Group-wise monitoring of multivariate data with missing values

Abstract

In this article, we aim to detect the changes in the process that generates multivariate data by monitoring their mean shift. To do this, we utilize a graphical tool known as a multivariate control chart. However, monitoring the mean of multivariate data poses two challenges: the grouped structure of individual observations and the presence of missing values. In this research, we introduce a novel method called HTC for monitoring group-wise multivariate data that includes missing values. HTC offers several advantages over existing methods. First, it is applicable to various types of dependence among individual observations within a group. Second, it provides a unique upper control limit (UCL) regardless of the missing data pattern. Lastly, HTC is computationally more efficient compared to resampling-based techniques. We conduct comprehensive numerical studies to evaluate the performance of the HTC method and compare it with the existing group-wise monitoring method, referred to as HTM. Compared to HTM, HTC achieves a higher true positive rate (TPR) while effectively controlling the in-control false alarm rate (${\text{FAR}}_0$) at a pre-determined level across various settings considered in our study. To illustrate its effectiveness, we applied HTC to monitoring multivariate environmental data collected from the manufacturing process of a semiconductor company in Korea.

Keywords:

• Cauchy combination test
• covariance matrix estimator
• group-wise monitoring
• Hotelling’s T²
• missing data
• statistics

Acknowledgements

The authors are grateful for the editors and two reviewers for several variable comments.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Research Foundation of Korea (No. NRF- 2021R1A2C1010786) and Samsung Electronics Co., Ltd. (No. IO201209-07860-01).

Notes on contributors

Kwangok Seo

Kwangok Seo is a graduate student Department of Statistics, Seoul National University. He did undergraduate study in Mathematics at Chonnam National University and obtained M.S. in Statistics from Yonsei University. His research interest lie in multiple testing and false discovery rate and hidden Markov model.

Johan Lim

Johan Lim has been a professor of Statistics, Seoul National University since 2008. He obtained his Ph.D. in Statistics from Stanford University. Before joining Seoul National University, he worked as a faculty member of Department of Statistics, Texas A&M University and Department of Applied Statistics, Yonsei University/ His research lies in inference on high dimensional covariance matrix, hidden Markov model, ranked set sampling and statistical process control.

Youngrae Kim

Youngrae Kim is a post-doc research of Yong Loo Lin School of Medicine, National University of Singapore. He obtained B.S. and Ph.D. in Statistics from Seoul National University. His research area is inference on high dimensional covariance matrix and statistical process control.