{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T22:19:04Z","timestamp":1767651544847,"version":"build-2065373602"},"reference-count":50,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,9]],"date-time":"2021-08-09T00:00:00Z","timestamp":1628467200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["302785\/2017-5","308621\/2019-0"],"award-info":[{"award-number":["302785\/2017-5","308621\/2019-0"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002322","name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior","doi-asserted-by":"publisher","award":["001"],"award-info":[{"award-number":["001"]}],"id":[{"id":"10.13039\/501100002322","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100010785","name":"Canada First Research Excellence Fund","doi-asserted-by":"publisher","award":["2015276"],"award-info":[{"award-number":["2015276"]}],"id":[{"id":"10.13039\/501100010785","id-type":"DOI","asserted-by":"publisher"}]},{"name":"NSF  NeuroNex","award":["2015276"],"award-info":[{"award-number":["2015276"]}]},{"DOI":"10.13039\/501100001655","name":"Deutscher Akademischer Austauschdienst","doi-asserted-by":"publisher","award":["x"],"award-info":[{"award-number":["x"]}],"id":[{"id":"10.13039\/501100001655","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004837","name":"Ministerio de Ciencia e Innovaci\u00f3n","doi-asserted-by":"publisher","award":["PGC2018-099443-B-I00"],"award-info":[{"award-number":["PGC2018-099443-B-I00"]}],"id":[{"id":"10.13039\/501100004837","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003741","name":"Instituci\u00f3 Catalana de Recerca i Estudis Avan\u00e7ats","doi-asserted-by":"publisher","award":["academia"],"award-info":[{"award-number":["academia"]}],"id":[{"id":"10.13039\/501100003741","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Time series analysis comprises a wide repertoire of methods for extracting information from data sets. Despite great advances in time series analysis, identifying and quantifying the strength of nonlinear temporal correlations remain a challenge. We have recently proposed a new method based on training a machine learning algorithm to predict the temporal correlation parameter, \u03b1, of flicker noise (FN) time series. The algorithm is trained using as input features the probabilities of ordinal patterns computed from FN time series, x\u03b1FN(t), generated with different values of \u03b1. Then, the ordinal probabilities computed from the time series of interest, x(t), are used as input features to the trained algorithm and that returns a value, \u03b1e, that contains meaningful information about the temporal correlations present in x(t). We have also shown that the difference, \u03a9, of the permutation entropy (PE) of the time series of interest, x(t), and the PE of a FN time series generated with \u03b1=\u03b1e, x\u03b1eFN(t), allows the identification of the underlying determinism in x(t). Here, we apply our methodology to different datasets and analyze how \u03b1e and \u03a9 correlate with well-known quantifiers of chaos and complexity. We also discuss the limitations for identifying determinism in highly chaotic time series and in periodic time series contaminated by noise. The open source algorithm is available on Github.<\/jats:p>","DOI":"10.3390\/e23081025","type":"journal-article","created":{"date-parts":[[2021,8,9]],"date-time":"2021-08-09T09:03:53Z","timestamp":1628499833000},"page":"1025","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Evaluating Temporal Correlations in Time Series Using Permutation Entropy, Ordinal Probabilities and Machine Learning"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0160-5338","authenticated-orcid":false,"given":"Bruno R. R.","family":"Boaretto","sequence":"first","affiliation":[{"name":"Department of Physics, Universidade Federal do Paran\u00e1, Curitiba 81531-980, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5189-2059","authenticated-orcid":false,"given":"Roberto C.","family":"Budzinski","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Western University, London, ON N6A 3K7, Canada"},{"name":"Brain and Mind Institute, Western University, London, ON N6A 3K7, Canada"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5055-3012","authenticated-orcid":false,"given":"Kalel L.","family":"Rossi","sequence":"additional","affiliation":[{"name":"Theoretical Physics\/Complex Systems, ICBM, Carl von Ossietzky University Oldenburg, 26129 Oldenburg, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6897-3034","authenticated-orcid":false,"given":"Thiago L.","family":"Prado","sequence":"additional","affiliation":[{"name":"Department of Physics, Universidade Federal do Paran\u00e1, Curitiba 81531-980, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7049-4902","authenticated-orcid":false,"given":"Sergio R.","family":"Lopes","sequence":"additional","affiliation":[{"name":"Department of Physics, Universidade Federal do Paran\u00e1, Curitiba 81531-980, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0768-2019","authenticated-orcid":false,"given":"Cristina","family":"Masoller","sequence":"additional","affiliation":[{"name":"Department of Physics, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1098\/rspa.1998.0193","article-title":"The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis","volume":"454","author":"Huang","year":"1998","journal-title":"Proc. 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