{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:14:58Z","timestamp":1760058898507,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,6]],"date-time":"2025-05-06T00:00:00Z","timestamp":1746489600000},"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>Permutation entropy is customarily implemented to quantify the intrinsic indeterminacy of complex time series, under the assumption that determinism manifests itself by lowering the (permutation) entropy of the resulting symbolic sequence. We expect this to be roughly true, but, in general, it is not clear to what extent a given ordinal pattern indeed provides a faithful reconstruction of the original signal. Here, we address this question by attempting the reconstruction of the original time series by invoking an ergodic Markov approximation of the symbolic dynamics, thereby inverting the encoding procedure. Using the H\u00e9non map as a testbed, we show that a meaningful reconstruction can also be made in the presence of a small observational noise.<\/jats:p>","DOI":"10.3390\/e27050499","type":"journal-article","created":{"date-parts":[[2025,5,6]],"date-time":"2025-05-06T04:28:22Z","timestamp":1746505702000},"page":"499","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Improved Reconstruction of Chaotic Signals from Ordinal Networks"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8688-1870","authenticated-orcid":false,"given":"Antonio","family":"Politi","sequence":"first","affiliation":[{"name":"Department of Physics, University of Aberdeen, Aberdeen AB24 3UE, UK"},{"name":"Institute for Complex Systems, National Research Council, (ISC-CNR), Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9454-0988","authenticated-orcid":false,"given":"Leonardo","family":"Ricci","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Trento, 38123 Trento, Italy"},{"name":"Center for Mind\/Brain Sciences, CIMeC, University of Trento, 38068 Rovereto, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2554","DOI":"10.1073\/pnas.79.8.2554","article-title":"Neural networks and physical systems with emergent collective computational abilities","volume":"79","author":"Hopfield","year":"1982","journal-title":"Proc. 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