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To obtain reliable results, one (often overlooked) prerequisite involves the stationarity of an analyzed time series, without which spurious functional connections may emerge. Here, we show how ordinal patterns and metrics derived from them can be used to assess the effectiveness of detrending methods. We apply this approach to data representing the evolution of delays in major European and US airports, and to synthetic versions of the same, obtaining operational conclusions about how these propagate in the two systems.<\/jats:p>","DOI":"10.3390\/e27030230","type":"journal-article","created":{"date-parts":[[2025,2,24]],"date-time":"2025-02-24T06:47:17Z","timestamp":1740379637000},"page":"230","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Evaluating Methods for Detrending Time Series Using Ordinal Patterns, with an Application to Air Transport Delays"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6212-8865","authenticated-orcid":false,"given":"Felipe","family":"Olivares","sequence":"first","affiliation":[{"name":"Instituto de F\u00edsica Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus UIB, 07122 Palma, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-1751-9443","authenticated-orcid":false,"given":"F. Javier","family":"Mar\u00edn-Rodr\u00edguez","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus UIB, 07122 Palma, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3542-6119","authenticated-orcid":false,"given":"Kishor","family":"Acharya","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus UIB, 07122 Palma, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5839-0393","authenticated-orcid":false,"given":"Massimiliano","family":"Zanin","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus UIB, 07122 Palma, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1126\/science.177.4047.393","article-title":"More Is Different: Broken symmetry and the nature of the hierarchical structure of science","volume":"177","author":"Anderson","year":"1972","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1038\/35065725","article-title":"Exploring complex networks","volume":"410","author":"Strogatz","year":"2001","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1038\/nrn2575","article-title":"Complex brain networks: Graph theoretical analysis of structural and functional systems","volume":"10","author":"Bullmore","year":"2009","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"247","DOI":"10.31887\/DCNS.2013.15.3\/osporns","article-title":"Structure and function of complex brain networks","volume":"15","author":"Sporns","year":"2013","journal-title":"Dialogues Clin. Neurosci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1238411","DOI":"10.1126\/science.1238411","article-title":"Structural and functional brain networks: From connections to cognition","volume":"342","author":"Park","year":"2013","journal-title":"Science"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1555","DOI":"10.1126\/science.1099511","article-title":"A probabilistic functional network of yeast genes","volume":"306","author":"Lee","year":"2004","journal-title":"Science"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"19033","DOI":"10.1073\/pnas.0609152103","article-title":"Using the principle of entropy maximization to infer genetic interaction networks from gene expression patterns","volume":"103","author":"Lezon","year":"2006","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1038\/nrg2085","article-title":"Exploring genetic interactions and networks with yeast","volume":"8","author":"Boone","year":"2007","journal-title":"Nat. Rev. Genet."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1016\/j.physa.2003.10.045","article-title":"The architecture of the climate network","volume":"333","author":"Tsonis","year":"2004","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"48007","DOI":"10.1209\/0295-5075\/87\/48007","article-title":"The backbone of the climate network","volume":"87","author":"Donges","year":"2009","journal-title":"Europhys. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1016\/j.cja.2017.01.012","article-title":"Network analysis of Chinese air transport delay propagation","volume":"30","author":"Zanin","year":"2017","journal-title":"Chin. J. Aeronaut."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"015001","DOI":"10.1088\/2632-072X\/ac4003","article-title":"Air delay propagation patterns in Europe from 2015 to 2018: An information processing perspective","volume":"3","author":"Pastorino","year":"2021","journal-title":"J. Phys. Complex."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"102585","DOI":"10.1016\/j.tre.2021.102585","article-title":"Detecting delay propagation in regional air transport systems using convergent cross mapping and complex network theory","volume":"157","author":"Guo","year":"2022","journal-title":"Transp. Res. Part E Logist. Transp. Rev."},{"key":"ref_14","unstructured":"Carlier, S., De L\u00e9pinay, I., Hustache, J.C., and Jelinek, F. (2007, January 2\u20135). Environmental impact of air traffic flow management delays. Proceedings of the 7th USA\/Europe air traffic management research and development seminar (ATM2007), Barcelona, Spain."},{"key":"ref_15","first-page":"107","article-title":"The economic cost of airline flight delay","volume":"47","author":"Peterson","year":"2013","journal-title":"J. Transp. Econ. Policy (JTEP)"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.physrep.2005.10.009","article-title":"Complex networks: Structure and dynamics","volume":"424","author":"Boccaletti","year":"2006","journal-title":"Phys. Rep."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1080\/00018730601170527","article-title":"Characterization of complex networks: A survey of measurements","volume":"56","author":"Costa","year":"2007","journal-title":"Adv. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/0304-4076(92)90104-Y","article-title":"Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root?","volume":"54","author":"Kwiatkowski","year":"1992","journal-title":"J. Econom."},{"key":"ref_19","first-page":"427","article-title":"Distribution of the estimators for autoregressive time series with a unit root","volume":"74","author":"Dickey","year":"1979","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1038\/s42005-021-00696-z","article-title":"Ordinal patterns-based methodologies for distinguishing chaos from noise in discrete time series","volume":"4","author":"Zanin","year":"2021","journal-title":"Commun. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"31001","DOI":"10.1209\/0295-5075\/ac6a72","article-title":"20 years of ordinal patterns: Perspectives and challenges","volume":"138","author":"Leyva","year":"2022","journal-title":"Europhys. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"646","DOI":"10.1111\/j.1467-9892.2007.00528.x","article-title":"Order patterns in time series","volume":"28","author":"Bandt","year":"2007","journal-title":"J. Time Ser. Anal."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"093107","DOI":"10.1063\/5.0094943","article-title":"Non-parametric tests for serial dependence in time series based on asymptotic implementations of ordinal-pattern statistics","volume":"32","year":"2022","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"033114","DOI":"10.1063\/5.0136492","article-title":"Continuous ordinal patterns: Creating a bridge between ordinal analysis and deep learning","volume":"33","author":"Zanin","year":"2023","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"043133","DOI":"10.1063\/5.0189642","article-title":"How representative are air transport functional complex networks? A quantitative validation","volume":"34","author":"Acharya","year":"2024","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Lee, T.W., and Lee, T.W. (1998). Independent Component Analysis, Springer.","DOI":"10.1007\/978-1-4757-2851-4"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"174102","DOI":"10.1103\/PhysRevLett.88.174102","article-title":"Permutation entropy: A natural complexity measure for time series","volume":"88","author":"Bandt","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"50001","DOI":"10.1209\/0295-5075\/79\/50001","article-title":"True and false forbidden patterns in deterministic and random dynamics","volume":"79","author":"Zambrano","year":"2007","journal-title":"Europhys. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"046212","DOI":"10.1103\/PhysRevE.82.046212","article-title":"Permutation-information-theory approach to unveil delay dynamics from time-series analysis","volume":"82","author":"Zunino","year":"2010","journal-title":"Phys. Rev. E\u2014Stat. Nonlinear Soft Matter Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.physa.2011.07.030","article-title":"Causality and the entropy\u2013complexity plane: Robustness and missing ordinal patterns","volume":"391","author":"Rosso","year":"2012","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"154102","DOI":"10.1103\/PhysRevLett.99.154102","article-title":"Distinguishing noise from chaos","volume":"99","author":"Rosso","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"046210","DOI":"10.1103\/PhysRevE.86.046210","article-title":"Distinguishing chaotic and stochastic dynamics from time series by using a multiscale symbolic approach","volume":"86","author":"Zunino","year":"2012","journal-title":"Phys. Rev. E\u2014Stat. Nonlinear Soft Matter Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"080401","DOI":"10.1063\/5.0167263","article-title":"Ordinal methods: Concepts, applications, new developments, and challenges-In memory of Karsten Keller (1961\u20132022)","volume":"33","author":"Rosso","year":"2023","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.physleta.2006.01.093","article-title":"Order patterns and chaos","volume":"355","author":"Kocarev","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2020","DOI":"10.1016\/j.physa.2010.01.030","article-title":"Missing ordinal patterns in correlated noises","volume":"389","author":"Carpi","year":"2010","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"122100","DOI":"10.1016\/j.physa.2019.122100","article-title":"Revisiting the decay of missing ordinal patterns in long-term correlated time series","volume":"534","author":"Olivares","year":"2019","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2854","DOI":"10.1016\/j.physa.2009.03.042","article-title":"Forbidden patterns, permutation entropy and stock market inefficiency","volume":"388","author":"Zunino","year":"2009","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_38","first-page":"367","article-title":"Quantifying stochasticity in the dynamics of delay-coupled semiconductor lasers via forbidden patterns","volume":"368","author":"Buldu","year":"2010","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"042215","DOI":"10.1103\/PhysRevE.100.042215","article-title":"Unraveling the decay of the number of unobserved ordinal patterns in noisy chaotic dynamics","volume":"100","author":"Olivares","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"033124","DOI":"10.1063\/5.0132602","article-title":"Statistics and contrasts of order patterns in univariate time series","volume":"33","author":"Bandt","year":"2023","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"123111","DOI":"10.1063\/1.5055855","article-title":"Detection of time reversibility in time series by ordinal patterns analysis","volume":"28","author":"Chavez","year":"2018","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Zanin, M., Rodr\u00edguez-Gonz\u00e1lez, A., Menasalvas Ruiz, E., and Papo, D. (2018). Assessing time series reversibility through permutation patterns. Entropy, 20.","DOI":"10.20944\/preprints201808.0083.v1"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"045310","DOI":"10.1103\/PhysRevE.105.045310","article-title":"Permutation Jensen-Shannon distance: A versatile and fast symbolic tool for complex time-series analysis","volume":"105","author":"Zunino","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1109\/18.61115","article-title":"Divergence measures based on the Shannon entropy","volume":"37","author":"Lin","year":"1991","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"033142","DOI":"10.1063\/5.0134848","article-title":"Markov-modulated model for landing flow dynamics: An ordinal analysis validation","volume":"33","author":"Olivares","year":"2023","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"107606","DOI":"10.1016\/j.cnsns.2023.107606","article-title":"Augmenting granger causality through continuous ordinal patterns","volume":"128","author":"Zanin","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_47","first-page":"424","article-title":"Investigating causal relations by econometric models and cross-spectral methods","volume":"37","author":"Granger","year":"1969","journal-title":"Econom. J. Econom. Soc."},{"key":"ref_48","unstructured":"Diebold, F.X. (1998). Independent Component Analysis, Cengage Learning. Elements of Forecasting."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1002\/j.1538-7305.1948.tb00917.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"066138","DOI":"10.1103\/PhysRevE.69.066138","article-title":"Estimating mutual information","volume":"69","author":"Kraskov","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1103\/PhysRevLett.85.461","article-title":"Measuring Information Transfer","volume":"85","author":"Schreiber","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_52","unstructured":"Beckenbach, E.F. (1956). The theory of prediction. Modern Mathematics for the Engineer, McGraw-Hill."},{"key":"ref_53","first-page":"1958","article-title":"Assessing Coupling Dynamics from an Ensemble of Time Series","volume":"17","author":"Wu","year":"2010","journal-title":"Entropy"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1016\/j.cja.2017.01.001","article-title":"ATM performance measurement in Europe, the US and China","volume":"30","author":"Cook","year":"2017","journal-title":"Chin. J. Aeronaut."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1177\/0049124194022004003","article-title":"Detrending time series: A cautionary note","volume":"22","author":"Raffalovich","year":"1994","journal-title":"Sociol. Methods Res."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.cmpb.2016.02.008","article-title":"Amplitude-aware permutation entropy: Illustration in spike detection and signal segmentation","volume":"128","author":"Azami","year":"2016","journal-title":"Comput. Methods Programs Biomed."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"6842","DOI":"10.3934\/mbe.2019342","article-title":"Permutation entropy: Influence of amplitude information on time series classification performance","volume":"16","year":"2019","journal-title":"Math. Biosci. Eng."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"022911","DOI":"10.1103\/PhysRevE.87.022911","article-title":"Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information","volume":"87","author":"Fadlallah","year":"2013","journal-title":"Phys. Rev. E\u2014Stat. Nonlinear Soft Matter Phys."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Stosic, D., Stosic, D., Stosic, T., and Stosic, B. (2022). Generalized weighted permutation entropy. Chaos Interdiscip. J. Nonlinear Sci., 32.","DOI":"10.1063\/5.0107427"},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Cuesta-Frau, D. (2019). Slope entropy: A new time series complexity estimator based on both symbolic patterns and amplitude information. Entropy, 21.","DOI":"10.3390\/e21121167"},{"key":"ref_61","unstructured":"Knight, M.I., Nunes, M., and Nason, G. (2016). Modelling, detrending and decorrelation of network time series. arXiv."},{"key":"ref_62","unstructured":"Eurocontrol, and FAA (2024, December 25). Comparison of Air Traffic Management Related Operational and Economic Performance U.S.\u2014Europe; Technical Report; 2024, Available online: https:\/\/www.eurocontrol.int\/publication\/comparison-air-traffic-management-related-operational-and-economic-performance."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/3\/230\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:41:00Z","timestamp":1760028060000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/3\/230"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,23]]},"references-count":62,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["e27030230"],"URL":"https:\/\/doi.org\/10.3390\/e27030230","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,23]]}}}