Hierarchical sparse functional principal component analysis for multistage multivariate profile data

Abstract

Modern manufacturing systems typically involve multiple production stages, the real-time status of which can be tracked continuously using sensor networks that generate a large number of profiles associated with all process variables at all stages. The analysis of the collective behavior of the multistage multivariate profile data is essential for understanding the variance patterns of the entire manufacturing process. For this purpose, two major challenges regarding the high data dimensionality and low model interpretability have to be well addressed. This article proposes integrating Multivariate Functional Principal Component Analysis (MFPCA) with a three-level structured sparsity idea to develop a novel Hierarchical Sparse MFPCA (HSMFPCA), in which the stage-wise, profile-wise and element-wise sparsity are jointly investigated to clearly identify the informative stages and variables in each eigenvector. In this way, the derived principal components would be more interpretable. The proposed HSMFPCA employs the regression-type reformulation of the PCA and the reparameterization of the entries of eigenvectors, and enjoys an efficient optimization algorithm in high-dimensional settings. The extensive simulations and a real example study verify the superiority of the proposed HSMFPCA with respect to the estimation accuracy and interpretation clarity of the derived eigenvectors.

Keywords:

• Hierarchical sparsity
• process variation
• sensor data
• sparse PCA
• variance decomposition

Acknowledgments

The authors greatly acknowledge the valuable comments provided by the editor and three referees that have resulted in great improvements of this article.

Notes on contributors

Kai Wang is currently an Assistant Professor in the Department of Industrial Engineering, School of Management, at the Xi’an Jiaotong University, Xi’an, China. He received his Ph.D. in Industrial Engineering and Logistics Management in 2018 from the HKUST, Hong Kong, and his bachelor’s degree in Industrial Engineering in 2014 from Xi’an Jiaotong University, Shaanxi, China. His research focuses on industrial big data analytics, machine learning and transfer learning, statistical process control and monitoring.

Fugee Tsung is a Chair Professor in the Department of Industrial Engineering and Decision Analytics (IEDA), Director of the Quality and Data Analytics Lab (QLab), at the Hong Kong University of Science and Technology (HKUST), Hong Kong, China. He is a Fellow of the American Society for Quality, Fellow of the American Statistical Association, Academician of the International Academy for Quality, and Fellow of the Hong Kong Institution of Engineers. He received both his M.Sc. and Ph.D. from the University of Michigan, Ann Arbor, and his B.Sc. from the National Taiwan University. His research interests include quality analytics in advanced manufacturing and service processes, industrial big data and statistical process control, monitoring, and diagnosis.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China under Grant 71931006, 71602155; Hong Kong RGC General Research Funds under Grant 16201718 and 16203917; Hong Kong Innovation and Technology Fund Postdoctoral Hub Program under Grant PiH/246/18; and Hong Kong Innovation and Technology Fund Project under Grant ITS/184/18FX.