{"title":"用于插值和概率预测的时空深度克里格","authors":"Pratik Nag , Ying Sun , Brian J. Reich","doi":"10.1016/j.spasta.2023.100773","DOIUrl":null,"url":null,"abstract":"<div><p><span><span><span><span>Gaussian processes (GP) and Kriging are widely used in traditional spatio-temporal modelling and prediction. These techniques typically presuppose that the data are observed from a </span>stationary GP<span> with a parametric<span> covariance structure<span>. However, processes in real-world applications often exhibit non-Gaussianity and nonstationarity. Moreover, likelihood-based inference for GPs is computationally expensive and thus prohibitive for large datasets. In this paper, we propose a deep </span></span></span></span>neural network<span> (DNN) based two-stage model for spatio-temporal interpolation and forecasting. Interpolation is performed in the first step, which utilizes a dependent DNN with the embedding layer constructed with spatio-temporal basis functions. For the second stage, we use Long-Short Term Memory (LSTM) and convolutional LSTM to forecast future observations at a given location. We adopt the quantile-based loss function in the DNN to provide probabilistic forecasting. Compared to Kriging, the proposed method does not require specifying covariance functions or making </span></span>stationarity assumptions and is computationally efficient. Therefore, it is suitable for large-scale prediction of complex spatio-temporal processes. We apply our method to monthly </span><span><math><mrow><mi>P</mi><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn><mo>.</mo><mn>5</mn></mrow></msub></mrow></math></span> data at more than 200,000 space–time locations from January 1999 to December 2022 for fast imputation of missing values and forecasts with uncertainties.</p></div>","PeriodicalId":48771,"journal":{"name":"Spatial Statistics","volume":"57 ","pages":"Article 100773"},"PeriodicalIF":2.1000,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Spatio-temporal DeepKriging for interpolation and probabilistic forecasting\",\"authors\":\"Pratik Nag , Ying Sun , Brian J. Reich\",\"doi\":\"10.1016/j.spasta.2023.100773\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p><span><span><span><span>Gaussian processes (GP) and Kriging are widely used in traditional spatio-temporal modelling and prediction. These techniques typically presuppose that the data are observed from a </span>stationary GP<span> with a parametric<span> covariance structure<span>. However, processes in real-world applications often exhibit non-Gaussianity and nonstationarity. Moreover, likelihood-based inference for GPs is computationally expensive and thus prohibitive for large datasets. In this paper, we propose a deep </span></span></span></span>neural network<span> (DNN) based two-stage model for spatio-temporal interpolation and forecasting. Interpolation is performed in the first step, which utilizes a dependent DNN with the embedding layer constructed with spatio-temporal basis functions. For the second stage, we use Long-Short Term Memory (LSTM) and convolutional LSTM to forecast future observations at a given location. We adopt the quantile-based loss function in the DNN to provide probabilistic forecasting. Compared to Kriging, the proposed method does not require specifying covariance functions or making </span></span>stationarity assumptions and is computationally efficient. Therefore, it is suitable for large-scale prediction of complex spatio-temporal processes. We apply our method to monthly </span><span><math><mrow><mi>P</mi><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn><mo>.</mo><mn>5</mn></mrow></msub></mrow></math></span> data at more than 200,000 space–time locations from January 1999 to December 2022 for fast imputation of missing values and forecasts with uncertainties.</p></div>\",\"PeriodicalId\":48771,\"journal\":{\"name\":\"Spatial Statistics\",\"volume\":\"57 \",\"pages\":\"Article 100773\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2023-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Spatial Statistics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S2211675323000489\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"GEOSCIENCES, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Spatial Statistics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2211675323000489","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"GEOSCIENCES, MULTIDISCIPLINARY","Score":null,"Total":0}
Spatio-temporal DeepKriging for interpolation and probabilistic forecasting
Gaussian processes (GP) and Kriging are widely used in traditional spatio-temporal modelling and prediction. These techniques typically presuppose that the data are observed from a stationary GP with a parametric covariance structure. However, processes in real-world applications often exhibit non-Gaussianity and nonstationarity. Moreover, likelihood-based inference for GPs is computationally expensive and thus prohibitive for large datasets. In this paper, we propose a deep neural network (DNN) based two-stage model for spatio-temporal interpolation and forecasting. Interpolation is performed in the first step, which utilizes a dependent DNN with the embedding layer constructed with spatio-temporal basis functions. For the second stage, we use Long-Short Term Memory (LSTM) and convolutional LSTM to forecast future observations at a given location. We adopt the quantile-based loss function in the DNN to provide probabilistic forecasting. Compared to Kriging, the proposed method does not require specifying covariance functions or making stationarity assumptions and is computationally efficient. Therefore, it is suitable for large-scale prediction of complex spatio-temporal processes. We apply our method to monthly data at more than 200,000 space–time locations from January 1999 to December 2022 for fast imputation of missing values and forecasts with uncertainties.
期刊介绍:
Spatial Statistics publishes articles on the theory and application of spatial and spatio-temporal statistics. It favours manuscripts that present theory generated by new applications, or in which new theory is applied to an important practical case. A purely theoretical study will only rarely be accepted. Pure case studies without methodological development are not acceptable for publication.
Spatial statistics concerns the quantitative analysis of spatial and spatio-temporal data, including their statistical dependencies, accuracy and uncertainties. Methodology for spatial statistics is typically found in probability theory, stochastic modelling and mathematical statistics as well as in information science. Spatial statistics is used in mapping, assessing spatial data quality, sampling design optimisation, modelling of dependence structures, and drawing of valid inference from a limited set of spatio-temporal data.