{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T20:41:50Z","timestamp":1777322510585,"version":"3.51.4"},"reference-count":44,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2020,10,6]],"date-time":"2020-10-06T00:00:00Z","timestamp":1601942400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Pearl River S&amp;T Nova Program of Guangzhou","award":["201806010172"],"award-info":[{"award-number":["201806010172"]}]},{"name":"Scientific Research Fund of Institute of Engineering Mechanics, China Earthquake Administration","award":["2019EEEVL0401"],"award-info":[{"award-number":["2019EEEVL0401"]}]},{"name":"Science and Technology Program of Guangzhou","award":["201804020069"],"award-info":[{"award-number":["201804020069"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The problem of uncertainty quantification (UQ) for multi-sensor data is one of the main concerns in structural health monitoring (SHM). One important task is multivariate joint probability density function (PDF) modelling. Copula-based statistical inference has attracted significant attention due to the fact that it decouples inferences on the univariate marginal PDF of each random variable and the statistical dependence structure (called copula) among the random variables. This paper proposes the Copula-UQ, composing multivariate joint PDF modelling, inference on model class selection and parameter identification, and probabilistic prediction using incomplete information, for multi-sensor data measured from a SHM system. Multivariate joint PDF is modeled based on the univariate marginal PDFs and the copula. Inference is made by combing the idea of the inference functions for margins and the maximum likelihood estimate. Prediction on the PDF of the target variable, using the complete (from normal sensors) or incomplete information (due to missing data caused by sensor fault issue) of the predictor variable, are made based on the multivariate joint PDF. One example using simulated data and one example using temperature data of a multi-sensor of a monitored bridge are presented to illustrate the capability of the Copula-UQ in joint PDF modelling and target variable prediction.<\/jats:p>","DOI":"10.3390\/s20195692","type":"journal-article","created":{"date-parts":[[2020,10,6]],"date-time":"2020-10-06T10:46:17Z","timestamp":1601981177000},"page":"5692","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Copula-Based Uncertainty Quantification (Copula-UQ) for Multi-Sensor Data in Structural Health Monitoring"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8358-6032","authenticated-orcid":false,"given":"He-Qing","family":"Mu","sequence":"first","affiliation":[{"name":"School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510640, China"},{"name":"State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510640, China"},{"name":"Key Laboratory of Earthquake Engineering and Engineering Vibration, Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Han-Teng","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510640, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ji-Hui","family":"Shen","sequence":"additional","affiliation":[{"name":"School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510640, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Yuen, K.-V. (2010). Bayesian Methods for Structural Dynamics and Civil Engineering, John Wiley & Sons (Asia) Pte Ltd.","DOI":"10.1002\/9780470824566"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1958","DOI":"10.1109\/TIM.2008.919011","article-title":"Crack shape reconstruction in eddy current testing using machine learning systems for regression","volume":"57","author":"Bernieri","year":"2008","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"11007","DOI":"10.1117\/1.OE.55.1.011007","article-title":"Damage detection tomography based on guided waves in composite structures using a distributed sensor network","volume":"55","author":"Memmolo","year":"2015","journal-title":"Opt. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.proeng.2016.11.668","article-title":"Structural health monitoring in composites based on probabilistic reconstruction techniques","volume":"167","author":"Memmolo","year":"2016","journal-title":"Procedia Eng."},{"key":"ref_5","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_6","doi-asserted-by":"crossref","unstructured":"Kralovec, C., and Schagerl, M. (2020). Review of Structural Health Monitoring Methods Regarding a Multi-Sensor Approach for Damage Assessment of Metal and Composite Structures. Sensors, 20.","DOI":"10.3390\/s20030826"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"04014100","DOI":"10.1061\/(ASCE)EM.1943-7889.0000810","article-title":"Novel outlier-resistant extended Kalman filter for robust online structural identification","volume":"141","author":"Mu","year":"2015","journal-title":"J. Eng. Mech."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Mu, H.Q., Kuok, S.C., and Yuen, K.V. (2017). Stable robust extended Kalman filter. J. Aerosp. Eng., 30.","DOI":"10.1061\/(ASCE)AS.1943-5525.0000665"},{"key":"ref_9","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. Heal. Monit."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1115\/1.2718241","article-title":"Time Series Based Structural Damage Detection Algorithm Using Gaussian Mixtures Modeling","volume":"129","author":"Kiremidjian","year":"2007","journal-title":"J. Dyn. Syst. Meas. Control."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1177\/1475921718759344","article-title":"An enhanced dynamic Gaussian mixture model\u2013based damage monitoring method of aircraft structures under environmental and operational conditions","volume":"18","author":"Qiu","year":"2019","journal-title":"Struct. Heal. Monit."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"e1886","DOI":"10.1002\/stc.1886","article-title":"Genetic-based EM algorithm to improve the robustness of Gaussian mixture models for damage detection in bridges","volume":"24","author":"Santos","year":"2017","journal-title":"Struct. Control. Heal. Monit."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Qiu, L., Yuan, S., Mei, H., and Fang, F. (2016). An improved Gaussian mixture model for damage propagation monitoring of an aircraft wing spar under changing structural boundary conditions. Sensors, 16.","DOI":"10.3390\/s16030291"},{"key":"ref_14","unstructured":"Murphy, K.P. (2012). Machine Learning: A Probabilistic Perspective, MIT press."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.apor.2005.01.002","article-title":"Statistical properties of successive wave heights and successive wave periods","volume":"26","author":"Wist","year":"2004","journal-title":"Appl. Ocean. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/j.apor.2011.02.005","article-title":"On the long-term response of marine structures","volume":"33","author":"Sagrilo","year":"2011","journal-title":"Appl. Ocean. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1016\/j.probengmech.2008.08.001","article-title":"An innovating analysis of the Nataf transformation from the copula viewpoint","volume":"24","author":"Lebrun","year":"2009","journal-title":"Probabilistic Eng. Mech."},{"key":"ref_18","unstructured":"Zhang, Y., and Kim, C.W. (2017, January 6\u201310). Use of copula theory in the long-term health monitoring for deteriorated bridges. Proceedings of the 12th International Conference on Structural Safety and Reliability, Vienna, Austria."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"e2185","DOI":"10.1002\/stc.2185","article-title":"Time-variant reliability prediction of bridge system based on BDGCM and SHM data","volume":"25","author":"Fan","year":"2018","journal-title":"Struct. Control. Heal. Monit."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1016\/j.ress.2019.03.048","article-title":"Modeling risks in dependent systems: A Copula-Bayesian approach","volume":"188","author":"Pan","year":"2019","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4017123","DOI":"10.1061\/(ASCE)BE.1943-5592.0001152","article-title":"Fatigue reliability assessment for orthotropic steel deck details using copulas: Application to Nan-Xi Yangtze River Bridge","volume":"23","author":"Liu","year":"2018","journal-title":"J. Bridg. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1061\/(ASCE)0733-947X(2006)132:12(945)","article-title":"Multivariate simulation and multimodal dependence modeling of vehicle axle weights with copulas","volume":"132","author":"Srinivas","year":"2006","journal-title":"J. Transp. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.compstruc.2015.02.029","article-title":"Long-term performance assessment and design of offshore structures","volume":"154","author":"Zhang","year":"2015","journal-title":"Comput. Struct."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.solener.2019.11.079","article-title":"A copula-based Bayesian method for probabilistic solar power forecasting","volume":"196","author":"Panamtash","year":"2020","journal-title":"Sol. Energy"},{"key":"ref_25","unstructured":"Hollander, M., Wolfe, D.A., and Chicken, E. (2013). Nonparametric Statistical Methods, John Wiley & Sons."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Cherubini, U., Luciano, E., and Vecchiato, W. (2004). Copula Methods in Finance, John Wiley & Sons.","DOI":"10.1002\/9781118673331"},{"key":"ref_27","first-page":"229","article-title":"Fonctions de repartition an dimensions et leurs marges","volume":"8","author":"Sklar","year":"1959","journal-title":"Publ. inst. Stat. univ. Paris"},{"key":"ref_28","unstructured":"Nelsen, R.B. (2007). An Introduction to Copulas, Springer Science & Business Media."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Bouy\u00e9, E., Durrleman, V., Nikeghbali, A., Riboulet, G., and Roncalli, T. (2000). Copulas for finance-a reading guide and some applications. SSRN Electron. J.","DOI":"10.2139\/ssrn.1032533"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1142\/S0219455419400091","article-title":"Uncertainty Quantification of Load Effects under Stochastic Traffic Flows","volume":"19","author":"Mu","year":"2019","journal-title":"Int. J. Struct. Stab. Dyn."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Bowman, A.W., and Azzalini, A. (1997). Applied Smoothing Techniques for Data Analysis: The Kernel Approach with S-Plus Illustrations, Oxford University Press.","DOI":"10.1093\/oso\/9780198523963.001.0001"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Xu, Y.L., and Xia, Y. (2011). Structural Health Monitoring of Long-Span Suspension Bridges, CRC Press.","DOI":"10.1201\/b13182"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1177\/1687814016662228","article-title":"Relevance feature selection of modal frequency-ambient condition pattern recognition in structural health assessment for reinforced concrete buildings","volume":"8","author":"Mu","year":"2016","journal-title":"Adv. Mech. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1177\/1461348418786520","article-title":"Bayesian network-based modal frequency\u2013multiple environmental factors pattern recognition for the Xinguang Bridge using long-term monitoring data","volume":"39","author":"Mu","year":"2020","journal-title":"J. Low Freq. Noise Vib. Act. Control"},{"key":"ref_35","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":"Measurement"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2336","DOI":"10.1016\/j.ymssp.2011.03.005","article-title":"Environmental effects on the identified natural frequencies of the Dowling Hall Footbridge","volume":"25","author":"Moser","year":"2011","journal-title":"Mech. Syst. Signal. Process."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1002\/1096-9845(200102)30:2<149::AID-EQE1>3.0.CO;2-Z","article-title":"One-year monitoring of the Z24-Bridge: Environmental effects versus damage events","volume":"30","author":"Peeters","year":"2001","journal-title":"Earthq. Eng. Struct. Dyn."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/s13296-012-2006-x","article-title":"Thermal field characteristic analysis of steel box girder based on long-term measurement data","volume":"12","author":"Ding","year":"2012","journal-title":"Int. J. Steel Struct."},{"key":"ref_39","first-page":"632","article-title":"Monitoring temperature effect on a long suspension bridge","volume":"17","author":"Xu","year":"2010","journal-title":"Struct. Control. Heal. Monit."},{"key":"ref_40","first-page":"63","article-title":"Long-Term Temperature Monitoring and Statistical Analysis on the Flat Steel-Box Girder of Sutong Bridge","volume":"8","author":"Wang","year":"2014","journal-title":"J. Highw. Transp. Res. Dev. English Ed."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1007\/s13296-015-9011-9","article-title":"Research on monitoring temperature difference from cross sections of steel truss arch girder of Dashengguan Yangtze Bridge","volume":"15","author":"Gaoxin","year":"2015","journal-title":"Int. J. Steel Struct."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/j.measurement.2017.10.036","article-title":"Monitoring and analysis of thermal effect on tower displacement in cable-stayed bridge","volume":"115","author":"Yang","year":"2018","journal-title":"Measurement"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1111\/j.1747-1567.2011.00751.x","article-title":"Design and Deployment of a Continuous Monitoring System for the Dowling Hall Footbridge","volume":"37","author":"Moser","year":"2013","journal-title":"Exp. Tech."},{"key":"ref_44","unstructured":"(2020, August 01). Continuous Monitoring of the Dowling Hall Footbridge. Available online: https:\/\/engineering.tufts.edu\/cee\/shm\/research_BM_continuousMonitoring.asp."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/19\/5692\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:16:48Z","timestamp":1760177808000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/19\/5692"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,6]]},"references-count":44,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2020,10]]}},"alternative-id":["s20195692"],"URL":"https:\/\/doi.org\/10.3390\/s20195692","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,6]]}}}