{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T14:41:42Z","timestamp":1774276902170,"version":"3.50.1"},"reference-count":150,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,8]],"date-time":"2024-04-08T00:00:00Z","timestamp":1712534400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Jiangsu Province","award":["BK20211528"],"award-info":[{"award-number":["BK20211528"]}]},{"name":"Natural Science Foundation of Jiangsu Province","award":["JUSRP123063"],"award-info":[{"award-number":["JUSRP123063"]}]},{"name":"Natural Science Foundation of Jiangsu Province","award":["B23008"],"award-info":[{"award-number":["B23008"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["BK20211528"],"award-info":[{"award-number":["BK20211528"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["JUSRP123063"],"award-info":[{"award-number":["JUSRP123063"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["B23008"],"award-info":[{"award-number":["B23008"]}]},{"DOI":"10.13039\/501100013314","name":"111 Project","doi-asserted-by":"publisher","award":["BK20211528"],"award-info":[{"award-number":["BK20211528"]}],"id":[{"id":"10.13039\/501100013314","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013314","name":"111 Project","doi-asserted-by":"publisher","award":["JUSRP123063"],"award-info":[{"award-number":["JUSRP123063"]}],"id":[{"id":"10.13039\/501100013314","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013314","name":"111 Project","doi-asserted-by":"publisher","award":["B23008"],"award-info":[{"award-number":["B23008"]}],"id":[{"id":"10.13039\/501100013314","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>As industrial processes grow increasingly complex, fault identification becomes challenging, and even minor errors can significantly impact both productivity and system safety. Fault detection and diagnosis (FDD) has emerged as a crucial strategy for maintaining system reliability and safety through condition monitoring and abnormality recovery to manage this challenge. Statistical-based FDD methods that rely on large-scale process data and their features have been developed for detecting faults. This paper overviews recent investigations and developments in statistical-based FDD methods, focusing on probabilistic models. The theoretical background of these models is presented, including Bayesian learning and maximum likelihood. We then discuss various techniques and methodologies, e.g., probabilistic principal component analysis (PPCA), probabilistic partial least squares (PPLS), probabilistic independent component analysis (PICA), probabilistic canonical correlation analysis (PCCA), and probabilistic Fisher discriminant analysis (PFDA). Several test statistics are analyzed to evaluate the discussed methods. In industrial processes, these methods require complex matrix operation and cost computational load. Finally, we discuss the current challenges and future trends in FDD.<\/jats:p>","DOI":"10.3390\/sym16040455","type":"journal-article","created":{"date-parts":[[2024,4,8]],"date-time":"2024-04-08T10:11:55Z","timestamp":1712571115000},"page":"455","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["A Review of Statistical-Based Fault Detection and Diagnosis with Probabilistic Models"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4051-3842","authenticated-orcid":false,"given":"Yanting","family":"Zhu","sequence":"first","affiliation":[{"name":"Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5363-8305","authenticated-orcid":false,"given":"Shunyi","family":"Zhao","sequence":"additional","affiliation":[{"name":"Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, China"}]},{"given":"Yuxuan","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Management Science and Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3130-6497","authenticated-orcid":false,"given":"Chengxi","family":"Zhang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5930-4170","authenticated-orcid":false,"given":"Jin","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong 999077, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"108967","DOI":"10.1016\/j.ress.2022.108967","article-title":"Early fault diagnosis of rotating machinery based on composite zoom permutation entropy","volume":"230","author":"Ma","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"108981","DOI":"10.1016\/j.ress.2022.108981","article-title":"Correlation feature distribution matching for fault diagnosis of machines","volume":"231","author":"Tan","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"109256","DOI":"10.1016\/j.ress.2023.109256","article-title":"A digital twin-enhanced semi-supervised framework for motor fault diagnosis based on phase-contrastive current dot pattern","volume":"235","author":"Xia","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"108968","DOI":"10.1016\/j.ress.2022.108968","article-title":"A Tensor-based domain alignment method for intelligent fault diagnosis of rolling bearing in rotating machinery","volume":"230","author":"Liu","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1016\/j.jpowsour.2012.10.060","article-title":"A review on the key issues for lithium-ion battery management in electric vehicles","volume":"226","author":"Lu","year":"2013","journal-title":"J. Power Sources"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"109253","DOI":"10.1016\/j.ress.2023.109253","article-title":"Digital twin-assisted multiscale residual-self-attention feature fusion network for hypersonic flight vehicle fault diagnosis","volume":"235","author":"Dong","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"109300","DOI":"10.1016\/j.ress.2023.109300","article-title":"Improving fault tolerance in diagnosing power system failures with optimal hierarchical extreme learning machine","volume":"236","author":"Yuan","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2517","DOI":"10.1109\/TPEL.2014.2373390","article-title":"Study and handling methods of power IGBT module failures in power electronic converter systems","volume":"30","author":"Choi","year":"2014","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1109\/TPEL.2012.2192503","article-title":"Survey on reliability of power electronic systems","volume":"28","author":"Song","year":"2012","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_10","first-page":"1746","article-title":"Advances in electrical machine, power electronic, and drive condition monitoring and fault detection: State of the art","volume":"62","author":"Capolino","year":"2014","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"108648","DOI":"10.1016\/j.ress.2022.108648","article-title":"Out-of-distribution detection-assisted trustworthy machinery fault diagnosis approach with uncertainty-aware deep ensembles","volume":"226","author":"Han","year":"2022","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1109\/TIA.2010.2090839","article-title":"A survey of condition monitoring and protection methods for medium-voltage induction motors","volume":"47","author":"Zhang","year":"2010","journal-title":"IEEE Trans. Ind. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1842","DOI":"10.1109\/TIE.2006.885131","article-title":"Online diagnosis of induction motors using MCSA","volume":"53","author":"Jung","year":"2006","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1109\/TIE.2007.911960","article-title":"Fault detection in induction machines using power spectral density in wavelet decomposition","volume":"55","author":"Romeral","year":"2008","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.rser.2007.05.008","article-title":"Condition monitoring and fault detection of wind turbines and related algorithms: A review","volume":"13","author":"Hameed","year":"2009","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.renene.2012.03.003","article-title":"Condition monitoring of wind turbines: Techniques and methods","volume":"46","author":"Papaelias","year":"2012","journal-title":"Renew. Energy"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3196","DOI":"10.1109\/TIE.2018.2844805","article-title":"Multiscale convolutional neural networks for fault diagnosis of wind turbine gearbox","volume":"66","author":"Jiang","year":"2018","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.rser.2015.11.050","article-title":"Big data driven smart energy management: From big data to big insights","volume":"56","author":"Zhou","year":"2016","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_19","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_20","doi-asserted-by":"crossref","unstructured":"Oppenheimer, C.H., and Loparo, K.A. (2002, January 1\u20135). Physically based diagnosis and prognosis of cracked rotor shafts. Proceedings of the Component and Systems Diagnostics, Prognostics, and Health Management II, Orlando, FL, USA.","DOI":"10.1117\/12.475502"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1016\/j.jfranklin.2023.12.036","article-title":"Tuning-free filtering for stochastic systems with unmodeled measurement dynamics","volume":"361","author":"Zhu","year":"2023","journal-title":"J. Frankl. Inst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1007\/s42405-023-00633-0","article-title":"Actuator Fault detection, identification, and control of a multirotor air vehicle using residual generation and parameter estimation approaches","volume":"25","author":"Asadi","year":"2023","journal-title":"Int. J. Aeronaut. Space Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"6506109","DOI":"10.1109\/TIM.2023.3328083","article-title":"Robustification of Finite Impulse Response Filter for Nonlinear Systems With Model Uncertainties","volume":"72","author":"Zhao","year":"2023","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"105711","DOI":"10.1016\/j.conengprac.2023.105711","article-title":"Localization of underground pipe jacking machinery: A reliable, real-time and robust INS\/OD solution","volume":"141","author":"Zhao","year":"2023","journal-title":"Control Eng. Pract."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3768","DOI":"10.1109\/TIE.2015.2417501","article-title":"A Survey of Fault Diagnosis and Fault-Tolerant Techniques-Part II: Fault Diagnosis with Knowledge-Based and Hybrid\/Active Approaches","volume":"62","author":"Gao","year":"2015","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.arcontrol.2012.09.004","article-title":"Survey on data-driven industrial process monitoring and diagnosis","volume":"36","author":"Qin","year":"2012","journal-title":"Annu. Rev. Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1016\/j.jprocont.2013.08.011","article-title":"Data-driven design of monitoring and diagnosis systems for dynamic processes: A review of subspace technique based schemes and some recent results","volume":"24","author":"Ding","year":"2014","journal-title":"J. Process Control"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"109047","DOI":"10.1016\/j.ress.2022.109047","article-title":"Statistical identification guided open-set domain adaptation in fault diagnosis","volume":"232","author":"Yu","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_29","first-page":"480","article-title":"Statistical process monitoring: Basics and beyond","volume":"17","year":"2003","journal-title":"J. Chemom. J. Chemom. Soc."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1517","DOI":"10.1109\/TIM.2004.834070","article-title":"PCA-based feature selection scheme for machine defect classification","volume":"53","author":"Malhi","year":"2004","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2014","DOI":"10.1109\/TIE.2011.2160138","article-title":"Diagnosis of three-phase electrical machines using multidimensional demodulation techniques","volume":"59","author":"Choqueuse","year":"2011","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1281","DOI":"10.1016\/S0098-1354(02)00093-5","article-title":"Multivariate process monitoring and fault diagnosis by multi-scale PCA","volume":"26","author":"Misra","year":"2002","journal-title":"Comput. Chem. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3372","DOI":"10.1109\/TIE.2012.2202358","article-title":"A new framework of simultaneous-fault diagnosis using pairwise probabilistic multi-label classification for time-dependent patterns","volume":"60","author":"Vong","year":"2012","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.automatica.2009.10.030","article-title":"Geometric properties of partial least squares for process monitoring","volume":"46","author":"Li","year":"2010","journal-title":"Automatica"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1109\/TII.2009.2033181","article-title":"Decentralized fault diagnosis of large-scale processes using multiblock kernel partial least squares","volume":"6","author":"Zhang","year":"2009","journal-title":"IEEE Trans. Ind. Inf."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3167","DOI":"10.1109\/TIE.2011.2167110","article-title":"A PLS-based statistical approach for fault detection and isolation of robotic manipulators","volume":"59","author":"Muradore","year":"2011","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1670","DOI":"10.1109\/TII.2013.2251891","article-title":"Least-squares fault detection and diagnosis for networked sensing systems using a direct state estimation approach","volume":"9","author":"He","year":"2013","journal-title":"IEEE Trans. Ind. Inf."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/S0169-7439(03)00063-7","article-title":"Process monitoring based on probabilistic PCA","volume":"67","author":"Kim","year":"2003","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1025","DOI":"10.1252\/jcej.38.1025","article-title":"Calibration, prediction and process monitoring model based on factor analysis for incomplete process data","volume":"38","author":"Kim","year":"2005","journal-title":"J. Chem. Eng. Jpn."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2316","DOI":"10.1021\/ie049081o","article-title":"Fault detection based on a maximum-likelihood principal component analysis (PCA) mixture","volume":"44","author":"Choi","year":"2005","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3639","DOI":"10.1007\/s10462-020-09934-2","article-title":"A review on fault detection and diagnosis techniques: Basics and beyond","volume":"54","author":"Abid","year":"2021","journal-title":"Artif. Intell. Rev."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"112395","DOI":"10.1016\/j.rser.2022.112395","article-title":"A review of computing-based automated fault detection and diagnosis of heating, ventilation and air conditioning systems","volume":"161","author":"Chen","year":"2022","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/s00521-022-08017-3","article-title":"Challenges and opportunities of deep learning-based process fault detection and diagnosis: A review","volume":"35","author":"Yu","year":"2023","journal-title":"Neural Comput. Appl."},{"key":"ref_44","unstructured":"Bishop, C.M., and Nasrabadi, N.M. (2006). Pattern Recognition and Machine Learning, Springer."},{"key":"ref_45","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_46","first-page":"921","article-title":"Maximum-likelihood estimation of molecular haplotype frequencies in a diploid population","volume":"12","author":"Excoffier","year":"1995","journal-title":"Mol. Biol. Evol."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1080\/01621459.1995.10476572","article-title":"Bayes factors","volume":"90","author":"Kass","year":"1995","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"921","DOI":"10.1109\/TIE.2022.3153814","article-title":"Tuning-Free Bayesian Estimation Algorithms for Faulty Sensor Signals in State-Space","volume":"70","author":"Zhao","year":"2022","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"8853","DOI":"10.1109\/TIE.2020.3016254","article-title":"Online probabilistic estimation of sensor faulty signal in industrial processes and its applications","volume":"68","author":"Zhao","year":"2020","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"2294","DOI":"10.1109\/TIE.2019.2907505","article-title":"Probabilistic monitoring of correlated sensors for nonlinear processes in state space","volume":"67","author":"Zhao","year":"2019","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1208","DOI":"10.1016\/j.asoc.2010.02.019","article-title":"Bond graph based Bayesian network for fault diagnosis","volume":"11","author":"Lo","year":"2011","journal-title":"Appl. Soft Comput."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/j.ress.2004.06.004","article-title":"A discrete-time Bayesian network reliability modeling and analysis framework","volume":"87","author":"Boudali","year":"2005","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"2227","DOI":"10.1109\/TII.2017.2695583","article-title":"Bayesian networks in fault diagnosis","volume":"13","author":"Cai","year":"2017","journal-title":"IEEE Trans. Ind. Inf."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.ymssp.2016.04.019","article-title":"A real-time fault diagnosis methodology of complex systems using object-oriented Bayesian networks","volume":"80","author":"Cai","year":"2016","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1109\/TASE.2016.2574875","article-title":"A dynamic-Bayesian-network-based fault diagnosis methodology considering transient and intermittent faults","volume":"14","author":"Cai","year":"2016","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1016\/j.ress.2007.03.035","article-title":"A Bayesian Belief Network modelling of organisational factors in risk analysis: A case study in maritime transportation","volume":"93","author":"Trucco","year":"2008","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.ress.2005.11.037","article-title":"Bayesian networks in reliability","volume":"92","author":"Langseth","year":"2007","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1016\/j.ress.2011.03.012","article-title":"Safety analysis in process facilities: Comparison of fault tree and Bayesian network approaches","volume":"96","author":"Khakzad","year":"2011","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1290","DOI":"10.1016\/j.ress.2005.11.025","article-title":"Bayesian analysis of computer code outputs: A tutorial","volume":"91","year":"2006","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/S0951-8320(00)00077-6","article-title":"Improving the analysis of dependable systems by mapping fault trees into Bayesian networks","volume":"71","author":"Bobbio","year":"2001","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ress.2017.11.021","article-title":"Remaining useful life estimation in prognostics using deep convolution neural networks","volume":"172","author":"Li","year":"2018","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1109\/TIE.2015.2466557","article-title":"Performance-driven distributed PCA process monitoring based on fault-relevant variable selection and Bayesian inference","volume":"63","author":"Jiang","year":"2015","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"1310","DOI":"10.1109\/TIE.2007.909060","article-title":"Induction machine broken bar and stator short-circuit fault diagnostics based on three-phase stator current envelopes","volume":"55","author":"Povinelli","year":"2008","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"108865","DOI":"10.1016\/j.ress.2022.108865","article-title":"An uncertainty-informed framework for trustworthy fault diagnosis in safety-critical applications","volume":"229","author":"Zhou","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"108525","DOI":"10.1016\/j.ress.2022.108525","article-title":"Towards trustworthy machine fault diagnosis: A probabilistic Bayesian deep learning framework","volume":"224","author":"Zhou","year":"2022","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"2319","DOI":"10.1126\/science.290.5500.2319","article-title":"A global geometric framework for nonlinear dimensionality reduction","volume":"290","author":"Tenenbaum","year":"2000","journal-title":"Science"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"2037","DOI":"10.1109\/TPAMI.2006.244","article-title":"Face description with local binary patterns: Application to face recognition","volume":"28","author":"Ahonen","year":"2006","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"20150202","DOI":"10.1098\/rsta.2015.0202","article-title":"Principal component analysis: A review and recent developments","volume":"374","author":"Jolliffe","year":"2016","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1109\/TPAMI.2004.1261097","article-title":"Two-dimensional PCA: A new approach to appearance-based face representation and recognition","volume":"26","author":"Yang","year":"2004","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1111\/1467-9868.00196","article-title":"Probabilistic principal component analysis","volume":"61","author":"Tipping","year":"1999","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"1198","DOI":"10.1016\/j.jprocont.2010.07.007","article-title":"A branch and bound method for isolation of faulty variables through missing variable analysis","volume":"20","author":"Kariwala","year":"2010","journal-title":"J. Process Control"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"1357","DOI":"10.1016\/j.cjche.2015.01.014","article-title":"An aligned mixture probabilistic principal component analysis for fault detection of multimode chemical processes","volume":"23","author":"Yang","year":"2015","journal-title":"Chin. J. Chem. Eng."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"1228","DOI":"10.1016\/j.jprocont.2012.05.010","article-title":"Reconstruction-based multivariate contribution analysis for fault isolation: A branch and bound approach","volume":"22","author":"He","year":"2012","journal-title":"J. Process Control"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"2143","DOI":"10.1002\/aic.14419","article-title":"Robust modeling of mixture probabilistic principal component analysis and process monitoring application","volume":"60","author":"Zhu","year":"2014","journal-title":"AlChE J."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1016\/j.jprocont.2011.02.005","article-title":"Robust monitoring and fault reconstruction based on variational inference component analysis","volume":"21","author":"Ge","year":"2011","journal-title":"J. Process Control"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1002\/cem.2714","article-title":"Dynamic mixture probabilistic PCA classifier modeling and application for fault classification","volume":"29","author":"Zhu","year":"2015","journal-title":"J. Chemom."},{"key":"ref_77","first-page":"3814","article-title":"HMM-driven robust probabilistic principal component analyzer for dynamic process fault classification","volume":"62","author":"Zhu","year":"2015","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"107837","DOI":"10.1016\/j.ress.2021.107837","article-title":"A framework to automate fault detection and diagnosis based on moving window principal component analysis and Bayesian network","volume":"215","year":"2021","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.chemolab.2016.08.014","article-title":"Probabilistic learning of partial least squares regression model: Theory and industrial applications","volume":"158","author":"Zheng","year":"2016","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1016\/j.chemolab.2008.09.005","article-title":"Calculation of the reliability of classification in discriminant partial least-squares binary classification","volume":"95","year":"2009","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.jprocont.2018.01.008","article-title":"Semisupervised learning for probabilistic partial least squares regression model and soft sensor application","volume":"64","author":"Zheng","year":"2018","journal-title":"J. Process Control"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"e3196","DOI":"10.1002\/cem.3196","article-title":"Fault monitoring based on locally weighted probabilistic kernel partial least square for nonlinear time-varying processes","volume":"33","author":"Xie","year":"2019","journal-title":"J. Chemom."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/j.talanta.2009.06.072","article-title":"Classification from microarray data using probabilistic discriminant partial least squares with reject option","volume":"80","author":"Botella","year":"2009","journal-title":"Talanta"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.chemolab.2017.09.015","article-title":"Concurrent probabilistic PLS regression model and its applications in process monitoring","volume":"171","author":"Li","year":"2017","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1109\/TMI.2003.822821","article-title":"Probabilistic independent component analysis for functional magnetic resonance imaging","volume":"23","author":"Beckmann","year":"2004","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_86","unstructured":"Bach, F.R., and Jordan, M.I. (2005). A Probabilistic Interpretation of Canonical Correlation Analysis, University of California Berkeley. Technical Report."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1109\/TPAMI.2005.55","article-title":"Face recognition using laplacianfaces","volume":"27","author":"He","year":"2005","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"2067","DOI":"10.1016\/S0031-3203(00)00162-X","article-title":"A direct LDA algorithm for high-dimensional data\u2014With application to face recognition","volume":"34","author":"Yu","year":"2001","journal-title":"Pattern Recognit."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"1713","DOI":"10.1016\/S0031-3203(99)00139-9","article-title":"A new LDA-based face recognition system which can solve the small sample size problem","volume":"33","author":"Chen","year":"2000","journal-title":"Pattern Recognit."},{"key":"ref_90","first-page":"1027","article-title":"Dimensionality reduction of multimodal labeled data by local Fisher discriminant analysis","volume":"8","author":"Sugiyama","year":"2007","journal-title":"J. Mach. Learn. Res."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"109178","DOI":"10.1016\/j.ress.2023.109178","article-title":"Reliable composite fault diagnosis of hydraulic systems based on linear discriminant analysis and multi-output hybrid kernel extreme learning machine","volume":"234","author":"Liu","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.neucom.2011.11.027","article-title":"Probabilistic Fisher discriminant analysis: A robust and flexible alternative to Fisher discriminant analysis","volume":"90","author":"Bouveyron","year":"2012","journal-title":"Neurocomputing"},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.conengprac.2015.05.010","article-title":"Fault detection in dynamic systems using the Kullback\u2013Leibler divergence","volume":"43","author":"Xie","year":"2015","journal-title":"Control Eng. Pract."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.isatra.2018.05.007","article-title":"An improved incipient fault detection method based on Kullback-Leibler divergence","volume":"79","author":"Chen","year":"2018","journal-title":"ISA Trans."},{"key":"ref_95","doi-asserted-by":"crossref","unstructured":"Beran, R. (1977). Minimum Hellinger distance estimates for parametric models. Ann. Stat., 445\u2013463.","DOI":"10.1214\/aos\/1176343842"},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"2497","DOI":"10.1021\/ie403540b","article-title":"Multiblock independent component analysis integrated with Hellinger distance and Bayesian inference for non-Gaussian plant-wide process monitoring","volume":"54","author":"Jiang","year":"2015","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1109\/TCST.2019.2955042","article-title":"Sensor selection embedded in active fault diagnosis algorithms","volume":"29","author":"Palmer","year":"2019","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"2198","DOI":"10.1109\/TITS.2018.2865410","article-title":"A newly robust fault detection and diagnosis method for high-speed trains","volume":"20","author":"Chen","year":"2018","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_99","unstructured":"Barnett, V., and Lewis, T. (1994). Outliers in Statistical Data, Wiley."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.arcontrol.2018.09.003","article-title":"Review and big data perspectives on robust data mining approaches for industrial process modeling with outliers and missing data","volume":"46","author":"Zhu","year":"2018","journal-title":"Annu. Rev. Control"},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"6418","DOI":"10.1109\/TIE.2014.2301773","article-title":"A review on basic data-driven approaches for industrial process monitoring","volume":"61","author":"Yin","year":"2014","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_102","doi-asserted-by":"crossref","unstructured":"Ding, S.X. (2014). Data-Driven Design of Fault Diagnosis and Fault-Tolerant Control Systems, Springer.","DOI":"10.1007\/978-1-4471-6410-4"},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"1700","DOI":"10.1109\/TITS.2020.3029946","article-title":"Data-driven fault diagnosis for traction systems in high-speed trains: A survey, challenges, and perspectives","volume":"23","author":"Chen","year":"2020","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1002\/cem.2686","article-title":"A novel multi-mode data processing method and its application in industrial process monitoring","volume":"29","author":"Wang","year":"2015","journal-title":"J. Chemom."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1016\/j.rser.2019.01.013","article-title":"A data-driven algorithm for online detection of component and system faults in modern wind turbines at different operating zones","volume":"103","author":"Bakdi","year":"2019","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"1604","DOI":"10.1109\/TIE.2020.2970664","article-title":"A dual robustness projection to latent structure method and its application","volume":"68","author":"Zhou","year":"2020","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1177\/14759217221100708","article-title":"An MPPCA-based approach for anomaly detection of structures under multiple operational conditions and missing data","volume":"22","author":"Ma","year":"2023","journal-title":"Struct. Health Monit."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"2441","DOI":"10.1109\/TIE.2013.2273471","article-title":"Motor bearing fault diagnosis using trace ratio linear discriminant analysis","volume":"61","author":"Jin","year":"2013","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"1559","DOI":"10.1109\/TIE.2017.2733501","article-title":"Fault detection for non-Gaussian processes using generalized canonical correlation analysis and randomized algorithms","volume":"65","author":"Chen","year":"2017","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"3543","DOI":"10.1021\/ie302069q","article-title":"Review of recent research on data-based process monitoring","volume":"52","author":"Ge","year":"2013","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/S0098-1354(01)00773-6","article-title":"Dimension reduction of process dynamic trends using independent component analysis","volume":"26","author":"Li","year":"2002","journal-title":"Comput. Chem. Eng."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1002\/aic.690490414","article-title":"Monitoring independent components for fault detection","volume":"49","author":"Kano","year":"2003","journal-title":"AlChE J."},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1109\/TITS.2019.2897583","article-title":"A review of fault detection and diagnosis for the traction system in high-speed trains","volume":"21","author":"Chen","year":"2019","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_114","doi-asserted-by":"crossref","unstructured":"Chen, H., Liu, Z., Alippi, C., Huang, B., and Liu, D. (2022). Explainable intelligent fault diagnosis for nonlinear dynamic systems: From unsupervised to supervised learning. IEEE Trans. Neural Netw. Learn. Syst., 1\u201314.","DOI":"10.1109\/TNNLS.2022.3201511"},{"key":"ref_115","unstructured":"Chiang, L.H., Russell, E.L., and Braatz, R.D. (2000). Fault Detection and Diagnosis in Industrial Systems, Springer Science & Business Media."},{"key":"ref_116","doi-asserted-by":"crossref","unstructured":"Sch\u00f6lkopf, B., Smola, A.J., and Bach, F. (2002). Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press.","DOI":"10.7551\/mitpress\/4175.001.0001"},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"1389","DOI":"10.1016\/j.compchemeng.2003.10.002","article-title":"Fault diagnosis based on Fisher discriminant analysis and support vector machines","volume":"28","author":"Chiang","year":"2004","journal-title":"Comput. Chem. Eng."},{"key":"ref_118","doi-asserted-by":"crossref","first-page":"1991","DOI":"10.1162\/08997660360675116","article-title":"Variational Bayesian learning of ICA with missing data","volume":"15","author":"Chan","year":"2003","journal-title":"Neural Comput."},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.patcog.2009.05.013","article-title":"A general procedure for learning mixtures of independent component analyzers","volume":"43","author":"Salazar","year":"2010","journal-title":"Pattern Recognit."},{"key":"ref_120","first-page":"1783","article-title":"Probabilistic non-linear principal component analysis with Gaussian process latent variable models","volume":"6","author":"Lawrence","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"4792","DOI":"10.1021\/ie9019402","article-title":"Nonlinear probabilistic monitoring based on the Gaussian process latent variable model","volume":"49","author":"Ge","year":"2010","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"108921","DOI":"10.1016\/j.ress.2022.108921","article-title":"Intelligent fault diagnosis of bevel gearboxes using semi-supervised probability support matrix machine and infrared imaging","volume":"230","author":"Li","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"e2926","DOI":"10.1002\/cem.2926","article-title":"Fault detection based on weighted difference principal component analysis","volume":"31","author":"Guo","year":"2017","journal-title":"J. Chemom."},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"5308","DOI":"10.1109\/TII.2018.2810822","article-title":"Canonical variate dissimilarity analysis for process incipient fault detection","volume":"14","author":"Pilario","year":"2018","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_125","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.bej.2003.08.009","article-title":"Modeling and monitoring of batch processes using principal component analysis (PCA) assisted generalized regression neural networks (GRNN)","volume":"18","author":"Kulkarni","year":"2004","journal-title":"Biochem. Eng. J."},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1162\/089976699300016674","article-title":"A unifying review of linear Gaussian models","volume":"11","author":"Roweis","year":"1999","journal-title":"Neural Comput."},{"key":"ref_127","doi-asserted-by":"crossref","unstructured":"Zhao, S.Y., Shmaliy, Y.S., and Liu, F. (2022). Batch Optimal FIR Smoothing: Increasing State Informativity in Nonwhite Measurement Noise Environments. IEEE Trans. Ind. Inf.","DOI":"10.1109\/TII.2022.3193879"},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"6342","DOI":"10.1109\/TSP.2021.3127677","article-title":"Discrete Time q-Lag Maximum Likelihood FIR Smoothing and Iterative Recursive Algorithm","volume":"69","author":"Zhao","year":"2021","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_129","doi-asserted-by":"crossref","unstructured":"Zhang, T.Y., Zhao, S.Y., Luan, X.L., and Liu, F. (2022). Bayesian Inference for State-Space Models with Student-t Mixture Distributions. IEEE Trans. Cybern., 4435\u20134445.","DOI":"10.1109\/TCYB.2022.3183104"},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"109184","DOI":"10.1016\/j.automatica.2020.109184","article-title":"Trial-and-error or avoiding a guess? Initialization of the Kalman filter","volume":"121","author":"Zhao","year":"2020","journal-title":"Automatica"},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"5360","DOI":"10.1109\/TII.2020.3026999","article-title":"Multipass optimal FIR filtering for processes with unknown initial states and temporary mismatches","volume":"17","author":"Zhao","year":"2020","journal-title":"IEEE Trans. Ind. Inf."},{"key":"ref_132","doi-asserted-by":"crossref","first-page":"1372","DOI":"10.1109\/TCST.2020.2991609","article-title":"Self-tuning unbiased finite impulse response filtering algorithm for processes with unknown measurement noise covariance","volume":"29","author":"Zhao","year":"2020","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_133","doi-asserted-by":"crossref","first-page":"2995","DOI":"10.1016\/j.ces.2004.04.031","article-title":"Statistical monitoring of dynamic processes based on dynamic independent component analysis","volume":"59","author":"Lee","year":"2004","journal-title":"Chem. Eng. Sci."},{"key":"ref_134","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1016\/j.ces.2008.10.012","article-title":"Enhanced statistical analysis of nonlinear processes using KPCA, KICA and SVM","volume":"64","author":"Zhang","year":"2009","journal-title":"Chem. Eng. Sci."},{"key":"ref_135","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1198\/004017004000000563","article-title":"ROBPCA: A new approach to robust principal component analysis","volume":"47","author":"Hubert","year":"2005","journal-title":"Technometrics"},{"key":"ref_136","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1080\/01621459.1985.10478181","article-title":"Projection-pursuit approach to robust dispersion matrices and principal components: Primary theory and Monte Carlo","volume":"80","author":"Li","year":"1985","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1002\/cem.1180070606","article-title":"Robust principal component analysis by projection pursuit","volume":"7","author":"Xie","year":"1993","journal-title":"J. Chemom."},{"key":"ref_138","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/S0169-7439(01)00188-5","article-title":"A fast method for robust principal components with applications to chemometrics","volume":"60","author":"Hubert","year":"2002","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"ref_139","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1016\/j.jmva.2004.08.002","article-title":"High breakdown estimators for principal components: The projection-pursuit approach revisited","volume":"95","author":"Croux","year":"2005","journal-title":"J. Multivar. Anal."},{"key":"ref_140","doi-asserted-by":"crossref","first-page":"3419","DOI":"10.1109\/TIP.2011.2156801","article-title":"Bayesian robust principal component analysis","volume":"20","author":"Ding","year":"2011","journal-title":"IEEE Trans. Image Process."},{"key":"ref_141","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s11063-012-9230-4","article-title":"Bayesian robust PCA of incomplete data","volume":"36","author":"Luttinen","year":"2012","journal-title":"Neural Process. Lett."},{"key":"ref_142","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1162\/neco.2007.11-06-397","article-title":"Robust L1 principal component analysis and its Bayesian variational inference","volume":"20","author":"Gao","year":"2008","journal-title":"Neural Comput."},{"key":"ref_143","doi-asserted-by":"crossref","first-page":"3706","DOI":"10.1016\/j.csda.2009.03.014","article-title":"Robust probabilistic PCA with missing data and contribution analysis for outlier detection","volume":"53","author":"Chen","year":"2009","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_144","doi-asserted-by":"crossref","unstructured":"Chen, H., Luo, H., Huang, B., Jiang, B., and Kaynak, O. (2022). Transfer Learning-motivated Intelligent Fault Diagnosis Designs: A Survey, Insights, and Perspectives. IEEE Trans. Neural Netw. Learn. Syst., 2969\u20132983.","DOI":"10.36227\/techrxiv.21301533.v1"},{"key":"ref_145","doi-asserted-by":"crossref","unstructured":"Chen, H., and Huang, B. (2022). Fault-tolerant soft sensors for dynamic systems. IEEE Trans. Control. Syst. Technol., 2805\u20132818.","DOI":"10.1109\/TCST.2023.3287758"},{"key":"ref_146","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1109\/TIE.2006.888786","article-title":"Induction machine condition monitoring using neural network modeling","volume":"54","author":"Su","year":"2007","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_147","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1016\/S0307-904X(99)00020-7","article-title":"Automatic structure and parameter training methods for modeling of mechanical systems by recurrent neural networks","volume":"23","author":"Li","year":"1999","journal-title":"Appl. Math. Modell."},{"key":"ref_148","doi-asserted-by":"crossref","first-page":"1321","DOI":"10.1006\/mssp.2002.1578","article-title":"On-line condition monitoring of a power transmission unit of a rail vehicle","volume":"17","author":"Deuszkiewicz","year":"2003","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_149","doi-asserted-by":"crossref","first-page":"109076","DOI":"10.1016\/j.ress.2022.109076","article-title":"Fractional Fourier and time domain recurrence plot fusion combining convolutional neural network for bearing fault diagnosis under variable working conditions","volume":"232","author":"Bai","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_150","doi-asserted-by":"crossref","first-page":"109102","DOI":"10.1016\/j.ress.2023.109102","article-title":"Bioinspired membrane learnable spiking neural network for autonomous vehicle sensors fault diagnosis under open environments","volume":"233","author":"Wang","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/4\/455\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:25:03Z","timestamp":1760106303000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/4\/455"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,8]]},"references-count":150,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["sym16040455"],"URL":"https:\/\/doi.org\/10.3390\/sym16040455","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,8]]}}}