{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T05:19:04Z","timestamp":1788326344834,"version":"build-2803163510"},"reference-count":31,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2023,5,12]],"date-time":"2023-05-12T00:00:00Z","timestamp":1683849600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"NSF","doi-asserted-by":"publisher","award":["DMS-2014626"],"award-info":[{"award-number":["DMS-2014626"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"NIH","doi-asserted-by":"publisher","award":["Echo UH3OD023313"],"award-info":[{"award-number":["Echo UH3OD023313"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,7,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Series of univariate distributions indexed by equally spaced time points are ubiquitous in applications and their analysis constitutes one of the challenges of the emerging field of distributional data analysis. To quantify such distributional time series, we propose a class of intrinsic autoregressive models that operate in the space of optimal transport maps. The autoregressive transport models that we introduce here are based on regressing optimal transport maps on each other, where predictors can be transport maps from an overall barycenter to a current distribution or transport maps between past consecutive distributions of the distributional time series. Autoregressive transport models and their associated distributional regression models specify the link between predictor and response transport maps by moving along geodesics in Wasserstein space. These models emerge as natural extensions of the classical autoregressive models in Euclidean space. Unique stationary solutions of autoregressive transport models are shown to exist under a geometric moment contraction condition of Wu &amp; Shao [(2004) Limit theorems for iterated random functions. Journal of Applied Probability 41, 425\u2013436)], using properties of iterated random functions. We also discuss an extension to a varying coefficient model for first-order autoregressive transport models. In addition to simulations, the proposed models are illustrated with distributional time series of house prices across U.S. counties and annual summer temperature distributions.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkad051","type":"journal-article","created":{"date-parts":[[2023,5,12]],"date-time":"2023-05-12T08:48:02Z","timestamp":1683881282000},"page":"1012-1033","source":"Crossref","is-referenced-by-count":24,"title":["Autoregressive optimal transport models"],"prefix":"10.1093","volume":"85","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3348-8877","authenticated-orcid":false,"given":"Changbo","family":"Zhu","sequence":"first","affiliation":[{"name":"Department of Applied and Computational Mathematics and Statistics, University of Notre Dame , Notre Dame, IN 46556 , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6396-2552","authenticated-orcid":false,"given":"Hans-Georg","family":"M\u00fcller","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of California, Davis , Davis, CA 95616 , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2023,5,12]]},"reference":[{"issue":"3","key":"2023071211415179300_qkad051-B1","doi-asserted-by":"crossref","first-page":"496","DOI":"10.1093\/jjfinec\/nbz021","article-title":"A mixed frequency stochastic volatility model for intraday stock market returns","volume":"19","author":"Bekierman","year":"2021","journal-title":"Journal of Financial Econometrics"},{"key":"2023071211415179300_qkad051-B2","first-page":"A12","volume-title":"Why we are experiencing so many unusually hot summer nights","author":"Bhatia","year":"2021,  16"},{"key":"2023071211415179300_qkad051-B3","first-page":"1","article-title":"Geodesic PCA in the Wasserstein space by convex PCA","volume":"53","author":"Bigot","year":"2017","journal-title":"Annales de l\u2019Institut Henri Poincar\u00e9 B: Probability and Statistics"},{"issue":"2","key":"2023071211415179300_qkad051-B4","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1111\/1540-6229.12233","article-title":"Local house price dynamics: New indices and stylized facts","volume":"47","author":"Bogin","year":"2019","journal-title":"Real Estate Economics"},{"key":"2023071211415179300_qkad051-B5","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1154-9","volume-title":"Linear processes in function spaces: Theory and applications","author":"Bosq","year":"2000"},{"key":"2023071211415179300_qkad051-B6","article-title":"Wasserstein regression","author":"Chen","year":"2022","journal-title":"Journal of the American Statistical Association"},{"issue":"2","key":"2023071211415179300_qkad051-B7","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/S0378-3758(96)00110-3","article-title":"Unified estimators of smooth quantile and quantile density functions","volume":"59","author":"Cheng","year":"1997","journal-title":"Journal of Statistical Planning and Inference"},{"issue":"1","key":"2023071211415179300_qkad051-B8","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1214\/21-EJS1942","article-title":"Statistical inference on the Hilbert sphere with application to random densities","volume":"16","author":"Dai","year":"2022","journal-title":"Electronic Journal of Statistics"},{"issue":"1","key":"2023071211415179300_qkad051-B9","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1137\/S0036144598338446","article-title":"Iterated random functions","volume":"41","author":"Diaconis","year":"1999","journal-title":"SIAM Review"},{"issue":"2","key":"2023071211415179300_qkad051-B10","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1111\/j.1467-9574.1983.tb00802.x","article-title":"Relative efficiency and deficiency of kernel type estimators of smooth distribution functions","volume":"37","author":"Falk","year":"1983","journal-title":"Statistica Neerlandica"},{"issue":"4","key":"2023071211415179300_qkad051-B11","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1093\/biomet\/asac005","article-title":"Distribution-on-distribution regression via optimal transport maps","volume":"109","author":"Ghodrati","year":"2022","journal-title":"Biometrika"},{"issue":"2","key":"2023071211415179300_qkad051-B12","first-page":"297","article-title":"A geometric study of Wasserstein spaces: Euclidean spaces","volume":"9","author":"Kloeckner","year":"2010","journal-title":"Annali della Scuola Normale Superiore di Pisa-Classe di Scienze"},{"issue":"4","key":"2023071211415179300_qkad051-B13","doi-asserted-by":"crossref","first-page":"1304","DOI":"10.1016\/j.ijforecast.2019.05.007","article-title":"Forecasting of density functions with an application to cross-sectional and intraday returns","volume":"35","author":"Kokoszka","year":"2019","journal-title":"International Journal of Forecasting"},{"issue":"5","key":"2023071211415179300_qkad051-B14","doi-asserted-by":"crossref","first-page":"919","DOI":"10.1007\/s10463-011-0339-4","article-title":"On estimating distribution functions using Bernstein polynomials","volume":"64","author":"Leblanc","year":"2012","journal-title":"Annals of the Institute of Statistical Mathematics"},{"key":"2023071211415179300_qkad051-B15","author":"Matabuena","year":"2021"},{"issue":"1","key":"2023071211415179300_qkad051-B16","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1111\/rssa.12053","article-title":"Fitting age-specific fertility rates by a flexible generalized skew normal probability density function","volume":"178","author":"Mazzuco","year":"2015","journal-title":"Journal of the Royal Statistical Society Series A"},{"issue":"1","key":"2023071211415179300_qkad051-B17","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1006\/aima.1997.1634","article-title":"A convexity principle for interacting gases","volume":"128","author":"McCann","year":"1997","journal-title":"Advances in Mathematics"},{"issue":"4","key":"2023071211415179300_qkad051-B18","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1080\/00401706.2018.1437473","article-title":"Profile monitoring of probability density functions via simplicial functional PCA with application to image data","volume":"60","author":"Menafoglio","year":"2018","journal-title":"Technometrics"},{"issue":"4","key":"2023071211415179300_qkad051-B19","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.2307\/2938179","article-title":"Uniform convergence in probability and stochastic equicontinuity","volume":"59","author":"Newey","year":"1991","journal-title":"Econometrica"},{"issue":"May 2018","key":"2023071211415179300_qkad051-B20","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.jue.2018.03.001","article-title":"US metropolitan house price dynamics","volume":"105","author":"Oikarinen","year":"2018","journal-title":"Journal of Urban Economics"},{"issue":"July\u2013December 2011","key":"2023071211415179300_qkad051-B21","doi-asserted-by":"crossref","first-page":"595","DOI":"10.4054\/DemRes.2011.25.19","article-title":"Changes in the age-at-death distribution in four low mortality countries: A nonparametric approach","volume":"25","author":"Ouellette","year":"2011","journal-title":"Demographic Research"},{"issue":"2","key":"2023071211415179300_qkad051-B22","doi-asserted-by":"crossref","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":"The Annals of Statistics"},{"issue":"37","key":"2023071211415179300_qkad051-B23","first-page":"1","article-title":"Projected statistical methods for distributional data on the real line with the Wasserstein metric","volume":"23","author":"Pegoraro","year":"2022","journal-title":"Journal of Machine Learning Research"},{"issue":"1","key":"2023071211415179300_qkad051-B24","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1214\/15-AOS1363","article-title":"Functional data analysis for density functions by transformation to a Hilbert space","volume":"44","author":"Petersen","year":"2016","journal-title":"The Annals of Statistics"},{"issue":"2","key":"2023071211415179300_qkad051-B25","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1214\/17-AOS1624","article-title":"Fr\u00e9chet regression for random objects with Euclidean predictors","volume":"47","author":"Petersen","year":"2019","journal-title":"The Annals of Statistics"},{"issue":"2","key":"2023071211415179300_qkad051-B26","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1080\/10618600.2016.1237877","article-title":"Grouped functional time series forecasting: An application to age-specific mortality rates","volume":"26","author":"Shang","year":"2017","journal-title":"Journal of Computational and Graphical Statistics"},{"key":"2023071211415179300_qkad051-B27","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719017","volume-title":"Empirical processes with applications to statistics","author":"Shorack","year":"2009"},{"key":"2023071211415179300_qkad051-B28","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/058","volume-title":"Topics in optimal transportation","author":"Villani","year":"2003"},{"issue":"2","key":"2023071211415179300_qkad051-B29","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1239\/jap\/1082999076","article-title":"Limit theorems for iterated random functions","volume":"41","author":"Wu","year":"2004","journal-title":"Journal of Applied Probability"},{"issue":"1","key":"2023071211415179300_qkad051-B30","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1111\/jtsa.12590","article-title":"Wasserstein autoregressive models for density time series","volume":"43","author":"Zhang","year":"2022","journal-title":"Journal of Time Series Analysis"},{"key":"2023071211415179300_qkad051-B31","volume-title":"Modeling financial time series with S-PLUS\u00ae","author":"Zivot","year":"2007"}],"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\/85\/3\/1012\/50859838\/qkad051.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/85\/3\/1012\/50859838\/qkad051.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,12]],"date-time":"2023-07-12T07:42:44Z","timestamp":1689147764000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/85\/3\/1012\/7160826"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,12]]},"references-count":31,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,5,12]]},"published-print":{"date-parts":[[2023,7,12]]}},"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkad051","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2023,7]]},"published":{"date-parts":[[2023,5,12]]}}}