{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T05:13:14Z","timestamp":1773724394981,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,17]],"date-time":"2020-04-17T00:00:00Z","timestamp":1587081600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>For the modeling of categorical time series, both nominal or ordinal time series, an extension of the basic discrete autoregressive moving-average (ARMA) models is proposed. It uses an observation-driven regime-switching mechanism, leading to the family of RS-DARMA models. After having discussed the stochastic properties of RS-DARMA models in general, we focus on the particular case of the first-order RS-DAR model. This RS-DAR    ( 1 )     model constitutes a parsimoniously parameterized type of Markov chain, which has an easy-to-interpret data-generating mechanism and may also handle negative forms of serial dependence. Approaches for model fitting are elaborated on, and they are illustrated by two real-data examples: the modeling of a nominal sequence from biology, and of an ordinal time series regarding cloudiness. For future research, one might use the RS-DAR    ( 1 )     model for constructing parsimonious advanced models, and one might adapt techniques for smoother regime transitions.<\/jats:p>","DOI":"10.3390\/e22040458","type":"journal-article","created":{"date-parts":[[2020,4,21]],"date-time":"2020-04-21T03:23:06Z","timestamp":1587439386000},"page":"458","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Regime-Switching Discrete ARMA Models for Categorical Time Series"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8739-6631","authenticated-orcid":false,"given":"Christian H.","family":"Wei\u00df","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Helmut Schmidt University, 22043 Hamburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,17]]},"reference":[{"key":"ref_1","unstructured":"Box, G.E.P., and Jenkins, G.M. (1970). Time Series Analysis: Forecasting and Control, Holden-Day. [1st ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1214\/09-SS060","article-title":"The ARMA alphabet soup: A tour of ARMA model variants","volume":"4","author":"Holan","year":"2010","journal-title":"Stat. Surv."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1111\/j.2517-6161.1980.tb01126.x","article-title":"Threshold autoregression, limit cycles and cyclical data","volume":"42","author":"Tong","year":"1980","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"107","DOI":"10.4310\/SII.2011.v4.n2.a1","article-title":"Threshold models in time series analysis\u201330 years on","volume":"4","author":"Tong","year":"2011","journal-title":"Stat. Its Interface"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Wei\u00df, C.H. (2018). An Introduction to Discrete-Valued Time Series, John Wiley & Sons, Inc.","DOI":"10.1002\/9781119097013"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1080\/15326349.2015.1085319","article-title":"Self-exciting threshold models for time series of counts with a finite range","volume":"32","year":"2016","journal-title":"Stoch. Model."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/s10182-015-0264-6","article-title":"Self-exciting threshold binomial autoregressive processes","volume":"100","author":"Silva","year":"2016","journal-title":"AStA Adv. Stat. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2717","DOI":"10.1080\/03610926.2011.556292","article-title":"Integer-valued self-exciting threshold autoregressive processes","volume":"41","author":"Monteiro","year":"2012","journal-title":"Commun. Stat. Methods"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1002\/(SICI)1099-095X(199907\/08)10:4<395::AID-ENV364>3.0.CO;2-M","article-title":"Integer valued autoregressive models for tipping bucket rainfall measurements","volume":"10","author":"Thyregod","year":"1999","journal-title":"Environmetrics"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1080\/01621459.2013.872994","article-title":"Self-excited threshold Poisson autoregression","volume":"109","author":"Wang","year":"2014","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_11","unstructured":"Churchman, C.W., and Ratoosh, P. (1959). Measurement, psychophysics and utility. Measurement: Definitions and Theories, John Wiley & Sons, Inc."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Klein, I., Mangold, B., and Doll, M. (2016). Cumulative paired \u03d5-entropy. Entropy, 18.","DOI":"10.3390\/e18070248"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Wei\u00df, C.H. (2019). Measures of dispersion and serial dependence in categorical time series. Econometrics, 7.","DOI":"10.3390\/econometrics7020017"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Wei\u00df, C.H. (2019). Distance-based analysis of ordinal data and ordinal time series. J. Am. Stat. Assoc.","DOI":"10.1080\/01621459.2019.1604370"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1111\/j.1467-9892.1983.tb00354.x","article-title":"Stationary discrete autoregressive-moving average time series generated by mixtures","volume":"4","author":"Jacobs","year":"1983","journal-title":"J. Time Ser. Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s10182-008-0055-4","article-title":"Measuring serial dependence in categorical time series","volume":"92","year":"2008","journal-title":"AStA Adv. Stat. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.csda.2013.07.009","article-title":"Serial dependence of NDARMA processes","volume":"68","year":"2013","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Billingsley, P. (1999). Convergence of Probability Measures, John Wiley & Sons, Inc.. [2nd ed.].","DOI":"10.1002\/9780470316962"},{"key":"ref_19","unstructured":"M\u00f6ller, T.A., and Wei\u00df, C.H. (2020). Generalized discrete ARMA models. Appl. Stoch. Models Bus. Ind."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/S0022-5193(86)80075-3","article-title":"The classification of amino acid conservation","volume":"119","author":"Taylor","year":"1986","journal-title":"J. Theor. Biol."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Zucchini, W., MacDonald, I.L., and Langrock, R. (2016). Hidden Markov Models for Time Series: An Introduction Using R, Chapman & Hall\/CRC Press. [2nd ed.].","DOI":"10.1201\/b20790"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Mansour, T. (2013). Combinatorics of Set Partitions, Chapman & Hall\/CRC Press.","DOI":"10.1201\/b12691"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1111\/j.2517-6161.1985.tb01383.x","article-title":"A model for high-order Markov chains","volume":"47","author":"Raftery","year":"1985","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Billingsley, P. (1961). Statistical Inference for Markov Processes, University of Chicago Press.","DOI":"10.2307\/1401956"},{"key":"ref_25","unstructured":"Burnham, K.P., and Anderson, D.R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, Springer. [2nd ed.]."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"243","DOI":"10.2307\/1267787","article-title":"On some criteria for estimating the order of a Markov chain","volume":"23","author":"Katz","year":"1981","journal-title":"Technometrics"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/S0092-8240(89)80049-7","article-title":"Stochastic models for heterogeneous DNA sequences","volume":"51","author":"Churchill","year":"1989","journal-title":"Bull. Math. Biol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/S0378-4371(03)00399-6","article-title":"A discrete autoregressive process as a model for short-range correlations in DNA sequences","volume":"327","author":"Dehnert","year":"2003","journal-title":"Physica A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/458\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:31:29Z","timestamp":1760362289000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/458"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,17]]},"references-count":28,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["e22040458"],"URL":"https:\/\/doi.org\/10.3390\/e22040458","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,17]]}}}