{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,13]],"date-time":"2026-09-13T11:55:50Z","timestamp":1789300550149,"version":"build-2803163510"},"reference-count":41,"publisher":"Oxford University Press (OUP)","license":[{"start":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T00:00:00Z","timestamp":1778457600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004052","name":"King Abdullah University of Science and Technology","doi-asserted-by":"publisher","award":["ORFS-CRG12-2024-6399"],"award-info":[{"award-number":["ORFS-CRG12-2024-6399"]}],"id":[{"id":"10.13039\/501100004052","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004359","name":"Swedish Research Council","doi-asserted-by":"publisher","award":["2023-04170"],"award-info":[{"award-number":["2023-04170"]}],"id":[{"id":"10.13039\/501100004359","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Whittle\u2013Mat\u00e9rn fields are a recently introduced class of Gaussian processes on metric graphs, specified as solutions to a fractional-order stochastic differential equation. Unlike previous covariance-based methods, these fields are well-defined for any compact metric graph and can provide Gaussian processes with differentiable sample paths. We derive the main statistical properties, including the consistency and asymptotic normality of maximum likelihood estimators and the necessary and sufficient conditions for optimal prediction with misspecified parameters. The covariance function is generally unavailable in closed form, which makes statistical inference challenging. However, we show that for specific values of the fractional exponent where the fields exhibit Markov properties, likelihood-based inference and spatial prediction can be performed exactly and efficiently. This enables the use of Whittle\u2013Mat\u00e9rn fields in large datasets without approximations. The results and methods are illustrated through simulation studies and through an application to traffic data modelling, where allowing for differentiable processes significantly improves results.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkag074","type":"journal-article","created":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T11:42:55Z","timestamp":1776944575000},"source":"Crossref","is-referenced-by-count":1,"title":["Statistical inference for Gaussian Whittle\u2013Mat\u00e9rn fields on metric graphs"],"prefix":"10.1093","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2361-5465","authenticated-orcid":false,"given":"David","family":"Bolin","sequence":"first","affiliation":[{"name":"Statistics program, King Abdullah University of Science and Technology , Thuwal, SA 23955 ,","place":["Saudi Arabia"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2562-2829","authenticated-orcid":false,"given":"Alexandre B","family":"Simas","sequence":"additional","affiliation":[{"name":"Statistics program, King Abdullah University of Science and Technology , Thuwal, SA 23955 ,","place":["Saudi Arabia"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0381-6593","authenticated-orcid":false,"given":"Jonas","family":"Wallin","sequence":"additional","affiliation":[{"name":"Department of Statistics, Lund University , Tycho Brahes v\u00e4g 1, Lund, SE 223 63 ,","place":["Sweden"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,5,11]]},"reference":[{"issue":"4","key":"2026051107214893400_qkag074-B1","doi-asserted-by":"publisher","first-page":"2478","DOI":"10.1214\/19-AOS1896","article-title":"Isotropic covariance functions on graphs and their edges","volume":"48","author":"Anderes","year":"2020","journal-title":"Annals of Statistics"},{"issue":"1","key":"2026051107214893400_qkag074-B2","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1002\/sta4.135","article-title":"Stationary point processes are uncommon on linear networks","volume":"6","author":"Baddeley","year":"2017","journal-title":"Stat"},{"key":"2026051107214893400_qkag074-B3","doi-asserted-by":"publisher","first-page":"268","DOI":"10.1016\/j.spasta.2019.01.002","article-title":"Non-stationary Gaussian models with physical barriers","volume":"29","author":"Bakka","year":"2019","journal-title":"Spatial Statistics"},{"key":"2026051107214893400_qkag074-B4","volume-title":"Introduction to quantum graphs","author":"Berkolaiko","year":"2013"},{"key":"2026051107214893400_qkag074-B5","doi-asserted-by":"publisher","first-page":"1476","DOI":"10.3150\/22-BEJ1507","article-title":"Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators","volume":"29","author":"Bolin","year":"2023","journal-title":"Bernoulli: Official Journal of the Bernoulli Society for Mathematical Statistics and Probability"},{"key":"2026051107214893400_qkag074-B6","author":"Bolin","year":"2024"},{"key":"2026051107214893400_qkag074-B7","doi-asserted-by":"publisher","first-page":"1611","DOI":"10.3150\/23-BEJ1647","article-title":"Gaussian Whittle-Mat\u00e9rn fields on metric graphs","volume":"30","author":"Bolin","year":"2024","journal-title":"Bernoulli: Official Journal of the Bernoulli Society for Mathematical Statistics and Probability"},{"key":"2026051107214893400_qkag074-B8","author":"Bolin","year":"2024"},{"key":"2026051107214893400_qkag074-B9","doi-asserted-by":"publisher","first-page":"153","DOI":"10.3150\/25-BEJ1853","article-title":"Markov properties of Gaussian random fields on compact metric graphs","volume":"32","author":"Bolin","year":"2026","journal-title":"Bernoulli: Official Journal of the Bernoulli Society for Mathematical Statistics and Probability"},{"key":"2026051107214893400_qkag074-B10","author":"Bolin","year":"2026"},{"key":"2026051107214893400_qkag074-B11","first-page":"9882","article-title":"Efficient methods for Gaussian Markov random fields under sparse linear constraints","volume":"34","author":"Bolin","year":"2021","journal-title":"Advances in Neural Information Processing Systems"},{"issue":"1","key":"2026051107214893400_qkag074-B12","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1214\/22-STS864","article-title":"Local scale invariance and robustness of proper scoring rules","volume":"38","author":"Bolin","year":"2023","journal-title":"Statistical Science: A Review Journal of the Institute of Mathematical Statistics"},{"key":"2026051107214893400_qkag074-B13","author":"Borovitskiy","year":"2025"},{"key":"2026051107214893400_qkag074-B14","author":"Borovitskiy","year":"2021"},{"issue":"1","key":"2026051107214893400_qkag074-B15","doi-asserted-by":"publisher","first-page":"96","DOI":"10.3141\/1748-12","article-title":"Freeway performance measurement system: Mining loop detector data","volume":"1748","author":"Chen","year":"2001","journal-title":"Transportation Research Record"},{"issue":"5","key":"2026051107214893400_qkag074-B16","doi-asserted-by":"publisher","first-page":"1221","DOI":"10.1007\/s11222-020-09942-w","article-title":"Inhomogeneous higher-order summary statistics for point processes on linear networks","volume":"30","author":"Cronie","year":"2020","journal-title":"Statistics and Computing"},{"issue":"2","key":"2026051107214893400_qkag074-B17","doi-asserted-by":"publisher","first-page":"414","DOI":"10.1111\/rssb.12486","article-title":"Graph based Gaussian processes on restricted domains","volume":"84","author":"Dunson","year":"2022","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"key":"2026051107214893400_qkag074-B18","author":"Fey","year":"2025"},{"issue":"477","key":"2026051107214893400_qkag074-B19","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1198\/016214506000001437","article-title":"Strictly proper scoring rules, prediction, and estimation","volume":"102","author":"Gneiting","year":"2007","journal-title":"Journal of the American Statistical Association"},{"key":"2026051107214893400_qkag074-B20","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511526169","volume-title":"Gaussian Hilbert spaces","author":"Janson","year":"1997"},{"issue":"2","key":"2026051107214893400_qkag074-B21","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1093\/biomet\/ass079","article-title":"The role of the range parameter for estimation and prediction in geostatistics","volume":"100","author":"Kaufman","year":"2013","journal-title":"Biometrika"},{"issue":"2","key":"2026051107214893400_qkag074-B22","doi-asserted-by":"publisher","first-page":"1038","DOI":"10.1214\/21-AOS2138","article-title":"Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces","volume":"50","author":"Kirchner","year":"2022","journal-title":"Annals of Statistics"},{"key":"2026051107214893400_qkag074-B23","doi-asserted-by":"publisher","first-page":"104685","DOI":"10.1016\/j.spa.2025.104685","article-title":"Smoothness estimation for Whittle\u2013Mat\u00e9rn processes on closed Riemannian manifolds","volume":"189","author":"Korte-Stapff","year":"2025","journal-title":"Stochastic Processes and Their Applications"},{"key":"2026051107214893400_qkag074-B24","volume-title":"Advanced spatial modeling with stochastic partial differential equations using R and INLA","author":"Krainski","year":"2018"},{"key":"2026051107214893400_qkag074-B25","doi-asserted-by":"publisher","first-page":"Paper No. 100599","DOI":"10.1016\/j.spasta.2022.100599","article-title":"The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running","volume":"50","author":"Lindgren","year":"2022","journal-title":"Spatial Statistics"},{"issue":"4","key":"2026051107214893400_qkag074-B26","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1111\/j.1467-9868.2011.00777.x","article-title":"An explicit link between Gaussian fields and Gaussian Markov random fields: The stochastic partial differential equation approach","volume":"73","author":"Lindgren","year":"2011","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"key":"2026051107214893400_qkag074-B27","author":"Lippe","year":"2025"},{"key":"2026051107214893400_qkag074-B28","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-58647-2","volume-title":"Stochastic partial differential equations","author":"Lototsky","year":"2017"},{"issue":"1","key":"2026051107214893400_qkag074-B29","doi-asserted-by":"publisher","first-page":"92","DOI":"10.3141\/2136-11","article-title":"Spatial autocorrelation and Bayesian spatial statistical method for analyzing intersections prone to injury crashes","volume":"2136","author":"Mitra","year":"2009","journal-title":"Transportation Research Record"},{"issue":"3","key":"2026051107214893400_qkag074-B30","doi-asserted-by":"publisher","first-page":"1288","DOI":"10.1111\/sjos.12720","article-title":"Cox processes driven by transformed Gaussian processes on linear networks-a review and new contributions","volume":"51","author":"M\u00f8ller","year":"2024","journal-title":"Scandinavian Journal of Statistics"},{"key":"2026051107214893400_qkag074-B31","doi-asserted-by":"crossref","DOI":"10.1002\/9781119967101","volume-title":"Spatial analysis along networks: Statistical and computational methods","author":"Okabe","year":"2012"},{"key":"2026051107214893400_qkag074-B32","author":"OpenStreetMap contributors","year":"2017"},{"issue":"5","key":"2026051107214893400_qkag074-B33","doi-asserted-by":"publisher","first-page":"808","DOI":"10.1109\/JPROC.2018.2820126","article-title":"Graph signal processing: Overview, challenges, and applications","volume":"106","author":"Ortega","year":"2018","journal-title":"Proceedings of the IEEE: Institute of Electrical and Electronics Engineers"},{"issue":"5","key":"2026051107214893400_qkag074-B34","doi-asserted-by":"publisher","first-page":"1417","DOI":"10.1093\/jrsssb\/qkad082","article-title":"Stationary nonseparable space-time covariance functions on networks","volume":"85","author":"Porcu","year":"2023","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"key":"2026051107214893400_qkag074-B35","doi-asserted-by":"crossref","DOI":"10.1201\/9780203492024","volume-title":"Gaussian Markov random fields","author":"Rue","year":"2005"},{"key":"2026051107214893400_qkag074-B36","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1214\/21-STS838","article-title":"The SPDE approach to Mat\u00e9rn fields: Graph representations","volume":"37","author":"Sanz-Alonso","year":"2022","journal-title":"Statistical Science: A Review Journal of the Institute of Mathematical Statistics"},{"issue":"11","key":"2026051107214893400_qkag074-B37","doi-asserted-by":"publisher","first-page":"1240","DOI":"10.1080\/19439962.2020.1726542","article-title":"Spatial analysis of road crash frequency using Bayesian models with integrated nested Laplace approximation (INLA)","volume":"13","author":"Satria","year":"2021","journal-title":"Journal of Transportation Security"},{"key":"2026051107214893400_qkag074-B38","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1494-6","volume-title":"Interpolation of spatial data: Some theory for Kriging","author":"Stein","year":"1999"},{"issue":"4","key":"2026051107214893400_qkag074-B39","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1007\/s10651-006-0022-8","article-title":"Spatial statistical models that use flow and stream distance","volume":"13","author":"Ver Hoef","year":"2006","journal-title":"Environmental and Ecological Statistics"},{"key":"2026051107214893400_qkag074-B40","first-page":"974","article-title":"Stochastic processes in several dimensions","volume":"40","author":"Whittle","year":"1963","journal-title":"Bulletin of the International Statistical Institute"},{"issue":"465","key":"2026051107214893400_qkag074-B41","doi-asserted-by":"publisher","first-page":"250","DOI":"10.1198\/016214504000000241","article-title":"Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics","volume":"99","author":"Zhang","year":"2004","journal-title":"Journal of the American Statistical Association"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jrsssb\/advance-article-pdf\/doi\/10.1093\/jrsssb\/qkag074\/68266158\/qkag074.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/advance-article-pdf\/doi\/10.1093\/jrsssb\/qkag074\/68266158\/qkag074.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T11:22:38Z","timestamp":1778498558000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/advance-article\/doi\/10.1093\/jrsssb\/qkag074\/8675707"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,11]]},"references-count":41,"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkag074","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,11]]},"article-number":"qkag074"}}