{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T14:46:37Z","timestamp":1766155597723,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2023,11,9]],"date-time":"2023-11-09T00:00:00Z","timestamp":1699488000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Degradation stage prediction, which is crucial to monitoring the health condition of rolling bearings, can improve safety and reduce maintenance costs. In this paper, a novel degradation stage prediction method based on hierarchical grey entropy (HGE) and a grey bootstrap Markov chain (GBMC) is presented. Firstly, HGE is proposed as a new entropy that measures complexity, considers the degradation information embedded in both lower- and higher-frequency components and extracts the degradation features of rolling bearings. Then, the HGE values containing degradation information are fed to the prediction model, based on the GBMC, to obtain degradation stage prediction results more accurately. Meanwhile, three parameter indicators, namely the dynamic estimated interval, the reliability of the prediction result and dynamic uncertainty, are employed to evaluate the prediction results from different perspectives. The estimated interval reflects the upper and lower boundaries of the prediction results, the reliability reflects the credibility of the prediction results and the uncertainty reflects the dynamic fluctuation range of the prediction results. Finally, three rolling bearing run-to-failure experiments were conducted consecutively to validate the effectiveness of the proposed method, whose results indicate that HGE is superior to other entropies and the GBMC surpasses other existing rolling bearing degradation prediction methods; the prediction reliabilities are 90.91%, 90% and 83.87%, respectively.<\/jats:p>","DOI":"10.3390\/s23229082","type":"journal-article","created":{"date-parts":[[2023,11,10]],"date-time":"2023-11-10T01:17:51Z","timestamp":1699579071000},"page":"9082","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A New Approach to the Degradation Stage Prediction of Rolling Bearings Using Hierarchical Grey Entropy and a Grey Bootstrap Markov Chain"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5168-7768","authenticated-orcid":false,"given":"Li","family":"Cheng","sequence":"first","affiliation":[{"name":"School of Information Engineering, Henan University of Science and Technology, Luoyang 471023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6843-6594","authenticated-orcid":false,"given":"Wensuo","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Mechatronics Engineering, Henan University of Science and Technology, Luoyang 471003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zuobin","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Mechatronics Engineering, Henan University of Science and Technology, Luoyang 471003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4165","DOI":"10.1109\/TIM.2019.2948414","article-title":"Performance prediction using high-order differential mathematical morphology gradient spectrum entropy and extreme learning machine","volume":"69","author":"Zhao","year":"2019","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_2","first-page":"582","article-title":"Empirical mode decomposition of weak fault characteristic signals of rolling bearing under strong noise background","volume":"33","author":"Yang","year":"2020","journal-title":"J. Vib. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1617","DOI":"10.1177\/1077546320946633","article-title":"Bearing performance degradation assessment based on topological representation and hidden Markov model","volume":"27","author":"Wang","year":"2021","journal-title":"J. Vib. Control."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"6615920","DOI":"10.1155\/2021\/6615920","article-title":"Evaluation and prediction method of rolling bearing performance degradation based on attention-LSTM","volume":"2012","author":"Wang","year":"2021","journal-title":"Shock. Vib."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6188","DOI":"10.1109\/ACCESS.2020.3048492","article-title":"A new performance degradation evaluation method integrating PCA, PSR and KELM","volume":"9","author":"Lv","year":"2020","journal-title":"IEEE Access"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"104051","DOI":"10.1016\/j.mechmachtheory.2020.104051","article-title":"Compound fault identification of rolling element bearing based on adaptive resonant frequency band extraction","volume":"154","author":"Chen","year":"2020","journal-title":"Mech. Mach. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Chen, Y.S., Yuan, Z.C., Chen, J.H., and Sun, K. (2022). A Novel Fault Diagnosis Method for Rolling Bearing Based on Hierarchical Refined Composite Multiscale Fluctuation-Based Dispersion Entropy and PSO-ELM. Entropy, 24.","DOI":"10.3390\/e24111517"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"41","DOI":"10.20855\/ijav.2020.25.11717","article-title":"A Novel Degradation Feature Extraction Technique Based on Improved Base-Scale Entropy","volume":"26","author":"Chen","year":"2021","journal-title":"Int. J. Acoust. Vib."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"105022","DOI":"10.1016\/j.knosys.2019.105022","article-title":"Enhanced deep gated recurrent unit and complex wavelet packet energy moment entropy for early fault prognosis of bearing","volume":"188","author":"Shao","year":"2020","journal-title":"Knowl. Based Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"61710","DOI":"10.1109\/ACCESS.2021.3073708","article-title":"An improved bearing fault diagnosis scheme based on hierarchical fuzzy entropy and Alexnet network","volume":"9","author":"Shi","year":"2021","journal-title":"IEEE Access"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Li, Z., Cui, Y.H., Li, L.L., Chen, R.L., Dong, L., and Du, J. (2022). Hierarchical amplitude-aware permutation entropy-based fault feature extraction method for rolling bearings. Entropy, 24.","DOI":"10.3390\/e24030310"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Yan, X.A., Xu, Y.D., and Jia, M.P. (2021). Intelligent fault diagnosis of rolling-element bearings using a self-adaptive hierarchical multiscale fuzzy entropy. Entropy, 23.","DOI":"10.3390\/e23091128"},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"2338","DOI":"10.1103\/PhysRevA.14.2338","article-title":"Kolmogorov entropy and numerical experiments","volume":"14","author":"Benettin","year":"1976","journal-title":"Phys. Rev. A"},{"key":"ref_15","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_16","doi-asserted-by":"crossref","first-page":"2039","DOI":"10.1152\/ajpheart.2000.278.6.H2039","article-title":"Physiological time-series analysis using approximate and sample entropy","volume":"278","author":"Richman","year":"2000","journal-title":"Am. J. Physiol. Heart Circ. Physiol."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Bandt, C., and Pompe, B. (2002). Permutation entropy: A natural complexity measure for time series. Phys. Rev. Lett., 88.","DOI":"10.1103\/PhysRevLett.88.174102"},{"key":"ref_18","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_19","doi-asserted-by":"crossref","first-page":"108343","DOI":"10.1016\/j.ymssp.2021.108343","article-title":"Enhanced hierarchical symbolic dynamic entropy and maximum mean and covariance discrepancy-based transfer joint matching with Welsh loss for intelligent cross-domain bearing health monitoring","volume":"165","author":"Yang","year":"2022","journal-title":"Mech. Syst. Signal Proc."},{"key":"ref_20","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_21","first-page":"10775463221118035","article-title":"Degradation feature extraction of rolling bearing based on equalization symbol sequence entropy","volume":"29","author":"Wang","year":"2022","journal-title":"J. Vib. Control."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"71979","DOI":"10.1109\/ACCESS.2021.3078823","article-title":"A Fault Feature Extraction Method for Rolling Bearings Based on Refined Composite Multi-Scale Amplitude-Aware Permutation Entropy","volume":"9","author":"Song","year":"2021","journal-title":"IEEE Access"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"6982","DOI":"10.1109\/TIM.2020.2978966","article-title":"A novel health indicator based on information theory features for assessing rotating machinery performance degradation","volume":"69","author":"Rai","year":"2020","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Li, Y.X., Gao, P.Y., Tang, B.Z., Yi, Y.M., and Zhang, J.J. (2022). Double feature extraction method of ship-radiated noise signal based on slope entropy and permutation entropy. Entropy, 24.","DOI":"10.3390\/e24091265"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"8824901","DOI":"10.1155\/2021\/8824901","article-title":"A new method of fault feature extraction based on hierarchical dispersion entropy","volume":"2021","author":"Chen","year":"2021","journal-title":"Shock. Vib."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1016\/j.measurement.2013.09.019","article-title":"A roller bearing fault diagnosis method based on hierarchical entropy and support vector machine with particle swarm optimization algorithm","volume":"47","author":"Zhu","year":"2014","journal-title":"Measurement"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"32957","DOI":"10.1109\/ACCESS.2020.2970444","article-title":"Cascade fusion convolutional long-short time memory network for remaining useful life prediction of rolling bearing","volume":"8","author":"Wu","year":"2020","journal-title":"IEEE Access"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1016\/j.isatra.2019.08.058","article-title":"Bearing remaining useful life prediction using support vector machine and hybrid degradation tracking model","volume":"98","author":"Yan","year":"2020","journal-title":"ISA Trans."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"106968","DOI":"10.1016\/j.ress.2020.106968","article-title":"Operational reliability evaluation and prediction of rolling bearing based on isometric mapping and NoCuSa-LSSVM. Reliab","volume":"201","author":"Gao","year":"2020","journal-title":"Eng. Syst. Saf."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1106","DOI":"10.1177\/0954406220941037","article-title":"Intelligent fault prediction of rolling bearing based on gate recurrent unit and hybrid autoencoder","volume":"235","author":"Che","year":"2021","journal-title":"Proc. Inst. Mech. Eng. Part C J. Eng. Mech. Eng. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"6599","DOI":"10.1177\/09544062211009556","article-title":"Stages prediction of the remaining useful life of rolling bearing based on regularized extreme learning machine","volume":"235","author":"Wu","year":"2021","journal-title":"Proc. Inst. Mech. Eng. Part C J. Eng. Mech. Eng. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"109686","DOI":"10.1016\/j.asoc.2022.109686","article-title":"RUL prediction for rolling bearings based on Convolutional Autoencoder and status degradation model","volume":"130","author":"Xu","year":"2022","journal-title":"Appl. Soft. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"015016","DOI":"10.1088\/1361-6501\/ac90dc","article-title":"Performance degradation prediction model of rolling bearing based on self-checking long short-term memory network","volume":"34","author":"Lan","year":"2022","journal-title":"Meas. Sci. Technol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"111893","DOI":"10.1016\/j.measurement.2022.111893","article-title":"A remaining life prediction of rolling element bearings based on a bidirectional gate recurrent unit and convolution neural network","volume":"202","author":"Shang","year":"2022","journal-title":"Measurement"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1006\/mssp.2000.1297","article-title":"Toward helicopter gearbox diagnostics from a small number of examples","volume":"14","author":"Zacksenhouse","year":"2000","journal-title":"Mech. Syst. Signal Proc."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1687814020919241","DOI":"10.1177\/1687814020919241","article-title":"Prediction and analysis of bearing vibration signal with a novel gray combination model","volume":"12","author":"Yuan","year":"2020","journal-title":"Adv. Mech. Eng."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"108971","DOI":"10.1016\/j.spl.2020.108971","article-title":"A note on stationary bootstrap variance estimator under long-range dependence","volume":"169","author":"Kang","year":"2021","journal-title":"Stat. Probab. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1272","DOI":"10.1007\/s10955-019-02342-z","article-title":"Quantum Markov chains associated with open quantum random walks","volume":"176","author":"Dhahri","year":"2019","journal-title":"J. Stat. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"6878132221082867","DOI":"10.1177\/16878132221082867","article-title":"Rolling bearing performance degradation evaluation using grey entropy","volume":"14","author":"Cheng","year":"2022","journal-title":"Adv. Mech. Eng."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.isatra.2019.01.018","article-title":"A fault diagnosis scheme for rotating machinery using hierarchical symbolic analysis and convolutional neural network","volume":"91","author":"Yang","year":"2019","journal-title":"ISA Trans."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.mechmachtheory.2015.11.010","article-title":"Hierarchical fuzzy entropy and improved support vector machine based binary tree approach for rolling bearing fault diagnosis","volume":"98","author":"Li","year":"2016","journal-title":"Mech. Mach. Theory"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"4259","DOI":"10.1007\/s00500-019-04191-0","article-title":"A multivariate grey prediction model with grey relational analysis for bankruptcy prediction problems","volume":"24","author":"Hu","year":"2020","journal-title":"Soft Comput."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/j.cja.2013.07.023","article-title":"Gray bootstrap method for estimating frequency-varying random vibration signals with small samples","volume":"27","author":"Wang","year":"2014","journal-title":"Chin. J. Aeronaut."},{"key":"ref_44","unstructured":"(2023, October 23). IMS, University of Cincinnati, NASA Ames Prognostics Data Repository, Available online: http:\/\/ti.arc.nasa.gov\/project\/prognostic-data-repository."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/22\/9082\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:20:45Z","timestamp":1760131245000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/22\/9082"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,9]]},"references-count":44,"journal-issue":{"issue":"22","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["s23229082"],"URL":"https:\/\/doi.org\/10.3390\/s23229082","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2023,11,9]]}}}