{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T09:26:22Z","timestamp":1771233982345,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,5,9]],"date-time":"2019-05-09T00:00:00Z","timestamp":1557360000000},"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":["51541506"],"award-info":[{"award-number":["51541506"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The extraction of rolling bearings\u2019 degradation features has been developed for decades. However, the degradation features always present different trends of different run-to-failure data. To find a consistent indicator of different data will be helpful to establish a general model and explore the nature of bearings\u2019 degradation. In this study, we have found there is a trend of similarity between the energy and complexity features. By using the cointegration test, we found the two kinds of features exhibit a certain degree of cointegration relationship. Fused by the cointegration method, we have obtained a novel health indicator which can depict different run-to-failure data in a unified way. The difference between the energy features and complexity features can be explained by the novel health indicator. The indicator has \u201ctwo-stage\u201d characters. The first stage is the zero-line stage and the second stage is the quickly raise stage, which presents like an exponential function. It is easy to think about using an exponential degradation model to model this indicator. Next, we have compared the indicator to root mean square (RMS) by using the exponential degradation model. It shows that the indicator is more suitable for the exponential degradation model. In this paper, we used eleven run-to-failure data to verify the generality and \u201ctwo-stage\u201d characters of the proposed indicator. The result shows that the novel indicator is general and effective and that it will promote the development of bearings\u2019 prognostics.<\/jats:p>","DOI":"10.3390\/s19092151","type":"journal-article","created":{"date-parts":[[2019,5,9]],"date-time":"2019-05-09T11:22:35Z","timestamp":1557400955000},"page":"2151","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A Novel Health Indicator Based on Cointegration for Rolling Bearings\u2019 Run-To-Failure Process"],"prefix":"10.3390","volume":"19","author":[{"given":"Hongru","family":"Li","sequence":"first","affiliation":[{"name":"Army Engineering University, No. 97 Heping West Road, Shijiazhuang 050003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0099-8363","authenticated-orcid":false,"given":"Yaolong","family":"Li","sequence":"additional","affiliation":[{"name":"Army Engineering University, No. 97 Heping West Road, Shijiazhuang 050003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"He","family":"Yu","sequence":"additional","affiliation":[{"name":"Army Engineering University, No. 97 Heping West Road, Shijiazhuang 050003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1016\/j.ymssp.2017.11.016","article-title":"Machinery health prognostics: A systematic review from data acquisition to RUL prediction","volume":"104","author":"Lei","year":"2018","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1109\/ACCESS.2017.2774261","article-title":"Prognostics and Health Management: A Review of Vibration based Bearing and Gear Health Indicators","volume":"6","author":"Wang","year":"2018","journal-title":"IEEE Access"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1483","DOI":"10.1016\/j.ymssp.2005.09.012","article-title":"A review on machinery diagnostics and prognostics implementing condition-based maintenance","volume":"20","author":"Jardine","year":"2006","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1016\/S0301-679X(99)00077-8","article-title":"A review of vibration and acoustic measurement methods for the detection of defects in rolling element bearings","volume":"32","author":"Tandon","year":"1999","journal-title":"Tribol. Int."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1066","DOI":"10.1016\/j.jsv.2005.03.007","article-title":"Wavelet filter-based weak signature detection method and its application on rolling element bearing prognostics","volume":"289","author":"Qiu","year":"2006","journal-title":"J. Sound Vib."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.ymssp.2012.09.015","article-title":"A review on empirical mode decomposition in fault diagnosis of rotating machinery","volume":"35","author":"Lei","year":"2013","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1327","DOI":"10.1016\/j.ymssp.2008.11.005","article-title":"Application of the EEMD method to rotor fault diagnosis of rotating machinery","volume":"23","author":"Lei","year":"2009","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_8","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_9","first-page":"1","article-title":"Rolling Element Bearing Performance Degradation Assessment Using Variational Mode Decomposition and Gath-Geva Clustering Time Series Segmentation","volume":"2017","author":"Li","year":"2017","journal-title":"Int. J. Rotating Mach."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.ymssp.2016.02.049","article-title":"A comparative study between Empirical Wavelet Transforms and Empirical Mode Decomposition Methods: Application to bearing defect diagnosis","volume":"81","author":"Kedadouche","year":"2016","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_11","first-page":"150","article-title":"Accurate bearing remaining useful life prediction based on Weibull distribution and artificial neural network","volume":"56","author":"Ali","year":"2015","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2363","DOI":"10.1109\/TIE.2011.2167893","article-title":"Local and Nonlocal Preserving Projection for Bearing Defect Classification and Performance Assessment","volume":"59","author":"Yu","year":"2012","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"7440","DOI":"10.1016\/j.eswa.2010.12.079","article-title":"Bearing performance degradation assessment using locality preserving projections","volume":"38","author":"Yu","year":"2011","journal-title":"Expert Syst. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3143","DOI":"10.1016\/j.measurement.2013.06.038","article-title":"Bearing degradation process prediction based on the PCA and optimized LS-SVM model","volume":"46","author":"Dong","year":"2013","journal-title":"Measurement"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/j.triboint.2014.11.021","article-title":"Dynamic modelling of wear evolution in rolling bearings","volume":"84","author":"Jantunen","year":"2015","journal-title":"Tribol. Int."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.engfailanal.2014.06.004","article-title":"A descriptive model of wear evolution in rolling bearings","volume":"45","author":"Jantunen","year":"2014","journal-title":"Eng. Fail. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ymssp.2015.02.016","article-title":"A review on prognostic techniques for non-stationary and non-linear rotating systems","volume":"62","author":"Kan","year":"2015","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_18","unstructured":"Fuller, W.A. (1976). Introduction to Statistical Time Series, John Wiley and Sons."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1093\/biomet\/71.3.599","article-title":"Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order","volume":"71","author":"Dickey","year":"1984","journal-title":"Biometrika"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"251","DOI":"10.2307\/1913236","article-title":"Co-Integration and Error Correction: Representation, Estimation, and Testing","volume":"55","author":"Engle","year":"1987","journal-title":"Econometrica"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/0165-1889(88)90041-3","article-title":"Statistical analysis of cointegration vectors","volume":"12","author":"Johansen","year":"1988","journal-title":"J. Econ. Dyn. Control"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A Mathematical Theory of Communiation","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1109\/TIT.1976.1055501","article-title":"On the Complexity of Finite Sequences","volume":"22","author":"Lempel","year":"1976","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_24","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_25","doi-asserted-by":"crossref","first-page":"2039","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 Circulat. Physiol."},{"key":"ref_26","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_27","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_28","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1006\/jsvi.1993.1134","article-title":"Routes to Chaos in Ball Bearings","volume":"162","author":"Mevel","year":"1993","journal-title":"J. Sound Vib."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Kwuimy, C.A.K., Samadani, M., Kappaganthu, K., and Nataraj, C. (2015). Sequential Recurrence Analysis of Experimental Time Series of a Rotor Response with Bearing Outer Race Faults. Vibration Engineering and Technology of Machinery, Springer.","DOI":"10.1115\/DETC2015-48106"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1006\/jsvi.1999.3109","article-title":"Effect of radial internal clearance of a ball bearing on the dynamics of a balanced horizontal rotor","volume":"238","author":"Tiwari","year":"2000","journal-title":"J. Sound Vib."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1016\/j.engfailanal.2015.08.013","article-title":"Fault analysis of the wear fault development in rolling bearings","volume":"57","author":"Jantunen","year":"2015","journal-title":"Eng. Fail. Anal."},{"key":"ref_32","first-page":"252","article-title":"A summary of fault modelling and predictive health monitoring of rolling element bearings","volume":"60","author":"Jantunen","year":"2015","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1109\/TIE.2014.2327917","article-title":"Enabling Health Monitoring Approach Based on Vibration Data for Accurate Prognostics","volume":"62","author":"Javed","year":"2014","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1080\/07408170590929018","article-title":"Residual-life distributions from component degradation signals: A Bayesian approach","volume":"37","author":"Gebraeel","year":"2005","journal-title":"IIE Trans."},{"key":"ref_35","unstructured":"Nectoux, P., Gouriveau, R., Medjaher, K., Ramasso, E., Morello, B., Zerhouni, N., and Varnier, C. (2012, January 18\u201321). PRONOSTIA: An experimental platform for bearings accelerated life test. Proceedings of the IEEE International Conference on Prognostics and Health Management, Denver, CO, USA."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1016\/j.ymssp.2016.05.038","article-title":"The application of a general mathematical morphological particle as a novel indicator for the performance degradation assessment of a bearing","volume":"82","author":"Li","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.measurement.2017.05.033","article-title":"Rolling Bearing Performance Degradation Condition Recognition Based on Mathematical Morphological Fractal Dimension and Fuzzy C-Means","volume":"109","author":"Wang","year":"2017","journal-title":"Measurement"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/9\/2151\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:50:31Z","timestamp":1760187031000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/9\/2151"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,9]]},"references-count":37,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["s19092151"],"URL":"https:\/\/doi.org\/10.3390\/s19092151","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,9]]}}}