{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T14:43:44Z","timestamp":1791470624496,"version":"4.3.4"},"reference-count":48,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2018,9,3]],"date-time":"2018-09-03T00:00:00Z","timestamp":1535932800000},"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>Time irreversibility, i.e., the lack of invariance of the statistical properties of a system under time reversal, is a fundamental property of all systems operating out of equilibrium. Time reversal symmetry is associated with important statistical and physical properties and is related to the predictability of the system generating the time series. Over the past fifteen years, various methods to quantify time irreversibility in time series have been proposed, but these can be computationally expensive. Here, we propose a new method, based on permutation entropy, which is essentially parameter-free, temporally local, yields straightforward statistical tests, and has fast convergence properties. We apply this method to the study of financial time series, showing that stocks and indices present a rich irreversibility dynamics. We illustrate the comparative methodological advantages of our method with respect to a recently proposed method based on visibility graphs, and discuss the implications of our results for financial data analysis and interpretation.<\/jats:p>","DOI":"10.3390\/e20090665","type":"journal-article","created":{"date-parts":[[2022,5,19]],"date-time":"2022-05-19T21:56:59Z","timestamp":1652997419000},"page":"665","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":45,"title":["Assessing Time Series Reversibility through Permutation Patterns"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5839-0393","authenticated-orcid":false,"given":"Massimiliano","family":"Zanin","sequence":"first","affiliation":[{"name":"Center for Biomedical Technology, Universidad Polit\u00e9cnica de Madrid, 28223 Pozuelo de Alarc\u00f3n, 28040 Madrid, Spain"},{"name":"Department of Computer Science, Faculty of Science and Technology, Universidade Nova de Lisboa, 2829-516 Lisboa, Portugal"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8801-4762","authenticated-orcid":false,"given":"Alejandro","family":"Rodr\u00edguez-Gonz\u00e1lez","sequence":"additional","affiliation":[{"name":"Center for Biomedical Technology, Universidad Polit\u00e9cnica de Madrid, 28223 Pozuelo de Alarc\u00f3n, 28040 Madrid, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5615-6798","authenticated-orcid":false,"given":"Ernestina","family":"Menasalvas Ruiz","sequence":"additional","affiliation":[{"name":"Center for Biomedical Technology, Universidad Polit\u00e9cnica de Madrid, 28223 Pozuelo de Alarc\u00f3n, 28040 Madrid, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Papo","sequence":"additional","affiliation":[{"name":"SCALab UMR CNRS 9193, University of Lille, 59800 Villeneuve d\u2019Ascq, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"30004","DOI":"10.1209\/0295-5075\/88\/30004","article-title":"Irreversible effects of memory","volume":"88","author":"Puglisi","year":"2009","journal-title":"Europhys. 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