{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T03:43:32Z","timestamp":1785469412119,"version":"3.56.0"},"reference-count":39,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,7]],"date-time":"2024-11-07T00:00:00Z","timestamp":1730937600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Guangxi Science and Technology Base and Talent Project","award":["GuikeAD21220098"],"award-info":[{"award-number":["GuikeAD21220098"]}]},{"name":"Guangxi Science and Technology Base and Talent Project","award":["CRKL210102"],"award-info":[{"award-number":["CRKL210102"]}]},{"name":"Key Laboratory of Cognitive Radio and Information Processing of the Ministry of Education","award":["GuikeAD21220098"],"award-info":[{"award-number":["GuikeAD21220098"]}]},{"name":"Key Laboratory of Cognitive Radio and Information Processing of the Ministry of Education","award":["CRKL210102"],"award-info":[{"award-number":["CRKL210102"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>To enhance the performance of entropy algorithms in analyzing complex time series, generalized Gaussian distribution improved permutation entropy (GGDIPE) and its multiscale variant (MGGDIPE) are proposed in this paper. First, the generalized Gaussian distribution cumulative distribution function is employed for data normalization to enhance the algorithm\u2019s applicability across time series with diverse distributions. The algorithm further processes the normalized data using improved permutation entropy, which maintains both the absolute magnitude and temporal correlations of the signals, overcoming the equal value issue found in traditional permutation entropy (PE). Simulation results indicate that GGDIPE is less sensitive to parameter variations, exhibits strong noise resistance, accurately reveals the dynamic behavior of chaotic systems, and operates significantly faster than PE. Real-world data analysis shows that MGGDIPE provides markedly better separability for RR interval signals, EEG signals, bearing fault signals, and underwater acoustic signals compared to multiscale PE (MPE) and multiscale dispersion entropy (MDE). Notably, in underwater target recognition tasks, MGGDIPE achieves a classification accuracy of 97.5% across four types of acoustic signals, substantially surpassing the performance of MDE (70.5%) and MPE (62.5%). Thus, the proposed method demonstrates exceptional capability in processing complex time series.<\/jats:p>","DOI":"10.3390\/e26110960","type":"journal-article","created":{"date-parts":[[2024,11,7]],"date-time":"2024-11-07T12:12:13Z","timestamp":1730981533000},"page":"960","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Generalized Gaussian Distribution Improved Permutation Entropy: A New Measure for Complex Time Series Analysis"],"prefix":"10.3390","volume":"26","author":[{"given":"Kun","family":"Zheng","sequence":"first","affiliation":[{"name":"Institute of Visual Informatics, National University of Malaysia (UKM), Bangi 43600, Selangor, Malaysia"},{"name":"School of Information and Communication, Guilin University of Electronic Technology, Guilin 541004, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hong-Seng","family":"Gan","sequence":"additional","affiliation":[{"name":"School of AI and Advanced Computing, XJTLU Entrepreneur College (Taicang), Xi\u2019an Jiaotong\u2014Liverpool University, Suzhou 215400, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6839-0784","authenticated-orcid":false,"given":"Jun Kit","family":"Chaw","sequence":"additional","affiliation":[{"name":"Institute of Visual Informatics, National University of Malaysia (UKM), Bangi 43600, Selangor, Malaysia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sze-Hong","family":"Teh","sequence":"additional","affiliation":[{"name":"School of Intelligent Manufacturing Ecosystem, XJTLU Entrepreneur College (Taicang), Xi\u2019an Jiaotong\u2014Liverpool University, Suzhou 215400, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhe","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Information and Communication, Guilin University of Electronic Technology, Guilin 541004, China"},{"name":"Cognitive Radio and Information Processing Key Laboratory Authorized by China\u2019s Ministry of Education Foundation, Guilin University of Electronic Technology, Guilin 541004, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"112101","DOI":"10.1016\/j.chaos.2022.112101","article-title":"An improved multiscale distribution entropy for analyzing complexity of real-world signals","volume":"158","author":"Deka","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"109876","DOI":"10.1016\/j.knosys.2022.109876","article-title":"Ensemble entropy: A low bias approach for data analysis","volume":"256","author":"Hamed","year":"2022","journal-title":"Knowl. -Based Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1002\/joa3.12839","article-title":"Cardiac Arrhythmia Detection Using Cross-Sample Entropy Measure Based on Short and Long RR Interval Series","volume":"39","author":"Sharma","year":"2023","journal-title":"J. Arrhythm."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/s11571-016-9418-9","article-title":"Multivariate multi-scale weighted permutation entropy analysis of EEG complexity for Alzheimer\u2019s disease","volume":"11","author":"Deng","year":"2017","journal-title":"Cogn. Neurodyn."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Li, P., Karmakar, C., Yan, C., Palaniswami, M., and Liu, C. (2016). Classification of 5-S Epileptic EEG Recordings Using Distribution Entropy and Sample Entropy. Front. Physiol., 7.","DOI":"10.3389\/fphys.2016.00136"},{"key":"ref_6","first-page":"3517112","article-title":"A Novel Bearing Faults Detection Method Using Generalized Gaussian Distribution Refined Composite Multiscale Dispersion Entropy","volume":"71","author":"Ragavesh","year":"2022","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"77688","DOI":"10.1109\/ACCESS.2022.3193129","article-title":"Compressed Sensing Multiscale Sample Entropy Feature Extraction Method for Underwater Target Radiation Noise","volume":"10","author":"Lei","year":"2022","journal-title":"IEEE Access"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"22399","DOI":"10.1007\/s11071-023-09026-0","article-title":"Hierarchical Refined Composite Multiscale Fluctuation-Based Dispersion Entropy: Application to Feature Extraction of Underwater Target Signal","volume":"111","author":"Li","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"ref_9","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_10","first-page":"H2039","article-title":"Physiological time-series analysis using approximate entropy and sample entropy","volume":"278","author":"Richman","year":"2000","journal-title":"Am. J. Physiol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1109\/TNSRE.2007.897025","article-title":"Characterization of Surface EMG Signal Based on Fuzzy Entropy","volume":"15","author":"Chen","year":"2007","journal-title":"IEEE Trans. Neural Syst. Rehabilitation Eng."},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s11517-014-1216-0","article-title":"Assessing the complexity of short-term heartbeat interval series by distribution entropy","volume":"53","author":"Li","year":"2015","journal-title":"Med. Biol. Eng. Comput."},{"key":"ref_14","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":"88","author":"Fadlallah","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_15","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_16","doi-asserted-by":"crossref","first-page":"610","DOI":"10.1109\/LSP.2016.2542881","article-title":"Dispersion entropy: A measure for time-series analysis","volume":"23","author":"Rostaghi","year":"2016","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_17","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_18","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1109\/TAFFC.2020.3031004","article-title":"Classification of Interbeat Interval Time-Series Using Attention Entropy","volume":"14","author":"Yang","year":"2023","journal-title":"IEEE Trans. Affect. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Azami, H., and Escudero, J. (2018). Amplitude- and Fluctuation-Based Dispersion Entropy. Entropy, 20.","DOI":"10.3390\/e20030210"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1403829","DOI":"10.1155\/2019\/1403829","article-title":"Improved permutation entropy for measuring complexity of time series under noisy condition","volume":"2019","author":"Chen","year":"2019","journal-title":"Complexity"},{"key":"ref_21","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_22","doi-asserted-by":"crossref","first-page":"728","DOI":"10.1016\/j.cam.2011.06.007","article-title":"Hierarchical entropy analysis for biological signals","volume":"236","author":"Jiang","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2872","DOI":"10.1109\/TBME.2017.2679136","article-title":"Refined Composite Multiscale Dispersion Entropy and its Application to Biomedical Signals","volume":"64","author":"Azami","year":"2017","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_24","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_25","doi-asserted-by":"crossref","first-page":"062106","DOI":"10.1103\/PhysRevE.95.062106","article-title":"Characterizing time series via complexity-entropy curves","volume":"95","author":"Ribeiro","year":"2017","journal-title":"Phys. Rev. E."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.physa.2018.01.026","article-title":"Characterization of time series via R\u00e9nyi complexity\u2013entropy curves","volume":"498","author":"Jauregui","year":"2018","journal-title":"Phys. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"108464","DOI":"10.1016\/j.patcog.2021.108464","article-title":"Characterizing ordinal network of time series based on complexity-entropy curve","volume":"124","author":"Peng","year":"2022","journal-title":"Pattern Recognit."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"110517","DOI":"10.1016\/j.expthermflusci.2021.110517","article-title":"Early warning detection of thermoacoustic instability using three-dimensional complexity-entropy causality space","volume":"130","author":"Lee","year":"2022","journal-title":"Exp. Therm. Fluid Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"103118","DOI":"10.1063\/1.5094807","article-title":"Cumulative Tsallis entropy based on power spectrum of financial time series","volume":"29","author":"Zhang","year":"2019","journal-title":"Chaos An. Interdiscip. J. Nonlinear Sci."},{"key":"ref_30","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_31","doi-asserted-by":"crossref","first-page":"1577","DOI":"10.1016\/j.physleta.2012.03.039","article-title":"Contrasting Chaos with Noise via Local versus Global Information Quantifiers","volume":"376","author":"Olivares","year":"2012","journal-title":"Phys.Lett. A"},{"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"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"124640","DOI":"10.1016\/j.physa.2020.124640","article-title":"Contrasting Stochasticity with Chaos in a Permutation Lempel\u2013Ziv Complexity-Shannon Entropy Plane","volume":"554","author":"Mateos","year":"2020","journal-title":"Phys. A"},{"key":"ref_34","first-page":"9055","article-title":"A review of estimating the shape parameter of generalized Gaussian distribution","volume":"21","author":"Yu","year":"2012","journal-title":"J. Comput. Inf. Syst."},{"key":"ref_35","first-page":"1078","article-title":"Age-related alterations in the fractal scaling of cardiac interbeat interval dynamics","volume":"271","author":"Iyengar","year":"1996","journal-title":"Am. J. Physiol."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"061907","DOI":"10.1103\/PhysRevE.64.061907","article-title":"Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: Dependence on recording region and brain state","volume":"64","author":"Andrzejak","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"8013","DOI":"10.3390\/s130608013","article-title":"Sequential Fuzzy Diagnosis Method for Motor Roller Bearing in Variable Operating Conditions Based on Vibration Analysis","volume":"13","author":"Li","year":"2013","journal-title":"Sensors"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1016\/j.isatra.2020.08.010","article-title":"Deep Learning Algorithms for Rotating Machinery Intelligent Diagnosis: An Open Source Benchmark Study","volume":"107","author":"Zhao","year":"2020","journal-title":"ISA Trans."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.apacoust.2016.06.008","article-title":"ShipsEar: An underwater vessel noise database","volume":"113","year":"2016","journal-title":"Appl. Acoust."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/11\/960\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:28:23Z","timestamp":1760113703000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/11\/960"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,7]]},"references-count":39,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["e26110960"],"URL":"https:\/\/doi.org\/10.3390\/e26110960","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,7]]}}}