A Multivariate Sign EWMA Control Chart

Nonparametric control charts are useful in statistical process control (SPC) when there is a lack of or limited knowledge about the underlying process distribution, especially when the process measurement is multivariate. This article develops a new multivariate SPC methodology for monitoring location parameters. It is based on adapting a powerful multivariate sign test to online sequential monitoring. The weighted version of the sign test is used to formulate the charting statistic by incorporating the exponentially weighted moving average control (EWMA) scheme, which results in a nonparametric counterpart of the classical multivariate EWMA (MEWMA). It is affine-invariant and has a strictly distribution-free property over a broad class of population models. That is, the in-control (IC) run length distribution can attain (or is always very close to) the nominal one when using the same control limit designed for a multivariate normal distribution. Moreover, when the process distribution comes from the elliptical direction class, the IC average run length can be calculated via a one-dimensional Markov chain model. This control chart possesses some other favorable features: it is fast to compute with a similar computational effort to the MEWMA chart; it is easy to implement because only the multivariate median and the associated transformation matrix need to be specified (estimated) from the historical data before monitoring; it is also very efficient in detecting process shifts, particularly small or moderate shifts when the process distribution is heavy tailed or skewed. Two real-data examples from manufacturing show that it performs quite well in applications. This article has supplementary material online.

[1]  J. Ledolter,et al.  A Control Chart Based on Ranks , 1991 .

[2]  K. Mardia.  Measures of multivariate skewness and kurtosis with applications , 1970 .

[3]  Hannu Oja,et al.  Affine Invariant Multivariate Sign and Rank Tests and Corresponding Estimates: a Review , 1999 .

[4]  Hannu Oja,et al.  OPERATING TRANSFORMATION RETRANSFORMATION ON SPATIAL MEDIAN AND ANGLE TEST , 1998 .

[5]  George C. Runger,et al.  A Markov Chain Model for the Multivariate Exponentially Weighted Moving Averages Control Chart , 1996 .

[6]  S. Chakraborti,et al.  Nonparametric Control Charts: An Overview and Some Results , 2001 .

[7]  Peihua Qiu,et al.  Multivariate Statistical Process Control Using LASSO , 2009 .

[8]  Peihua Qiu,et al.  A Rank-Based Multivariate CUSUM Procedure , 2001, Technometrics.

[9]  York Marcel Dekker.  The State of Statistical Process Control as We Proceed into the 21st Century , 2000 .

[10]  Ronald H. Randles,et al.  A Simpler, Affine-Invariant, Multivariate, Distribution-Free Sign Test , 2000 .

[11]  Douglas C. Montgomery,et al.  Research Issues and Ideas in Statistical Process Control , 1999 .

[12]  Douglas M. Hawkins,et al.  A Multivariate Change-Point Model for Statistical Process Control , 2006, Technometrics.

[13]  Zengrong Wang,et al.  Structural damage detection using autoregressive-model-incorporating multivariate exponentially weighted moving average control chart , 2009 .

[14]  Charles W. Champ,et al.  The Performance of Exponentially Weighted Moving Average Charts With Estimated Parameters , 2001, Technometrics.

[15]  Douglas C. Montgomery,et al.  Introduction to Statistical Quality Control , 1986 .

[16]  T. Hettmansperger,et al.  Robust Nonparametric Statistical Methods , 1998 .

[17]  R. Randles,et al.  A practical affine equivariant multivariate median , 2002 .

[18]  D. A. Evans,et al.  An approach to the probability distribution of cusum run length , 1972 .

[19]  Regina Y. Liu.  Control Charts for Multivariate Processes , 1995 .

[20]  Fugee Tsung,et al.  A kernel-distance-based multivariate control chart using support vector methods , 2003 .

[21]  R. Randles.  A Distribution-Free Multivariate Sign Test Based on Interdirections , 1989 .

[22]  Douglas M. Hawkins,et al.  Self-Starting Multivariate Exponentially Weighted Moving Average Control Charting , 2007, Technometrics.

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

[24]  J. Healy.  A note on multivariate CUSUM procedures , 1987 .

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

[26]  B. M. Brown,et al.  Statistical Uses of the Spatial Median , 1983 .

[27]  D. Hawkins,et al.  A nonparametric multivariate cumulative sum procedure for detecting shifts in all directions , 2003 .

[28]  J. Macgregor,et al.  The exponentially weighted moving variance , 1993 .

[29]  D. Hawkins,et al.  Cumulative Sum Charts and Charting for Quality Improvement , 1998, Statistics for Engineering and Physical Science.

[30]  Zachary G. Stoumbos,et al.  Robustness to Non-Normality of the Multivariate EWMA Control Chart , 2002 .

[31]  Eric R. Ziegel,et al.  Biostatistics: A Foundation for Analysis in the Health Sciences , 1988 .

[32]  Charles W. Champ,et al.  Effects of Parameter Estimation on Control Chart Properties: A Literature Review , 2006 .

[33]  Hannu Oja,et al.  Multivariate Nonparametric Tests , 2004 .

[34]  L. Allison Jones,et al.  The Statistical Design of EWMA Control Charts with Estimated Parameters , 2002 .

[35]  C. Quesenberry.  On Properties of Q Charts for Variables , 1995 .

[36]  George C. Runger,et al.  Designing a Multivariate EWMA Control Chart , 1997 .

[37]  Regina Y. Liu,et al.  DDMA-charts: Nonparametric multivariate moving average control charts based on data depth , 2004 .

[38]  J. C. Gower,et al.  Algorithm AS 78: The Mediancentre , 1974 .

[39]  Chien-Wei Wu,et al.  Monitoring Multivariate Process Variability for Individual Observations , 2007 .

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

[41]  James M. Lucas,et al.  Exponentially weighted moving average control schemes: Properties and enhancements , 1990 .

[42]  Changliang Zou,et al.  Nonparametric control chart based on change-point model , 2009 .

[43]  David E. Tyler.  A Distribution-Free $M$-Estimator of Multivariate Scatter , 1987 .

[44]  R. Crosier.  Multivariate generalizations of cumulative sum quality-control schemes , 1988 .

[45]  Karen A. F. Copeland.  Cumulative Sum Charts and Charting for Quality Improvement , 1999 .

[46]  Roger M. Sauter,et al.  Introduction to Statistical Quality Control (2nd ed.) , 1992 .