Research Papers

Bayesian Network Learning for Data-Driven Design

[+] Author and Article Information
Zhen Hu

Department of Industrial and Manufacturing
Systems Engineering,
University of Michigan-Dearborn,
2340 Heinz Prechter Engineering
Complex (HPEC),
Dearborn, MI 48128
e-mail: zhennhu@umich.edu

Sankaran Mahadevan

Department of Civil and
Environmental Engineering,
Vanderbilt University,
272 Jacobs Hall VU, PMB 351831,
Nashville, TN 37235
e-mail: sankaran.mahadevan@vanderbilt.edu

1Corresponding author.

Manuscript received May 16, 2017; final manuscript received January 15, 2018; published online April 18, 2018. Assoc. Editor: Faisal Khan.

ASME J. Risk Uncertainty Part B 4(4), 041002 (Apr 18, 2018) (12 pages) Paper No: RISK-17-1063; doi: 10.1115/1.4039149 History: Received May 16, 2017; Revised January 15, 2018

Bayesian networks (BNs) are being studied in recent years for system diagnosis, reliability analysis, and design of complex engineered systems. In several practical applications, BNs need to be learned from available data before being used for design or other purposes. Current BN learning algorithms are mainly developed for networks with only discrete variables. Engineering design problems often consist of both discrete and continuous variables. This paper develops a framework to handle continuous variables in BN learning by integrating learning algorithms of discrete BNs with Gaussian mixture models (GMMs). We first make the topology learning more robust by optimizing the number of Gaussian components in the univariate GMMs currently available in the literature. Based on the BN topology learning, a new multivariate Gaussian mixture (MGM) strategy is developed to improve the accuracy of conditional probability learning in the BN. A method is proposed to address this difficulty of MGM modeling with data of mixed discrete and continuous variables by mapping the data for discrete variables into data for a standard normal variable. The proposed framework is capable of learning BNs without discretizing the continuous variables or making assumptions about their conditional probability densities (CPDs). The applications of the learned BN to uncertainty quantification and model calibration are also investigated. The results of a mathematical example and an engineering application example demonstrate the effectiveness of the proposed framework.

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Telenko, C. , and Seepersad, C. C. , 2014, “ Probabilistic Graphical Modeling of Use Stage Energy Consumption: A Lightweight Vehicle Example,” ASME J. Mech. Des., 136(10), p. 101403. [CrossRef]
Liang, C. , and Mahadevan, S. , 2017, “ Pareto Surface Construction for Multi-Objective Optimization Under Uncertainty,” Struct. Multidiscip. Optim., 55(5), pp. 1865–1882.
Du, X. , and Chen, W. , 2004, “ Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design,” ASME J. Mech. Des., 126(2), pp. 225–233.
Cheng, Y. , and Du, X. , 2016, “ System Reliability Analysis With Dependent Component Failures During Early Design Stage—A Feasibility Study,” ASME J. Mech. Des., 138(5), p. 051405. [CrossRef]
Hu, Z. , and Mahadevan, S. , 2016, “ Resilience Assessment Based on Time-Dependent System Reliability Analysis,” ASME J. Mech. Des., 138(11), p. 111404. [CrossRef]
Hu, Z. , Mahadevan, S. , and Du, X. , 2016, “ Uncertainty Quantification of Time-Dependent Reliability Analysis in the Presence of Parametric Uncertainty,” ASCE-ASME J. Risk Uncertainty Eng. Syst., Part B: Mech. Eng., 2(3), p. 031005. [CrossRef]
Shahan, D. W. , and Seepersad, C. C. , 2012, “ Bayesian Network Classifiers for Set-Based Collaborative Design,” ASME J. Mech. Des., 134(7), p. 071001. [CrossRef]
Khakzad, N. , Khan, F. , and Amyotte, P. , 2013, “ Dynamic Safety Analysis of Process Systems by Mapping Bow-Tie Into Bayesian Network,” Process Saf. Environ. Prot., 91(1–2), pp. 46–53. [CrossRef]
Khakzad, N. , Khan, F. , and Amyotte, P. , 2012, “ Dynamic Risk Analysis Using Bow-Tie Approach,” Reliab. Eng. Syst. Saf., 104, pp. 36–44. [CrossRef]
Yuan, Z. , Khakzad, N. , Khan, F. , and Amyotte, P. , 2015, “ Risk Analysis of Dust Explosion Scenarios Using Bayesian Networks,” Risk Anal., 35(2), pp. 278–291. [CrossRef] [PubMed]
Gradowska, P. L. , and Cooke, R. M. , 2014, “ Estimating Expected Value of Information Using Bayesian Belief Networks: A Case Study in Fish Consumption Advisory,” Environ. Syst. Decisions, 34(1), pp. 88–97. [CrossRef]
Liang, C. , and Mahadevan, S. , 2016, “ Multidisciplinary Optimization Under Uncertainty Using Bayesian Network,” SAE Int. J. Mater. Manf., 9(2), pp. 419–429.
Bartram, G. , and Mahadevan, S. , 2014, “ Integration of Heterogeneous Information in SHM Models,” Struct. Control Health Monit., 21(3), pp. 403–422. [CrossRef]
Groth, K. M. , and Swiler, L. P. , 2013, “ Bridging the Gap Between HRA Research and HRA Practice: A Bayesian Network Version of SPAR-H,” Reliab. Eng. Syst. Saf., 115, pp. 33–42. [CrossRef]
Groth, K. M. , and Mosleh, A. , 2012, “ A Data-Informed PIF Hierarchy for Model-Based Human Reliability Analysis,” Reliab. Eng. Syst. Saf., 108, pp. 154–174. [CrossRef]
Sankararaman, S. , and Mahadevan, S. , 2015, “ Integration of Model Verification, Validation, and Calibration for Uncertainty Quantification in Engineering Systems,” Reliab. Eng. Syst. Saf., 138, pp. 194–209. [CrossRef]
Hu, Z. , and Mahadevan, S. , 2017, “ Bayesian Network Learning for Uncertainty Quantification,” ASME Paper No. DETC2017-68187.
He, L. , Wang, M. , Chen, W. , and Conzelmann, G. , 2014, “ Incorporating Social Impact on New Product Adoption in Choice Modeling: A Case Study in Green Vehicles,” Transp. Res. Part D: Transp. Environ., 32, pp. 421–434. [CrossRef]
Vinh, N. X. , Chetty, M. , Coppel, R. , and Wangikar, P. P. , 2011, “ GlobalMIT: Learning Globally Optimal Dynamic Bayesian Network With the Mutual Information Test Criterion,” Bioinformatics, 27(19), pp. 2765–2766. [CrossRef] [PubMed]
Ziebarth, J. D. , Bhattacharya, A. , and Cui, Y. , 2013, “ Bayesian Network Webserver: A Comprehensive Tool for Biological Network Modeling,” Bioinformatics, 29(21), pp. 2801–2803. [CrossRef] [PubMed]
Murphy, K. P. , 2002, “ Dynamic Bayesian Networks: Representation, Inference and Learning,” Ph.D. dissertation, University of California, Berkeley, CA.
Karkera, K. R. , 2014, Building Probabilistic Graphical Models With Python, Packt Publishing, Birmingham, UK.
Hanea, A. , Kurowicka, D. , Cooke, R. M. , and Ababei, D. , 2010, “ Mining and Visualising Ordinal Data With Non-Parametric Continuous BBNs,” Comput. Stat. Data Anal., 54(3), pp. 668–687. [CrossRef]
Bedford, T. , and Cooke, R. M. , 2002, “ Vines: A New Graphical Model for Dependent Random Variables,” Ann. Stat., 30(4), pp. 1031–1068. [CrossRef]
Shenoy, P. P. , and West, J. C. , 2011, “ Inference in Hybrid Bayesian Networks Using Mixtures of Polynomials,” Int. J. Approximate Reasoning, 52(5), pp. 641–657. [CrossRef]
Dojer, N. , Bednarz, P. , Podsiadło, A. , and Wilczyński, B. , 2013, “ BNFinder2: Faster Bayesian Network Learning and Bayesian Classification,” Bioinformatics, 29(16), pp. 2068–2070. [CrossRef] [PubMed]
McGeachie, M. J. , Chang, H.-H. , and Weiss, S. T. , 2014, “ CGBayesNets: Conditional Gaussian Bayesian Network Learning and Inference With Mixed Discrete and Continuous Data,” PLoS Comput. Biol., 10(6), p. e1003676. [CrossRef] [PubMed]
Hu, Z. , and Mahadevan, S. , 2017, “ Uncertainty Quantification and Management in Additive Manufacturing: Current Status, Needs, and Opportunities,” Int. J. Adv. Manuf. Technol., 93(5–8), pp. 2855–2874. [CrossRef]
Hu, Z. , Mahadevan, S. , and Ao, D. , 2017, “ Uncertainty Aggregation and Reduction in Structure–Material Performance Prediction,” Comput. Mech., epub.
Scutari, M. , 2009, “ Learning Bayesian Networks With the bnlearn R Package,” preprint arXiv: 0908.3817.
Bonissone, P. , Henrion, M. , Kanal, L. , and Lemmer, J. , “ Equivalence and Synthesis of Causal Models,” Uncertainty Artificial Intelligence, Elsevier, Amsterdam, The Netherlands, pp. 255–270.
Wilczyński, B. , and Dojer, N. , 2009, “ BNFinder: Exact and Efficient Method for Learning Bayesian Networks,” Bioinformatics, 25(2), pp. 286–287. [CrossRef] [PubMed]
Bilmes, J. A. , 1998, “ A Gentle Tutorial of the EM Algorithm and Its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models,” Int. Comput. Sci. Inst., 4(510), p. 126.
Sahin, F. , Yavuz, M. Ç. , Arnavut, Z. , and Uluyol, Ö. , 2007, “ Fault Diagnosis for Airplane Engines Using Bayesian Networks and Distributed Particle Swarm Optimization,” Parallel Comput., 33(2), pp. 124–143. [CrossRef]
Yang, L. , and Lee, J. , 2012, “ Bayesian Belief Network-Based Approach for Diagnostics and Prognostics of Semiconductor Manufacturing Systems,” Rob. Comput.-Integr. Manuf., 28(1), pp. 66–74. [CrossRef]
Rodriguez‐Zas, S. , and Ko, Y. , 2011, “ Elucidation of General and Condition‐Dependent Gene Pathways Using Mixture Models and Bayesian Networks,” Applied Statistics for Network Biology: Methods in Systems Biology, Wiley-Blackwell, Weinheim, Germany, pp. 91–103. [CrossRef]
Sun, S. , Zhang, C. , and Yu, G. , 2006, “ A Bayesian Network Approach to Traffic Flow Forecasting,” IEEE Trans. Intell. Transp. Syst., 7(1), pp. 124–132. [CrossRef]
Zhang, H. , Giles, C. L. , Foley, H. C. , and Yen, J. , 2017, “ Probabilistic Community Discovery Using Hierarchical Latent Gaussian Mixture Model,” 22nd National conference on Artificial Intelligence (AAAI'07), Vancouver, BC, Canada, July 22–26, pp. 663–668.
Hu, Z. , and Mahadevan, S. , 2017, “ Time-Dependent Reliability Analysis Using a Vine-ARMA Load Model,” ASCE-ASME J. Risk Uncertainty Eng. Syst., Part B: Mech. Eng., 3(1), p. 011007. [CrossRef]
Davies, S. , and Moore, A. , “ Mix-Nets: Factored Mixtures of Gaussians in Bayesian Networks With Mixed Continuous and Discrete Variables,” Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI), Stanford, CA, June 30–July 3, pp. 168–175.
Morlini, I. , 2012, “ A Latent Variables Approach for Clustering Mixed Binary and Continuous Variables Within a Gaussian Mixture Model,” Adv. Data Anal. Classif., 6(1), pp. 5–28. [CrossRef]
Bartram, G. W. , 2013, “ System Health Diagnosis and Prognosis Using Dynamic Bayesian Networks,” Ph.D. thesis, Vanderbilt University, Nashville, TN.


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Fig. 1

Concept of the bnfinder method

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Fig. 2

Overview of the proposed method

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Fig. 3

Concept of the mix-nets method

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Fig. 4

A continuous node with discrete and continuous parents

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Fig. 5

Results comparison of learned CPDs, given x1 = 14 (Note: Gray background implies mix-nets failed to model the CPD)

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Fig. 6

Bayesian network of Example 1

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Fig. 7

Learned BN topology from bnfinder with different numbers of samples

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Fig. 8

Learned BN topology from OUGM with different numbers of samples

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Fig. 9

Learned CPDs of C3 with different values of parent nodes: (a) D3 = 0 and (b) D3 = 1

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Fig. 10

Model calibration results based on different methods: (a) five observations and (b) ten observations

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Fig. 11

A beam with possible crack and support damage: (a) a beam with crack and (b) exact BN

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Fig. 12

Learned BN from different methods with 1000–5000 samples

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Fig. 13

Learned CPD of F under given values of its parents

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Fig. 14

Model calibration results using different methods: (a) four observations and (b) eight observations




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