Phase I analysis of high-dimensional covariance matrices based on sparse leading eigenvalues

Abstract

In statistical process control (SPC), a proper Phase I analysis is essential to the success of Phase II monitoring. With recent advances in sensing technology and data acquisition systems, Phase I analysis of high-dimensional data is increasingly encountered. However, the high dimensionality presents a new challenge to the traditional Phase I techniques. A literature review reveals nearly no Phase I techniques in existence for analyzing high-dimensional process variability. Motivated by this, this paper develops a sparse-leading-eigenvalue-driven control chart for retrospectively monitoring high-dimensional covariance matrices in Phase I, denoted as the SLED control chart. The key idea of it is to track changes in the sparse leading eigenvalue between two covariance matrices. Compared to the L₂-type and $L_{\infty }$-type methods, the proposed method can extract stronger signal with less noise. It is shown that the proposed method can gain high detection power, especially when the shift is weak and is not very dense, which is often the case in practical applications.

Keywords:

• statistical process control
• two-sample tests
• eigenvalues
• control chart

Additional information

Funding

The authors would like to thank the editor and two referees for their valuable comments. Dr. Fan’s work was support by Big data and Educational Statistics Application Laboratory (No. 2017WSYS001/510320). Prof. Shu’s work was funded in part by the Research Committee under the grant MYRG2018-00087-FBA, and by the Science and Technology Development Fund, Macau SAR (File No. FDCT/0064/2018/A2). Prof. Li’s work was supported by the grant from the National Natural Science Foundation of China (No. 71672109).

Notes on contributors

Jinyu Fan

Jinyu Fan is Lecturer in School of Mathematics and Statistics, Guangdong University of Finance & Economics. She received her B.Sc. degree in Statistics from Hubei Normal University, M.Sc. degree in Probability and Mathematical Statistics from Guangzhou University, and Ph.D. in Decision Science from the University of Macau, respectively. Her current research interests include statistical process control, high dimensional data analysis and image-based process monitoring, and diagnostics.

Lianjie Shu

Lianjie Shu is Professor in Faculty of Business Administration at University of Macau. He received his B.Sc. degree in Mechanical Engineering and Automation from Xi’an Jiao Tong University, and his Ph.D. in Industrial Engineering and Engineering Management from the Hong Kong University of Science and Technology (HKUST). His current research interests include statistical process control, health care surveillance, and computational statistics. He is an Associate Editor of Journal of Statistical Computation and Simulation, and a senior member of both Institute of and Industrial and Systems Engineers (ISYE) and American Society for Quality (ASQ).

Aijun Yang

Aijun Yang is Professor in the College of Economics and Management of Nanjing Forestry University, China. He received his M.Sc. degree in Financial Statistics from Southeast University, and Ph.D. in Financial Statistics from the Chinese University of Hong Kong (CUHK). His current research interests include high-dimensional data analysis, Computational Economics, and Financial Econometrics.

Yanting Li

Yanting Li is Associate Professor in the Department of Industrial Engineering and Logistics Engineering of Shanghai Jiao Tong University, China. She is also a member of China Quality Development Research Institute. She received her Ph.D. in Industrial Engineering and Logistics Management, Hong Kong University of Science and Technology. Her current research interests include the modeling, prediction and process control of multivariate data, spatio-temporal data in manufacturing and service industry.