{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T10:20:33Z","timestamp":1773656433637,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,17]],"date-time":"2024-06-17T00:00:00Z","timestamp":1718582400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Special Program of Guangxi Science and Technology Base and Talent","award":["GuiKeAD21220098"],"award-info":[{"award-number":["GuiKeAD21220098"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Due to its capacity to unveil the dynamic characteristics of time series data, entropy has attracted growing interest. However, traditional entropy feature extraction methods, such as permutation entropy, fall short in concurrently considering both the absolute amplitude information of signals and the temporal correlation between sample points. Consequently, this limitation leads to inadequate differentiation among different time series and susceptibility to noise interference. In order to augment the discriminative power and noise robustness of entropy features in time series analysis, this paper introduces a novel method called Tsallis entropy-based complexity-improved permutation entropy casualty plane (TC-IPE-CP). TC-IPE-CP adopts a novel symbolization approach that preserves both absolute amplitude information and inter-point correlations within sequences, thereby enhancing feature separability and noise resilience. Additionally, by incorporating Tsallis entropy and weighting the probability distribution with parameter q, it integrates with statistical complexity to establish a feature plane of complexity and entropy, further enriching signal features. Through the integration of multiscale algorithms, a multiscale Tsallis-improved permutation entropy algorithm is also developed. The simulation results indicate that TC-IPE-CP requires a small amount of data, exhibits strong noise resistance, and possesses high separability for signals. When applied to the analysis of heart rate signals, fault diagnosis, and underwater acoustic signal recognition, experimental findings demonstrate that TC-IPE-CP can accurately differentiate between electrocardiographic signals of elderly and young subjects, achieve precise bearing fault diagnosis, and identify four types of underwater targets. Particularly in underwater acoustic signal recognition experiments, TC-IPE-CP achieves a recognition rate of 96.67%, surpassing the well-known multi-scale dispersion entropy and multi-scale permutation entropy by 7.34% and 19.17%, respectively. This suggests that TC-IPE-CP is highly suitable for the analysis of complex time series.<\/jats:p>","DOI":"10.3390\/e26060521","type":"journal-article","created":{"date-parts":[[2024,6,17]],"date-time":"2024-06-17T11:14:16Z","timestamp":1718622856000},"page":"521","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Tsallis Entropy-Based Complexity-IPE Casualty Plane: A Novel Method for Complex Time Series Analysis"],"prefix":"10.3390","volume":"26","author":[{"given":"Zhe","family":"Chen","sequence":"first","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":[{"role":"author","vocabulary":"crossref"}]},{"given":"Changling","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Information and Communication, Guilin University of Electronic Technology, Guilin 541004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junyi","family":"Wang","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":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongbing","family":"Qiu","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":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1016\/j.physleta.2016.12.037","article-title":"Weighted fractional permutation entropy and fractional sample entropy for nonlinear Potts financial dynamics","volume":"381","author":"Xu","year":"2017","journal-title":"Phys. Lett. A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/j.ymssp.2017.12.008","article-title":"A fault diagnosis scheme for planetary gearboxes using adaptive multi-scale morphology filter and modified hierarchical permutation entropy","volume":"105","author":"Li","year":"2018","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Zhou, S., Qian, S., and Chang, W. (2018). A novel bearing multi-fault diagnosis approach based on weighted permutation entropy and an improved SVM ensemble classifier. Sensors, 18.","DOI":"10.3390\/s18061934"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1063\/1.166092","article-title":"Approximate entropy (ApEn) as a complexity measure","volume":"5","author":"Pincus","year":"1995","journal-title":"Chaos"},{"key":"ref_6","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_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","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_9","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_10","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_11","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_12","doi-asserted-by":"crossref","unstructured":"Costa, M., Goldberger, A.L., and Peng, C.K. (2002). Multiscale entropy to distinguish physiologic and synthetic RR time series. Computers in Cardiology, IEEE.","DOI":"10.1109\/CIC.2002.1166726"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Aziz, W., and Arif, M. (2005, January 23\u201324). Multiscale Permutation Entropy of Physiological Time Series. Proceedings of the 2005 Pakistan Section Multitopic Conference, Karachi, Pakistan.","DOI":"10.1109\/INMIC.2005.334494"},{"key":"ref_14","unstructured":"Azami, H., Rostaghi, M., Ab\u00e1solo, D., and Escudero, J. (2017). Refined Composite Multiscale Dispersion Entropy and its Application to Biomedical Signals. IEEE Trans. Biomed. Eng., 2017."},{"key":"ref_15","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_16","doi-asserted-by":"crossref","first-page":"105660","DOI":"10.1016\/j.cnsns.2020.105660","article-title":"Transition-based complexity-entropy causality diagram: A novel method to characterize complex systems","volume":"95","author":"Zhang","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1515\/zna-2023-0115","article-title":"Refined composite multivariate multiscale complexity-entropy causality plane analysis for gas-liquid two-phase flow","volume":"78","author":"Li","year":"2023","journal-title":"Z. Naturforschung A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_19","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_20","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 Stat. Mech. Its Appl."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Freitas, C.G., Rosso, O.A., and Aquino, A.L. (2020, January 7\u201310). Mapping Network Traffic Dynamics in the Complexity-Entropy Plane. Proceedings of the 2020 IEEE Symposium on Computers and Communications, Rennes, France.","DOI":"10.1109\/ISCC50000.2020.9219559"},{"key":"ref_22","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_23","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_24","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_25","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 Interdiscip. J. Nonlinear Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1078","DOI":"10.1152\/ajpregu.1996.271.4.R1078","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. Regul. Integr. Comp. Physiol."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Chen, Z., Ma, X., Fu, J., and Li, Y. (2023). Ensemble Improved Permutation Entropy: A New Approach for Time Series Analysis. Entropy, 25.","DOI":"10.3390\/e25081175"},{"key":"ref_28","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_29","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\/6\/521\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:00:09Z","timestamp":1760108409000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/6\/521"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,17]]},"references-count":29,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["e26060521"],"URL":"https:\/\/doi.org\/10.3390\/e26060521","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,17]]}}}