An efficient charting scheme for multivariate categorical process with a sparse contingency table

Abstract Multivariate categorical quality characteristics, whose distribution can be displayed by a contingency table, are routinely encountered in many applications. When most of the cell entries in the contingency table are very small or zeros counts, which is so-called sparse contingency table in the literature, existing methods developed in the literature are often inadequate for use, due to the inaccuracy of the maximum likelihood estimate of its probability distribution, and the inflation of online charting statistics. This paper studies the multivariate statistical process control problem for such sparse contingency table. We integrate the group least absolute shrinkage and selection operator (LASSO) method with the Ridge method to estimate the in-control distribution of a contingency table and propose an efficient EWMA control chart, based on a modified Pearson χ2 statistic, to monitor the changes in it. Numerical results show that our proposed approach has the best overall performance, compared with its competitors. Finally, a real data example is used to demonstrate the effectiveness of the proposed control chart.

[1]  Jian Li,et al.  Causation-based process monitoring and diagnosis for multivariate categorical processes , 2017 .

[2]  Armin Shmilovici,et al.  Context-Based Statistical Process Control , 2003, Technometrics.

[3]  A. Rinaldo,et al.  The Log-Linear Group Lasso Estimator and Its Asymptotic Properties , 2007, 0709.3526.

[4]  Dongdong Xiang,et al.  Univariate Dynamic Screening System: An Approach For Identifying Individuals With Irregular Longitudinal Behavior , 2014, Technometrics.

[5]  Wei Jiang,et al.  A context tree method for multistage fault detection and isolation with applications to commercial video broadcasting systems , 2009 .

[6]  P. Bühlmann,et al.  The group lasso for logistic regression , 2008 .

[7]  Chi Zhang,et al.  Monitoring censored lifetime data with a weighted‐likelihood scheme , 2016 .

[8]  Jian Li,et al.  Directional change‐point detection for process control with multivariate categorical data , 2013 .

[9]  M. Yuan,et al.  Model selection and estimation in regression with grouped variables , 2006 .

[10]  H. Zou.  The Adaptive Lasso and Its Oracle Properties , 2006 .

[11]  M. Marcucci.  MONITORING MULTINOMIAL PROCESSES , 1985 .

[12]  Seyed Taghi Akhavan Niaki,et al.  New Approaches in Monitoring Multivariate Categorical Processes based on Contingency Tables in Phase II , 2017, Qual. Reliab. Eng. Int..

[13]  Dongdong Xiang,et al.  Surveillance of cardiovascular diseases using a multivariate dynamic screening system , 2015, Statistics in medicine.

[14]  Fugee Tsung,et al.  Multivariate binomial/multinomial control chart , 2014 .

[15]  Fugee Tsung,et al.  A Multivariate Sign EWMA Control Chart , 2011, Technometrics.

[16]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[17]  D. Hawkins.  Multivariate quality control based on regression-adjusted variables , 1991 .

[18]  Seyed Taghi Akhavan Niaki,et al.  Phase-II monitoring and diagnosing of multivariate categorical processes using generalized linear test-based control charts , 2017, Commun. Stat. Simul. Comput..

[19]  D. Edwards,et al.  Collapsibility and response variables in contingency tables , 1983 .

[20]  D. Zelterman.  Goodness-of-Fit Tests for Large Sparse Multinomial Distributions , 1987 .

[21]  Douglas M. Hawkins,et al.  Regression Adjustment for Variables in Multivariate Quality Control , 1993 .

[22]  Jian Li,et al.  Multivariate Ordinal Categorical Process Control Based on Log-Linear Modeling , 2017 .

[23]  Amirhossein Amiri,et al.  Phase-I monitoring of log-linear model-based processes (a case study in health care: Kidney patients) , 2019, Qual. Reliab. Eng. Int..

[24]  Stephen E. Fienberg,et al.  Three centuries of categorical data analysis: Log-linear models and maximum likelihood estimation , 2007 .

[25]  Qiang Zhou,et al.  A General Approach for Monitoring Serially Dependent Categorical Processes , 2017 .

[26]  H. B. Lawal.  Comparing Power Behaviours for Some Goodness-of-Fit Test Statistics in Sparse Multinomials , 1989 .

[27]  George C. Runger,et al.  Comparison of multivariate CUSUM charts , 1990 .

[28]  M. Reiser,et al.  3. A Goodness-of-Fit Test for the Latent Class Model When Expected Frequencies are Small , 1999 .

[29]  Wei Jiang,et al.  A Cluster-Based Context-Tree Model for Multivariate Data Streams with Applications to Anomaly Detection , 2011, INFORMS J. Comput..

[30]  Charles W. Champ,et al.  A multivariate exponentially weighted moving average control chart , 1992 .

[31]  Peter Bühlmann,et al.  Penalized likelihood for sparse contingency tables with an application to full-length cDNA libraries , 2007, BMC Bioinformatics.

[32]  Fugee Tsung,et al.  A Reference-Free Cuscore Chart for Dynamic Mean Change Detection and a Unified Framework for Charting Performance Comparison , 2006 .

[33]  Dongdong Xiang,et al.  A Robust Multivariate EWMA Control Chart for Detecting Sparse Mean Shifts , 2016 .

[34]  Dongdong Xiang,et al.  Statistical Process Control for Latent Quality Characteristics Using the Up-and-Down Test , 2017, Technometrics.

[35]  J. Chiu,et al.  Attribute Control Chart for Multivariate Poisson Distribution , 2007 .

[36]  A. Agresti.  Categorical data analysis , 1993 .

[37]  Jian Li,et al.  Directional Control Schemes for Multivariate Categorical Processes , 2012 .

[38]  H. I. Patel.  QUALITY CONTROL METHODS FOR MULTIVARIATE BINOMIAL AND POISSON DISTRIBUTIONS , 1973 .

[39]  Peihua Qiu,et al.  Distribution-free multivariate process control based on log-linear modeling , 2008 .