{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T18:43:55Z","timestamp":1777574635578,"version":"3.51.4"},"reference-count":68,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,6,22]],"date-time":"2022-06-22T00:00:00Z","timestamp":1655856000000},"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>In the last decade permutation entropy (PE) has become a popular tool to analyze the degree of randomness within a time series. In typical applications, changes in the dynamics of a source are inferred by observing changes of PE computed on different time series generated by that source. However, most works neglect the crucial question related to the statistical significance of these changes. The main reason probably lies in the difficulty of assessing, out of a single time series, not only the PE value, but also its uncertainty. In this paper we propose a method to overcome this issue by using generation of surrogate time series. The analysis conducted on both synthetic and experimental time series shows the reliability of the approach, which can be promptly implemented by means of widely available numerical tools. The method is computationally affordable for a broad range of users.<\/jats:p>","DOI":"10.3390\/e24070853","type":"journal-article","created":{"date-parts":[[2022,6,22]],"date-time":"2022-06-22T21:31:06Z","timestamp":1655933466000},"page":"853","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Estimating Permutation Entropy Variability via Surrogate Time Series"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9454-0988","authenticated-orcid":false,"given":"Leonardo","family":"Ricci","sequence":"first","affiliation":[{"name":"Department of Physics, University of Trento, 38123 Trento, Italy"},{"name":"CIMeC, Center for Mind\/Brain Sciences, University of Trento, 38068 Rovereto, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5603-3950","authenticated-orcid":false,"given":"Alessio","family":"Perinelli","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Trento, 38123 Trento, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Verd\u00fa, S. (2019). Empirical Estimation of Information Measures: A Literature Guide. Entropy, 21.","DOI":"10.3390\/e21080720"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1016\/0960-0779(92)90058-U","article-title":"Word frequency and entropy of symbolic sequences: A dynamical perspective","volume":"2","author":"Ebeling","year":"1992","journal-title":"Chaos Solitons Fractals"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2265","DOI":"10.1142\/S0218127408021695","article-title":"Symbolic dynamics generated by a combination of graphs","volume":"18","author":"Basios","year":"2008","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3465","DOI":"10.1142\/S0218127411030660","article-title":"Symbolic dynamics, coarse graining and the monitoring of complex systems","volume":"21","author":"Basios","year":"2011","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5793","DOI":"10.1103\/PhysRevE.53.5793","article-title":"Asymptotic scaling behavior of block entropies for an intermittent process","volume":"53","author":"Freund","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"053122","DOI":"10.1063\/5.0048396","article-title":"Symbolic dynamics of music from Europe and Japan","volume":"31","author":"Basios","year":"2021","journal-title":"Chaos"},{"key":"ref_7","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_8","doi-asserted-by":"crossref","unstructured":"Ricci, L., and Politi, A. (2022). Permutation Entropy of Weakly Noise-Affected Signals. Entropy, 24.","DOI":"10.3390\/e24010054"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/j.compbiomed.2011.03.017","article-title":"Classifying cardiac biosignals using ordinal pattern statistics and symbolic dynamics","volume":"42","author":"Parlitz","year":"2012","journal-title":"Comput. Biol. Med."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"021906","DOI":"10.1103\/PhysRevE.85.021906","article-title":"Modified permutation-entropy analysis of heartbeat dynamics","volume":"85","author":"Bian","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Zanin, M., Zunino, L., Rosso, O.A., and Papo, D. (2012). Permutation Entropy and Its Main Biomedical and Econophysics Applications: A Review. Entropy, 14.","DOI":"10.3390\/e14081553"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Shumbayawonda, E., Fern\u00e1ndez, A., Hughes, M.P., and Ab\u00e1solo, D. (2017). Permutation Entropy for the Characterisation of Brain Activity Recorded with Magnetoencephalograms in Healthy Ageing. Entropy, 19.","DOI":"10.3390\/e19040141"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Redelico, F.O., Traversaro, F., Garc\u00eda, M.D.C., Silva, W., Rosso, O.A., and Risk, M. (2017). Classification of Normal and Pre-Ictal EEG Signals Using Permutation Entropies and a Generalized Linear Model as a Classifier. Entropy, 19.","DOI":"10.3390\/e19020072"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1016\/j.bspc.2017.08.012","article-title":"Unbiased estimation of permutation entropy in EEG analysis for Alzheimer\u2019s disease classification","volume":"39","author":"Kukal","year":"2018","journal-title":"Biomed. Signal Process. Control"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"106116","DOI":"10.1016\/j.cmpb.2021.106116","article-title":"Complexity of EEG Dynamics for Early Diagnosis of Alzheimer\u2019s Disease Using Permutation Entropy Neuromarker","volume":"206","author":"Yener","year":"2021","journal-title":"Comput. Methods Programs Biomed."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5022","DOI":"10.1016\/j.physa.2010.07.006","article-title":"Entropy analysis of the dynamics of El Ni\u00f1o\/Southern Oscillation during the Holocene","volume":"389","author":"Saco","year":"2010","journal-title":"Physics A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2061","DOI":"10.1142\/S021798491102725X","article-title":"Complexity Furthermore, Synchronization In Chaotic Injection Locking Semiconductor Lasers","volume":"25","author":"Yang","year":"2011","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"17840","DOI":"10.1364\/OE.22.017840","article-title":"Complexity in pulsed nonlinear laser systems interrogated by permutation entropy","volume":"22","author":"Toomey","year":"2014","journal-title":"Opt. Express"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"124485","DOI":"10.1016\/j.physa.2020.124485","article-title":"Multiscale complexity analysis on airport air traffic flow volume time series","volume":"548","author":"Liu","year":"2020","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"144101","DOI":"10.1103\/PhysRevLett.118.144101","article-title":"Quantifying the Dynamical Complexity of Chaotic Time Series","volume":"118","author":"Politi","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"046217","DOI":"10.1103\/PhysRevE.70.046217","article-title":"Detecting dynamical changes in time series using the permutation entropy","volume":"70","author":"Cao","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1140\/epjst\/e2013-01861-8","article-title":"Segmentation and classification of time series using ordinal pattern distributions","volume":"222","author":"Sinn","year":"2013","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Gao, J., Hou, Y., Fan, F., and Liu, F. (2020). Complexity Changes in the US and China\u2019s Stock Markets: Differences, Causes, and Wider Social Implications. Entropy, 22.","DOI":"10.3390\/e22010075"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3049","DOI":"10.3390\/e16063049","article-title":"Using Permutation Entropy to Measure the Changes in EEG Signals during Absence Seizures","volume":"16","author":"Li","year":"2014","journal-title":"Entropy"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"9254309","DOI":"10.1155\/2019\/9254309","article-title":"Effects of Ageing and Sex on Complexity in the Human Sleep EEG: A Comparison of Three Symbolic Dynamic Analysis Methods","volume":"2019","author":"Tosun","year":"2019","journal-title":"Complexity"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Rubega, M., Scarpa, F., Teodori, D., Sejling, A.-S., Frandsen, C.S., and Sparacino, G. (2020). Detection of Hypoglycemia Using Measures of EEG Complexity in Type 1 Diabetes Patients. Entropy, 22.","DOI":"10.3390\/e22010081"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"zsaa226","DOI":"10.1093\/sleep\/zsaa226","article-title":"Changes in EEG permutation entropy in the evening and in the transition from wake to sleep","volume":"44","author":"Hou","year":"2020","journal-title":"Sleep"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.jastp.2013.11.005","article-title":"Permutation entropy analysis of complex magnetospheric dynamics","volume":"115\u2013116","author":"Consolini","year":"2014","journal-title":"J. Atmos.-Sol.-Terr. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"37733","DOI":"10.1038\/srep37733","article-title":"Reduction of randomness in seismic noise as a short-term precursor to a volcanic eruption","volume":"6","author":"Glynn","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"106307","DOI":"10.1063\/1.5036959","article-title":"Differentiating resting brain states using ordinal symbolic analysis","volume":"28","author":"Montesano","year":"2018","journal-title":"Chaos"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Vecchio, F., Miraglia, F., Pappalettera, C., Orticoni, A., Al\u00f9, F., Judica, E., Cotelli, M., and Rossini, P.M. (2021). Entropy as Measure of Brain Networks\u2019 Complexity in Eyes Open and Closed Conditions. Symmetry, 13.","DOI":"10.3390\/sym13112178"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"052126","DOI":"10.1103\/PhysRevE.95.052126","article-title":"Variance of permutation entropy and the influence of ordinal pattern selection","volume":"95","author":"Little","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.chaos.2018.12.039","article-title":"Permutation entropy revisited","volume":"120","author":"Watt","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"022215","DOI":"10.1103\/PhysRevE.103.022215","article-title":"Asymptotic distribution of sample Shannon entropy in the case of an underlying finite, regular Markov chain","volume":"103","author":"Ricci","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"024220","DOI":"10.1103\/PhysRevE.104.024220","article-title":"Estimating the variance of Shannon entropy","volume":"104","author":"Ricci","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"388","DOI":"10.1016\/j.cnsns.2017.10.013","article-title":"Confidence intervals and hypothesis testing for the permutation entropy with an application to epilepsy","volume":"57","author":"Traversaro","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/0167-2789(92)90102-S","article-title":"Testing for nonlinearity in time series: The method of surrogate data","volume":"58","author":"Theiler","year":"1992","journal-title":"Phys. D"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1016\/S0167-2789(00)00043-9","article-title":"Surrogate time series","volume":"142","author":"Schreiber","year":"2000","journal-title":"Physics D"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1103\/PhysRevLett.77.635","article-title":"Improved Surrogate Data for Nonlinearity Tests","volume":"77","author":"Schreiber","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1137\/1104033","article-title":"On a statistical estimate for the entropy of a sequence of independent random variables","volume":"4","author":"Basharin","year":"1959","journal-title":"Theor. Probab. Appl."},{"key":"ref_41","unstructured":"Miller, G. (1955). Note on the bias of information estimates. Information Theory in Psychology II-B, Free Press."},{"key":"ref_42","first-page":"323","article-title":"The statistical estimation of entropy in the non-parametric case","volume":"16","author":"Harris","year":"1975","journal-title":"Topics. Inf. Theory"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"051139","DOI":"10.1103\/PhysRevE.85.051139","article-title":"Estimation of the entropy based on its polynomial representation","volume":"85","author":"Vinck","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"063110","DOI":"10.1063\/5.0049901","article-title":"ordpy: A Python package for data analysis with permutation entropy and ordinal network methods","volume":"31","author":"Pessa","year":"2021","journal-title":"Chaos"},{"key":"ref_45","unstructured":"(2022, May 23). Function Implementing Permutation Entropy in R. Available online: https:\/\/rdrr.io\/cran\/statcomp\/man\/permutation_entropy.html."},{"key":"ref_46","unstructured":"(2022, May 23). Function Implementing Permutation Entropy in Matlab. Available online: https:\/\/it.mathworks.com\/matlabcentral\/fileexchange\/44161-permutation-entropy-fast-algorithm."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1126","DOI":"10.1152\/jn.00093.2008","article-title":"Data-Driven Significance Estimation for Precise Spike Correlation","volume":"101","year":"2009","journal-title":"J. Neurophysiol."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2018.06.001","article-title":"Surrogate data for hypothesis testing of physical systems","volume":"748","author":"Lancaster","year":"2018","journal-title":"Phys. Rep."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"063127","DOI":"10.1063\/1.5025242","article-title":"Correlation in brain networks at different time scale resolution","volume":"28","author":"Perinelli","year":"2018","journal-title":"Chaos"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"765332","DOI":"10.3389\/fnetp.2021.765332","article-title":"Measuring the Rate of Information Exchange in Point-Process Data With Application to Cardiovascular Variability","volume":"1","author":"Mijatovic","year":"2022","journal-title":"Front. Netw. Physiol."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"2105","DOI":"10.1103\/PhysRevLett.80.2105","article-title":"Constrained Randomization of Time Series Data","volume":"80","author":"Schreiber","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"073120","DOI":"10.1063\/5.0011328","article-title":"SpiSeMe: A multi-language package for spike train surrogate generation","volume":"30","author":"Perinelli","year":"2020","journal-title":"Chaos"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1063\/1.166424","article-title":"Practical implementation of nonlinear time series methods: The TISEAN package","volume":"9","author":"Hegger","year":"1999","journal-title":"Chaos"},{"key":"ref_54","unstructured":"(2022, May 23). TISEAN Website. Available online: https:\/\/www.pks.mpg.de\/%7Etisean\/Tisean_3.0.1\/."},{"key":"ref_55","unstructured":"(2022, May 23). Function Implementing the IAAFT Algorithm in, R. Available online: https:\/\/rdrr.io\/github\/dpabon\/ecofunr\/src\/R\/iAAFT.R."},{"key":"ref_56","unstructured":"(2022, May 23). Function Implementing a Modified Version of the IAAFT Algorithm in Python. Available online: https:\/\/github.com\/mlcs\/iaaft."},{"key":"ref_57","unstructured":"(2022, May 23). Function Implementing the IAAFT Algorithm in Matlab. Available online: https:\/\/github.com\/nmitrou\/Simulations\/blob\/master\/matlab_codes\/IAAFT.m."},{"key":"ref_58","unstructured":"(2022, May 23). IAAFT Functions. Available online: https:\/\/github.com\/LeonardoRicci\/iaaft or https:\/\/osf.io\/emkpj."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"180308","DOI":"10.1038\/sdata.2018.308","article-title":"A mind-brain-body dataset of MRI, EEG, cognition, emotion, and peripheral physiology in young and old adults","volume":"6","author":"Babayan","year":"2019","journal-title":"Sci. Data"},{"key":"ref_60","unstructured":"(2022, May 23). LEMON Public Database. Available online: http:\/\/fcon_1000.projects.nitrc.org\/indi\/retro\/MPI_LEMON.html."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"073106","DOI":"10.1063\/5.0053857","article-title":"Relationship between mutual information and cross-correlation time scale of observability as measures of connectivity strength","volume":"31","author":"Perinelli","year":"2021","journal-title":"Chaos"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"119247","DOI":"10.1016\/j.neuroimage.2022.119247","article-title":"Power shift and connectivity changes in healthy aging during resting-state EEG","volume":"256","author":"Perinelli","year":"2022","journal-title":"NeuroImage"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1038\/nature18933","article-title":"A multi-modal parcellation of human cerebral cortex","volume":"536","author":"Glasser","year":"2016","journal-title":"Nature"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1073\/pnas.88.6.2297","article-title":"Approximate entropy as a measure of system complexity","volume":"88","author":"Pincus","year":"1991","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Delgado-Bonal, A., and Marshak, A. (2019). Approximate Entropy and Sample Entropy: A Comprehensive Tutorial. Entropy, 21.","DOI":"10.3390\/e21060541"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"H2039","DOI":"10.1152\/ajpheart.2000.278.6.H2039","article-title":"Physiological time-series analysis using approximate entropy and sample entropy","volume":"278","author":"Richman","year":"2000","journal-title":"Am. J. Physiol.-Heart Circ. Physiol."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"127","DOI":"10.3389\/fncom.2010.00127","article-title":"Surrogate Spike Train Generation Through Dithering in Operational Time","volume":"4","author":"Louis","year":"2010","journal-title":"Front. Comput. Neurosci."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"121102","DOI":"10.1063\/1.5138250","article-title":"Generation of surrogate event sequences via joint distribution of successive inter-event intervals","volume":"29","author":"Ricci","year":"2019","journal-title":"Chaos"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/7\/853\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:37:23Z","timestamp":1760139443000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/7\/853"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,22]]},"references-count":68,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["e24070853"],"URL":"https:\/\/doi.org\/10.3390\/e24070853","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,22]]}}}