{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T04:11:16Z","timestamp":1773547876260,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"20","license":[{"start":{"date-parts":[[2022,10,19]],"date-time":"2022-10-19T00:00:00Z","timestamp":1666137600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific Research Fund of Institute of Engineering Mechanics, China Earthquake Administration","award":["2019EEEVL0401"],"award-info":[{"award-number":["2019EEEVL0401"]}]},{"name":"Scientific Research Fund of Institute of Engineering Mechanics, China Earthquake Administration","award":["2021B1212040003"],"award-info":[{"award-number":["2021B1212040003"]}]},{"name":"Scientific Research Fund of Institute of Engineering Mechanics, China Earthquake Administration","award":["20212B01"],"award-info":[{"award-number":["20212B01"]}]},{"name":"Scientific Research Fund of Institute of Engineering Mechanics, China Earthquake Administration","award":["JCYJ20190806143618723"],"award-info":[{"award-number":["JCYJ20190806143618723"]}]},{"name":"Guangdong Provincial Key Laboratory of Modern Civil Engineering Technology","award":["2019EEEVL0401"],"award-info":[{"award-number":["2019EEEVL0401"]}]},{"name":"Guangdong Provincial Key Laboratory of Modern Civil Engineering Technology","award":["2021B1212040003"],"award-info":[{"award-number":["2021B1212040003"]}]},{"name":"Guangdong Provincial Key Laboratory of Modern Civil Engineering Technology","award":["20212B01"],"award-info":[{"award-number":["20212B01"]}]},{"name":"Guangdong Provincial Key Laboratory of Modern Civil Engineering Technology","award":["JCYJ20190806143618723"],"award-info":[{"award-number":["JCYJ20190806143618723"]}]},{"name":"Natural Science Foundation of Shenzhen","award":["2019EEEVL0401"],"award-info":[{"award-number":["2019EEEVL0401"]}]},{"name":"Natural Science Foundation of Shenzhen","award":["2021B1212040003"],"award-info":[{"award-number":["2021B1212040003"]}]},{"name":"Natural Science Foundation of Shenzhen","award":["20212B01"],"award-info":[{"award-number":["20212B01"]}]},{"name":"Natural Science Foundation of Shenzhen","award":["JCYJ20190806143618723"],"award-info":[{"award-number":["JCYJ20190806143618723"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Measurement error is non-negligible and crucial in SHM data analysis. In many applications of SHM, measurement errors are statistically correlated in space and\/or in time for data from sensor networks. Existing works solely consider spatial correlation for measurement error. When both spatial and temporal correlation are considered simultaneously, the existing works collapse, as they do not possess a suitable form describing spatially and temporally correlated measurement error. In order to tackle this burden, this paper generalizes the form of correlated measurement error from spatial correlation only or temporal correlation only to spatial-temporal correlation. A new form of spatial-temporal correlation and the corresponding likelihood function are proposed, and multiple candidate model classes for the measurement error are constructed, including no correlation, spatial correlation, temporal correlation, and the proposed spatial-temporal correlation. Bayesian system identification is conducted to achieve not only the posterior probability density function (PDF) for the model parameters, but also the posterior probability of each candidate model class for selecting the most suitable\/plausible model class for the measurement error. Examples are presented with applications to model updating and modal frequency prediction under varying environmental conditions, ensuring the necessity of considering correlated measurement error and the capability of the proposed Bayesian system identification in the uncertainty quantification at the parameter and model levels.<\/jats:p>","DOI":"10.3390\/s22207981","type":"journal-article","created":{"date-parts":[[2022,10,19]],"date-time":"2022-10-19T22:19:53Z","timestamp":1666217993000},"page":"7981","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Analysis of Structural Health Monitoring Data with Correlated Measurement Error by Bayesian System Identification: Theory and Application"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8358-6032","authenticated-orcid":false,"given":"He-Qing","family":"Mu","sequence":"first","affiliation":[{"name":"Key Laboratory of Earthquake Engineering and Engineering Vibration, Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, China"},{"name":"Key Laboratory of Earthquake Disaster Mitigation, Ministry of Emergency Management, Harbin 150080, China"},{"name":"School of Civil Engineering and Transportation, State Key Laboratory of Subtropical Building Science, Guangzhou 510640, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xin-Xiong","family":"Liang","sequence":"additional","affiliation":[{"name":"School of Civil Engineering and Transportation, State Key Laboratory of Subtropical Building Science, Guangzhou 510640, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ji-Hui","family":"Shen","sequence":"additional","affiliation":[{"name":"School of Civil Engineering and Transportation, State Key Laboratory of Subtropical Building Science, Guangzhou 510640, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Feng-Liang","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1177\/1475921710365269","article-title":"Structural health monitoring in mainland China: Review and future trends","volume":"9","author":"Ou","year":"2010","journal-title":"Struct. Health Monit."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Farrar, C.R., and Worden, K. (2013). Structural Health Monitoring a Machine Learning Perspective, Wiley.","DOI":"10.1002\/9781118443118"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1999","DOI":"10.1260\/1369-4332.18.12.1999","article-title":"Structural health monitoring based on vehicle-bridge interaction: Accomplishments and challenges","volume":"18","author":"Zhu","year":"2015","journal-title":"Adv. Struct. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1508","DOI":"10.1109\/JPROC.2016.2588818","article-title":"Structural health monitoring: Technological advances to practical implementations [scanning the issue]","volume":"104","author":"Lynch","year":"2016","journal-title":"Proc. IEEE"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"7103039","DOI":"10.1155\/2016\/7103039","article-title":"A Review of Machine Vision-Based Structural Health Monitoring: Methodologies and Applications","volume":"2016","author":"Ye","year":"2016","journal-title":"J. Sens."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1177\/1475921716653278","article-title":"Applications of structural health monitoring technology in Asia","volume":"16","author":"Annamdas","year":"2017","journal-title":"Struct. Health. Monit."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1016\/j.eng.2018.11.027","article-title":"The State of the Art of Data Science and Engineering in Structural Health Monitoring","volume":"5","author":"Bao","year":"2019","journal-title":"Engineering"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Azimi, M., Eslamlou, A.D., and Pekcan, G. (2020). Data-driven structural health monitoring and damage detection through deep learning: State-of-the-art review. Sensors, 20.","DOI":"10.3390\/s20102778"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Burgos, D.A.T., Vargas, R.C.G., Pedraza, C., Agis, D., and Pozo, F. (2020). Damage identification in structural health monitoring: A brief review from its implementation to the use of data-driven applications. Sensors, 20.","DOI":"10.3390\/s20030733"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"e2526","DOI":"10.1002\/stc.2526","article-title":"Novel vibration structural health monitoring technology for deep foundation piles by non-stationary higher order frequency response function","volume":"27","author":"Gelman","year":"2020","journal-title":"Struct. Control Health Monit."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2275","DOI":"10.1016\/j.ymssp.2010.10.012","article-title":"The sensitivity method in finite element model updating: A tutorial","volume":"25","author":"Mottershead","year":"2011","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2467","DOI":"10.1016\/j.cma.2007.05.030","article-title":"Multivariate significance testing and model calibration under uncertainty","volume":"197","author":"Mcfarland","year":"2008","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1016\/j.ress.2005.09.004","article-title":"Validation of models with multivariate output","volume":"91","author":"Rebba","year":"2006","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1061\/(ASCE)0733-9399(1998)124:4(455)","article-title":"Updating models and their uncertainties. I: Bayesian statistical framework","volume":"124","author":"Beck","year":"1998","journal-title":"J. Eng. Mech."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Yuen, K.-V. (2010). Bayesian Methods for Structural Dynamics and Civil Engineering, Wiley.","DOI":"10.1002\/9780470824566"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1002\/stc.424","article-title":"Bayesian system identification based on probability logic","volume":"17","author":"Beck","year":"2010","journal-title":"Struct. Control Health Monit."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"010802","DOI":"10.1115\/1.4004479","article-title":"Bayesian Methods for Updating Dynamic Models","volume":"64","author":"Yuen","year":"2011","journal-title":"Appl. Mech. Rev."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Jaynes, E.T. (2003). Probability Theory: The Logic of Science, Cambridge University Press.","DOI":"10.1017\/CBO9780511790423"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1583","DOI":"10.1002\/nme.2385","article-title":"Construction of probability distributions in high dimension using the maximum entropy principle: Applications to stochastic processes, random fields and random matrices","volume":"76","author":"Soize","year":"2008","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.comnet.2004.03.007","article-title":"Spatio-temporal correlation: Theory and applications for wireless sensor networks","volume":"45","author":"Vuran","year":"2004","journal-title":"Comput. Netw."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1137","DOI":"10.1016\/j.ress.2010.09.013","article-title":"Bayesian uncertainty analysis with applications to turbulence modeling","volume":"96","author":"Cheung","year":"2011","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.ymssp.2011.05.019","article-title":"The effect of prediction error correlation on optimal sensor placement in structural dynamics","volume":"28","author":"Papadimitriou","year":"2012","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4136","DOI":"10.1016\/j.jsv.2013.03.019","article-title":"On prediction error correlation in Bayesian model updating","volume":"332","author":"Simoen","year":"2013","journal-title":"J. Sound Vib."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"04017031","DOI":"10.1061\/(ASCE)CP.1943-5487.0000668","article-title":"Novel Sparse Bayesian Learning and Its Application to Ground Motion Pattern Recognition","volume":"31","author":"Mu","year":"2017","journal-title":"J. Comput. Civ. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1111\/j.1467-8667.2006.00431.x","article-title":"Structural health monitoring via measured Ritz vectors utilizing artificial neural networks","volume":"21","author":"Lam","year":"2006","journal-title":"Comput. Civ. Infrastruct. Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.measurement.2018.08.022","article-title":"Modal frequency-environmental condition relation development using long-term structural health monitoring measurement: Uncertainty quantification, sparse feature selection and multivariate prediction","volume":"130","author":"Mu","year":"2018","journal-title":"Meas. J. Int. Meas. Confed."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1940009","DOI":"10.1142\/S0219455419400091","article-title":"Uncertainty Quantification of Load Effects under Stochastic Traffic Flows","volume":"19","author":"Mu","year":"2018","journal-title":"Int. J. Struct. Stab. Dyn."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"113571","DOI":"10.1016\/j.engstruct.2021.113571","article-title":"A novel generative approach for modal frequency probabilistic prediction under varying environmental condition using incomplete information","volume":"252","author":"Mu","year":"2022","journal-title":"Eng. Struct."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"113548","DOI":"10.1016\/j.engstruct.2021.113548","article-title":"Sensor placement for model identification of multi-story buildings under unknown earthquake ground motion","volume":"251","author":"Yin","year":"2022","journal-title":"Eng. Struct."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1061\/(ASCE)0733-9399(2004)130:2(192)","article-title":"Model selection using response measurements: Bayesian probabilistic approach","volume":"130","author":"Beck","year":"2004","journal-title":"J. Eng. Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1111\/mice.12146","article-title":"Real-Time System Identification: An Algorithm for Simultaneous Model Class Selection and Parametric Identification","volume":"30","author":"Yuen","year":"2015","journal-title":"Comput. Civ. Infrastruct. Eng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"806","DOI":"10.1111\/mice.12441","article-title":"Self-calibrating Bayesian real-time system identification","volume":"34","author":"Yuen","year":"2019","journal-title":"Comput. Civ. Infrastruct. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"111118","DOI":"10.1016\/j.engstruct.2020.111118","article-title":"Bayesian model selection for the nonlinear hysteretic model of CLT connections","volume":"223","author":"Cao","year":"2020","journal-title":"Eng. Struct."},{"key":"ref_34","first-page":"30","article-title":"Peak ground acceleration estimation by linear and nonlinear models with reduced order Monte Carlo simulation","volume":"26","author":"Yuen","year":"2011","journal-title":"Comput. Civ. Infrastruct. Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1111\/mice.12215","article-title":"Ground Motion Prediction Equation Development by Heterogeneous Bayesian Learning","volume":"31","author":"Mu","year":"2016","journal-title":"Comput. Civ. Infrastruct. Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11803-014-0207-3","article-title":"Seismic attenuation relationship with homogeneous and heterogeneous prediction-error variance models","volume":"13","author":"Mu","year":"2014","journal-title":"Earthq. Eng. Eng. Vib."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1007\/s12205-020-0983-4","article-title":"Bayesian finite element model updating of a long-span suspension bridge utilizing hybrid Monte Carlo simulation and kriging predictor","volume":"24","author":"Mao","year":"2020","journal-title":"KSCE J. Civ. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"04020297","DOI":"10.1061\/(ASCE)ST.1943-541X.0002881","article-title":"Probabilistic Framework with Bayesian Optimization for Predicting Typhoon-Induced Dynamic Responses of a Long-Span Bridge","volume":"147","author":"Zhang","year":"2021","journal-title":"J. Struct. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"e3030","DOI":"10.1002\/stc.3030","article-title":"A Bayesian neural network approach for probabilistic model updating using incomplete modal data","volume":"29","author":"Zhang","year":"2022","journal-title":"Struct. Control Health Monit."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Gao, K., Zhang, Z., Weng, S., Zhu, H., Yu, H., and Peng, T. (2022). Review of Flexible Piezoresistive Strain Sensors in Civil Structural Health Monitoring. Appl. Sci., 12.","DOI":"10.3390\/app12199750"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Park, S., and Jun, S. (2022). Cognitive Artificial Intelligence Using Bayesian Computing Based on Hybrid Monte Carlo Algorithm. Appl. Sci., 12.","DOI":"10.3390\/app12189270"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Cai, H., Wang, Y., Lin, Y., Li, S., Wang, M., and Teng, F. (2022). Systematic Comparison of Objects Classification Methods Based on ALS and Optical Remote Sensing Images in Urban Areas. Electronics, 11.","DOI":"10.3390\/electronics11193041"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2250051","DOI":"10.1142\/S0219455422500511","article-title":"Bayesian Operational Modal Analysis with Genetic Optimization for Structural Health Monitoring of the Long-Span Bridge","volume":"22","author":"Mao","year":"2022","journal-title":"Int. J. Struct. Stab. Dyn."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1016\/j.ymssp.2016.05.025","article-title":"Environmental effects on natural frequencies of the San Pietro bell tower in Perugia, Italy, and their removal for structural performance assessment","volume":"82","author":"Ubertini","year":"2017","journal-title":"Mech. Syst. Signal Process."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/20\/7981\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:57:32Z","timestamp":1760144252000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/20\/7981"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,19]]},"references-count":44,"journal-issue":{"issue":"20","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["s22207981"],"URL":"https:\/\/doi.org\/10.3390\/s22207981","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,19]]}}}