{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T21:59:17Z","timestamp":1765231157052,"version":"3.37.3"},"reference-count":37,"publisher":"Wiley","license":[{"start":{"date-parts":[[2020,1,31]],"date-time":"2020-01-31T00:00:00Z","timestamp":1580428800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100007219","name":"Natural Science Foundation of Shanghai","doi-asserted-by":"publisher","award":["14ZR141850","17ZR1411900","18XJC002"],"award-info":[{"award-number":["14ZR141850","17ZR1411900","18XJC002"]}],"id":[{"id":"10.13039\/100007219","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007219","name":"Natural Science Foundation of Shanghai","doi-asserted-by":"publisher","award":["14ZR141850","17ZR1411900","18XJC002"],"award-info":[{"award-number":["14ZR141850","17ZR1411900","18XJC002"]}],"id":[{"id":"10.13039\/100007219","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009002","name":"Shanghai University","doi-asserted-by":"publisher","award":["14ZR141850","17ZR1411900","18XJC002"],"award-info":[{"award-number":["14ZR141850","17ZR1411900","18XJC002"]}],"id":[{"id":"10.13039\/501100009002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2020,1,31]]},"abstract":"<jats:p>Gradual degradation of the bearing vibration signal is usually studied as a nonstationary stochastic time series. Roller bearings are working at high speed in a heavy load environment so that the combination of bearing faults gradually degraded during the rotation might lead to unpredicted catastrophic accidents. The degradation process has the property of long-range dependence (LRD), so that the fractional Brownian motion (fBm) is taken into account for a prediction model. Because of the dramatic changes in the bearing degradation process, the Hurst exponent that describes the fBm will change during the degradation process. A priori Hurst value of the conventional fBm in the prediction is fixed, thus inducing a minor accuracy of the prediction. To avoid this problem, we propose an improved prediction method. Based on the following steps, at the initial data processing, a skip-over factor is selected as the characteristics parameter of the bearing degradation process. A multifractional Brownian motion (mfBm) replaces the fBm for the degradation modeling. We will show that also our mfBm has the same property of long-range dependence as the fBm. Moreover, a time-varying Hurst exponent <jats:italic>H<\/jats:italic>(<jats:italic>t<\/jats:italic>) is taken to replace the constant <jats:italic>H<\/jats:italic> in fBm. Finally, we apply the quantum-behaved partial swarm optimization (QPSO) to optimize <jats:italic>H<\/jats:italic>(<jats:italic>t<\/jats:italic>) for a finite interval. Some tests and corresponding experimental results will show that our model QPSO\u2009+\u2009mfBm have a much better performance on the prediction effect than fBm.<\/jats:p>","DOI":"10.1155\/2020\/8543131","type":"journal-article","created":{"date-parts":[[2020,1,31]],"date-time":"2020-01-31T18:33:51Z","timestamp":1580495631000},"page":"1-9","source":"Crossref","is-referenced-by-count":20,"title":["Multifractional Brownian Motion and Quantum-Behaved Partial Swarm Optimization for Bearing Degradation Forecasting"],"prefix":"10.1155","volume":"2020","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0561-3258","authenticated-orcid":true,"given":"Song","family":"Wanqing","sequence":"first","affiliation":[{"name":"School of Electronic and Electrical Engineering, Shanghai University of Engineering Science, No. 333 Longteng Road, Songjiang District, Shanghai 201620, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6519-035X","authenticated-orcid":true,"given":"Xiaoxian","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Electronic and Electrical Engineering, Shanghai University of Engineering Science, No. 333 Longteng Road, Songjiang District, Shanghai 201620, China"}]},{"given":"Carlo","family":"Cattani","sequence":"additional","affiliation":[{"name":"Engineering School (DEIM), University of Tuscia, Viterbo, Italy"}]},{"given":"Enrico","family":"Zio","sequence":"additional","affiliation":[{"name":"Energy Department, Politecnico di Milano, Campus Bovisa, Via La Masa 34\/3, Milano, MI 20156, Italy"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1155\/2019\/4031795"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2017.02.045"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2015.12.021"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1155\/2019\/1564243"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.3390\/s130505542"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.3390\/e19040176"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1109\/tii.2017.2755064"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1155\/2019\/6943234"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.21595\/jve.2016.16910"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1016\/j.ymssp.2014.07.011"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1109\/tie.2017.2733469"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1109\/tie.2017.2733487"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1109\/tie.2015.2455055"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1109\/tie.2014.2336616"},{"year":"1994","key":"15"},{"year":"1999","key":"16"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1109\/tie.2016.2522941"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1109\/tip.2014.2312284"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1140\/epjb\/e2015-60515-5"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1109\/tr.2017.2717488"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1109\/tr.2017.2720752"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.3390\/w11010164"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1016\/j.ymssp.2017.12.017"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1016\/j.energy.2019.116847"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfranklin.2018.09.023"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-018-3331-6"},{"key":"28","doi-asserted-by":"publisher","DOI":"10.1007\/s00209-015-1606-5"},{"key":"29","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2010.05.025"},{"issue":"3","key":"30","first-page":"679","volume":"59","year":"1997","journal-title":"Journal of the Royal Statistical Society. 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