{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,6]],"date-time":"2026-07-06T13:41:39Z","timestamp":1783345299317,"version":"3.54.6"},"reference-count":30,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,27]],"date-time":"2025-03-27T00:00:00Z","timestamp":1743033600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11861049"],"award-info":[{"award-number":["11861049"]}]},{"name":"National Natural Science Foundation of China","award":["12361058"],"award-info":[{"award-number":["12361058"]}]},{"name":"National Natural Science Foundation of China","award":["2022MS01006"],"award-info":[{"award-number":["2022MS01006"]}]},{"name":"National Natural Science Foundation of China","award":["JY20220083"],"award-info":[{"award-number":["JY20220083"]}]},{"name":"Natural Science Foundation of Inner Mongolia","award":["11861049"],"award-info":[{"award-number":["11861049"]}]},{"name":"Natural Science Foundation of Inner Mongolia","award":["12361058"],"award-info":[{"award-number":["12361058"]}]},{"name":"Natural Science Foundation of Inner Mongolia","award":["2022MS01006"],"award-info":[{"award-number":["2022MS01006"]}]},{"name":"Natural Science Foundation of Inner Mongolia","award":["JY20220083"],"award-info":[{"award-number":["JY20220083"]}]},{"name":"Basic Research Funds for Universities Directly under the Inner Mongolia Autonomous Region","award":["11861049"],"award-info":[{"award-number":["11861049"]}]},{"name":"Basic Research Funds for Universities Directly under the Inner Mongolia Autonomous Region","award":["12361058"],"award-info":[{"award-number":["12361058"]}]},{"name":"Basic Research Funds for Universities Directly under the Inner Mongolia Autonomous Region","award":["2022MS01006"],"award-info":[{"award-number":["2022MS01006"]}]},{"name":"Basic Research Funds for Universities Directly under the Inner Mongolia Autonomous Region","award":["JY20220083"],"award-info":[{"award-number":["JY20220083"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Recent advancements in science and technology have resulted in products with enhanced reliability and extended lifespans across the aerospace and related sectors. Traditional statistical models struggle to assess their reliability accurately, prompting increased interest in predicting product lifespans during service. These products, characterized by intricate structures and diverse functionalities, exhibit complex, multistage, multiperformance, and nonlinear degradation processes. To address these challenges, this paper proposes a framework for multiperformance, multi-phase Wiener process modeling and reliability analysis. It introduces a two-phase nonlinear Wiener degradation model and identifies change points via the Schwarz information criterion (SIC). The analytical formula for remaining useful life (RUL) is obtained from the concept of the first hitting time (FHT), which considers the stochastic nature of the degradation amount at the change point. The Akaike information criterion (AIC) is then utilized, and an appropriate copula function is chosen to analyze the correlation between two performance indices, given an established complexity with parameters in the degradation model. A two-step method for estimating these uncertain parameters is presented in this paper. Validation through a turbine engine case study underscores its potential to advance reliability theory and engineering practices.<\/jats:p>","DOI":"10.3390\/e27040349","type":"journal-article","created":{"date-parts":[[2025,3,28]],"date-time":"2025-03-28T04:36:48Z","timestamp":1743136608000},"page":"349","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Remaining Useful Life (RUL) Prediction Based on the Bivariant Two-Phase Nonlinear Wiener Degradation Process"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-4998-6253","authenticated-orcid":false,"given":"Lijun","family":"Sun","sequence":"first","affiliation":[{"name":"College of Science, Inner Mongolia University of Technology, Hohhot 010051, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuying","family":"Liang","sequence":"additional","affiliation":[{"name":"College of Science, Inner Mongolia University of Technology, Hohhot 010051, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5889-1326","authenticated-orcid":false,"given":"Zaizai","family":"Yan","sequence":"additional","affiliation":[{"name":"College of Science, Inner Mongolia University of Technology, Hohhot 010051, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,27]]},"reference":[{"key":"ref_1","first-page":"2119","article-title":"Data-model interactive remaining useful life prediction technologies for stochastic degrading devices with big data","volume":"48","author":"Li","year":"2022","journal-title":"Acta Autom. Sin."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ross, R.W. (2016). 1\u2014Integrated vehicle health management in aerospace structures. Structural Health Monitoring (SHM) in Aerospace Structures, Woodhead Publishing.","DOI":"10.1016\/B978-0-08-100148-6.00001-9"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"100693","DOI":"10.1016\/j.paerosci.2020.100693","article-title":"Gas turbine aero-engines real time on-board modelling: A review, research challenges, and exploring the future","volume":"121","author":"Wei","year":"2020","journal-title":"Prog. Aerosp. Sci."},{"key":"ref_4","first-page":"1285","article-title":"A Review of Data Driven-based Incipient Fault Diagnosis","volume":"42","author":"Wen","year":"2016","journal-title":"Acta. Autom. Sin."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.jmsy.2018.04.018","article-title":"A failure-dependency modeling and state discretization approach for condition-based maintenance optimization of multi-component systems","volume":"47","author":"Xu","year":"2018","journal-title":"J. Manuf. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/j.ieri.2012.06.034","article-title":"Life Prediction based on Degradation Amount Distribution using Composite Time Series Analysis","volume":"1","author":"Wang","year":"2012","journal-title":"IERI Procedia"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.cja.2014.12.015","article-title":"Lifetime prediction based on Gamma processes from accelerated degradation data","volume":"28","author":"Wang","year":"2015","journal-title":"Chin. J. Aeronaut."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1016\/j.jmva.2008.12.007","article-title":"Wiener processes with random effects for degradation data","volume":"101","author":"Wang","year":"2010","journal-title":"J. Multivar. Anal."},{"key":"ref_9","first-page":"3727","article-title":"A Bivariate Degradation Model for Reliability Analysis based on Inverse Gaussian Process","volume":"39","author":"An","year":"2024","journal-title":"Control. Decis."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"e34625","DOI":"10.1016\/j.heliyon.2024.e34625","article-title":"Reliability analysis of inverse Gaussian processes with two-stage degenerate paths","volume":"10","author":"Liang","year":"2024","journal-title":"Heliyon"},{"key":"ref_11","first-page":"152","article-title":"A Gamma wear Process, Energy Conversion","volume":"24","year":"1975","journal-title":"IEEE Trans. Reliab."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1016\/j.cja.2020.12.001","article-title":"Reliability estimation of mechanical seals based on bivariate dependence analysis and considering mode uncertainty","volume":"34","author":"Chen","year":"2021","journal-title":"Chin. J. Aeronaut."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"29774","DOI":"10.1109\/JSEN.2023.3322135","article-title":"An Online Remaining Useful Life Prediction Method with Adaptive Degradation Model Calibration","volume":"23","author":"Ren","year":"2023","journal-title":"IEEE Sens. J."},{"key":"ref_14","first-page":"1006","article-title":"Performance reliability prediction for civil aviation aircraft engine based on Wiener process","volume":"28","author":"Zhu","year":"2013","journal-title":"J. Aerosp."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.ress.2012.11.022","article-title":"Remaining useful life estimation based on stochastic deterioration models: A comparative study","volume":"112","author":"Khanh","year":"2013","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_16","first-page":"807","article-title":"Reliability Analysis of Turbine engines Based on Multivariate Degradation Modeling Under Competitive Failure","volume":"2","author":"Wang","year":"2021","journal-title":"J. Northeast Univ. Nat."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"107450","DOI":"10.1016\/j.cie.2021.107450","article-title":"On Multivariate Copula Modeling of Dependent Degradation Processes","volume":"159","author":"Fang","year":"2021","journal-title":"Comput. Ind. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.aej.2024.04.036","article-title":"A novel compound G family with properties, copulas, modelling the asymmetric and positive-skewed engineering and medical data","volume":"98","author":"Alqasem","year":"2024","journal-title":"Alex. Eng. J."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"109020","DOI":"10.1016\/j.ress.2022.109020","article-title":"Bayesian reliability analysis of complex k-out-of-n: L systems under degradation performance","volume":"231","author":"Saberzadeh","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1177\/1687814018784201","article-title":"Remaining useful life prediction for power-shift steering transmission based on fusion of multiple oil spectra","volume":"10","author":"Yan","year":"2018","journal-title":"Adv. Mech. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3621","DOI":"10.1080\/03610918.2020.1720734","article-title":"Reliability assessment for products with two performance characteristics based on marginal stochastic processes and copulas","volume":"21","author":"Chen","year":"2022","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1109\/TR.2011.2182221","article-title":"Remaining Useful Life Estimation Based on a Nonlinear Diffusion Degradation Process","volume":"61","author":"Si","year":"2012","journal-title":"IEEE Trans. Reliab."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"102422","DOI":"10.1016\/j.aei.2024.102422","article-title":"Degradation path approximation for remaining useful life estimation","volume":"60","author":"Fan","year":"2024","journal-title":"Adv. Eng. Inform."},{"key":"ref_24","unstructured":"Cox, D.R., and Miller, H.D. (1977). The Theory of Stochastic Process, Routledge."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1316","DOI":"10.1080\/00949655.2012.658805","article-title":"Bivariate degradation analysis of products based on Wiener processes and copulas","volume":"83","author":"Pan","year":"2013","journal-title":"J. Stat. Comput. Sim."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1088\/1361-6501\/acb808","article-title":"Remaining useful life prediction for nonlinear two-phase degradation process with measurement errors and imperfect prior information","volume":"34","author":"Lin","year":"2023","journal-title":"Meas. Sci. Technol."},{"key":"ref_27","unstructured":"Frederick, D., Decastro, J., and Litt, J.S. (2024, December 10). User\u2019s Guide for the Commercial Modular Aero-Propulsion System Simulation (C-MAPSS), Available online: https:\/\/ntrs.nasa.gov\/api\/citations\/20070034949\/downloads\/20070034949.pdf."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1002\/qre.2058","article-title":"Multivariate Degradation Modeling of Smart Electricity Meter with Multiple Performance Characteristics via Vine Copulas","volume":"33","author":"Xu","year":"2017","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"102066","DOI":"10.1016\/j.aei.2023.102066","article-title":"A wiener-based remaining useful life prediction method with multiple degradation patterns","volume":"57","author":"Li","year":"2023","journal-title":"Adv. Eng. Inform."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"118727","DOI":"10.1016\/j.compstruct.2024.118727","article-title":"Fatigue life prognosis of composite structures using a transferable deep reinforcement learning-based approach","volume":"353","author":"Liu","year":"2025","journal-title":"Compos. Struct."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/349\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:03:32Z","timestamp":1760029412000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/349"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,27]]},"references-count":30,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["e27040349"],"URL":"https:\/\/doi.org\/10.3390\/e27040349","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,27]]}}}