{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T04:55:44Z","timestamp":1791435344647,"version":"4.3.3"},"reference-count":44,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2024,9,13]],"date-time":"2024-09-13T00:00:00Z","timestamp":1726185600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001711","name":"Swiss National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,2,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Let X={Xu}u\u2208U be a real-valued Gaussian process indexed by a set U. We show that X can be viewed as a graphical model with an uncountably infinite graph, where each Xu is a vertex. This graph is characterized by the reproducing property of X\u2019s covariance kernel, without restricting U to be finite or countable, allowing the modelling of stochastic processes in continuous time\/space. Unlike traditional methods, this characterization is not based on zero entries of an inverse covariance, posing challenges for structure estimation. We propose a plug-in methodology that targets graph recovery up to a finite resolution and shows consistency for graphs which are sufficiently regular and that can be applied to virtually any measurement regime. Furthermore, we derive convergence rates and finite-sample guarantees for the method, and demonstrate its performance through a simulation study and two data analyses.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkae086","type":"journal-article","created":{"date-parts":[[2024,9,14]],"date-time":"2024-09-14T08:18:46Z","timestamp":1726301926000},"page":"211-231","source":"Crossref","is-referenced-by-count":2,"title":["Continuously indexed graphical models"],"prefix":"10.1093","volume":"87","author":[{"given":"Kartik G","family":"Waghmare","sequence":"first","affiliation":[{"name":"Institut de Math\u00e9matiques, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne , Lausanne ,","place":["Switzerland"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Victor M","family":"Panaretos","sequence":"additional","affiliation":[{"name":"Institut de Math\u00e9matiques, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne , Lausanne ,","place":["Switzerland"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2024,9,13]]},"reference":[{"issue":"3","key":"2025021015285432100_qkae086-B1","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1090\/tran\/1950-068-03","article-title":"Theory of reproducing kernels","volume":"68","author":"Aronszajn","year":"1950","journal-title":"Transactions of the American Mathematical Society"},{"key":"2025021015285432100_qkae086-B2","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1090\/S0002-9947-1973-0336795-3","article-title":"Joint measures and cross-covariance operators","volume":"186","author":"Baker","year":"1973","journal-title":"Transactions of the American Mathematical Society"},{"issue":"6","key":"2025021015285432100_qkae086-B3","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1109\/MSP.2010.938028","article-title":"Graphical models for time-series","volume":"27","author":"Barber","year":"2010","journal-title":"IEEE Signal Processing Magazine"},{"key":"2025021015285432100_qkae086-B4","volume-title":"Reproducing kernel Hilbert spaces in probability and statistics","author":"Berlinet","year":"2011"},{"issue":"3","key":"2025021015285432100_qkae086-B5","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1016\/0022-247X(65)90108-3","article-title":"The operator theory of the pseudo-inverse I. 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