{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T12:00:35Z","timestamp":1772884835305,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2016,1,8]],"date-time":"2016-01-08T00:00:00Z","timestamp":1452211200000},"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":["51175077"],"award-info":[{"award-number":["51175077"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007219","name":"Natural Science Foundation of Shanghai","doi-asserted-by":"publisher","award":["14ZR1418500"],"award-info":[{"award-number":["14ZR1418500"]}],"id":[{"id":"10.13039\/100007219","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>According to the chaotic features and typical fractional order characteristics of the bearing vibration intensity time series, a forecasting approach based on long range dependence (LRD) is proposed. In order to reveal the internal chaotic properties, vibration intensity time series are reconstructed based on chaos theory in phase-space, the delay time is computed with C-C method and the optimal embedding dimension and saturated correlation dimension are calculated via the Grassberger\u2013Procaccia (G-P) method, respectively, so that the chaotic characteristics of vibration intensity time series can be jointly determined by the largest Lyapunov exponent and phase plane trajectory of vibration intensity time series, meanwhile, the largest Lyapunov exponent is calculated by the Wolf method and phase plane trajectory is illustrated using Duffing-Holmes Oscillator (DHO). The Hurst exponent and long range dependence prediction method are proposed to verify the typical fractional order features and improve the prediction accuracy of bearing vibration intensity time series, respectively. Experience shows that the vibration intensity time series have chaotic properties and the LRD prediction method is better than the other prediction methods (largest Lyapunov, auto regressive moving average (ARMA) and BP neural network (BPNN) model) in prediction accuracy and prediction performance, which provides a new approach for running tendency predictions for rotating machinery and provide some guidance value to the engineering practice.<\/jats:p>","DOI":"10.3390\/e18010023","type":"journal-article","created":{"date-parts":[[2016,1,8]],"date-time":"2016-01-08T23:38:27Z","timestamp":1452296307000},"page":"23","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["Long Range Dependence Prognostics for Bearing Vibration Intensity Chaotic Time Series"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7170-4679","authenticated-orcid":false,"given":"Qing","family":"Li","sequence":"first","affiliation":[{"name":"College of Mechanical Engineering, Donghua University, Shanghai 201620, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Steven","family":"Liang","sequence":"additional","affiliation":[{"name":"College of Mechanical Engineering, Donghua University, Shanghai 201620, China"},{"name":"George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0560, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianguo","family":"Yang","sequence":"additional","affiliation":[{"name":"College of Mechanical Engineering, Donghua University, Shanghai 201620, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Beizhi","family":"Li","sequence":"additional","affiliation":[{"name":"College of Mechanical Engineering, Donghua University, Shanghai 201620, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,1,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.dsp.2014.08.006","article-title":"An adaptive method for health trend prediction of rotating bearings","volume":"35","author":"Hong","year":"2014","journal-title":"Digit. Signal Process."},{"key":"ref_2","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_3","first-page":"150","article-title":"Accurate bearing remaining useful life prediction based on Weibull distribution and artificial neural network","volume":"56\u201357","author":"Ali","year":"2015","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1016\/j.ress.2009.10.009","article-title":"Time series methods applied to failure prediction and detection","volume":"95","author":"Pedregal","year":"2010","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"343","DOI":"10.4028\/www.scientific.net\/AMM.493.343","article-title":"Degradation trend estimation and prognosis of large low speed slewing bearing lifetime","volume":"493","author":"Kosasih","year":"2014","journal-title":"Appl. Mech. Mater."},{"key":"ref_6","first-page":"130","article-title":"Prediction of selected Indian stock using a partitioning\u2014Interpolation based ARIMA\u2013GARCH model","volume":"11","author":"Badu","year":"2015","journal-title":"Appl. Comput. Inf."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1016\/j.engappai.2003.10.004","article-title":"Elman\u2019s recurrent neural network applications to condition monitoring in nuclear power plant and rotating machinery","volume":"16","author":"Ayaz","year":"2003","journal-title":"Eng. Appl. Artif. Intel."},{"key":"ref_8","unstructured":"Malhi, A., and Gao, R.X. (2004, January 18\u201320). Recurrent neural networks for long-term prediction in machine condition monitoring. Proceedings of the 21st IEEE Instrumentation and Measurement Technology Conference, Como, Italy."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5021","DOI":"10.4028\/www.scientific.net\/AMM.110-116.5021","article-title":"Machine fault diagnosis using MLPs and RBF Neural Networks","volume":"110","author":"Payganeh","year":"2011","journal-title":"Appl. Mech. Mater."},{"key":"ref_10","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_11","doi-asserted-by":"crossref","first-page":"5200","DOI":"10.1016\/j.eswa.2011.11.019","article-title":"Bearing fault prognosis based on health state probability estimation","volume":"39","author":"Kim","year":"2012","journal-title":"Expert Syst. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4301","DOI":"10.1109\/TIE.2012.2183834","article-title":"Prognostics health management of electronic systems under mechanical shock and vibration using Kalman filter models and metrics","volume":"59","author":"Lall","year":"2012","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2495","DOI":"10.4028\/www.scientific.net\/AMR.945-949.2495","article-title":"Chaotic time series adaptive prediction based on volterra series","volume":"945\u2013949","author":"Dai","year":"2014","journal-title":"Adv. Mater. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1120","DOI":"10.4028\/www.scientific.net\/AMM.44-47.1120","article-title":"Dynamic Prediction of Rolling Bearing Friction Torque Using Lyapunov Exponent Method","volume":"44\u201347","author":"Xia","year":"2010","journal-title":"Appl. Mech. Mater."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s11071-014-1857-4","article-title":"Distance-based analysis of dynamical systems reconstructed from vibrations for bearing diagnostics","volume":"80","author":"Selina","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"397454","DOI":"10.1155\/2010\/397454","article-title":"On the predictability of long-range dependent series","volume":"2010","author":"Li","year":"2010","journal-title":"Math. Probl. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1016\/j.jfranklin.2003.09.002","article-title":"A correlation-based computational model for synthesizing long-range dependent data","volume":"340","author":"Li","year":"2004","journal-title":"J. Frankl. Inst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1016\/S0895-7177(03)90068-9","article-title":"Software Agents Architecture for Controlling Long-Range Dependent Network Traffic","volume":"38","author":"Gyires","year":"2003","journal-title":"Math. Comput. Model."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/j.automatica.2011.07.012","article-title":"A regularised estimator for long-range dependent processes","volume":"48","author":"Vivero","year":"2012","journal-title":"Automatica"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1443","DOI":"10.1177\/1077546307087438","article-title":"Fractional order signal processing of electrochemical noises","volume":"14","author":"Chen","year":"2008","journal-title":"J. Vib. Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1016\/j.sigpro.2010.01.023","article-title":"FARIMA with stable innovations model of Great Salt Lake elevation time series","volume":"91","author":"Sheng","year":"2011","journal-title":"Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0167-2789(85)90011-9","article-title":"Determining Lyapunov exponents form a time series","volume":"16","author":"Wolf","year":"1985","journal-title":"Phys. D"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"510406","DOI":"10.1155\/2008\/510406","article-title":"Tool Wear Detection Based on Duffing-Holmes Oscillator","volume":"2008","author":"Song","year":"2008","journal-title":"Math. Probl. Eng."},{"key":"ref_24","unstructured":"Takens, F. (1981). Dynamical Systems and Turbulence, Springer."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/S0167-2789(98)00240-1","article-title":"Nonlinear dynamics, delay times, and embedding windows","volume":"127","author":"Kim","year":"1999","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/0167-2789(84)90269-0","article-title":"Dimensions and entropies of strange attractors from a fluctuating dynamics approach","volume":"13","author":"Grassberger","year":"1984","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/0167-2789(83)90298-1","article-title":"Measuring the strangeness of strange attractors","volume":"9","author":"Grassberger","year":"1983","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.aop.2015.05.031","article-title":"Nonlinear power spectral densities for the harmonic oscillator","volume":"361","author":"Hauer","year":"2015","journal-title":"Ann. Phys."},{"key":"ref_29","unstructured":"Beran, J. (1994). Statistics for Long-Memory Processes, Chapman & Hall. [1st ed.]."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1093\/biomet\/68.1.165","article-title":"Fractional differencing","volume":"68","author":"Hosking","year":"1981","journal-title":"Biometrika"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1388","DOI":"10.1016\/j.renene.2008.09.006","article-title":"Day-ahead wind speed forecasting using f-ARIMA models","volume":"34","author":"Kavasseri","year":"2009","journal-title":"Renew. Energy"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.comnet.2011.07.027","article-title":"Non-asymptotic end-to-end performance bounds for networks with long range dependent fBm cross traffic","volume":"56","author":"Rizk","year":"2012","journal-title":"Comput. Netw."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1061\/TACEAT.0006518","article-title":"Long-term storage capacity of reservoirs","volume":"116","author":"Hurst","year":"1951","journal-title":"Trans. Am. Soc. Civil. Eng."},{"key":"ref_34","unstructured":"Center for Intelligent Maintenance Systems. Available online: http:\/\/www.imscenter.net\/."},{"key":"ref_35","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 Applicationon Roller Bearing Prognostics","volume":"289","author":"Qiu","year":"2006","journal-title":"J. Sound Vib."},{"key":"ref_36","first-page":"4787","article-title":"The Turbine Machine Fault Prediction Based on Kernel Principal Component Analysis","volume":"383","author":"Liu","year":"2011","journal-title":"Adv. Mater. Res."},{"key":"ref_37","first-page":"1","article-title":"Acceleration signal processing by numerical integration","volume":"38","author":"Chen","year":"2010","journal-title":"J. Huazhong Univ. Sci. Technol. (Nat. Sci. Ed.)"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/1\/23\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:17:27Z","timestamp":1760210247000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/1\/23"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1,8]]},"references-count":37,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,1]]}},"alternative-id":["e18010023"],"URL":"https:\/\/doi.org\/10.3390\/e18010023","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,1,8]]}}}