{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T11:17:55Z","timestamp":1772623075493,"version":"3.50.1"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,1,15]],"date-time":"2025-01-15T00:00:00Z","timestamp":1736899200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2025,1,15]],"date-time":"2025-01-15T00:00:00Z","timestamp":1736899200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["72301293"],"award-info":[{"award-number":["72301293"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Complex Intell. Syst."],"published-print":{"date-parts":[[2025,2]]},"DOI":"10.1007\/s40747-024-01773-w","type":"journal-article","created":{"date-parts":[[2025,1,15]],"date-time":"2025-01-15T07:11:01Z","timestamp":1736925061000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A generalized diffusion model for remaining useful life prediction with uncertainty"],"prefix":"10.1007","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3449-5841","authenticated-orcid":false,"given":"Bincheng","family":"Wen","sequence":"first","affiliation":[]},{"given":"Xin","family":"Zhao","sequence":"additional","affiliation":[]},{"given":"Xilang","family":"Tang","sequence":"additional","affiliation":[]},{"given":"Mingqing","family":"Xiao","sequence":"additional","affiliation":[]},{"given":"Haizhen","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Jianfeng","family":"Li","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,1,15]]},"reference":[{"key":"1773_CR1","volume":"151","author":"B Cai","year":"2021","unstructured":"Cai B, Fan H, Shao X, Liu Y, Liu G, Liu Z, Ji R (2021) Remaining useful life re-prediction methodology based on wiener process: Subsea Christmas tree system as a case study. Comput Ind Eng 151:106983","journal-title":"Comput Ind Eng"},{"key":"1773_CR2","volume":"244","author":"L Chen","year":"2022","unstructured":"Chen L, Ding Y, Liu B, Wu S, Wang Y, Pan H (2022) Remaining useful life prediction of lithium-ion battery using a novel particle filter framework with grey neural network. Energy 244:122581","journal-title":"Energy"},{"key":"1773_CR3","doi-asserted-by":"crossref","first-page":"3498","DOI":"10.1109\/ACCESS.2019.2961951","volume":"8","author":"X Chen","year":"2019","unstructured":"Chen X, Sun X, Si X, Li G (2019) Remaining useful life prediction based on an adaptive inverse gaussian degradation process with measurement errors. IEEE Access 8:3498\u20133510","journal-title":"IEEE Access"},{"issue":"3","key":"1773_CR4","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1109\/TSMCA.2012.2207109","volume":"43","author":"MJ Daigle","year":"2012","unstructured":"Daigle MJ, Goebel K (2012) Model-based prognostics with concurrent damage progression processes. IEEE Trans Syst Man Cybern Syst 43(3):535\u2013546","journal-title":"IEEE Trans Syst Man Cybern Syst"},{"key":"1773_CR5","doi-asserted-by":"crossref","DOI":"10.1016\/j.ymssp.2021.107974","volume":"161","author":"S Duan","year":"2021","unstructured":"Duan S, Song W, Zio E, Cattani C, Li M (2021) Product technical life prediction based on multi-modes and fractional l\u00e9vy stable motion. Mech Syst Signal Process 161:107974","journal-title":"Mech Syst Signal Process"},{"issue":"4","key":"1773_CR6","doi-asserted-by":"crossref","first-page":"1072","DOI":"10.1109\/TASE.2012.2227960","volume":"10","author":"L Feng","year":"2013","unstructured":"Feng L, Wang H, Si X, Zou H (2013) A state-space-based prognostic model for hidden and age-dependent nonlinear degradation process. IEEE Trans Autom Sci Eng 10(4):1072\u20131086","journal-title":"IEEE Trans Autom Sci Eng"},{"key":"1773_CR7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ress.2017.11.006","volume":"171","author":"H Gao","year":"2018","unstructured":"Gao H, Cui L, Kong D (2018) Reliability analysis for a wiener degradation process model under changing failure thresholds. Reliab Eng Syst Saf 171:1\u20138","journal-title":"Reliab Eng Syst Saf"},{"issue":"3","key":"1773_CR8","doi-asserted-by":"crossref","first-page":"724","DOI":"10.1016\/j.ymssp.2008.06.009","volume":"23","author":"A Heng","year":"2009","unstructured":"Heng A, Zhang S, Tan AC, Mathew J (2009) Rotating machinery prognostics: state of the art, challenges and opportunities. Mech Syst Signal Process 23(3):724\u2013739","journal-title":"Mech Syst Signal Process"},{"issue":"5","key":"1773_CR9","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1080\/24725854.2020.1793036","volume":"53","author":"T Huang","year":"2021","unstructured":"Huang T, Zhao Y, Coit DW, Tang LC (2021) Reliability assessment and lifetime prediction of degradation processes considering recoverable shock damages. IISE Trans 53(5):614\u2013628","journal-title":"IISE Trans"},{"key":"1773_CR10","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1016\/j.ymssp.2017.11.016","volume":"104","author":"Y Lei","year":"2018","unstructured":"Lei Y, Li N, Guo L, Li N, Yan T, Lin J (2018) Machinery health prognostics: a systematic review from data acquisition to rul prediction. Mech Syst Signal Process 104:799\u2013834","journal-title":"Mech Syst Signal Process"},{"issue":"7","key":"1773_CR11","doi-asserted-by":"crossref","first-page":"5709","DOI":"10.1109\/TIE.2017.2677334","volume":"64","author":"N Li","year":"2017","unstructured":"Li N, Lei Y, Guo L, Yan T, Lin J (2017) Remaining useful life prediction based on a general expression of stochastic process models. IEEE Trans Ind Electron 64(7):5709\u20135718","journal-title":"IEEE Trans Ind Electron"},{"issue":"3","key":"1773_CR12","doi-asserted-by":"crossref","first-page":"2092","DOI":"10.1109\/TIE.2018.2838078","volume":"66","author":"N Li","year":"2019","unstructured":"Li N, Lei Y, Yan T, Li N, Han T (2019) A wiener-process-model-based method for remaining useful life prediction considering unit-to-unit variability. IEEE Trans Ind Electron 66(3):2092\u20132101","journal-title":"IEEE Trans Ind Electron"},{"key":"1773_CR13","doi-asserted-by":"publisher","first-page":"5471","DOI":"10.1109\/ACCESS.2019.2962502","volume":"8","author":"T Li","year":"2020","unstructured":"Li T, Pei H, Pang Z, Si X, Zheng J (2020) A sequential Bayesian updated wiener process model for remaining useful life prediction. IEEE Access 8:5471\u20135480. https:\/\/doi.org\/10.1109\/ACCESS.2019.2962502","journal-title":"IEEE Access"},{"issue":"1","key":"1773_CR14","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1109\/TR.2014.2299152","volume":"63","author":"L Liao","year":"2014","unstructured":"Liao L, Koettig F (2014) Review of hybrid prognostics approaches for remaining useful life prediction of engineered systems, and an application to battery life prediction. IEEE Trans Reliab 63(1):191\u2013207","journal-title":"IEEE Trans Reliab"},{"key":"1773_CR15","volume":"214","author":"CP Lin","year":"2021","unstructured":"Lin CP, Ling MH, Cabrera J, Yang F, Yu DYW, Tsui KL (2021) Prognostics for lithium-ion batteries using a two-phase gamma degradation process model. Reliab Eng Syst Saf 214:107797","journal-title":"Reliab Eng Syst Saf"},{"issue":"15","key":"1773_CR16","doi-asserted-by":"publisher","first-page":"12010","DOI":"10.3390\/su151512010","volume":"15","author":"M Liu","year":"2023","unstructured":"Liu M, Dong Z, Shi H (2023) Multi-sensor information fusion and multi-model fusion-based remaining useful life prediction of fan slewing bearings with the nonlinear wiener process. Sustainability 15(15):12010. https:\/\/doi.org\/10.3390\/su151512010","journal-title":"Sustainability"},{"key":"1773_CR17","volume":"218","author":"S Liu","year":"2022","unstructured":"Liu S, Fan L (2022) An adaptive prediction approach for rolling bearing remaining useful life based on multistage model with three-source variability. Reliab Eng Syst Saf 218:108182","journal-title":"Reliab Eng Syst Saf"},{"key":"1773_CR18","volume":"217","author":"X Liu","year":"2022","unstructured":"Liu X, Matias J, J\u00e4schke J, Vatn J (2022) Gibbs sampler for noisy transformed gamma process: inference and remaining useful life estimation. Reliab Eng Syst Saf 217:108084","journal-title":"Reliab Eng Syst Saf"},{"key":"1773_CR19","volume":"485","author":"YH Lui","year":"2021","unstructured":"Lui YH, Li M, Downey A, Shen S, Nemani VP, Ye H, VanElzen C, Jain G, Hu S, Laflamme S, Hu C (2021) Physics-based prognostics of implantable-grade lithium-ion battery for remaining useful life prediction. J Power Sources 485:229327","journal-title":"J Power Sources"},{"key":"1773_CR20","volume":"195","author":"PR de Oliveira da Costa","year":"2020","unstructured":"de Oliveira da Costa PR, Ak\u00e7ay A, Zhang Y, Kaymak U (2020) Remaining useful lifetime prediction via deep domain adaptation. Reliab Eng Syst Saf 195:106682","journal-title":"Reliab Eng Syst Saf"},{"issue":"5","key":"1773_CR21","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1109\/JAS.2021.1003859","volume":"9","author":"Z Pang","year":"2022","unstructured":"Pang Z, Si X, Hu C, Zhang Z (2022) An age-dependent and state-dependent adaptive prognostic approach for hidden nonlinear degrading system. IEEE\/CAA J Automat Sin 9(5):907\u2013921","journal-title":"IEEE\/CAA J Automat Sin"},{"issue":"8","key":"1773_CR22","doi-asserted-by":"crossref","first-page":"5239","DOI":"10.1109\/TII.2020.3032690","volume":"17","author":"M Ragab","year":"2020","unstructured":"Ragab M, Chen Z, Wu M, Foo CS, Kwoh CK, Yan R, Li X (2020) Contrastive adversarial domain adaptation for machine remaining useful life prediction. IEEE Trans Ind Inform 17(8):5239\u20135249","journal-title":"IEEE Trans Ind Inform"},{"issue":"1","key":"1773_CR23","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/0022-247X(76)90244-4","volume":"54","author":"LM Ricciardi","year":"1976","unstructured":"Ricciardi LM (1976) On the transformation of diffusion processes into the wiener process. J Math Anal Appl 54(1):185\u2013199","journal-title":"J Math Anal Appl"},{"issue":"8","key":"1773_CR24","doi-asserted-by":"crossref","first-page":"5082","DOI":"10.1109\/TIE.2015.2393840","volume":"62","author":"XS Si","year":"2015","unstructured":"Si XS (2015) An adaptive prognostic approach via nonlinear degradation modeling: application to battery data. IEEE Trans Ind Electron 62(8):5082\u20135096","journal-title":"IEEE Trans Ind Electron"},{"issue":"1","key":"1773_CR25","first-page":"219","volume":"35","author":"XS Si","year":"2013","unstructured":"Si XS, Wang W, Hu CH, Chen MY, Zhou DH (2013) A wiener-process-based degradation model with a recursive filter algorithm for remaining useful life estimation. Mech Syst Signal Process 35(1):219\u2013237","journal-title":"Mech Syst Signal Process"},{"issue":"1","key":"1773_CR26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ejor.2010.11.018","volume":"213","author":"XS Si","year":"2011","unstructured":"Si XS, Wang W, Hu CH, Zhou DH (2011) Remaining useful life estimation - a review on the statistical data driven approaches. Eur J Oper Res 213(1):1\u201314","journal-title":"Eur J Oper Res"},{"issue":"1","key":"1773_CR27","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1109\/TR.2014.2299151","volume":"63","author":"XS Si","year":"2014","unstructured":"Si XS, Wang W, Hu CH, Zhou DH (2014) Estimating remaining useful life with three-source variability in degradation modeling. IEEE Trans Reliab 63(1):167\u2013190","journal-title":"IEEE Trans Reliab"},{"issue":"1","key":"1773_CR28","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1109\/TR.2011.2182221","volume":"61","author":"XS Si","year":"2012","unstructured":"Si XS, Wang W, Hu CH, Zhou DH, Pecht MG (2012) Remaining useful life estimation based on a nonlinear diffusion degradation process. IEEE Trans Reliab 61(1):50\u201367","journal-title":"IEEE Trans Reliab"},{"key":"1773_CR29","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1016\/j.apm.2021.09.041","volume":"102","author":"W Song","year":"2022","unstructured":"Song W, Liu H, Zio E (2022) Long-range dependence and heavy tail characteristics for remaining useful life prediction in rolling bearing degradation. Appl Math Model 102:268\u2013284","journal-title":"Appl Math Model"},{"key":"1773_CR30","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.microrel.2017.06.045","volume":"75","author":"Y Song","year":"2017","unstructured":"Song Y, Liu D, Yang C, Peng Y (2017) Data-driven hybrid remaining useful life estimation approach for spacecraft lithium-ion battery. Microelectron Reliab 75:142\u2013153","journal-title":"Microelectron Reliab"},{"key":"1773_CR31","volume":"173","author":"B Sun","year":"2021","unstructured":"Sun B, Li Y, Wang Z, Ren Y, Feng Q, Yang D (2021) An improved inverse gaussian process with random effects and measurement errors for rul prediction of hydraulic piston pump. Measurement 173:108604","journal-title":"Measurement"},{"issue":"11","key":"1773_CR32","doi-asserted-by":"crossref","first-page":"4493","DOI":"10.1007\/s00170-020-05264-3","volume":"107","author":"H Sun","year":"2020","unstructured":"Sun H, Pan J, Zhang J, Cao D (2020) Non-linear wiener process-based cutting tool remaining useful life prediction considering measurement variability. Int J Adv Manuf Technol 107(11):4493\u20134502","journal-title":"Int J Adv Manuf Technol"},{"issue":"2","key":"1773_CR33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3390\/en7020520","volume":"7","author":"S Tang","year":"2014","unstructured":"Tang S, Yu C, Wang X, Guo X, Si X (2014) Remaining useful life prediction of lithium-ion batteries based on the wiener process with measurement error. Energies 7(2):1\u201328","journal-title":"Energies"},{"issue":"8","key":"1773_CR34","doi-asserted-by":"crossref","first-page":"5741","DOI":"10.1007\/s00500-019-04311-w","volume":"24","author":"X Tang","year":"2020","unstructured":"Tang X, Xiao M, Hu B (2020) Application of Kalman filter to model-based prognostics for solenoid valve. Soft Comput 24(8):5741\u20135753","journal-title":"Soft Comput"},{"issue":"1","key":"1773_CR35","doi-asserted-by":"crossref","first-page":"523","DOI":"10.3233\/JIFS-169608","volume":"35","author":"X Tang","year":"2018","unstructured":"Tang X, Xiao M, Liang Y, Hu B, Zhang L (2018) Application of particle filter technique to online prognostics for solenoid valve. J Intell Fuzzy Syst 35(1):523\u2013532","journal-title":"J Intell Fuzzy Syst"},{"key":"1773_CR36","doi-asserted-by":"crossref","unstructured":"Usynin A, Hines JW, Urmanov A (2008) Uncertain failure thresholds in cumulative damage models. In: 2008 Annual Reliability and Maintainability Symposium, pp. 334\u2013340","DOI":"10.1109\/RAMS.2008.4925818"},{"key":"1773_CR37","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1016\/j.ymssp.2017.07.015","volume":"99","author":"D Wang","year":"2018","unstructured":"Wang D, Tsui KL (2018) Brownian motion with adaptive drift for remaining useful life prediction: revisited. Mech Syst Signal Process 99:691\u2013701","journal-title":"Mech Syst Signal Process"},{"key":"1773_CR38","doi-asserted-by":"crossref","unstructured":"Wang H, Ma X, Zhao Y (2020) A mixed-effects model of two-phase degradation process for reliability assessment and rul prediction. Microelectron Reliab 107:113622","DOI":"10.1016\/j.microrel.2020.113622"},{"issue":"12","key":"1773_CR39","doi-asserted-by":"crossref","first-page":"12760","DOI":"10.1109\/TIE.2020.3047037","volume":"68","author":"P Wang","year":"2020","unstructured":"Wang P, Gao RX, Woyczynski WA (2020) Levy process-based stochastic modeling for machine performance degradation prognosis. IEEE Trans Ind Electron 68(12):12760\u201312770","journal-title":"IEEE Trans Ind Electron"},{"key":"1773_CR40","doi-asserted-by":"crossref","first-page":"5562","DOI":"10.1016\/j.egyr.2021.08.182","volume":"7","author":"S Wang","year":"2021","unstructured":"Wang S, Jin S, Bai D, Fan Y, Shi H, Fernandez C (2021) A critical review of improved deep learning methods for the remaining useful life prediction of lithium-ion batteries. Energy Reports 7:5562\u20135574","journal-title":"Energy Reports"},{"issue":"11","key":"1773_CR41","doi-asserted-by":"crossref","first-page":"2734","DOI":"10.1007\/s12555-018-0558-z","volume":"17","author":"G Xie","year":"2019","unstructured":"Xie G, Li X, Peng X, Qian F, Hei X (2019) Estimating the probability density function of remaining useful life for wiener degradation process with uncertain parameters. Int J Control Autom Syst 17(11):2734\u20132745","journal-title":"Int J Control Autom Syst"},{"key":"1773_CR42","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.ress.2015.02.005","volume":"139","author":"ZS Ye","year":"2015","unstructured":"Ye ZS, Chen N, Shen Y (2015) A new class of wiener process models for degradation analysis. Reliab Eng Syst Saf 139:58\u201367","journal-title":"Reliab Eng Syst Saf"},{"issue":"9","key":"1773_CR43","doi-asserted-by":"publisher","first-page":"2919","DOI":"10.1007\/s11771-021-4816-5","volume":"28","author":"D Yi","year":"2021","unstructured":"Yi D, Shu C, Qin-jie G, Tian-jian Y, Xun W, Fu-liang B (2021) Life prediction of Ni-Cd battery based on linear wiener process. J Central South Univ 28(9):2919\u20132930. https:\/\/doi.org\/10.1007\/s11771-021-4816-5","journal-title":"J Central South Univ"},{"key":"1773_CR44","volume":"217","author":"W Yu","year":"2022","unstructured":"Yu W, Shao Y, Xu J, Mechefske C (2022) An adaptive and generalized wiener process model with a recursive filtering algorithm for remaining useful life estimation. Reliab Eng Syst Saf 217:108099","journal-title":"Reliab Eng Syst Saf"},{"key":"1773_CR45","volume":"212","author":"W Yu","year":"2021","unstructured":"Yu W, Tu W, Kim IY, Mechefske C (2021) A nonlinear-drift-driven wiener process model for remaining useful life estimation considering three sources of variability. Reliab Eng Syst Saf 212:107631","journal-title":"Reliab Eng Syst Saf"},{"issue":"6","key":"1773_CR46","doi-asserted-by":"crossref","first-page":"2911","DOI":"10.1109\/TII.2017.2684821","volume":"13","author":"Q Zhai","year":"2017","unstructured":"Zhai Q, Ye ZS (2017) Rul prediction of deteriorating products using an adaptive wiener process model. IEEE Trans Ind Inform 13(6):2911\u20132921","journal-title":"IEEE Trans Ind Inform"},{"issue":"6","key":"1773_CR47","doi-asserted-by":"crossref","first-page":"3847","DOI":"10.1109\/TSMC.2021.3073052","volume":"52","author":"Y Zhang","year":"2022","unstructured":"Zhang Y, Yang Y, Li H, Xiu X, Liu W (2022) A data-driven modeling method for stochastic nonlinear degradation process with application to rul estimation. IEEE Trans Syst Man Cybern Syst 52(6):3847\u20133858","journal-title":"IEEE Trans Syst Man Cybern Syst"},{"issue":"3","key":"1773_CR48","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1016\/j.ejor.2018.02.033","volume":"271","author":"Z Zhang","year":"2018","unstructured":"Zhang Z, Si X, Hu C, Lei Y (2018) Degradation data analysis and remaining useful life estimation: a review on wiener-process-based methods. Eur J Oper Res 271(3):775\u2013796","journal-title":"Eur J Oper Res"}],"container-title":["Complex &amp; Intelligent Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-024-01773-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40747-024-01773-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-024-01773-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,7]],"date-time":"2025-02-07T16:32:31Z","timestamp":1738945951000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40747-024-01773-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,15]]},"references-count":48,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,2]]}},"alternative-id":["1773"],"URL":"https:\/\/doi.org\/10.1007\/s40747-024-01773-w","relation":{},"ISSN":["2199-4536","2198-6053"],"issn-type":[{"value":"2199-4536","type":"print"},{"value":"2198-6053","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,15]]},"assertion":[{"value":"25 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 December 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 January 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"140"}}