Kesen Wang , Sameh Abdulah , Ying Sun , Marc G. Genton
{"title":"哪一种参数化的matsamn协方差函数?","authors":"Kesen Wang , Sameh Abdulah , Ying Sun , Marc G. Genton","doi":"10.1016/j.spasta.2023.100787","DOIUrl":null,"url":null,"abstract":"<div><p><span><span>The Matérn family of covariance functions is currently the most popularly used model in spatial </span>statistics, geostatistics, and machine learning to specify the correlation between two geographical locations based on spatial distance. Compared to existing covariance functions, the Matérn family has more flexibility in data fitting because it allows the control of the field smoothness through a dedicated parameter. Moreover, it generalizes other popular covariance functions. However, fitting the smoothness parameter is computationally challenging since it complicates the optimization process. As a result, some practitioners set the smoothness parameter at an arbitrary value to reduce the optimization convergence time. In the literature, studies have used various parameterizations of the Matérn covariance function, assuming they are equivalent. This work aims at studying the effectiveness of different parameterizations under various settings. We demonstrate the feasibility of inferring all parameters simultaneously and quantifying their uncertainties on large-scale data using the </span><em>ExaGeoStat</em><span><span><span> parallel software. We also highlight the importance of the smoothness parameter by analyzing the Fisher information of the statistical parameters. We show that the various parameterizations have different properties and differ from several perspectives. In particular, we study the three most popular parameterizations in terms of parameter estimation accuracy, modeling accuracy and efficiency, prediction efficiency, </span>uncertainty quantification, and </span>asymptotic properties. We further demonstrate their differing performances under nugget effects and approximated covariance. Lastly, we give recommendations for parameterization selection based on our experimental results.</span></p></div>","PeriodicalId":48771,"journal":{"name":"Spatial Statistics","volume":"58 ","pages":"Article 100787"},"PeriodicalIF":2.1000,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Which parameterization of the Matérn covariance function?\",\"authors\":\"Kesen Wang , Sameh Abdulah , Ying Sun , Marc G. Genton\",\"doi\":\"10.1016/j.spasta.2023.100787\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p><span><span>The Matérn family of covariance functions is currently the most popularly used model in spatial </span>statistics, geostatistics, and machine learning to specify the correlation between two geographical locations based on spatial distance. Compared to existing covariance functions, the Matérn family has more flexibility in data fitting because it allows the control of the field smoothness through a dedicated parameter. Moreover, it generalizes other popular covariance functions. However, fitting the smoothness parameter is computationally challenging since it complicates the optimization process. As a result, some practitioners set the smoothness parameter at an arbitrary value to reduce the optimization convergence time. In the literature, studies have used various parameterizations of the Matérn covariance function, assuming they are equivalent. This work aims at studying the effectiveness of different parameterizations under various settings. We demonstrate the feasibility of inferring all parameters simultaneously and quantifying their uncertainties on large-scale data using the </span><em>ExaGeoStat</em><span><span><span> parallel software. We also highlight the importance of the smoothness parameter by analyzing the Fisher information of the statistical parameters. We show that the various parameterizations have different properties and differ from several perspectives. In particular, we study the three most popular parameterizations in terms of parameter estimation accuracy, modeling accuracy and efficiency, prediction efficiency, </span>uncertainty quantification, and </span>asymptotic properties. We further demonstrate their differing performances under nugget effects and approximated covariance. Lastly, we give recommendations for parameterization selection based on our experimental results.</span></p></div>\",\"PeriodicalId\":48771,\"journal\":{\"name\":\"Spatial Statistics\",\"volume\":\"58 \",\"pages\":\"Article 100787\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2023-10-12\",\"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/S2211675323000623\",\"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/S2211675323000623","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"GEOSCIENCES, MULTIDISCIPLINARY","Score":null,"Total":0}
Which parameterization of the Matérn covariance function?
The Matérn family of covariance functions is currently the most popularly used model in spatial statistics, geostatistics, and machine learning to specify the correlation between two geographical locations based on spatial distance. Compared to existing covariance functions, the Matérn family has more flexibility in data fitting because it allows the control of the field smoothness through a dedicated parameter. Moreover, it generalizes other popular covariance functions. However, fitting the smoothness parameter is computationally challenging since it complicates the optimization process. As a result, some practitioners set the smoothness parameter at an arbitrary value to reduce the optimization convergence time. In the literature, studies have used various parameterizations of the Matérn covariance function, assuming they are equivalent. This work aims at studying the effectiveness of different parameterizations under various settings. We demonstrate the feasibility of inferring all parameters simultaneously and quantifying their uncertainties on large-scale data using the ExaGeoStat parallel software. We also highlight the importance of the smoothness parameter by analyzing the Fisher information of the statistical parameters. We show that the various parameterizations have different properties and differ from several perspectives. In particular, we study the three most popular parameterizations in terms of parameter estimation accuracy, modeling accuracy and efficiency, prediction efficiency, uncertainty quantification, and asymptotic properties. We further demonstrate their differing performances under nugget effects and approximated covariance. Lastly, we give recommendations for parameterization selection based on our experimental results.
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.