{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T20:35:17Z","timestamp":1783024517017,"version":"3.54.6"},"reference-count":43,"publisher":"MDPI AG","issue":"13","license":[{"start":{"date-parts":[[2022,6,30]],"date-time":"2022-06-30T00:00:00Z","timestamp":1656547200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51975433"],"award-info":[{"award-number":["51975433"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51975430"],"award-info":[{"award-number":["51975430"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2019CFB133"],"award-info":[{"award-number":["2019CFB133"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003819","name":"Natural Science Foundation of Hubei Province","doi-asserted-by":"publisher","award":["51975433"],"award-info":[{"award-number":["51975433"]}],"id":[{"id":"10.13039\/501100003819","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003819","name":"Natural Science Foundation of Hubei Province","doi-asserted-by":"publisher","award":["51975430"],"award-info":[{"award-number":["51975430"]}],"id":[{"id":"10.13039\/501100003819","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003819","name":"Natural Science Foundation of Hubei Province","doi-asserted-by":"publisher","award":["2019CFB133"],"award-info":[{"award-number":["2019CFB133"]}],"id":[{"id":"10.13039\/501100003819","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Aiming at non-stationary signals with complex components, the performance of a variational mode decomposition (VMD) algorithm is seriously affected by the key parameters such as the number of modes K, the quadratic penalty parameter \u03b1 and the update step \u03c4. In order to solve this problem, an adaptive empirical variational mode decomposition (EVMD) method based on a binary tree model is proposed in this paper, which can not only effectively solve the problem of VMD parameter selection, but also effectively reduce the computational complexity of searching the optimal VMD parameters using intelligent optimization algorithm. Firstly, the signal noise ratio (SNR) and refined composite multi-scale dispersion entropy (RCMDE) of the decomposed signal are calculated. The RCMDE is used as the setting basis of the \u03b1, and the SNR is used as the parameter value of the \u03c4. Then, the signal is decomposed into two components based on the binary tree mode. Before decomposing, the \u03b1 and \u03c4 need to be reset according to the SNR and MDE of the new signal. Finally, the cycle iteration termination condition composed of the least squares mutual information and reconstruction error of the components determines whether to continue the decomposition. The components with large least squares mutual information (LSMI) are combined, and the LSMI threshold is set as 0.8. The simulation and experimental results indicate that the proposed empirical VMD algorithm can decompose the non-stationary signals adaptively, with lower complexity, which is O(n2), good decomposition effect and strong robustness.<\/jats:p>","DOI":"10.3390\/s22134961","type":"journal-article","created":{"date-parts":[[2022,7,1]],"date-time":"2022-07-01T01:40:36Z","timestamp":1656639636000},"page":"4961","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Empirical Variational Mode Decomposition Based on Binary Tree Algorithm"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7830-849X","authenticated-orcid":false,"given":"Huipeng","family":"Li","sequence":"first","affiliation":[{"name":"Engineering Research Center for Metallurgical Automation and Measurement Technology, Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China"},{"name":"School of Physics and Electronic Information, Huanggang Normal University, Huanggang 438000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4510-7402","authenticated-orcid":false,"given":"Bo","family":"Xu","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Metallurgical Automation and Measurement Technology, Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China"},{"name":"School of Physics and Electronic Information, Huanggang Normal University, Huanggang 438000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fengxing","family":"Zhou","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Metallurgical Automation and Measurement Technology, Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Baokang","family":"Yan","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Metallurgical Automation and Measurement Technology, Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fengqi","family":"Zhou","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Metallurgical Automation and Measurement Technology, Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3057","DOI":"10.1109\/TIA.2017.2661250","article-title":"Deep Learning Based Approach for Bearing Fault Diagnosis","volume":"53","author":"He","year":"2017","journal-title":"IEEE Trans. Ind. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1109\/JSEN.2010.2049103","article-title":"Fuzzy Diagnosis Method for Rotating Machinery in Variable Rotating Speed","volume":"11","author":"Wang","year":"2010","journal-title":"IEEE Sens. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.renene.2015.12.010","article-title":"Generator bearing fault diagnosis for wind turbine via empirical wavelet transform using measured vibration signals","volume":"89","author":"Chen","year":"2016","journal-title":"Renew. Energy"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.sigpro.2013.04.015","article-title":"Wavelets for fault diagnosis of rotary machines: A review with applications","volume":"96","author":"Yan","year":"2014","journal-title":"Signal Process."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1793","DOI":"10.21595\/jve.2017.17680","article-title":"Application of EMD-WVD and particle filter for gearbox fault feature extraction and remaining useful life prediction","volume":"19","author":"Liu","year":"2017","journal-title":"J. Vibroeng."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Faysal, A., Ngui, W.K., Lim, M.H., and Leong, M.S. (2021). Noise Eliminated Ensemble Empirical Mode Decomposition Scalogram Analysis for Rotating Machinery Fault Diagnosis. Sensors, 21.","DOI":"10.3390\/s21238114"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"V25","DOI":"10.1190\/geo2019-0347.1","article-title":"Multichannel maximum-entropy method for the Wigner-Ville distribution","volume":"85","author":"Wang","year":"2020","journal-title":"Geophysics"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Zhang, X.H., Zhao, J.M., Bajri\u0107, R., and Wang, L.L. (2017). Application of the DC Offset Cancellation Method and S Transform to Gearbox Fault Diagnosis. Appl. Sci., 7.","DOI":"10.3390\/app7020207"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1740","DOI":"10.1109\/LCOMM.2018.2846229","article-title":"High-Order Statistics for the Channel Capacity of EGC Receivers Over Generalized Fading Channels","volume":"22","author":"Peppas","year":"2018","journal-title":"IEEE Commun. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.apacoust.2017.05.018","article-title":"An improved EMD method based on the multi-objective optimization and its application to fault feature extraction of rolling bearing","volume":"127","author":"Guo","year":"2017","journal-title":"Appl. Acoust."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5965","DOI":"10.3390\/e17095965","article-title":"Friction Signal Denoising Using Complete Ensemble EMD with Adaptive Noise and Mutual Information","volume":"17","author":"Li","year":"2015","journal-title":"Entropy"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.bspc.2014.06.009","article-title":"Improved complete ensemble EMD: A suitable tool for biomedical signal processing","volume":"14","author":"Colominas","year":"2014","journal-title":"Biomed. Signal Process. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2646","DOI":"10.1109\/TIM.2016.2598019","article-title":"A Morphological Hilbert-Huang Transform Technique for Bearing Fault Detection","volume":"27","author":"Osman","year":"2016","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107550","DOI":"10.1016\/j.epsr.2021.107550","article-title":"Fault Location of Hybrid Three-terminal HVDC Transmission Line Based on Improved LMD","volume":"201","author":"Gao","year":"2021","journal-title":"Electr. Pow. Syst. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1714","DOI":"10.1007\/s11668-021-01226-3","article-title":"Research on Rolling Bearing Fault Diagnosis Method Based on Improved LMD and CMWPE","volume":"21","author":"Song","year":"2021","journal-title":"J. Fail. Anal. Prev."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2747","DOI":"10.1007\/s00202-021-01261-1","article-title":"Detection of Islanding and Non-islanding Fault Disturbances in Microgrid Using LMD and Deep Stacked RVFLN Based Auto-encode","volume":"103","author":"Priyadarshini","year":"2021","journal-title":"Electr. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"107736","DOI":"10.1016\/j.measurement.2020.107736","article-title":"A Novel ITD-GSP-based Characteristic Extraction Method for Compound Faults of Rolling Bearing","volume":"159","author":"Yu","year":"2020","journal-title":"Measurement"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"171313","DOI":"10.1109\/ACCESS.2019.2956077","article-title":"Adaptive Signal Processing Algorithms Based on EMD and ITD","volume":"7","author":"Voznesensky","year":"2019","journal-title":"IEEE Access"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1109\/TSP.2013.2288675","article-title":"Variational Mode Decomposition","volume":"62","author":"Dragomiretskiy","year":"2014","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Li, K., Su, L., Wu, J.J., Wang, H.Q., and Chen, P. (2017). A Rolling Bearing Fault Diagnosis Method Based on Variational Mode Decomposition and an Improved Kernel Extreme Learning Machine. Appl. Sci., 7.","DOI":"10.3390\/app7101004"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1016\/j.ymssp.2017.02.013","article-title":"Research on variational mode decomposition in rolling bearings fault diagnosis of the multistage centrifugal pump","volume":"93","author":"Zhang","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"24601","DOI":"10.1109\/JSEN.2021.3116252","article-title":"Adaptive Signal Enhancement Based on Improved VMD-SVD for Leak Location in Water-Supply Pipeline","volume":"21","author":"Mei","year":"2021","journal-title":"IEEE Sens. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.ymssp.2017.11.046","article-title":"Chatter detection in milling process based on VMD and energy entropy","volume":"105","author":"Liu","year":"2018","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1016\/j.isatra.2020.10.060","article-title":"Adaptive Variational Mode Decomposition and Its Application to Multi-fault Detection Using Mechanical Vibration Signals","volume":"111","author":"He","year":"2021","journal-title":"ISA Trans."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4096","DOI":"10.1049\/iet-gtd.2017.0577","article-title":"Identification method for power system low-frequency oscillations based on improved VMD and Teager-Kaiser energy operator","volume":"11","author":"Xiao","year":"2017","journal-title":"IET Gener. Transm. Distrib."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"102337","DOI":"10.1016\/j.bspc.2020.102337","article-title":"EEG Signal denoising using hybrid approach of Variational Mode Decomposition and wavelets for depression","volume":"65","author":"Kaur","year":"2021","journal-title":"Biomed. Signal Process. Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1049\/iet-smt.2016.0510","article-title":"Denoising of UHF PD signals based on optimised VMD and wavelet transform","volume":"11","author":"Long","year":"2017","journal-title":"IET Sci. Meas. Technol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"108514","DOI":"10.1016\/j.ymssp.2021.108514","article-title":"Acoustic emission sources localization of laser cladding metallic panels using improved fruit fly optimization algorithm-based independent variational mode decomposition","volume":"166","author":"Li","year":"2022","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1109\/TMECH.2017.2787686","article-title":"Novel particle swarm optimization-based variational mode decomposition method for the fault diagnosis of complex rotating machinery","volume":"23","author":"Wang","year":"2017","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/j.isatra.2018.11.010","article-title":"Early fault feature extraction of bearings based on Teager energy operator and optimal VMD","volume":"86","author":"Xu","year":"2019","journal-title":"ISA Trans."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1059","DOI":"10.1109\/JBHI.2017.2734074","article-title":"Variational Mode Extraction: A New Efficient Method to Derive Respiratory Signals from ECG","volume":"22","author":"Nazari","year":"2018","journal-title":"IEEE J. Biomed. Health Inform."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"107610","DOI":"10.1016\/j.sigpro.2020.107610","article-title":"Successive Variational Mode Decomposition","volume":"174","author":"Nazari","year":"2020","journal-title":"Signal Process."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"108311","DOI":"10.1016\/j.sigpro.2021.108311","article-title":"Successive Multivariate Variational Mode Decomposition Based on Instantaneous Linear Mixing Model","volume":"190","author":"Liu","year":"2022","journal-title":"Signal Process."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"7825","DOI":"10.1016\/j.jfranklin.2021.07.021","article-title":"Self-tuning Variational Mode Decomposition","volume":"358","author":"Chen","year":"2021","journal-title":"J. Frankl. Inst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"774","DOI":"10.1007\/s11707-016-0611-2","article-title":"A mutual information Dempster-Shafer based decision ensemble system for land cover classification of hyperspectral data","volume":"11","author":"Pahlavani","year":"2017","journal-title":"Front. Earth Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1103\/PhysRevLett.119.225301","article-title":"Quantifying complexity in quantum phase transitions via mutual information complex networks","volume":"119","author":"Valdez","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.neunet.2017.07.009","article-title":"A multivariate extension of mutual information for growing neural networks","volume":"95","author":"Ball","year":"2017","journal-title":"Neural Netw."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"101171","DOI":"10.1016\/j.pmedr.2020.101171","article-title":"Lifestyle and subjective musculoskeletal symptoms in young male Japanese workers: A 16-year retrospective cohort study","volume":"20","author":"Tani","year":"2020","journal-title":"Prev. Med. Rep."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"110939","DOI":"10.1016\/j.chaos.2021.110939","article-title":"Automated detection of epileptic seizures using multiscale and refined composite multiscale dispersion entropy","volume":"146","author":"Sukriti","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_40","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_41","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_42","unstructured":"(2020, October 17). CWRU Bearing Data Center. Available online: http:\/\/csegroups.case.edu\/bearingdatacenter\/home."},{"key":"ref_43","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","author":"Smith","year":"2015","journal-title":"Mech. Syst. Signal Process."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/13\/4961\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:41:26Z","timestamp":1760139686000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/13\/4961"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,30]]},"references-count":43,"journal-issue":{"issue":"13","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["s22134961"],"URL":"https:\/\/doi.org\/10.3390\/s22134961","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,30]]}}}