{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,3]],"date-time":"2026-01-03T06:28:51Z","timestamp":1767421731903,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T00:00:00Z","timestamp":1638316800000},"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>Multiscale entropy (MSE) analysis is a fundamental approach to access the complexity of a time series by estimating its information creation over a range of temporal scales. However, MSE may not be accurate or valid for short time series. This is why previous studies applied different kinds of algorithm derivations to short-term time series. However, no study has systematically analyzed and compared their reliabilities. This study compares the MSE algorithm variations adapted to short time series on both human and rat heart rate variability (HRV) time series using long-term MSE as reference. The most used variations of MSE are studied: composite MSE (CMSE), refined composite MSE (RCMSE), modified MSE (MMSE), and their fuzzy versions. We also analyze the errors in MSE estimations for a range of incorporated fuzzy exponents. The results show that fuzzy MSE versions\u2014as a function of time series length\u2014present minimal errors compared to the non-fuzzy algorithms. The traditional multiscale entropy algorithm with fuzzy counting (MFE) has similar accuracy to alternative algorithms with better computing performance. For the best accuracy, the findings suggest different fuzzy exponents according to the time series length.<\/jats:p>","DOI":"10.3390\/e23121620","type":"journal-article","created":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T05:02:36Z","timestamp":1638334956000},"page":"1620","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Multiscale Entropy Analysis of Short Signals: The Robustness of Fuzzy Entropy-Based Variants Compared to Full-Length Long Signals"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9303-8291","authenticated-orcid":false,"suffix":"Jr.","given":"Airton","family":"Borin","sequence":"first","affiliation":[{"name":"Federal Institute of Education, Science and Technology of Triangulo Mineiro, Uberaba 38064-790, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6289-0040","authenticated-orcid":false,"given":"Anne","family":"Humeau-Heurtier","sequence":"additional","affiliation":[{"name":"LARIS\u2014Laboratoire Angevin de Recherche en Ing\u00e9nierie des Syst\u00e8mes, University of Angers, 49035 Angers, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0801-1209","authenticated-orcid":false,"given":"Luiz","family":"Virg\u00edlio Silva","sequence":"additional","affiliation":[{"name":"Department of Internal Medicine, Ribeir\u00e3o Preto Medical School, University of S\u00e3o Paulo, Ribeir\u00e3o Preto 14049-900, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2197-6008","authenticated-orcid":false,"suffix":"Jr.","given":"Luiz","family":"Murta","sequence":"additional","affiliation":[{"name":"Department of Computing and Mathematics, School of Philosophy, Sciences and Languages of Ribeir\u00e3o Preto, University of S\u00e3o Paulo, Ribeir\u00e3o Preto 14040-901, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Boccara, N. (2010). Cellular Automata. Modeling Complex Systems, Springer.","DOI":"10.1007\/978-1-4419-6562-2"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Klamut, J., Kutner, R., and Struzik, Z.R. (2020). Towards a Universal Measure of Complexity. arXiv.","DOI":"10.3390\/e22080866"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Delgado-Bonal, A., and Marshak, A. (2019). Approximate Entropy and Sample Entropy: A Comprehensive Tutorial. Entropy, 21.","DOI":"10.3390\/e21060541"},{"key":"ref_4","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_5","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. Rehabil. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"068102","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":"5865","DOI":"10.1016\/j.physa.2013.07.075","article-title":"Modified multiscale entropy for short-term time series analysis","volume":"392","author":"Wu","year":"2013","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.physa.2003.08.022","article-title":"Multiscale entropy analysis of human gait dynamics","volume":"330","author":"Costa","year":"2003","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4032","DOI":"10.3390\/e16074032","article-title":"Application of a modified entropy computational method in assessing the complexity of pulse wave velocity signals in healthy and diabetic subjects","volume":"16","author":"Chang","year":"2014","journal-title":"Entropy"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.3390\/e15031069","article-title":"Time series analysis using composite multiscale entropy","volume":"15","author":"Wu","year":"2013","journal-title":"Entropy"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.1016\/j.physleta.2014.03.034","article-title":"Analysis of complex time series using refined composite multiscale entropy","volume":"378","author":"Wu","year":"2014","journal-title":"Phys. Lett. A"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Wang, F., Lu, B., Kang, X., and Fu, R. (2021). Research on driving fatigue alleviation using interesting auditory stimulation based on VMD-MMSE. Entropy, 23.","DOI":"10.3390\/e23091209"},{"key":"ref_13","first-page":"145","article-title":"Multiscale fuzzy entropy and its application in rolling bearing fault diagnosis","volume":"27","author":"Zheng","year":"2014","journal-title":"J. Vib. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/j.jsv.2015.09.016","article-title":"A fault diagnosis scheme for rolling bearing based on local mean decomposition and improved multiscale fuzzy entropy","volume":"360","author":"Li","year":"2016","journal-title":"J. Sound Vib."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"746","DOI":"10.1016\/j.ymssp.2016.09.010","article-title":"Rolling bearing fault detection and diagnosis based on composite multiscale fuzzy entropy and ensemble support vector machines","volume":"85","author":"Zheng","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2037","DOI":"10.1007\/s11517-017-1647-5","article-title":"Refined multiscale fuzzy entropy based on standard deviation for biomedical signal analysis","volume":"55","author":"Azami","year":"2017","journal-title":"Med. Biol. Eng. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Zheng, J., Pan, H., Tong, J., and Liu, Q. (2021). Generalized refined composite multiscale fuzzy entropy and multi-cluster feature selection based intelligent fault diagnosis of rolling bearing. ISA Trans.","DOI":"10.1016\/j.isatra.2021.05.042"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Tom\u010dala, J. (2020). New Fast ApEn and SampEn Entropy Algorithms Implementation and Their Application to Supercomputer Power Consumption. Entropy, 22.","DOI":"10.3390\/e22080863"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"083135","DOI":"10.1063\/5.0010330","article-title":"Modified multiscale fuzzy entropy: A robust method for short-term physiologic signals","volume":"30","author":"Borin","year":"2020","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1016\/j.physa.2006.10.077","article-title":"Revisiting sample entropy analysis","volume":"376","author":"Govindan","year":"2007","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_21","first-page":"338","article-title":"Information and control","volume":"8","author":"Zadeh","year":"1965","journal-title":"Fuzzy Sets"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Mayer, C., Bachler, M., Holzinger, A., Stein, P.K., and Wassertheurer, S. (2016). The effect of threshold values and weighting factors on the association between entropy measures and mortality after myocardial infarction in the cardiac arrhythmia suppression trial (CAST). Entropy, 18.","DOI":"10.3390\/e18040129"},{"key":"ref_23","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_24","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.bspc.2015.08.004","article-title":"Improved multiscale permutation entropy for biomedical signal analysis: Interpretation and application to electroencephalogram recordings","volume":"23","author":"Azami","year":"2016","journal-title":"Biomed. Signal Process Control"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"R150","DOI":"10.1152\/ajpregu.00076.2016","article-title":"Multiscale entropy analysis of heart rate variability in heart failure, hypertensive, and sinoaortic-denervated rats: Classical and refined approaches","volume":"311","author":"Silva","year":"2016","journal-title":"Am. J.-Physiol.-Regul. Integr. Comp. Physiol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"e215","DOI":"10.1161\/01.CIR.101.23.e215","article-title":"PhysioBank, PhysioToolkit, and PhysioNet","volume":"101","author":"Goldberger","year":"2000","journal-title":"Circulation"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1016\/j.medengphy.2012.06.020","article-title":"Comparison of different methods of heart rate entropy analysis during acute anoxia superimposed on a chronic rat model of pulmonary hypertension","volume":"35","author":"Rocha","year":"2013","journal-title":"Med. Eng. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"104833","DOI":"10.1109\/ACCESS.2019.2930625","article-title":"Fuzzy entropy metrics for the analysis of biomedical signals: Assessment and comparison","volume":"7","author":"Azami","year":"2019","journal-title":"IEEE Access"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/12\/1620\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:38:24Z","timestamp":1760168304000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/12\/1620"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,1]]},"references-count":28,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["e23121620"],"URL":"https:\/\/doi.org\/10.3390\/e23121620","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,12,1]]}}}