{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T09:56:47Z","timestamp":1787479007723,"version":"build-2736575974"},"reference-count":32,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"name":"NSF DMS","award":["2346292"],"award-info":[{"award-number":["2346292"]}]},{"DOI":"10.13039\/100000002","name":"NIH","doi-asserted-by":"publisher","award":["5 R01 CA280970-03"],"award-info":[{"award-number":["5 R01 CA280970-03"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,2,13]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>Functional principal component analysis is a key tool in the study of functional data, driving both exploratory analyses and feature construction for use in formal modelling and testing procedures. However, existing methods do not apply when functional observations are censored; for example, when the measurement instrument only supports recordings within a prespecified interval, thereby truncating values outside this range to the nearest boundary. A na\u00efve application of existing methods, without correction for instrument-induced censoring, introduces bias into the estimators of the mean, covariance and functional principal component scores. We extend the functional principal component analysis framework to accommodate noisy and potentially sparse censored functional data. Local loglikelihood maximization is used to recover smooth estimates of the mean and covariance surface that are representative of the latent process\u2019s mean and covariance functions. The covariance-smoothing procedure yields a positive semidefinite covariance surface, computed without the need to retroactively remove negative eigenvalues in the covariance-operator decomposition. Additionally, we construct a predictor of the scores, conditional on the censored functional data, and demonstrate its use in the generalized functional linear model. Convergence rates for the proposed estimators are established. In simulation experiments, the proposed method yields improved predictive performance and lower bias than existing alternatives. We illustrate its practical value in a study aimed at classifying eating disorder diagnoses in individuals with type 1 diabetes, using censored functional blood glucose data.<\/jats:p>","DOI":"10.1093\/biomet\/asag023","type":"journal-article","created":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T11:42:09Z","timestamp":1776253329000},"source":"Crossref","is-referenced-by-count":1,"title":["Functional principal component analysis forsparse censored data"],"prefix":"10.1093","volume":"113","author":[{"given":"Caitrin","family":"Murphy","sequence":"first","affiliation":[{"name":"Department of Statistical Science, Duke University , 214 Old Chemistry , Durham, North Carolina 27708,","place":["U.S.A"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2640-7696","authenticated-orcid":false,"given":"Eric","family":"Laber","sequence":"additional","affiliation":[{"name":"Department of Statistical Science, Duke University , 214 Old Chemistry , Durham, North Carolina 27708,","place":["U.S.A"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rhonda","family":"Merwin","sequence":"additional","affiliation":[{"name":"Department of Psychiatry & Behavioral Sciences, Duke University , 905 W Main St , Durham, North Carolina 27701,","place":["U.S.A"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5473-120X","authenticated-orcid":false,"given":"Brian","family":"Reich","sequence":"additional","affiliation":[{"name":"Department of Statistics, North Carolina State University , 2311 Stinson Drive , Raleigh, North Carolina 27695,","place":["U.S.A"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jake","family":"Koerner","sequence":"additional","affiliation":[{"name":"Department of Statistics, North Carolina State University , 2311 Stinson Drive , Raleigh, North Carolina 27695,","place":["U.S.A"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,4,16]]},"reference":[{"key":"2026060914565293100_asag023-B1","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1007\/BF02293986","article-title":"Principal components analysis of sampled functions","volume":"51","author":"Besse","year":"1986","journal-title":"Psychometrika"},{"key":"2026060914565293100_asag023-B2","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/0047-259X(82)90088-4","article-title":"Asymptotic theory for the principal component analysis of a vector random function: some applications to statistical inference","volume":"12","author":"Dauxois","year":"1982","journal-title":"J. Mult. Anal"},{"key":"2026060914565293100_asag023-B3","doi-asserted-by":"crossref","first-page":"1269","DOI":"10.1080\/01621459.2013.824893","article-title":"Classification using censored functional data","volume":"108","author":"Delaigle","year":"2013","journal-title":"J. Am. Statist. Assoc"},{"key":"2026060914565293100_asag023-B4","doi-asserted-by":"crossref","first-page":"1383","DOI":"10.1080\/01621459.2020.1723597","article-title":"Estimating the covariance of fragmented and other related types of functional data","volume":"116","author":"Delaigle","year":"2020","journal-title":"J. Am. Statist. Assoc"},{"key":"2026060914565293100_asag023-B5","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1111\/rssb.12125","article-title":"Truncated linear models for functional data","volume":"78","author":"Hall","year":"2016","journal-title":"J. R. Statist. Soc. B"},{"key":"2026060914565293100_asag023-B6","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1111\/1467-9868.00342","article-title":"Generalized linear models with functional predictors","volume":"64","author":"James","year":"2002","journal-title":"J. R. Statist. Soc. B"},{"key":"2026060914565293100_asag023-B7","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1093\/biomet\/87.3.587","article-title":"Principal component models for sparse functional data","volume":"87","author":"James","year":"2000","journal-title":"Biometrika"},{"key":"2026060914565293100_asag023-B8","first-page":"1","article-title":"\u00dcber lineare Methoden in der Wahrscheinlichkeitsrechnung","volume":"37","author":"Karhunen","year":"1947","journal-title":"Ann. Acad. Sci. Fenn"},{"key":"2026060914565293100_asag023-B9","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1146\/annurev-statistics-032921-033726","article-title":"Second-generation functional data","volume":"10","author":"Koner","year":"2023","journal-title":"Ann. Rev. Statist. Applic"},{"key":"2026060914565293100_asag023-B10","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1093\/biomet\/asv055","article-title":"Semiparametric approach to regression with a covariate subject to a detection limit","volume":"103","author":"Kong","year":"2016","journal-title":"Biometrika"},{"key":"2026060914565293100_asag023-B11","doi-asserted-by":"crossref","first-page":"1219","DOI":"10.1080\/01621459.2017.1321545","article-title":"Functional feature construction for individualized treatment regimes","volume":"113","author":"Laber","year":"2018","journal-title":"J. Am. Statist. Assoc"},{"key":"2026060914565293100_asag023-B12","doi-asserted-by":"crossref","first-page":"3321","DOI":"10.1214\/10-AOS813","article-title":"Uniform convergence rates for nonparametric regression and principal component analysis in functional\/longitudinal data","volume":"38","author":"Li","year":"2010","journal-title":"Ann. Statist."},{"key":"2026060914565293100_asag023-B13","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1093\/biomet\/asaa088","article-title":"Basis expansions for functional snippets","volume":"108","author":"Lin","year":"2021","journal-title":"Biometrika"},{"key":"2026060914565293100_asag023-B14","doi-asserted-by":"crossref","first-page":"e467","DOI":"10.1002\/sta4.467","article-title":"Fast estimators for the mean function for functional data with detection limits","volume":"11","author":"Liu","year":"2022","journal-title":"Stat"},{"key":"2026060914565293100_asag023-B15","doi-asserted-by":"crossref","first-page":"730","DOI":"10.1080\/10485252.2023.2258999","article-title":"On estimation of covariance function for functional data with detection limits","volume":"36","author":"Liu","year":"2023","journal-title":"J. Nonparam. Statist"},{"key":"2026060914565293100_asag023-B16","doi-asserted-by":"crossref","first-page":"107421","DOI":"10.1016\/j.csda.2022.107421","article-title":"Truncated estimation in functional generalized linear regression models","volume":"169","author":"Liu","year":"2022","journal-title":"Comp. Statist. Data Anal"},{"key":"2026060914565293100_asag023-B17","first-page":"159","article-title":"Fonctions al\u00e9atoires \u00e0 d\u00e9composition orthogonale exponentielle","volume":"84","author":"Lo\u00e8ve","year":"1946","journal-title":"La Revue Scientifique"},{"key":"2026060914565293100_asag023-B18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/00401706.1999.10485591","article-title":"Generalized linear regression on sampled signals and curves: a p-spline approach","volume":"41","author":"Marx","year":"1999","journal-title":"Technometrics"},{"key":"2026060914565293100_asag023-B19","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1146\/annurev-statistics-010814-020413","article-title":"Functional regression","volume":"2","author":"Morris","year":"2015","journal-title":"Ann. Rev. Statist. Appl"},{"key":"2026060914565293100_asag023-B20","doi-asserted-by":"crossref","first-page":"774","DOI":"10.1214\/009053604000001156","article-title":"Generalized functional linear models","volume":"33","author":"M\u00fcller","year":"2005","journal-title":"Ann. Statist."},{"key":"2026060914565293100_asag023-B21","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1111\/pedi.13160","article-title":"Performance of three different continuous glucose monitoring systems in children with type 1 diabetes during a diabetes summer camp","volume":"22","author":"Nagl","year":"2021","journal-title":"Pediatr. Diabetes"},{"key":"2026060914565293100_asag023-B22","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1111\/j.2517-6161.1991.tb01844.x","article-title":"Some tools for functional data analysis","volume":"53","author":"Ramsay","year":"1991","journal-title":"J. R. Statist. Soc. B"},{"key":"2026060914565293100_asag023-B23","doi-asserted-by":"crossref","DOI":"10.1007\/b98888","volume-title":"Functional Data Analysis","author":"Ramsay","year":"2005","edition":"2nd"},{"key":"2026060914565293100_asag023-B24","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1111\/j.2517-6161.1991.tb01821.x","article-title":"Estimating the mean and covariance structure nonparametrically when the data are curves","volume":"53","author":"Rice","year":"1991","journal-title":"J. R. Statist. Soc. B"},{"key":"2026060914565293100_asag023-B25","doi-asserted-by":"crossref","first-page":"1441","DOI":"10.1111\/j.1468-2354.2007.00470.x","article-title":"Estimation with censored regressors: basic issues","volume":"48","author":"Rigobon","year":"2007","journal-title":"Int. Econ. Rev"},{"key":"2026060914565293100_asag023-B26","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1198\/jbes.2009.06119","article-title":"Bias from censored regressors","volume":"27","author":"Rigobon","year":"2009","journal-title":"J. Bus. Econ. Statist"},{"key":"2026060914565293100_asag023-B27","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1002\/sim.8798","article-title":"Functional principal component analysis for longitudinal data with informative dropout","volume":"40","author":"Shi","year":"2021","journal-title":"Statist. Med"},{"key":"2026060914565293100_asag023-B28","doi-asserted-by":"crossref","first-page":"2821","DOI":"10.1002\/sim.8930","article-title":"Censored functional data for incomplete follow-up studies","volume":"40","author":"Strzalkowska-Kominiak","year":"2021","journal-title":"Statist. Med"},{"key":"2026060914565293100_asag023-B29","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1080\/01621459.1987.10478466","article-title":"Local likelihood estimation","volume":"82","author":"Tibshirani","year":"1987","journal-title":"J. Am. Statist. Assoc"},{"key":"2026060914565293100_asag023-B30","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1146\/annurev-statistics-041715-033624","article-title":"Functional data analysis","volume":"3","author":"Wang","year":"2016","journal-title":"Ann. Rev. Statist. Appl"},{"key":"2026060914565293100_asag023-B31","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1198\/016214504000001745","article-title":"Functional data analysis for sparse longitudinal data","volume":"100","author":"Yao","year":"2005","journal-title":"J. Am. Statist. Assoc"},{"key":"2026060914565293100_asag023-B32","doi-asserted-by":"crossref","first-page":"676","DOI":"10.1111\/1541-0420.00078","article-title":"Shrinkage estimation for functional principal component scores with application to the population kinetics of plasma folate","volume":"59","author":"Yao","year":"2003","journal-title":"Biometrics"}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asag023\/68098329\/asag023.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/113\/2\/asag023\/68098329\/asag023.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/113\/2\/asag023\/68098329\/asag023.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T18:57:02Z","timestamp":1781031422000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/doi\/10.1093\/biomet\/asag023\/8656836"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026]]},"references-count":32,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,2,13]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asag023","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026]]},"published":{"date-parts":[[2026]]},"article-number":"asag023"}}