{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T06:25:17Z","timestamp":1771914317751,"version":"3.50.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,11,1]],"date-time":"2021-11-01T00:00:00Z","timestamp":1635724800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,11,1]],"date-time":"2021-11-01T00:00:00Z","timestamp":1635724800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Stat"],"published-print":{"date-parts":[[2022,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Mixture autoregressive (MAR) models provide a flexible way to model time series with predictive distributions which depend on the recent history of the process and are able to accommodate asymmetry and multimodality. Bayesian inference for such models offers the additional advantage of incorporating the uncertainty in the estimated models into the predictions. We introduce a new way of sampling from the posterior distribution of the parameters of MAR models which allows for covering the complete parameter space of the models, unlike previous approaches. We also propose a relabelling algorithm to deal a posteriori with label switching. We apply our new method to simulated and real datasets, discuss the accuracy and performance of our new method, as well as its advantages over previous studies. The idea of density forecasting using MCMC output is also introduced.<\/jats:p>","DOI":"10.1007\/s00180-021-01162-8","type":"journal-article","created":{"date-parts":[[2021,11,1]],"date-time":"2021-11-01T21:04:31Z","timestamp":1635800671000},"page":"1399-1433","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Bayesian analysis of mixture autoregressive models covering the complete parameter space"],"prefix":"10.1007","volume":"37","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7146-7685","authenticated-orcid":false,"given":"Davide","family":"Ravagli","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Georgi N.","family":"Boshnakov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,11,1]]},"reference":[{"issue":"15","key":"1162_CR1","doi-asserted-by":"publisher","first-page":"1704","DOI":"10.1016\/j.spl.2009.04.009","volume":"79","author":"GN Boshnakov","year":"2009","unstructured":"Boshnakov GN (2009) Analytic expressions for predictive distributions in mixture autoregressive models. Stat Probab Lett 79(15):1704\u20131709","journal-title":"Stat Probab Lett"},{"issue":"2","key":"1162_CR2","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1016\/j.laa.2010.09.023","volume":"434","author":"GN Boshnakov","year":"2011","unstructured":"Boshnakov GN (2011) On first and second order stationarity of random coefficient models. Linear Algebra Appl 434(2):415\u2013423","journal-title":"Linear Algebra Appl"},{"key":"1162_CR3","doi-asserted-by":"crossref","unstructured":"Boshnakov GN, Ravagli D (2020) mixAR: mixture autoregressive models. R package version 0.22.4. https:\/\/CRAN.R-project.org\/package=mixAR","DOI":"10.32614\/CRAN.package.mixAR"},{"key":"1162_CR4","unstructured":"Box GEP, Jenkins GM (1976) Time series analysis: forecasting and control\/George E.P. Box and Gwilym M. Jenkins, rev. ed. edn, Holden-Day San Francisco"},{"key":"1162_CR5","unstructured":"Celeux G (2000) Bayesian inference of mixture: the label switching problem. In: Payne R, Green P (eds) COMPSTAT. Physica, Heidelberg"},{"issue":"432","key":"1162_CR6","doi-asserted-by":"publisher","first-page":"1313","DOI":"10.1080\/01621459.1995.10476635","volume":"90","author":"S Chib","year":"1995","unstructured":"Chib S (1995) Marginal likelihood from the Gibbs output. J Am Stat Assoc 90(432):1313\u20131321","journal-title":"J Am Stat Assoc"},{"issue":"453","key":"1162_CR7","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1198\/016214501750332848","volume":"96","author":"S Chib","year":"2001","unstructured":"Chib S, Jeliazkov I (2001) Marginal likelihood from the Metropolis\u2013Hastings output. J Am Stat Assoc 96(453):270\u2013281","journal-title":"J Am Stat Assoc"},{"key":"1162_CR8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","volume":"39","author":"AP Dempster","year":"1977","unstructured":"Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the em algorithm. J R Stat Soc Ser B (Methodol) 39:1\u201338","journal-title":"J R Stat Soc Ser B (Methodol)"},{"key":"1162_CR9","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1111\/j.2517-6161.1994.tb01985.x","volume":"56","author":"J Diebolt","year":"1994","unstructured":"Diebolt J, Robert CP (1994) Estimation of finite mixture distributions through Bayesian sampling. J R Stat Soc Ser B (Methodol) 56:363\u2013375","journal-title":"J R Stat Soc Ser B (Methodol)"},{"issue":"2","key":"1162_CR10","doi-asserted-by":"publisher","first-page":"215","DOI":"10.2307\/1358","volume":"11","author":"C Elton","year":"1942","unstructured":"Elton C, Nicholson M (1942) The ten-year cycle in numbers of the lynx in Canada. J Anim Ecol 11(2):215\u2013244","journal-title":"J Anim Ecol"},{"issue":"2","key":"1162_CR11","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1214\/aos\/1176342360","volume":"1","author":"TS Ferguson","year":"1973","unstructured":"Ferguson TS (1973) A Bayesian analysis of some nonparametric problems. Ann Stat 1(2):209\u2013230","journal-title":"Ann Stat"},{"issue":"4","key":"1162_CR12","doi-asserted-by":"publisher","first-page":"711","DOI":"10.1093\/biomet\/82.4.711","volume":"82","author":"PJ Green","year":"1995","unstructured":"Green PJ (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4):711\u2013732","journal-title":"Biometrika"},{"key":"1162_CR13","unstructured":"Hossain AS (2012) Complete Bayesian analysis of some mixture time series models. PhD thesis, Probability and Statistics Group, School of Mathematics, University of Manchester"},{"key":"1162_CR14","unstructured":"Hyndman RJ (2017) fma: data sets from \u201cForecasting: Methods and Applications\u201d by Makridakis, Wheelwright & Hyndman (1998). R package version 2.3. https:\/\/CRAN.R-project.org\/package=fma"},{"issue":"2","key":"1162_CR15","first-page":"134","volume":"36","author":"MC Jones","year":"1987","unstructured":"Jones MC (1987) Randomly choosing parameters from the stationarity and invertibility region of autoregressive-moving average models. J R Stat Soc Ser C (Appl Stat) 36(2):134\u2013138","journal-title":"J R Stat Soc Ser C (Appl Stat)"},{"issue":"1","key":"1162_CR16","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1016\/j.csda.2008.06.001","volume":"53","author":"JW Lau","year":"2008","unstructured":"Lau JW, So MK (2008) Bayesian mixture of autoregressive models. Comput Stat Data Anal 53(1):38\u201360","journal-title":"Comput Stat Data Anal"},{"issue":"3","key":"1162_CR17","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1093\/biomet\/92.3.529","volume":"92","author":"JF Lawless","year":"2005","unstructured":"Lawless JF, Fredette M (2005) Frequentist prediction intervals and predictive distributions. Biometrika 92(3):529\u2013542","journal-title":"Biometrika"},{"issue":"436","key":"1162_CR18","first-page":"1504","volume":"91","author":"ND Le","year":"1996","unstructured":"Le ND, Martin R, Raftery AE (1996) Modeling flat stretches, bursts, and outliers in time series using mixture transition distribution models. J Am Stat Assoc 91(436):1504\u20131515","journal-title":"J Am Stat Assoc"},{"issue":"01","key":"1162_CR19","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1142\/S0219607705000073","volume":"01","author":"AY Lo","year":"2005","unstructured":"Lo AY (2005) Weighted Chinese restaurant processes. COSMOS 01(01):107\u2013111. https:\/\/doi.org\/10.1142\/S0219607705000073","journal-title":"COSMOS"},{"key":"1162_CR20","unstructured":"Met Office Centre for Environmental Data Analysis: 2019, Met office midas open: UK land surface stations data (1853-current). data retrieved from the CEDA Archive, http:\/\/catalogue.ceda.ac.uk\/uuid\/dbd451271eb04662beade68da43546e1"},{"key":"1162_CR21","first-page":"347","volume":"59","author":"DB Nelson","year":"1991","unstructured":"Nelson DB (1991) Conditional heteroskedasticity in asset returns: a new approach. Econom J Econom Soc 59:347\u2013370","journal-title":"Econom J Econom Soc"},{"issue":"4","key":"1162_CR22","doi-asserted-by":"publisher","first-page":"731","DOI":"10.1111\/1467-9868.00095","volume":"59","author":"S Richardson","year":"1997","unstructured":"Richardson S, Green PJ (1997) On Bayesian analysis of mixtures with an unknown number of components. J R Stat Soc Ser B Stat Methodol 59(4):731\u2013792","journal-title":"J R Stat Soc Ser B Stat Methodol"},{"issue":"5","key":"1162_CR23","doi-asserted-by":"publisher","first-page":"689","DOI":"10.1111\/j.1467-9868.2011.00781.x","volume":"73","author":"J Rousseau","year":"2011","unstructured":"Rousseau J, Mengersen K (2011) Asymptotic behaviour of the posterior distribution in overfitted mixture models. J R Stat Soc Ser B (Stat Methodol) 73(5):689\u2013710. https:\/\/doi.org\/10.1111\/j.1467-9868.2011.00781.x","journal-title":"J R Stat Soc Ser B (Stat Methodol)"},{"issue":"3","key":"1162_CR24","doi-asserted-by":"publisher","first-page":"242","DOI":"10.1002\/asmb.613","volume":"22","author":"S Sampietro","year":"2006","unstructured":"Sampietro S (2006) Bayesian analysis of mixture of autoregressive components with an application to financial market volatility. Appl Stoch Models Bus Ind 22(3):242","journal-title":"Appl Stoch Models Bus Ind"},{"key":"1162_CR25","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198522249.001.0001","volume-title":"Non-linear time series: a dynamical system approach","author":"H Tong","year":"1990","unstructured":"Tong H (1990) Non-linear time series: a dynamical system approach. Oxford University Press, Oxford"},{"issue":"1","key":"1162_CR26","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1111\/1467-9868.00222","volume":"62","author":"CS Wong","year":"2000","unstructured":"Wong CS, Li WK (2000) On a mixture autoregressive model. J R Stat Soc Ser B Stat Methodol 62(1):95\u2013115","journal-title":"J R Stat Soc Ser B Stat Methodol"},{"issue":"1","key":"1162_CR27","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1198\/jcgs.2010.09174","volume":"20","author":"S Wood","year":"2011","unstructured":"Wood S, Rosen O, Kohn R (2011) Bayesian mixtures of autoregressive models. J Comput Graph Stat 20(1):174\u2013195. https:\/\/doi.org\/10.1198\/jcgs.2010.09174","journal-title":"J Comput Graph Stat"}],"container-title":["Computational Statistics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-021-01162-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00180-021-01162-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-021-01162-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,11]],"date-time":"2024-09-11T06:34:34Z","timestamp":1726036474000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00180-021-01162-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,1]]},"references-count":27,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,7]]}},"alternative-id":["1162"],"URL":"https:\/\/doi.org\/10.1007\/s00180-021-01162-8","relation":{},"ISSN":["0943-4062","1613-9658"],"issn-type":[{"value":"0943-4062","type":"print"},{"value":"1613-9658","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,1]]},"assertion":[{"value":"30 July 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 September 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 November 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"The analysis uses functions from the R package <b>mixAR<\/b> (Boshnakov and Ravagli), available on CRAN.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Code availability"}}]}}