Laplace approximation in sparse Bayesian learning for structural damage detection

Abstract The Bayesian theorem has been demonstrated as a rigorous method for uncertainty assessment and system identification. Given that damage usually occurs at limited positions in the preliminary stage of structural failure, the sparse Bayesian learning has been developed for solving the structural damage detection problem. However, in most cases an analytical posterior probability density function (PDF) of the damage index is not available due to the nonlinear relationship between the measured modal data and structural parameters. This study tackles the nonlinear problem using the Laplace approximation technique. By assuming that the item in the integration follows a Gaussian distribution, the asymptotic solution of the evidence is obtained. Consequently the most probable values of the damage index and hyper-parameters are expressed in a coupled closed form, and then solved sequentially through iterations. The effectiveness of the proposed algorithm is validated using a laboratory tested frame. As compared with other techniques, the present technique results in the analytical solutions of the damage index and hyper-parameters without using hierarchical models or numerical sampling. Consequently, the computation is more efficient.

[1]  Hongping Zhu,et al.  Substructuring approach to the calculation of higher-order eigensensitivity , 2013 .

[2]  E. Peter Carden,et al.  Vibration Based Condition Monitoring: A Review , 2004 .

[3]  Aggelos K. Katsaggelos,et al.  Bayesian Compressive Sensing Using Laplace Priors , 2010, IEEE Transactions on Image Processing.

[4]  Costas Papadimitriou,et al.  Accounting for amplitude of excitation in model updating through a hierarchical Bayesian approach: Application to a two-story reinforced concrete building , 2019, Mechanical Systems and Signal Processing.

[5]  Raul Tempone,et al.  Fast estimation of expected information gains for Bayesian experimental designs based on Laplace approximations , 2013 .

[6]  S. Au,et al.  Bayesian model updating of a coupled-slab system using field test data utilizing an enhanced Markov chain Monte Carlo simulation algorithm , 2015 .

[7]  Michael E. Tipping.  Sparse Bayesian Learning and the Relevance Vector Machine , 2001, J. Mach. Learn. Res..

[8]  Jun Lu,et al.  An introduction to Bayesian hierarchical models with an application in the theory of signal detection , 2005, Psychonomic bulletin & review.

[9]  John E. Mottershead,et al.  Model Updating In Structural Dynamics: A Survey , 1993 .

[10]  Costas Papadimitriou,et al.  Hierarchical Bayesian model updating for structural identification , 2015 .

[11]  Arzu Onar.  Laplace Approximations in Bayesian Lifetime Analysis , 2008 .

[12]  H. Hao,et al.  Long term vibration monitoring of an RC slab: Temperature and humidity effect , 2006 .

[13]  Matthew J. Beal,et al.  The variational Bayesian EM algorithm for incomplete data: with application to scoring graphical model structures , 2003 .

[14]  Yong Huang,et al.  Hierarchical sparse Bayesian learning for structural damage detection: Theory, computation and application , 2017 .

[15]  Richard B. Nelson,et al.  Simplified calculation of eigenvector derivatives , 1976 .

[16]  Yong Xia,et al.  Experimental Investigation on Statistical Moment-based Structural Damage Detection Method , 2009 .

[17]  Yong Xia,et al.  An iterative substructuring approach to the calculation of eigensolution and eigensensitivity , 2011 .

[18]  Heung-Fai Lam,et al.  Markov chain Monte Carlo‐based Bayesian method for structural model updating and damage detection , 2018 .

[19]  O. S. Salawu.  Detection of structural damage through changes in frequency: a review , 1997 .

[20]  You-Lin Xu,et al.  A new statistical moment-based structural damage detection method , 2008 .

[21]  Lambros S. Katafygiotis,et al.  Efficient model updating and health monitoring methodology using incomplete modal data without mode matching , 2006 .

[22]  Li Cheng,et al.  Development in vibration-based structural damage detection technique , 2007 .

[23]  Qi Xia,et al.  Genetic algorithm based optimal sensor placement for L1‐regularized damage detection , 2018, Structural Control and Health Monitoring.

[24]  Hui Li,et al.  The State of the Art of Data Science and Engineering in Structural Health Monitoring , 2019, Engineering.

[25]  Yong Huang,et al.  Sparse Bayesian learning for structural damage detection using expectation–maximization technique , 2019, Structural Control and Health Monitoring.

[26]  L. Tierney,et al.  Accurate Approximations for Posterior Moments and Marginal Densities , 1986 .

[27]  James L. Beck,et al.  Hierarchical sparse Bayesian learning for structural health monitoring with incomplete modal data , 2014 .

[28]  J. Beck,et al.  Updating Models and Their Uncertainties. I: Bayesian Statistical Framework , 1998 .

[29]  P. Laplace.  Memoir on the Probability of the Causes of Events , 1986 .

[30]  J. Beck,et al.  UPDATING MODELS AND THEIR UNCERTAINTIES. II: MODEL IDENTIFIABILITY. TECHNICAL NOTE , 1998 .

[31]  Costas Papadimitriou,et al.  Probabilistic hierarchical Bayesian framework for time-domain model updating and robust predictions , 2019, Mechanical Systems and Signal Processing.

[32]  Nasser M. Nasrabadi,et al.  Pattern Recognition and Machine Learning , 2006, Technometrics.

[33]  Yong Huang,et al.  Bayesian system identification based on hierarchical sparse Bayesian learning and Gibbs sampling with application to structural damage assessment , 2017, Computer Methods in Applied Mechanics and Engineering.

[34]  J. Beck.  Bayesian system identification based on probability logic , 2010 .

[35]  Shun Weng,et al.  L1 regularization approach to structural damage detection using frequency data , 2015 .