{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:44:33Z","timestamp":1788223473767,"version":"build-2803163510"},"reference-count":28,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T00:00:00Z","timestamp":1581379200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,6,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>We consider estimating a functional graphical model from multivariate functional observations. In functional data analysis, the classical assumption is that each function has been measured over a densely sampled grid. However, in practice the functions have often been observed, with measurement error, at a relatively small number of points. We propose a class of doubly functional graphical models to capture the evolving conditional dependence relationship among a large number of sparsely or densely sampled functions. Our approach first implements a nonparametric smoother to perform functional principal components analysis for each curve, then estimates a functional covariance matrix and finally computes sparse precision matrices, which in turn provide the doubly functional graphical model. We derive some novel concentration bounds, uniform convergence rates and model selection properties of our estimator for both sparsely and densely sampled functional data in the high-dimensional large-$p$, small-$n$ regime. We demonstrate via simulations that the proposed method significantly outperforms possible competitors. Our proposed method is applied to a brain imaging dataset.<\/jats:p>","DOI":"10.1093\/biomet\/asz072","type":"journal-article","created":{"date-parts":[[2019,11,18]],"date-time":"2019-11-18T14:07:38Z","timestamp":1574086058000},"page":"415-431","source":"Crossref","is-referenced-by-count":35,"title":["Doubly functional graphical models in high dimensions"],"prefix":"10.1093","volume":"107","author":[{"given":"Xinghao","family":"Qiao","sequence":"first","affiliation":[{"name":"Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cheng","family":"Qian","sequence":"first","affiliation":[{"name":"Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gareth M","family":"James","sequence":"first","affiliation":[{"name":"Department of Data Sciences and Operations, University of Southern California, 3670 Trousdale Parkway, Los Angeles, California 90089, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shaojun","family":"Guo","sequence":"first","affiliation":[{"name":"Institute of Statistics and Big Data, Renmin University of China, 59 Zhongguancun Street, Beijing 100872, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2020,2,11]]},"reference":[{"key":"2020051801580515500_B1","doi-asserted-by":"crossref","first-page":"2577","DOI":"10.1214\/08-AOS600","article-title":"Covariance regularization by thresholding","volume":"136","author":"Bickel,","year":"2008","journal-title":"Ann. 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