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 o...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/18/3899 |
| _version_ | 1797578965436596224 |
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| author | Hongxia Wang Xiao Jin Jianian Wang Hongxia Hao |
| author_facet | Hongxia Wang Xiao Jin Jianian Wang Hongxia Hao |
| author_sort | Hongxia Wang |
| collection | DOAJ |
| description | 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 (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mi>S</mi><mi>E</mi></mrow></semantics></math></inline-formula>). Simulations and empirical studies demonstrate that the convergence speed of neural network methods is significantly better than that of local linear methods. |
| first_indexed | 2024-03-10T22:30:07Z |
| format | Article |
| id | doaj.art-89f5b7f7ef3248a3b541ea2ea89a7e72 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T22:30:07Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-89f5b7f7ef3248a3b541ea2ea89a7e722023-11-19T11:49:08ZengMDPI AGMathematics2227-73902023-09-011118389910.3390/math11183899Nonparametric Estimation for High-Dimensional Space Models Based on a Deep Neural NetworkHongxia Wang0Xiao Jin1Jianian Wang2Hongxia Hao3School of Statistics and Data Science, Nanjing Audit University, Nanjing 211815, ChinaSchool of Statistics and Data Science, Nanjing Audit University, Nanjing 211815, ChinaSchool of Statistics and Data Science, Nanjing Audit University, Nanjing 211815, ChinaSchool of Statistics and Data Science, Nanjing Audit University, Nanjing 211815, ChinaWith 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 (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mi>S</mi><mi>E</mi></mrow></semantics></math></inline-formula>). Simulations and empirical studies demonstrate that the convergence speed of neural network methods is significantly better than that of local linear methods.https://www.mdpi.com/2227-7390/11/18/3899deep neural networkspatial dependencespatial heterogeneityReLU activation function |
| spellingShingle | Hongxia Wang Xiao Jin Jianian Wang Hongxia Hao Nonparametric Estimation for High-Dimensional Space Models Based on a Deep Neural Network Mathematics deep neural network spatial dependence spatial heterogeneity ReLU activation function |
| title | Nonparametric Estimation for High-Dimensional Space Models Based on a Deep Neural Network |
| title_full | Nonparametric Estimation for High-Dimensional Space Models Based on a Deep Neural Network |
| title_fullStr | Nonparametric Estimation for High-Dimensional Space Models Based on a Deep Neural Network |
| title_full_unstemmed | Nonparametric Estimation for High-Dimensional Space Models Based on a Deep Neural Network |
| title_short | Nonparametric Estimation for High-Dimensional Space Models Based on a Deep Neural Network |
| title_sort | nonparametric estimation for high dimensional space models based on a deep neural network |
| topic | deep neural network spatial dependence spatial heterogeneity ReLU activation function |
| url | https://www.mdpi.com/2227-7390/11/18/3899 |
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