{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T12:04:10Z","timestamp":1753877050697,"version":"3.41.2"},"reference-count":41,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T00:00:00Z","timestamp":1735603200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,7,14]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We consider separating and joint modelling amplitude and phase variations for functional data in an identifiable manner. To rigorously address this separability issue, we introduce the notion of \u03b1-separability upon constructing a family of \u03b1-indexed metrics. We bridge \u03b1-separability with the uniqueness of Fr\u00e9chet mean, leading to the proposed adjustable combination of amplitude and phase model. The parameter \u03b1 allows user-defined modelling emphasis between vertical and horizontal features and provides a novel viewpoint on the identifiability issue. We prove the consistency of the sample Fr\u00e9chet mean and variance, and the proposed estimators. Our method is illustrated in simulations and COVID-19 infection rate data.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkae112","type":"journal-article","created":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T08:30:21Z","timestamp":1735633821000},"page":"746-771","source":"Crossref","is-referenced-by-count":0,"title":["<i>\u03b1<\/i>-separability and adjustable combination of amplitude and phase model for functional data"],"prefix":"10.1093","volume":"87","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3975-4729","authenticated-orcid":false,"given":"Tian","family":"Wang","sequence":"first","affiliation":[{"name":"Department of Biostatistics, Mailman School of Public Health, Columbia University , New York, NY ,","place":["USA"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jimin","family":"Ding","sequence":"additional","affiliation":[{"name":"Department of Statistics and Data Science, Washington University in St. Louis , St. Louis, MO ,","place":["USA"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2024,12,31]]},"reference":[{"volume-title":"Dynamic programming and optimal control","year":"2012","author":"Bertsekas","key":"2025071319530961200_qkae112-B1"},{"issue":"1","key":"2025071319530961200_qkae112-B2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1214\/aos\/1046294456","article-title":"Large sample theory of intrinsic and extrinsic sample means on manifolds","volume":"31","author":"Bhattacharya","year":"2003","journal-title":"Annals of Statistics"},{"issue":"1","key":"2025071319530961200_qkae112-B3","doi-asserted-by":"publisher","first-page":"e404","DOI":"10.1002\/sta4.404","article-title":"Covariance-based low-dimensional registration for function-on-function regression","volume":"10","author":"Boschi","year":"2021","journal-title":"Stat"},{"issue":"2","key":"2025071319530961200_qkae112-B4","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1111\/j.0006-341X.2004.00191.x","article-title":"Self modeling with flexible, random time transformations","volume":"60","author":"Brumback","year":"2004","journal-title":"Biometrics"},{"issue":"3","key":"2025071319530961200_qkae112-B5","doi-asserted-by":"publisher","first-page":"839","DOI":"10.1111\/biom.13340","article-title":"Cross-component registration for multivariate functional data, with application to growth curves","volume":"77","author":"Carroll","year":"2021","journal-title":"Biometrics"},{"issue":"2","key":"2025071319530961200_qkae112-B6","doi-asserted-by":"publisher","first-page":"1103","DOI":"10.3150\/20-BEJ1267","article-title":"Functional registration and local variations: Identifiability, rank, and tuning","volume":"27","author":"Chakraborty","year":"2021","journal-title":"Bernoulli"},{"key":"2025071319530961200_qkae112-B7","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1214\/15-BA957","article-title":"Bayesian registration of functions and curves","volume":"11","author":"Cheng","year":"2016","journal-title":"Bayesian Analysis"},{"issue":"5","key":"2025071319530961200_qkae112-B8","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1111\/j.1467-9868.2010.00752.x","article-title":"A multiresolution approach to time warping achieved by a Bayesian prior\u2013posterior transfer fitting strategy","volume":"72","author":"Claeskens","year":"2010","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"key":"2025071319530961200_qkae112-B9","doi-asserted-by":"crossref","DOI":"10.1002\/9781119072492","volume-title":"Statistical shape analysis: With applications in R","author":"Dryden","year":"2016"},{"issue":"2","key":"2025071319530961200_qkae112-B10","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1080\/10618600.2016.1164709","article-title":"Combining functional data registration and factor analysis","volume":"26","author":"Earls","year":"2017","journal-title":"Journal of Computational and Graphical Statistics"},{"volume-title":"Nonparametric functional data analysis: Theory and practice","year":"2006","author":"Ferraty","key":"2025071319530961200_qkae112-B11"},{"key":"2025071319530961200_qkae112-B12","first-page":"215","article-title":"Les \u00e9l\u00e9ments al\u00e9atoires de nature quelconque dans un espace distanci\u00e9","volume":"10","author":"Fr\u00e9chet","year":"1948","journal-title":"Annales de l\u2019institut Henri Poincar\u00e9 D"},{"key":"2025071319530961200_qkae112-B13","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1016\/j.csda.2018.06.010","article-title":"Model-based curve registration via stochastic approximation EM algorithm","volume":"131","author":"Fu","year":"2019","journal-title":"Computational Statistics & Data Analysis"},{"issue":"4","key":"2025071319530961200_qkae112-B14","doi-asserted-by":"publisher","first-page":"801","DOI":"10.1093\/biomet\/92.4.801","article-title":"Nonparametric maximum likelihood estimation of the structural mean of a sample of curves","volume":"92","author":"Gervini","year":"2005","journal-title":"Biometrika"},{"volume-title":"Mathematical foundations of infinite-dimensional statistical models","year":"2021","author":"Gin\u00e9","key":"2025071319530961200_qkae112-B15"},{"issue":"51","key":"2025071319530961200_qkae112-B16","doi-asserted-by":"publisher","first-page":"2376","DOI":"10.21105\/joss.02376","article-title":"Covid-19 data hub","volume":"5","author":"Guidotti","year":"2020","journal-title":"Journal of Open Source Software"},{"issue":"510","key":"2025071319530961200_qkae112-B17","doi-asserted-by":"publisher","first-page":"545","DOI":"10.1080\/01621459.2015.1006729","article-title":"Unifying amplitude and phase analysis: A compositional data approach to functional multivariate mixed-effects modeling of mandarin Chinese","volume":"110","author":"Hadjipantelis","year":"2015","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"2025071319530961200_qkae112-B18","doi-asserted-by":"publisher","first-page":"e220","DOI":"10.1002\/sta4.220","article-title":"A general framework for multivariate functional principal component analysis of amplitude and phase variation","volume":"8","author":"Happ","year":"2019","journal-title":"Stat"},{"key":"2025071319530961200_qkae112-B19","doi-asserted-by":"crossref","DOI":"10.1002\/9781118762547","volume-title":"Theoretical foundations of functional data analysis, with an introduction to linear operators","author":"Hsing","year":"2015"},{"issue":"2","key":"2025071319530961200_qkae112-B20","doi-asserted-by":"publisher","first-page":"480","DOI":"10.1214\/07-AOAS127","article-title":"Curve alignment by moments","volume":"1","author":"James","year":"2007","journal-title":"The Annals of Applied Statistics"},{"issue":"3","key":"2025071319530961200_qkae112-B21","doi-asserted-by":"publisher","first-page":"1266","DOI":"10.1214\/aos\/1176348769","article-title":"Statistical tools to analyze data representing a sample of curves","volume":"20","author":"Kneip","year":"1992","journal-title":"Annals of Statistics"},{"issue":"483","key":"2025071319530961200_qkae112-B22","doi-asserted-by":"publisher","first-page":"1155","DOI":"10.1198\/016214508000000517","article-title":"Combining registration and fitting for functional models","volume":"103","author":"Kneip","year":"2008","journal-title":"Journal of the American Statistical Association"},{"key":"2025071319530961200_qkae112-B23","doi-asserted-by":"crossref","DOI":"10.1201\/9781315117416","volume-title":"Introduction to functional data analysis","author":"Kokoszka","year":"2017"},{"issue":"467","key":"2025071319530961200_qkae112-B24","doi-asserted-by":"publisher","first-page":"687","DOI":"10.1198\/016214504000000999","article-title":"Functional convex averaging and synchronization for time-warped random curves","volume":"99","author":"Liu","year":"2004","journal-title":"Journal of the American Statistical Association"},{"issue":"540","key":"2025071319530961200_qkae112-B25","doi-asserted-by":"publisher","first-page":"1964","DOI":"10.1080\/01621459.2021.1893179","article-title":"Bayesian framework for simultaneous registration and estimation of noisy, sparse, and fragmented functional data","volume":"117","author":"Matuk","year":"2022","journal-title":"Journal of the American Statistical Association"},{"issue":"2","key":"2025071319530961200_qkae112-B26","doi-asserted-by":"publisher","first-page":"771","DOI":"10.1214\/15-AOS1387","article-title":"Amplitude and phase variation of point processes","volume":"44","author":"Panaretos","year":"2016","journal-title":"Annals of Statistics"},{"issue":"2","key":"2025071319530961200_qkae112-B27","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1111\/1467-9868.00129","article-title":"Curve registration","volume":"60","author":"Ramsay","year":"1998","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"key":"2025071319530961200_qkae112-B28","doi-asserted-by":"crossref","DOI":"10.1007\/b98888","volume-title":"Functional data analysis","author":"Ramsay","year":"2005"},{"issue":"2","key":"2025071319530961200_qkae112-B29","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1111\/1467-9868.00283","article-title":"Nonparametric maximum likelihood estimation for shifted curves","volume":"63","author":"R\u00f8nn","year":"2001","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"issue":"485","key":"2025071319530961200_qkae112-B30","doi-asserted-by":"publisher","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":"Journal of the American Statistical Association"},{"issue":"4","key":"2025071319530961200_qkae112-B31","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1111\/j.2517-6161.1995.tb02055.x","article-title":"Incorporating parametric effects into functional principal components analysis","volume":"57","author":"Silverman","year":"1995","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"key":"2025071319530961200_qkae112-B32","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4939-4020-2","volume-title":"Functional and shape data analysis","author":"Srivastava","year":"2016"},{"issue":"4","key":"2025071319530961200_qkae112-B33","doi-asserted-by":"publisher","first-page":"875","DOI":"10.1093\/biomet\/asn047","article-title":"Pairwise curve synchronization for functional data","volume":"95","author":"Tang","year":"2008","journal-title":"Biometrika"},{"issue":"481","key":"2025071319530961200_qkae112-B34","doi-asserted-by":"publisher","first-page":"328","DOI":"10.1198\/016214507000001139","article-title":"Bayesian hierarchical curve registration","volume":"103","author":"Telesca","year":"2008","journal-title":"Journal of the American Statistical Association"},{"key":"2025071319530961200_qkae112-B35","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1016\/j.csda.2012.12.001","article-title":"Generative models for functional data using phase and amplitude separation","volume":"61","author":"Tucker","year":"2013","journal-title":"Computational Statistics & Data Analysis"},{"issue":"4","key":"2025071319530961200_qkae112-B36","doi-asserted-by":"publisher","first-page":"676","DOI":"10.1007\/s11749-011-0268-9","article-title":"On the definition of phase and amplitude variability in functional data analysis","volume":"21","author":"Vantini","year":"2012","journal-title":"Test"},{"key":"2025071319530961200_qkae112-B37","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/j.csda.2019.03.004","article-title":"Nonparametric registration to low-dimensional function spaces","volume":"138","author":"Wagner","year":"2019","journal-title":"Computational Statistics & Data Analysis"},{"issue":"1","key":"2025071319530961200_qkae112-B38","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1146\/annurev-statistics-041715-033624","article-title":"Functional data analysis","volume":"3","author":"Wang","year":"2016","journal-title":"Annual Review of Statistics and Its Application"},{"issue":"3","key":"2025071319530961200_qkae112-B39","doi-asserted-by":"publisher","first-page":"1251","DOI":"10.1214\/aos\/1069362747","article-title":"Alignment of curves by dynamic time warping","volume":"25","author":"Wang","year":"1997","journal-title":"Annals of Statistics"},{"issue":"1","key":"2025071319530961200_qkae112-B40","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1111\/biom.12963","article-title":"Registration for exponential family functional data","volume":"75","author":"Wrobel","year":"2019","journal-title":"Biometrics"},{"issue":"7","key":"2025071319530961200_qkae112-B41","doi-asserted-by":"publisher","first-page":"2234","DOI":"10.1016\/j.csda.2011.01.007","article-title":"Functional density synchronization","volume":"55","author":"Zhang","year":"2011","journal-title":"Computational Statistics & Data Analysis"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/87\/3\/746\/61301030\/qkae112.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/87\/3\/746\/61301030\/qkae112.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,13]],"date-time":"2025-07-13T23:53:17Z","timestamp":1752450797000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/87\/3\/746\/7935314"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,31]]},"references-count":41,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,12,31]]},"published-print":{"date-parts":[[2025,7,14]]}},"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkae112","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"type":"print","value":"1369-7412"},{"type":"electronic","value":"1467-9868"}],"subject":[],"published-other":{"date-parts":[[2025,7]]},"published":{"date-parts":[[2024,12,31]]}}}