Bayesian learning of structures of ordered block graphical models with an application on multistage manufacturing processes

Abstract

The Ordered Block Model (OBM) is a special form of directed graphical models and is widely used in various fields. In this article, we focus on learning of structures of OBM based on prior knowledge obtained from historical data. The proposed learning method is applied to a multistage car body assembly process to validate the learning efficiency. In this approach, Bayesian score is used to learn the graph structure and a novel informative structure prior distribution is constructed to help the learning process. Specifically, the graphical structure is represented by a categorical random variable and its distribution is treated as the informative prior. In this way, the informative prior distribution construction is equivalent to the parameter estimation of the graph random variable distribution using historical data. Since the historical OBMs may not contain the same nodes as those in the new OBM, the sample space of the graphical structure of the historical OBMs and the new OBM may be inconsistent. We deal with this issue by adding pseudo nodes with probability normalization, then removing extra nodes through marginalization to align the sample space between historical OBMs and the new OBM. The performance of the proposed method is illustrated and compared to conventional methods through numerical studies and a real car assembly process. The results show the proposed informative structure prior can effectively boost the performance of the graph structure learning procedure, especially when the data from the new OBM is small.

Keywords:

• Graphical models
• structure learning
• ordered block model
• Bayesian score
• multistage manufacturing process

Acknowledgment

The authors thank the editors and reviewers for their valuable comments and suggestions. This research is supported by the National Science Foundation Award 1561512.

Additional information

Notes on contributors

Chao Wang

Chao Wang is an assistant professor in the Department of Industrial and Systems Engineering at the University of Iowa. He received his B.S. from the Hefei University of Technology in 2012, and MS from the University of Science and Technology of China in 2015, both in mechanical engineering, and his MS in statistics and PhD in industrial and systems engineering from the University of Wisconsin-Madison in 2018 and 2019, respectively. His research interests include statistical modeling, analysis, monitoring and control for complex systems. He is member of INFORMS, IISE, and SME.

Xiaojin Zhu

Xiaojin Zhu is the Sheldon & Marianne Lubar Professor in the Department of Computer Sciences at the University of Wisconsin-Madison. His research focuses on machine learning, in particular machine teaching and semi-supervised learning. He received his PhD from Carnegie Mellon University in 2005. He is a recipient of a National Science Foundation CAREER Award in 2010, ICML classic paper prize in 2013, and several best paper awards. He is co-chair for CogSci 2018 and AISTATS 2017.

Shiyu Zhou

Shiyu Zhou is the Vilas Distinguished Achievement Professor in the Department of Industrial and Systems Engineering at the University of Wisconsin‐Madison. He received his BS degree from the University of Science and Technology of China in 1993, and his master's degree and PhD from the University of Michigan in 2000. His research focuses on industrial analytics and system informatics methodologies for quality and productivity improvement and operation optimization. He has received numerous research awards and grants from various federal agencies. He is currently a Fellow of IISE, ASME, and SME.

Yingqing Zhou

Yingqing Zhou earned his PhD in systems engineering from the Case Western Reserve University, Cleveland, Ohio, in 1992. After graduation, he was a senior developer in Trikon Design from 1992 to 1995. Since 1995, he has been the Director of Research and Development with Dimensional Control Systems, Inc. His current research includes dimensional variation simulation and optimization in kinematic motion systems and deformable components. Dr. Zhou has been the chief editor for Dimensional Engineering News, a monthly newsletter, since May 2003.