Numerical identification of a linear oscillator stiffness using Bayesian inference and Markov chain Monte Carlo


  • Marcela Machado Universidade de Brasília
  • Gustavo Taffarel Oliveira Freire Universidade de Brasília
  • Vitória Carolina Silva Duarte Universidade de Brasília


Bayesian Inference; Maximum Likelihood; Markov Chain Monte Carlo (MCMC); Inverse problem; Parameters estimation; Dynamic system.


Inverse problem techniques have been used in different engineering application aiming to convert
observed measurements or data acquired together to the prior knowledge of the system into information about
material properties, geometry, locations of anomalies, e.g. cracks and structural damage, excitation force, among
others. The present papers aim to estimate parameters of a dynamic system with the inverse problem using
Bayesian Inference technique. Multiples studies are presented to analyse the statistical significance of the catches
for the settings, making a critical analysis between a solution via Bayesian Inference linked to minimising the
objective function with stochastic methods. It applied through stochastic strategies as the Maximum Likelihood
(MLE), Least Squares (LSE) and Markov Chains Monte Carlo (MCMC), implemented with the Metropolis-Hastings
algorithm (MH). In the estimation, the random parameters assumed distribution inference of Gaussian and Uniform
types for different standard deviations. The results demonstrated the efficacy of Bayesian inference to estimates
parameter of the oscillator systems from its dynamic response and the statistical parameter information.


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Albuquerque, Emerson B., Cynthia Guzman, Lavinia A. Borges, and Daniel A. Castello. 2018. “A Bayesian Framework for the Calibration of Cohesive Zone Models.” Journal of Adhesion 94 (4): 255”“77.

Castello, Daniel Alves, and Thiago Gamboa Ritto. 2015. Quantificação De Incertezas E Estimação De Parâmetros Em Dinâmica Estrutural: Uma Introdução A Partir De Exemplos Computacionais. Vol. 79.

Christen, J Andrés, and Colin Fox. 2005. “Markov Chain Monte Carlo Using an Approximation.” Journal of Computational and Graphical Statistics 14 (4): 795”“810.

Dashti, Masoumeh, and Andrew M Stuart. 2016. Handbook of Uncertainty Quantification. Handbook of Uncertainty Quantification.

Green, P. L., and K. Worden. 2015. “Bayesian and Markov Chain Monte Carlo Methods for Identifying Nonlinear Systems in the Presence of Uncertainty.” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373 (2051).

Hatch, Michael R. 2001. Vibration Simulation Using Matlab Abd Ansys.

Kaipio, Jari P., and Erkki Somersalo. 2006. “Statistical and Computational Inverse Problems.” Applied Mathematical Sciences (Switzerland) 160: i”“339.

Kruschke, John K. 2010. “Bayesian Data Analysis.” Wiley Interdisciplinary Reviews: Cognitive Science 1 (5): 658”“76.

Lataniotis, C, S Marelli, and B Sudret. 2019. “UQLAB USER MANUAL THE INPUT MODULE.” Szwitzerland.

Lynch, Scott M. 2008. “Introduction to Applied Bayesian Statistics and Estimation for Social Scientists.” Journal of the American Statistical Association 103 (483): 1322”“23.

Meirovitch, Leonard. 2000. Fundamentals of Vibrations. Handbook of Machinery Dynamics. Virginia: MCGraw Hill.

Mohammad-Djafar, Ali. 1998. “From Deterministic To Probabilistic Approaches To Solve Inverse Problems” 3459 (July): 2”“11.

Myung, In Jae. 2003. “Tutorial on Maximum Likelihood Estimation.” Journal of Mathematical Psychology 47 (1): 90”“100.

Oliveira, C., J. Lugon Junior, D.C Knupp, AJ Silva Neto, A Prieto-Moreno, and O Llanes-Santiago. 2018. “Estimation of Kinetic Parameters in a Chromatographic Separation Model via Bayesian Inference.” Revista Internacional de Métodos Numéricos Para Cálculo y Diseño En Ingeniería, 2018.

Rouchier, Simon. 2018. “Solving Inverse Problems in Building Physics: An Overview of Guidelines for a Careful and Optimal Use of Data.” Energy and Buildings 166: 178”“95.

Schevenels, M., G. Lombaert, and G. Degrande. 2004. “Application of the Stochastic Finite Element Method for Gaussian and Non-Gaussian Systems.” In Proceedings of the 2004 International Conference on Noise and Vibration Engineering, ISMA.

FOX, J.-P. Bayesian item response modeling: Theory and applications. [S.l.]: Springer Science & Business Media, 2010.

OLIVEIRA, C. et al. Estimation of kinetic parameters in a chromatographic separation model via bayesian inference. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, v. 34, n. 1, 2018.

SONG, M. et al. Bayesian model updating of nonlinear systems using nonlinear normal modes. Structural control and Health Monitoring, Wiley Online Library, v. 25, n. 12, p. e2258, 2018.

VRUGT, J. A. Markov chain monte carlo simulation using the dream software package: Theory, concepts, and matlab implementation. Environmental Modelling & Software, v. 75, p. 273”“316, 2016.

JIN, S.-S.; JU, H.; JUNG, H.-J. Adaptive markov chain monte carlo algorithms for bayesian inference: recent advances and comparative study. Structure and Infrastructure Engineering, Taylor & Francis, p. 1”“18, 2019.

HAARIO, H. et al. Dram: efficient adaptive mcmc. Statistics and computing, Springer, v. 16, n. 4, p. 339”“354, 2006.

BESAG, J. et al. Bayesian computation and stochastic systems. Statistical science, Institute of Mathematical Statistics, v. 10, n. 1, p. 3”“41, 1995.

DAHLIN, J.; SCHÖN, T. B. Getting started with particle metropolis - hastings for inference in nonlinear dynamical models. arXiv preprint arXiv:1511.01707, 2015.

KHALIL, M. et al. The estimation of time-invariant parameters of noisy nonlinear oscillatory systems. Journal of Sound and Vibration, Elsevier, v. 344, p. 81”“100, 2015.




Como Citar

Machado, M. R., Freire, . G. T. O. ., & Duarte, V. C. S. . (2020). Numerical identification of a linear oscillator stiffness using Bayesian inference and Markov chain Monte Carlo. Revista Interdisciplinar De Pesquisa Em Engenharia, 6(1), 28–37. Recuperado de