Two approaches to monitoring multivariate Poisson counts: Simple and accurate

Abstract

We consider the monitoring of multivariate correlated count data, which have many applications in practice. Although there are quite a few methods for the statistical process control of Multivariate Poisson (MP) counts, they are either too complicated or too simple to provide a satisfactory tool for efficient online monitoring. In addition, they mostly focus on only the mean vector of multivariate counts and ignore the correlations among them. In this article, we adopt the multivariate Poisson distribution with a two-way covariance structure for modeling MP counts, which has marginal Poisson distributions in each dimension and allows for pairwise correlations. Based on this, we develop two control charts to simultaneously monitor the mean vector and covariance matrix of MP counts. The first chart enjoys a simple charting statistic and is computationally fast, whereas the second one is accurate and provides a gold standard for monitoring MP counts. We also give recommendations on choice between them. Numerical simulations have demonstrated the advantages of the proposed two charts, and in non-Poisson cases we also test their robustness against underdispersion and overdispersion that are encountered often in count data.

Keywords:

• Likelihood ratio test
• multivariate Poisson distribution
• normal approximation
• overdispersion
• statistical process control

Data availability statement

The data that support the findings of this study are openly available at http://www9.health.gov.au/cda/source/cda-index.cfm.

Acknowledgments

The authors would like to thank the Department Editor and three referees for their many helpful comments that have resulted in significant improvements in this article.

Additional information

Funding

This work was supported by the National Key R&D Program of China under Grant 2021YFF0901300; the National Natural Science Foundation of China under Grants 72271193, 71772147; and the Key Science Foundation of the Education Department of Shaanxi Provincial Government under Grant 21JZ030.

Notes on contributors

Kaizong Bai

Kaizong Bai is currently pursuing the PhD degree in the School of Management, Xi’an Jiaotong University, Xi’an, China. He received his BE degree in Metallurgical Engineering from Northeastern University, Shenyang, China, and his MS degree in Management Science and Engineering from Xi’an Jiaotong University, Xi’an, China. His current research interests include quality engineering and statistical process control.

Jian Li

Jian Li is a Professor in the School of Management, Xi’an Jiaotong University, China. He received his BS degree in Automation from Tsinghua University, Beijing, China, and his PhD degree in Industrial Engineering and Decision Analytics from the Hong Kong University of Science and Technology, Hong Kong. His current research interests include quality management and quality engineering, Six Sigma implementation, and statistical process control.

Dong Ding

Dong Ding is an Associate Professor in the School of Management, Xi’an Polytechnic University, China. She received her BS degree in Statistics from Nankai University, Tianjin, China, and her PhD degree in Industrial Engineering and Decision Analytics from the Hong Kong University of Science and Technology, Hong Kong. Her current research interests include quality management, quality engineering, and statistical process control.