Bayesian Updating of Structural Model with a Conditionally Heteroscedastic Error Distribution
Publication: Journal of Engineering Mechanics
Volume 145, Issue 12
Abstract
The existing literature on Bayesian updating of structural models has assigned equal variances (homoscedasticity) in the measured observables across all modes by assuming a Gaussian error distribution. This paper relaxes the assumption by allowing the error distribution to be conditionally heteroscedastic but marginally follow the Student’s -distribution. Modeling heteroscedasticity in structures is necessary because higher modal parameters are prone to higher uncertainty—an idea that is supported by experimental data obtained from a scaled building model. We adopted a hierarchical modeling framework for system equations and employed Bayesian techniques to update the parameters. Estimation used Gibbs sampling, a well-known Markov chain Monte Carlo (MCMC) algorithm. The proposed framework is illustrated in various simulated data obtained from a 10-story shear building for varying levels of noise contamination. The model was also implemented for experimental data from an eight-degrees-of-freedom spring-mass model tested at the Los Alamos National Laboratory. We show that our framework provides several advantages over the homoscedastic model, including a more stable and accurate inference about the stiffness and modal parameters and a noticeable improvement in noise immunity compared to those reported in the literature. The importance of the heteroscedasticity is also highlighted because it leads to a better exploration of the associated uncertainty in the observables, not captured by its homoscedastic counterpart. The paper also discusses some limitations and provides possible extensions for future research.
Get full access to this article
View all available purchase options and get full access to this article.
References
Bansal, S. 2015. “A new Gibbs sampling based Bayesian model updating approach using modal data from multiple setups.” Int. J. Uncertainty Quantif. 5 (4): 361–374. https://doi.org/10.1615/Int.J.UncertaintyQuantification.2015013581.
Beck, J. L., and S. K. Au. 2002. “Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation.” J. Eng. Mech. 128 (4): 380–391. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:4(380).
Beck, J. L., S. K. Au, and M. W. Vanik. 1999. “A Bayesian probabilistic approach to structural health monitoring.” In Proc., American Control Conf., 1119–1123. Piscataway, NY: IEEE.
Beck, J. L., S. K. Au, and M. W. Vanik. 2001. “Monitoring structural health using a probabilistic measure.” Comput. Aided Civ. Infrastruct. Eng. 16(1), 1–11. https://doi.org/10.1111/0885-9507.00209.
Beck, J. L., and L. S. Katafygiotis. 1998. “Updating models and their uncertainties. I: Bayesian statistical framework.” J. Eng. Mech. 124 (4): 455–461. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:4(455).
Beck, J. L., and K. V. Yuen. 2004. “Model selection using response measurements: Bayesian probabilistic approach.” J. Eng. Mech. 130 (2): 192–203. https://doi.org/10.1061/(ASCE)0733-9399(2004)130:2(192).
Behmanesh, I., B. Moaveni, G. Lombaertt, and C. Papadimitriou. 2015. “Hierarchical Bayesian model updating for structural identification.” Mech. Syst. Signal Processing 64 (Dec): 360–376. https://doi.org/10.1016/j.ymssp.2015.03.026.
Benjamin, J. R., and C. A. Cornell. 1970. Probability, statistics, and decisions for civil engineers. New York: McGraw-Hill.
Boulkaibet, I., L. Mthembu, T. Marwala, M. I. Friswell, and S. Adhikari. 2015. “Finite element model updating using the shadow hybrid Monte Carlo technique.” Mech. Syst. Signal Processing 52–53 (Feb): 115–132. https://doi.org/10.1016/j.ymssp.2014.06.005.
Brincker, R., L. Zhang, and P. Andersen. 2000. “Modal identification from ambient responses using frequency domain decomposition.” In Proc., 18th Int. Modal Analysis Conf. (IMAC), 625–630. San Antonio.
Cheung, S. H., and J. L. Beck. 2009. “Bayesian model updating using hybrid Monte Carlo simulation with application to structural dynamic models with many uncertain parameters.” J. Eng. Mech. 135 (4): 243–255. https://doi.org/10.1061/(ASCE)0733-9399(2009)135:4(243).
Chib, S. 1995. “Marginal likelihood from the Gibbs output.” J. Am. Stat. Assoc. 90 (432): 1313–1321. https://doi.org/10.1080/01621459.1995.10476635.
Chib, S., and E. Greenberg. 1995. “Understanding the Metropolis-Hastings algorithm.” Am. Statistician 49 (4): 327–335. https://doi.org/10.1080/00031305.1995.10476177.
Chib, S., and I. Jeliazkov. 2001. “Marginal likelihood from the Metropolis-Hastings output.” J. Am. Stat. Assoc. 96 (453): 270–281. https://doi.org/10.1198/016214501750332848.
Chib, S., and I. Jeliazkov. 2005. “Accept-reject Metropolis-Hastings sampling and marginal likelihood estimation.” Stat. Neerl. 59 (1): 30–44. https://doi.org/10.1111/j.1467-9574.2005.00277.x.
Ching, J., and Y. C. Chen. 2007. “Transitional Markov chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging.” J. Eng. Mech. 133 (7): 816–832. https://doi.org/10.1061/(ASCE)0733-9399(2007)133:7(816).
Ching, J., M. Muto, and J. L. Beck. 2006. “Structural model updating and health monitoring with incomplete modal data using Gibbs sampler.” Comput. Aided Civ. Infrastruct. Eng. 21 (4): 242–257. https://doi.org/10.1111/j.1467-8667.2006.00432.x.
Christodoulou, K., and C. Papadimitriou. 2007. “Structural identification based on optimally weighted modal residuals.” Mech. Syst. Signal Processing 21 (1): 4–23. https://doi.org/10.1016/j.ymssp.2006.05.011.
Collins, J. D., G. C. Hart, T. Haselman, and B. Kennedy. 1974. “Statistical identification of structures.” AIAA J. 12 (2): 185–190.
Ewins, D. J. 2009. Modal testing: Theory, practice and application. New York: Wiley.
Gardoni, P., K. F. Reinschmidt, and R. Kumar. 2007. “A probabilistic framework for Bayesian adaptive forecasting of project progress.” Comput. Aided Civ. Infrastruct. Eng. 22 (3): 182–196. https://doi.org/10.1111/j.1467-8667.2007.00478.x.
Gelfand, A. E., S. E. Hills, A. Racine-Poon, and A. F. M. Smith. 1990. “Illustration of Bayesian inference in normal data models using Gibbs sampling.” J. Am. Stat. Assoc. 85 (412): 972–985. https://doi.org/10.1080/01621459.1990.10474968.
Geman, S., and D. Geman. 1984. “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images.” IEEE Trans. Pattern Anal. Mach. Intell. 6 (6): 721–741. https://doi.org/10.1109/TPAMI.1984.4767596.
Goller, B., J. L. Beck, and G. I. Schuëller. 2012. “Evidence-based identification of weighting factors in Bayesian model updating using modal data.” J. Eng. Mech. 138 (5): 430–440. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000351.
Goller, B., and G. I. Schuëller. 2011. “Investigation of model uncertainties in Bayesian structural model updating.” J. Sound Vib. 330 (25): 6122–6136. https://doi.org/10.1016/j.jsv.2011.07.036.
Greenberg, E. 2012. Introduction to Bayesian econometrics. New York: Cambridge University Press.
Grewal, M. S., and A. P. Andrews. 2011. Kalman filtering: Theory and practice using MATLAB. New York: Wiley.
Jeliazkov, I., and M. A. Rahman. 2012. “Binary and ordinal data analysis in economics: Modeling and estimation.” In Mathematical modeling with multidisciplinary applications, edited by X. S. Yang, 123–150. New York: Wiley.
Katafygiotis, L. S., and K. V. Yuen. 2001. “Bayesian spectral density approach for modal updating using ambient data.” Earthquake Eng. Struct. Dyn. 30 (8): 1103–1123. https://doi.org/10.1002/eqe.5310.1002/eqe.53.
Khodaparast, H. H., J. E. Mottershead, and M. I. Friswell. 2008. “Perturbation methods for the estimation of parameter variability in stochastic model updating.” Mech. Syst. Signal Processing 22 (8): 1751–1773. https://doi.org/10.1016/j.ymssp.2008.03.001.
Los Alamos National Laboratory. n.d. “Eight degree of freedom system.” Accessed July 31, 2019. https://www.lanl.gov/projects/national-security-education-center/engineering/ei-software-download/downloads/8-cof-system-data/Eightdof.pdf.
Marwala, T. 2002. “Finite element model updating using wavelet data and genetic algorithm.” J. Aircr. 39 (4): 709–711. https://doi.org/10.2514/2.2985.
Marwala, T., I. Boulkaibet, and S. Adhikari. 2016. “Probabilistic finite element model updating using Bayesian statistics: Applications to aeronautical and mechanical engineering. New York: Wiley.
Maybeck, P. S. 1990. “The Kalman filter: An introduction to concepts.” In Autonomous robot vehicles, edited by I. J. Cox and G. T. Wolfgong, 194–204. New York: Springer.
Mottershead, J., and M. Friswell. 1993. “Model updating in structural dynamics: A survey.” J Sound Vib. 167 (2): 347–375. https://doi.org/10.1006/jsvi.1993.134010.1006/jsvi.1993.1340.
Natke, H. G. 1988. “Updating computational models in the frequency domain based on measured data: A survey.” Probab. Eng. Mech. 3 (1): 28–35. https://doi.org/10.1016/0266-8920(88)90005-7.
Papadimitriou, C. 2004. “Optimal sensor placement methodology for parametric identification of structural systems.” J. Sound Vib. 278 (4–5): 923–947. https://doi.org/10.1016/j.jsv.2003.10.063.
Papadimitriou, C., C. Argyris, D. C. Papadioti, and P. Panetsos. 2014. “Uncertainty calibration of large-order models of bridges using ambient vibration measurements.” In Proc., EWSHM 7th European Workshop on Structural Health Monitoring. Nantes, France: Université de Nantes.
Papadimitriou, C., J. L. Beck, and L. S. Katafygiotis. 2001. “Updating robust reliability using structural test data.” Probab. Eng. Mech. 16 (2): 103–113. https://doi.org/10.1016/S0266-8920(00)00012-6.
Papadimitriou, C., and D. C. Papadioti. 2013. “Component mode synthesis techniques for finite element model updating.” Comput. Struct. 126 (Sep): 15–28. https://doi.org/10.1016/j.compstruc.2012.10.018.
Prajapat, K., and S. Ray-Chaudhuri. 2016. “Prediction error variances in Bayesian model updating employing data sensitivity.” J. Eng. Mech. 142 (12): 04016096. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001158.
Saito, T., and J. L. Beck. 2010. “Bayesian model selection for ARX models and its application to structural health monitoring.” Earthquake Eng. Struct. Dyn. 39 (15): 1737–1759. https://doi.org/10.1002/eqe.100610.1002/eqe.1006.
Sanayei, M., G. R. Imbaro, J. A. S. McClain, and L. C. Brown. 1997. “Structural model updating using experimental static measurements.” J. Struct. Eng. 123 (6): 792–798. https://doi.org/10.1061/(ASCE)0733-9445(1997)123:6(792).
Sanayei, M., J. E. Phelps, J. D. Sipple, E. S. Bell, and B. R. Brenner. 2012. “Instrumentation, nondestructive testing, and finite-element model updating for bridge evaluation using strain measurements.” J. Bridge Eng. 17 (1): 130–138. https://doi.org/10.1061/(ASCE)BE.1943-5592.0000228.
Simoen, E., C. Papadimitriou, and G. Lombaert. 2013. “On prediction error correlation in Bayesian model updating.” J. Sound Vib. 332 (18): 4136–4152. https://doi.org/10.1016/j.jsv.2013.03.019.
Sohn, H., and K. H. Law. 1997. “A Bayesian probabilistic approach for structure damage detection.” Earthquake Eng. Struct. Dyn. 26 (12): 1259–1281. https://doi.org/10.1002/(SICI)1096-9845(199712)26:12%3C1259::AID-EQE709%3E3.0.CO;2-3.
Sun, H., and O. Büyüköztürk. 2016. “Probabilistic updating of building models using incomplete modal data.” Mech. Syst. Signal Processing 75 (Jun): 27–40. https://doi.org/10.1016/j.ymssp.2015.12.024.
Yang, J. N., S. Linn, H. Huang, and L. Zhou. 2006. “An adaptive extended Kalman filter for structural damage identification.” Struct. Control Health Monit. 13 (4): 849–867. https://doi.org/10.1002/stc.84.
Yuen, K. V. 2010. Bayesian methods for structural dynamics and civil engineering. Singapore: Wiley.
Yuen, K. V., J. L. Beck, and L. S. Katafygiotis. 2002. “Probabilistic approach for modal identification using non-stationary noisy response measurements only.” Earthquake Eng. Struct. Dyn. 31 (4): 1007–1023. https://doi.org/10.1002/eqe.13510.1002/eqe.135.
Yuen, K. V., J. L. Beck, and L. S. Katafygiotis. 2006a. “Efficient model updating and health monitoring methodology using incomplete modal data without mode matching.” Struct. Control Health Monit. 13 (1): 91–107. https://doi.org/10.1002/stc.144.
Yuen, K. V., J. L. Beck, and L. S. Katafygiotis. 2006b. “Unified probabilistic approach for model updating and damage detection.” J. Appl. Mech. 73 (4): 555–564. https://doi.org/10.1115/1.2150235.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 145 • Issue 12 • December 2019
Copyright
©2019 American Society of Civil Engineers.
History
Received: Oct 3, 2018
Accepted: Mar 19, 2019
Published online: Sep 17, 2019
Published in print: Dec 1, 2019
Discussion open until: Feb 17, 2020
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.