{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,14]],"date-time":"2026-09-14T22:49:46Z","timestamp":1789426186569,"version":"build-2803163510"},"reference-count":37,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,6]],"date-time":"2020-02-06T00:00:00Z","timestamp":1580947200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Xiaofeng Zhang","award":["11574191"],"award-info":[{"award-number":["11574191"]}]},{"name":"Xiaofeng Zhang","award":["11874252"],"award-info":[{"award-number":["11874252"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>An improved permutation entropy (PE) algorithm named coded permutation entropy (CPE) is proposed in this paper to optimize the problems existing in PE based on the secondary partitioning. The principle of CPE algorithm is given, and the performance of it for dynamical change detection is analyzed using synthetic signal, logistic map and Lorenz map. The detection ability of CPE algorithm in different signal-to-noise ratios (SNR) is studied and the algorithm complexity is discussed. The results show that CPE can accurately capture minor feature information and amplify the detection results of dynamical changes compared with PE, weighted permutation entropy (WPE) and amplitude-aware permutation entropy (AAPE), but it has less robustness to noise and requires a higher computation cost than the others. Finally, we use the new algorithm to analyze the rolling bearing fault signals. The application of actual signals illustrates that CPE performs better in detecting abnormal pulse of the rolling bearing when the embedded dimension is small. From all the analyses in this paper, we find that CPE has a better performance for dynamical change detection compared with the other three algorithms when there is a larger repetition rate of permutation pattern in the position sequences.<\/jats:p>","DOI":"10.3390\/e22020187","type":"journal-article","created":{"date-parts":[[2020,2,7]],"date-time":"2020-02-07T03:13:27Z","timestamp":1581045207000},"page":"187","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Coded Permutation Entropy: A Measure for Dynamical Changes Based on the Secondary Partitioning of Amplitude Information"],"prefix":"10.3390","volume":"22","author":[{"given":"Huan","family":"Kang","sequence":"first","affiliation":[{"name":"School of Physics and Information Technology, Shaanxi Normal University, Xi\u2019an 710119, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9933-7495","authenticated-orcid":false,"given":"Xiaofeng","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Physics and Information Technology, Shaanxi Normal University, Xi\u2019an 710119, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guangbin","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Physics and Information Technology, Shaanxi Normal University, Xi\u2019an 710119, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,6]]},"reference":[{"key":"ref_1","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_2","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_3","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1016\/j.neuroimage.2013.08.056","article-title":"Phase transfer entropy: A novel phase-based measure for directed connectivity in networks coupled by oscillatory interactions","volume":"85","author":"Lobier","year":"2014","journal-title":"NeuroImage"},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"052902","DOI":"10.1103\/PhysRevE.73.052902","article-title":"Dynamical complexity detection in short-term physiological series using base-scale entropy","volume":"73","author":"Li","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1103\/PhysRevLett.89.068102","article-title":"Multiscale entropy analysis of complex physiologic time series","volume":"89","author":"Costa","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"021906","DOI":"10.1103\/PhysRevE.71.021906","article-title":"Multiscale entropy analysis of biological signals","volume":"71","author":"Costa","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0167-2789(85)90011-9","article-title":"Determining Lyapunov exponents from a time series","volume":"16","author":"Wolf","year":"1985","journal-title":"Physica D"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.physleta.2006.11.094","article-title":"Nonlinear finite-time Lyapunov exponent and predictability","volume":"364","author":"Ding","year":"2007","journal-title":"Phys. Lett. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/0375-9601(95)00867-5","article-title":"A statistical measure of complexity","volume":"209","author":"Mancini","year":"1995","journal-title":"Phys. Lett. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/j.physa.2005.11.053","article-title":"Generalized statistical complexity measures: Geometrical and analytical properties","volume":"369","author":"Martin","year":"2006","journal-title":"Physica A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1016\/j.ymssp.2004.08.001","article-title":"A method for the correlation dimension estimation for online condition monitoring of large rotating machinery","volume":"19","year":"2005","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hao, B., and Zheng, W. (1998). Applied symbolic dynamics and chaos. Dir. Chaos, 7.","DOI":"10.1142\/3830"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1595","DOI":"10.1088\/0951-7715\/15\/5\/312","article-title":"Entropy of interval maps via permutations","volume":"15","author":"Bandt","year":"2002","journal-title":"Nonlinearity"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/j.ecolmodel.2004.04.003","article-title":"Ordinal time series analysis","volume":"182","author":"Bandt","year":"2005","journal-title":"Ecol. Model."},{"key":"ref_16","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_17","doi-asserted-by":"crossref","unstructured":"Aziz, W., and Arif, M. (2005, January 24\u201325). Multiscale permutation entropy of physiological time series. Proceedings of the 2005 Pakistan Section Multitopic Conference, Karachi, Pakiston.","DOI":"10.1109\/INMIC.2005.334494"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1016\/j.physa.2016.11.102","article-title":"Multiscale permutation entropy analysis of electrocardiogram","volume":"471","author":"Liu","year":"2017","journal-title":"Physica A"},{"key":"ref_19","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"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Gao, J., Chen, K., Wu, P., Chen, L., and Lin, P. (2017, January 28\u201330). Leakage detection of pipeline based on weighted-permutation entropy. Proceedings of the 2017 29th Chinese Control And Decision Conference, Chongqing, China.","DOI":"10.1109\/CCDC.2017.7978992"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.cnsns.2015.07.011","article-title":"Permutation and weighted-permutation entropy analysis for the complexity of nonlinear time series","volume":"31","author":"Xia","year":"2016","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"40005","DOI":"10.1209\/0295-5075\/102\/40005","article-title":"Permutation complexity and dependence measures of time series","volume":"102","author":"Zhao","year":"2013","journal-title":"EPL"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4627","DOI":"10.3390\/e17074627","article-title":"Differentiating interictal and ictal states in childhood absence epilepsy through permutation R\u00e9nyi entropy","volume":"17","author":"Mammone","year":"2015","journal-title":"Entropy"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"053114","DOI":"10.1063\/1.4921552","article-title":"Generalized permutation entropy analysis based on the two-index entropic form Sq,\u03b4","volume":"25","author":"Xu","year":"2015","journal-title":"Chaos"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"812","DOI":"10.1016\/j.physa.2016.06.012","article-title":"Multivariate permutation entropy and its application for complexity analysis of chaotic systems","volume":"461","author":"He","year":"2016","journal-title":"Physica A"},{"key":"ref_26","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. Meth. Prog. Biol."},{"key":"ref_27","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_28","doi-asserted-by":"crossref","first-page":"2690","DOI":"10.1088\/1674-1056\/18\/7\/011","article-title":"Fine-grained permutation entropy as a measure of natural complexity for time series","volume":"18","author":"Liu","year":"2009","journal-title":"Chin. Phys. B"},{"key":"ref_29","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_30","doi-asserted-by":"crossref","first-page":"2517","DOI":"10.1007\/s11071-018-4339-2","article-title":"Operation conditions monitoring of flood discharge structure based on variance dedication rate and permutation entropy","volume":"93","author":"Zhang","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1360\/N112018-00197","article-title":"Generalized phase permutation entropy algorithm based on two-index entropy","volume":"49","author":"Kang","year":"2019","journal-title":"Sci. Sin. Inform."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/S0375-9601(00)00304-2","article-title":"On the structures and quantification of recurrence plots","volume":"270","author":"Gao","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1016\/j.bspc.2015.04.002","article-title":"A permutation Lempel-Ziv complexity measure for EEG analysis","volume":"19","author":"Bai","year":"2015","journal-title":"Biomed. Signal Process."},{"key":"ref_34","unstructured":"(2019, May 30). Bearing Data Center of Case Western Reserve University. Available online: http:\/\/csegroups.case.edu\/bearingdatacenter\/pages\/12k-drive-end-bearing-fault-data."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1343","DOI":"10.3390\/e14081343","article-title":"Bearing Fault Diagnosis Based on Multiscale Permutation Entropy and Support Vector Machine","volume":"14","author":"Wu","year":"2012","journal-title":"Entropy"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1016\/j.physa.2016.01.044","article-title":"Refined scale-dependent permutation entropy to analyze systems complexity","volume":"450","author":"Wu","year":"2016","journal-title":"Physica A"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.ymssp.2015.04.021","article-title":"Rolling Element Bearing Diagnostics Using the Case Western Reserve University Data: A Benchmark Study","volume":"64\u201365","author":"Smith","year":"2015","journal-title":"Mech. Syst. Signal Process."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/2\/187\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:55:29Z","timestamp":1760172929000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/2\/187"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,6]]},"references-count":37,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2]]}},"alternative-id":["e22020187"],"URL":"https:\/\/doi.org\/10.3390\/e22020187","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,6]]}}}