The role of Bhattacharyya distance in stochastic model updating


Generally, various measurements and data mining methods are susceptible to uncertainties that threaten the accuracy of data validation models if not properly managed. This is why taking into account the uncertainties in the model updating and validation has been highly encouraged. Presently, stochastic model updating methods are widely used together with uncertainty quantification metrics to provide accurate uncertainty measurements. The system parameters in uncertainty quantification are generally grouped into three: those without uncertainties, those with only epistemic uncertainties and those with both epistemic and aleatoric uncertainties.

Several methodologies for stochastic model updating have been developed. Regardless of the method used, the uncertainty quantification metric definition is of great importance in defining the uncertainty discrepancy in a data sample. However, uncertainty quantification metric with the ability to capture higher level data has remained a great challenge. Alternatively, researchers have recently identified Bhattacharyya distance, a measurement method between two sample data taking into account the probability distribution, as a promising solution for capturing high-level statistical information. Even though it has high potential for improving the uncertainty treatment, its application in stochastic model updating has not been fully explored due to its complexity and time-consuming nature.

To this note, Leibniz University Hannover scientists: Dr. Sifeng Bi, Dr. Matteo Broggi and Professor Michael Beer looked at the feasibility of using Bhattacharyya distance as an uncertainty quantification matrix in the stochastic model updating approach. Fundamentally, they aimed at solving the NASA uncertainty quantification challenge problems including the uncertainty characterization based on the approximate Bayesian computational approach. Additionally, they investigated the advantages of using the Bhattacharyya distance metric in solving real-world practical problems. Their work is currently published in the journal, Mechanical Systems and Signal Processing.

In brief, the research team scrutinized the available stochastic measurements methods including the Euclidian, Mahalanobis, and Bhattacharyya distances model updating metrics. Secondly, they developed a fully embedded Bhattacharyya distance-based updating framework using the transitional Markov chain Monte Carlo algorithm. On the other hand, a binning algorithm was used to evaluate the Bhattacharyya distance between various sets of data samples. Additionally, Euclidian and Bhattacharyya distances were combined in the initial steps to developed a two-step Bayesian updating framework to enhance the results accuracy. Eventually, they performed a stochastic sensitivity analysis by ranking the input parameters based on the uncertainty properties of their corresponding outputs.

The Bhattacharyya distance was observed to demonstrate powerful uncertainty quantification metric abilities for the model updating framework. This was attributed to the effective connection between the updating framework and the Bhattacharyya distance. In addition, it was worth noting that the Bhattacharyya distance metric was an ideal alternative for replacing the Euclidian distance in stochastic model updating frameworks.

In summary, Professor Michael Beer and his research team demonstrated the role of Bhattacharyya distance in stochastic model updating thus enabling efficient stochastic sensitivity analysis. To actualize their study, they accessed the performance of the developed framework in two different applications: simulated mass-spring problem and benchmark-based uncertainty treatment problem. Interestingly, in all the two cases, a quality gain for stochastic updating was obtained. Therefore, the study will advance future uncertainty characterization and stochastic model updating.

 role of Bhattacharyya distance in stochastic model updating - Advances in Engineering The role of Bhattacharyya distance in stochastic model updating - Advances in Engineering The role of Bhattacharyya distance in stochastic model updating - Advances in Engineering The role of Bhattacharyya distance in stochastic model updating - Advances in Engineering

About the author

Dr. Sifeng Bi is currently an Alexander von Humboldt Research Fellow in the Institute for Risk and Reliability, Leibniz Universität Hannover, Germany. He got his doctoral degree on aircraft design from Beihang University (also known as the Beijing University of Aeronautics and Astronautics), in 2015, and subsequently joined the Femto-ST Institute, France, as a post-doctoral researcher of the French National Center for Scientific Research (CNRS). Then he was granted the Humboldt Research Fellowship, and joined the Institute for Risk and Reliability since 2017.

His research topics are uncertainty quantification, stochastic model updating and validation, especially in the application of vibroacoustics and complex structural dynamics.

He was recognized as the “2018 REVIEWER OF THE YEAR” by the American Society of Mechanical Engineers (ASME), and he is also the recipient of the best paper award of the 2017 International Conference in Aerospace for Young Scientists.

About the author

Dr. Matteo Broggi is Senior Research Associate of the Institute for Risk and Reliability, Leibniz Universität Hannover, Germany, since 2016, and since 2018 he is the deputy head of the institute. He obtained a doctoral degree in October 2011 at the unit of Engineering Mechnics of the Leopold-Franzens-Universität Innsbruck, Austria, under the supervision of Prof. Schuëller. From 2012 to 2014 he pursued a post-doctoral research at the Virtual Engineering Centre, University of Liverpool, UK, working on industrial projects as well as research activities.

From 2014 to 2015 he held two positions at University of Liverpool, working as a part-time lecturer and research at the Institute for Risk and Uncertainty as well as continuing his work at the Virtual Engineering Centre while leading an industrial project.

Dr. Broggi’s research is focused on uncertainty quantification in engineering. Specifically, he is working on simulation tools for uncertainty quantification of large models and systems with high performance computing, analysis with imprecise probabilities, model updating and system reliability.

Dr. Broggi is the recipient of the 2011 best paper award of the International Journal of Structural Stability and Dynamics and of the best presented paper award at the 2015 NAFEMS world congress.

About the author

Michael Beer is Professor and Head of the Institute for Risk and Reliability, Leibniz Universität Hannover, Germany, since 2015. He is also part time Professor in the Institute for Risk and Uncertainty at the University of Liverpool and in the International Joint Research Center for Engineering Reliability and Stochastic Mechanics at Tongji University, Shanghai, China. He obtained a doctoral degree from Technische Universität Dresden and pursued post-doctoral research at Rice University. From 2007 to 2011 Dr. Beer worked as an Assistant Professor at National University of Singapore. In 2011 he joined the University of Liverpool as Chair in Uncertainty in Engineering and Founding Director of the Institute for Risk and Uncertainty and established a large Doctoral Training Center on Quantification and Management of Risk & Uncertainty.

Dr. Beer’s research is focused on uncertainty quantification in engineering with emphasis on imprecise probabilities. Dr. Beer is Editor in Chief (joint) of the Encyclopedia of Earthquake Engineering, Associate Editor of the ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Associate Editor of the International Journal of Reliability and Safety, and Member of thirteen Editorial Boards including Probabilistic Engineering Mechanics, Computers & Structures, Structural Safety, Mechanical Systems and Signal Processing, and International Journal for Uncertainty Quantification.

He has won several awards including the CADLM PRIZE 2007 – Intelligent Optimal Design and a Certificate for Highly Cited Research in Structural Safety. His publications include a book, several monographs and a large number of journal and conference papers. He is a Fellow of the Alexander von Humboldt-Foundation, Chair of the C(PS)2 of the Bernoulli Society and Member of ASCE (EMI), ASME, IACM, ESRA, EASD, and GACM.


Bi, S., Broggi, M., & Beer, M. (2019). The role of the Bhattacharyya distance in stochastic model updating. Mechanical Systems and Signal Processing, 117, 437-452.

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