{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T15:45:36Z","timestamp":1774626336210,"version":"3.50.1"},"reference-count":11,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,28]],"date-time":"2021-12-28T00:00:00Z","timestamp":1640649600000},"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>We analyze the permutation entropy of deterministic chaotic signals affected by a weak observational noise. We investigate the scaling dependence of the entropy increase on both the noise amplitude and the window length used to encode the time series. In order to shed light on the scenario, we perform a multifractal analysis, which allows highlighting the emergence of many poorly populated symbolic sequences generated by the stochastic fluctuations. We finally make use of this information to reconstruct the noiseless permutation entropy. While this approach works quite well for H\u00e9non and tent maps, it is much less effective in the case of hyperchaos. We argue about the underlying motivations.<\/jats:p>","DOI":"10.3390\/e24010054","type":"journal-article","created":{"date-parts":[[2021,12,28]],"date-time":"2021-12-28T06:55:03Z","timestamp":1640674503000},"page":"54","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Permutation Entropy of Weakly Noise-Affected Signals"],"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":"Center for Mind\/Brain Sciences (CIMeC), University of Trento, 38068 Rovereto, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8688-1870","authenticated-orcid":false,"given":"Antonio","family":"Politi","sequence":"additional","affiliation":[{"name":"Institute for Pure and Applied Mathematics and Department of Physics, University of Aberdeen, Aberdeen AB24 3UE, UK"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.physa.2005.05.022","article-title":"Ordinal analysis of time series","volume":"356","author":"Keller","year":"2005","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_2","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_3","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_4","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 Soliton. Fract."},{"key":"ref_5","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_6","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_7","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_8","doi-asserted-by":"crossref","first-page":"1154","DOI":"10.1515\/zna-1988-1221","article-title":"Fractals, Multifractals, and Thermodynamics: An Introductory Review","volume":"43","year":"1988","journal-title":"Z. Naturforschung A"},{"key":"ref_9","unstructured":"Compagnoni, A., Casey, W., Cai, Y., and Mishra, B. (2019). Classification of Permutation Distance Metrics for Fitness Landscape Analysis. Bio-Inspired Information and Communication Technologies. BICT 2019. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, Springer."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1148","DOI":"10.1137\/110851390","article-title":"Sorting by Transpositions is Difficult","volume":"26","author":"Bulteau","year":"2012","journal-title":"SIAM J. Discret. Math."},{"key":"ref_11","unstructured":"Ronald, S. (1998, January 4\u20139). More Distance Functions for Order-Based Encodings. Proceedings of the 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence (Cat. 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