{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T04:26:50Z","timestamp":1791606410667,"version":"4.3.4"},"reference-count":43,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2014,9,25]],"date-time":"2014-09-25T00:00:00Z","timestamp":1411603200000},"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\/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":[[2015,9,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>Functional data are traditionally assumed to be observed on the same domain. Motivated by a data set of heart rate temporal profiles, we develop methodology for the analysis of incomplete functional samples where each curve may be observed on a subset of the domain and unobserved elsewhere. We formalize this observation regime and develop the fundamental procedures of functional data analysis for this framework: estimation of parameters (mean and covariance operator) and principal component analysis. Principal scores of a partially observed function cannot be computed directly and we solve this challenging issue by estimating their best predictions as linear functionals of the observed part of the trajectory. Next, we propose a functional completion procedure that recovers the missing part by using the observed part of the curve. We construct prediction intervals for principal scores and bands for missing parts of trajectories. The prediction problems are seen to be ill-posed inverse problems; regularization techniques are used to obtain a stable solution. A simulation study shows the good performance of our methods. We illustrate the methods on the heart rate data and provide practical computational algorithms and theoretical arguments and proofs of all results.<\/jats:p>","DOI":"10.1111\/rssb.12087","type":"journal-article","created":{"date-parts":[[2014,9,25]],"date-time":"2014-09-25T03:15:51Z","timestamp":1411614951000},"page":"777-801","source":"Crossref","is-referenced-by-count":64,"title":["Components and Completion of Partially Observed Functional Data"],"prefix":"10.1093","volume":"77","author":[{"given":"David","family":"Kraus","sequence":"first","affiliation":[{"name":"University Hospital Lausanne \u00a0 Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2014,9,25]]},"reference":[{"key":"2023021709291353900_rssb12087-cit-0001","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/S0047-259X(03)00028-9","article-title":"Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes","volume":"87","author":"Antoniadis","year":"2003","journal-title":"J. Multiv. Anal."},{"key":"2023021709291353900_rssb12087-cit-0002","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.jmva.2012.03.006","article-title":"Detecting and estimating changes in dependent functional data","volume":"109","author":"Aston","year":"2012","journal-title":"J. Multiv. Anal."},{"key":"2023021709291353900_rssb12087-cit-0003","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/07-AOS516","article-title":"Common functional principal components","volume":"37","author":"Benko","year":"2009","journal-title":"Ann. Statist."},{"key":"2023021709291353900_rssb12087-cit-0004","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1154-9","volume-title":"Linear Processes in Function Spaces","author":"Bosq","year":"2000"},{"key":"2023021709291353900_rssb12087-cit-0005","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1017\/S0266466612000023","article-title":"Specification test for missing functional data","volume":"28","author":"Bugni","year":"2012","journal-title":"Econmetr. Theor."},{"key":"2023021709291353900_rssb12087-cit-0006","doi-asserted-by":"crossref","first-page":"2159","DOI":"10.1214\/009053606000000830","article-title":"Prediction in functional linear regression","volume":"34","author":"Cai","year":"2006","journal-title":"Ann. Statist."},{"key":"2023021709291353900_rssb12087-cit-0007","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/S0167-7152(99)00036-X","article-title":"Functional linear model","volume":"45","author":"Cardot","year":"1999","journal-title":"Statist. Probab. Lett."},{"key":"2023021709291353900_rssb12087-cit-0008","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/s00440-006-0025-2","article-title":"CLT in functional linear regression models","volume":"138","author":"Cardot","year":"2007","journal-title":"Probab. Theor. Reltd Flds"},{"key":"2023021709291353900_rssb12087-cit-0009","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. Multiv. Anal."},{"key":"2023021709291353900_rssb12087-cit-0010","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. Ass."},{"key":"2023021709291353900_rssb12087-cit-0011","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1007\/s00180-011-0256-2","article-title":"Empirical properties of forecasts with the functional autoregressive model","volume":"27","author":"Didericksen","year":"2012","journal-title":"Computnl Statist."},{"key":"2023021709291353900_rssb12087-cit-0012","volume-title":"The Oxford Handbook of Functional Data Analysis","author":"Ferraty","year":"2011"},{"key":"2023021709291353900_rssb12087-cit-0013","volume-title":"Nonparametric Functional Data Analysis","author":"Ferraty","year":"2006"},{"key":"2023021709291353900_rssb12087-cit-0014","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.jspi.2013.11.006","article-title":"Predicting the continuation of a function with applications to call center data","volume":"147","author":"Goldberg","year":"2014","journal-title":"J. Statist. Planng Inf."},{"key":"2023021709291353900_rssb12087-cit-0015","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-322-99202-4","volume-title":"Inverse Problems in the Mathematical Sciences","author":"Groetsch","year":"1993"},{"key":"2023021709291353900_rssb12087-cit-0016","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1214\/009053606000000957","article-title":"Methodology and convergence rates for functional linear regression","volume":"35","author":"Hall","year":"2007","journal-title":"Ann. Statist."},{"key":"2023021709291353900_rssb12087-cit-0017","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1111\/j.1467-9868.2005.00535.x","article-title":"On properties of functional principal components analysis","volume":"68","author":"Hall","year":"2006","journal-title":"J. R. Statist. Soc. B"},{"key":"2023021709291353900_rssb12087-cit-0018","doi-asserted-by":"crossref","first-page":"1493","DOI":"10.1214\/009053606000000272","article-title":"Properties of principal component methods for functional and longitudinal data analysis","volume":"34","author":"Hall","year":"2006","journal-title":"Ann. Statist."},{"key":"2023021709291353900_rssb12087-cit-0019","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/S0047-259X(02)00056-8","article-title":"Functional canonical analysis for square integrable stochastic processes","volume":"85","author":"He","year":"2003","journal-title":"J. Multiv. Anal."},{"key":"2023021709291353900_rssb12087-cit-0020","doi-asserted-by":"crossref","first-page":"705","DOI":"10.3150\/09-BEJ228","article-title":"Functional linear regression via canonical analysis","volume":"16","author":"He","year":"2010","journal-title":"Bernoulli"},{"key":"2023021709291353900_rssb12087-cit-0021","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1016\/j.jmva.2008.12.008","article-title":"Testing the stability of the functional autoregressive process","volume":"101","author":"Horv\u00e1th","year":"2010","journal-title":"J. Multiv. Anal."},{"key":"2023021709291353900_rssb12087-cit-0022","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4614-3655-3","volume-title":"Inference for Functional Data with Applications","author":"Horv\u00e1th","year":"2012"},{"key":"2023021709291353900_rssb12087-cit-0023","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1111\/j.1467-9868.2012.01032.x","article-title":"Estimation of the mean of functional time series and a two-sample problem","volume":"75","author":"Horv\u00e1th","year":"2013","journal-title":"J. R. Statist. Soc. B"},{"key":"2023021709291353900_rssb12087-cit-0024","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1111\/1467-9868.00297","article-title":"Functional linear discriminant analysis for irregularly sampled curves","volume":"63","author":"James","year":"2001","journal-title":"J. R. Statist. Soc. B"},{"key":"2023021709291353900_rssb12087-cit-0025","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":"2023021709291353900_rssb12087-cit-0026","doi-asserted-by":"crossref","first-page":"1500","DOI":"10.1016\/j.jspi.2013.04.011","article-title":"Testing for a change in covariance operator","volume":"143","author":"Jaru\u0161kov\u00e1","year":"2013","journal-title":"J. Statist. Planng Inf."},{"key":"2023021709291353900_rssb12087-cit-0027","volume-title":"Principal Component Analysis","author":"Jolliffe","year":"2002"},{"key":"2023021709291353900_rssb12087-cit-0028","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-4015-8","volume-title":"Foundations of Modern Probability","author":"Kallenberg","year":"2002"},{"key":"2023021709291353900_rssb12087-cit-0029","doi-asserted-by":"crossref","first-page":"2508","DOI":"10.1016\/j.jmva.2008.03.001","article-title":"Curve forecasting by functional autoregression","volume":"99","author":"Kargin","year":"2008","journal-title":"J. Multiv. Anal."},{"key":"2023021709291353900_rssb12087-cit-0030","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1093\/biomet\/ass037","article-title":"Dispersion operators and resistant second-order functional data analysis","volume":"99","author":"Kraus","year":"2012","journal-title":"Biometrika"},{"key":"2023021709291353900_rssb12087-cit-0031","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198507086.001.0001","volume-title":"Principles of Multivariate Analysis","author":"Krzanowski","year":"2000"},{"key":"2023021709291353900_rssb12087-cit-0133","doi-asserted-by":"crossref","first-page":"1562","DOI":"10.1214\/13-AOAS652","article-title":"Modeling and forecasting electricity spot prices: a functional data perspective","volume":"7","author":"Liebl","year":"2013","journal-title":"Ann. Appl. Statist"},{"key":"2023021709291353900_rssb12087-cit-0032","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1198\/jasa.2009.0115","article-title":"Estimating derivatives for samples of sparsely observed functions, with application to online auction dynamics","volume":"104","author":"Liu","year":"2009","journal-title":"J. Am. Statist. Ass."},{"key":"2023021709291353900_rssb12087-cit-0033","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s11203-005-0754-3","article-title":"Testing for the mean of random curves: a penalization approach","volume":"10","author":"Mas","year":"2007","journal-title":"Statist. Inf. Stoch. Processes"},{"key":"2023021709291353900_rssb12087-cit-0034","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":"2023021709291353900_rssb12087-cit-0035","doi-asserted-by":"crossref","first-page":"670","DOI":"10.1198\/jasa.2010.tm09239","article-title":"Second-order comparison of Gaussian random functions and the geometry of DNA minicircles","volume":"105","author":"Panaretos","year":"2010","journal-title":"J. Am. Statist. Ass."},{"key":"2023021709291353900_rssb12087-cit-0036","doi-asserted-by":"crossref","first-page":"2899","DOI":"10.1007\/s00330-013-2900-4","article-title":"Heritability, determinants and reference values of renal length: a family-based population study","volume":"23","author":"Pruijm","year":"2013","journal-title":"Eur. Radiol."},{"key":"2023021709291353900_rssb12087-cit-0037","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-98185-7","volume-title":"Functional Data Analysis with R and MATLAB","author":"Ramsay","year":"2009"},{"key":"2023021709291353900_rssb12087-cit-0038","volume-title":"Applied Functional Data Analysis","author":"Ramsay","year":"2002"},{"key":"2023021709291353900_rssb12087-cit-0039","doi-asserted-by":"crossref","DOI":"10.1007\/b98888","volume-title":"Functional Data Analysis","author":"Ramsay","year":"2005"},{"key":"2023021709291353900_rssb12087-cit-0040","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1198\/jasa.2009.0002","article-title":"A case study in exploratory functional data analysis: geometrical features of the internal carotid artery","volume":"104","author":"Sangalli","year":"2009","journal-title":"J. Am. Statist. Ass."},{"key":"2023021709291353900_rssb12087-cit-0041","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. Ass."},{"key":"2023021709291353900_rssb12087-cit-0042","doi-asserted-by":"crossref","first-page":"2873","DOI":"10.1214\/009053605000000660","article-title":"Functional linear regression analysis for longitudinal data","volume":"33","author":"Yao","year":"2005","journal-title":"Ann. Statist."}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Frssb.12087","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/77\/4\/777\/49231957\/jrsssb_77_4_777.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/77\/4\/777\/49231957\/jrsssb_77_4_777.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,2]],"date-time":"2024-06-02T18:58:07Z","timestamp":1717354687000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/77\/4\/777\/7040596"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,9,25]]},"references-count":43,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2014,9,25]]},"published-print":{"date-parts":[[2015,9,1]]}},"URL":"https:\/\/doi.org\/10.1111\/rssb.12087","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2015,9]]},"published":{"date-parts":[[2014,9,25]]}}}