[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-127814-en":3,"doc-seo-127814-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},127814,1099523882367,"Hazel","https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc",8,"Research & Report","The Matérn Model - A Journey through Statistics, Numerical Analysis and Machine Learning","The Matérn model is a foundational tool in spatial statistics for estimation and prediction, and its influence extends far beyond covariance modeling. This article connects spatial statistics to numerical analysis, approximation theory, computational statistics, machine learning, and probability theory. It surveys scalable computation methods such as the SPDE approach and Vecchia likelihood approximation, highlights Bayesian computational applications, and reviews flexible alternatives to the Matérn model, comparing estimation, prediction, screening effect, computation, and Sobolev regularity properties.","Submitted to Statistical Science  \nThe Matérn Model:  \nA Journey through Statistics, Numerical Analysis and Machine Learning  \nEmilio Porcu 1 , Moreno Bevilacqua, Robert Schaback and Chris J. Oates  \nAbstract. The Matérn model has been a cornerstone of spatial statistics for more than half a century. More recently, the Matérn model has been exploited in disciplines as diverse as numerical analysis, approximation theory, computational statistics, machine learning, and probability theory. In this article we take a Matérn-based journey across these disciplines. First, we reflect on the importance of the Matérn model for estimation and prediction in spatial statistics, establishing also connections to other disciplines in which the Matérn model has been influential. Then, we position the Matérn model within the literature on big data and scalable computation: the SPDE approach, the Vecchia likelihood approximation, and recent applications in Bayesian computation are all discussed. Finally, we review recent devlopments, including flexible alternatives to the Matérn model, whose performance we compare in terms of estimation, prediction, screening effect, computation, and Sobolev regularity properties.  \nKeywords: Approximation Theory, Compact Support, Covariance, Kernel, Kriging, Machine Learning, Maximum Likelihood, Reproducing Kernel Hilbert Spaces, Spatial Statistics, Sobolev Spaces.  \n1. INTRODUCTION  \n1 This paper serves two purposes: On the one hand, we 2 provide a panoramic view, across several disciplines, of 3 the Matérn model. On the other hand, the paper illustrates 4 the role of the Matérn model in several disciplines, while 5 discussing alternative or more general models and their 6 relevance to many aspects of statistical modeling, estima- 7 tion, prediction, computational statistics, numerical anal- 8 ysis, and machine learning.  \nEmilio Porcu is Professor, Department of Mathematics, Khalifa University, and Research Fellow atADIA Lab, both institutions located in Abu Dhabi, The United Arab Emirates (e-mail: [emilio.porcu@bku.ac.ae](emilio.porcu@bku.ac.ae)). Moreno Bevilacqua is Professor, Department of Statistics, Universidad Adolfo Ibanez ([e-mail:](e-mail: moreno.bevilacqua@uai.cl)[ moreno.bevilacqua@uai.cl](e-mail: moreno.bevilacqua@uai.cl)). Robert Schaback is Professor, Department of Mathematics, University of Göttingen, Germany (e-mail:  \n[schaback@math.uni-goettingen.de](schaback@math.uni-goettingen.de)). Chris J. Oates is Professor, School of Mathematics, Statistics & Physics, Newcastle University, UK (e-mail: [chris.oates@ncl.ac.uk](chris.oates@ncl.ac.uk)).  \n1Corresponding Author.  \n9 A historical account of the Matérn model is provided  \n10 by Guttorp and Gneiting [69] . The Matérn model – also  \n11 called the Matérn covariance function, or the Matérn ker- 12 nel, depending on context – is commonly attributed to  \n13 Matérn [109], but can be found under alternative names  \n14 in different branches of the scientific literature. The use  \n15 of the Matérn model is widespread, and it is impossible to  \n16 cover all its diverse applications here; our review focuses  \n17 on a selection of applications that are of especial interest  \n18 and significance. Specifically, we aim to cover  \n19 1. estimation and prediction using the Matérn model  \n20 in statistics, with emphasis on maximum likelihood  \n21 estimation, Kriging prediction, and the associated  \n22 screening effect;  \n23 2. applications of the Matérn model in  \n24 a) computational statistics, including the stochas- 25 tic differential equation (SDE) and stochas- 26 tic partial differential equation (SPDE) ap- 27 proaches, likelihood approximation, inference  \n28 of partial differential equations (PDEs) and  \n29 Charles Stein’s method;  \n30 b) statistical modeling, including non-standard  \n31 scenarios, for instance when isotropy and sta-  \n2  \n32 tionarity cannot be assumed, or to model di- 33 rections and curves;  \n34 c) approximation theory and numerical ","cbCairaMF0DIcAJa","https://ap.wps.com/l/cbCairaMF0DIcAJa","pdf",549700,1,23,"English","en",105,"# Introduction\n## Setting and Notation\n# Scope and Goals\n## Estimation and Prediction\n## Computational Statistics and Scalable Computation\n## Statistical Modeling Beyond Isotropy/Stationarity\n## Approximation Theory and Numerical Analysis\n## Machine Learning and Gaussian Processes\n## Probability Theory and Stochastic Processes\n# Comparison With Flexible Alternatives","[{\"question\":\"What disciplines does the Matérn model connect to beyond spatial statistics?\",\"answer\":\"The article links spatial statistics to numerical analysis, approximation theory, computational statistics, machine learning, and probability theory, using the Matérn model as a common lens across fields.\"},{\"question\":\"Which scalable computation methods are discussed for Matérn-based modeling?\",\"answer\":\"The review covers the SPDE approach, Vecchia likelihood approximation, and applications related to Bayesian computation.\"},{\"question\":\"How does the paper evaluate alternatives to the Matérn model?\",\"answer\":\"Recent flexible alternatives are compared in terms of estimation, prediction, screening effect, computation, and Sobolev regularity properties.\"}]","The Matérn Model - 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