanjing Audit University, Nanjing 211815, China)

  • Xiao Jin

    (School of Statistics and Data Science, Nanjing Audit University, Nanjing 211815, China)

  • Jianian Wang

    (School of Statistics and Data Science, Nanjing Audit University, Nanjing 211815, China)

  • Hongxia Hao

    (School of Statistics and Data Science, Nanjing Audit University, Nanjing 211815, China)

Abstract

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 ( M S E ). Simulations and empirical studies demonstrate that the convergence speed of neural network methods is significantly better than that of local linear methods.

Suggested Citation

  • Hongxia Wang & Xiao Jin & Jianian Wang & Hongxia Hao, 2023. "Nonparametric Estimation for High-Dimensional Space Models Based on a Deep Neural Network," Mathematics, MDPI, vol. 11(18), pages 1-37, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:18:p:3899-:d:1239229
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    References listed on IDEAS

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    1. Bentsen, Lars Ødegaard & Warakagoda, Narada Dilp & Stenbro, Roy & Engelstad, Paal, 2023. "Spatio-temporal wind speed forecasting using graph networks and novel Transformer architectures," Applied Energy, Elsevier, vol. 333(C).
    2. Gérard Biau & Benoît Cadre, 2004. "Nonparametric Spatial Prediction," Statistical Inference for Stochastic Processes, Springer, vol. 7(3), pages 327-349, October.
    3. Michael Hamers & Michael Kohler, 2006. "Nonasymptotic Bounds on the L 2 Error of Neural Network Regression Estimates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(1), pages 131-151, March.
    4. Marc Hallin & Zudi Lu & Lanh T. Tran, 2004. "Local linear spatial regression," ULB Institutional Repository 2013/2131, ULB -- Universite Libre de Bruxelles.
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