A Distribution‐Free Multivariate Control Chart for Phase I Applications

The purpose of this paper is to provide a novel distribution-free control chart for monitoring the location parameter vector of a multivariate process in phase I analysis. To be robust to the process distribution, the spatial sign statistic that defines the multivariate direction of an observation is used to construct a Shewhart-type control chart for detecting out-of-control observations in historical phase I data. The proposed control chart is distribution free in the sense that the false-positive rate (or false alarm rate), the proportion of wrongly classified in-control samples, can be controlled at the specified value for elliptical-direction distributions. In addition, we demonstrate through simulation studies that the false-positive rate of the proposed chart is robust to the shift size of the out-of-control condition if we only delete the most extreme out-of-control observation at each iteration of phase I analysis. Compared with the traditional Hotelling's T2 control chart and some of its robust versions, the proposed chart is generally more powerful in detecting out-of-control observations and more robust to the normality assumption. Copyright © 2014 John Wiley & Sons, Ltd.

[1]  H. Oja.  Multivariate Nonparametric Methods with R , 2010 .

[2]  Douglas C. Montgomery,et al.  Some Current Directions in the Theory and Application of Statistical Process Monitoring , 2014 .

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

[4]  Zhonghua Li,et al.  Multivariate Change Point Control Chart Based on Data Depth for Phase I Analysis , 2014, Commun. Stat. Simul. Comput..

[5]  Peter J. Rousseeuw,et al.  Robust regression and outlier detection , 1987 .

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

[7]  Peter J. Rousseeuw,et al.  Robust Regression and Outlier Detection , 2005, Wiley Series in Probability and Statistics.

[8]  A. B. Yeh,et al.  A likelihood-ratio-based EWMA control chart for monitoring variability of multivariate normal processes , 2004 .

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

[10]  Fugee Tsung,et al.  A spatial rank‐based multivariate EWMA control chart , 2012 .

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

[12]  Jyh-Jen Horng Shiau,et al.  A new strategy for Phase I analysis in SPC , 2010, Qual. Reliab. Eng. Int..

[13]  Douglas M. Hawkins,et al.  Multivariate Exponentially Weighted Moving Covariance Matrix , 2008, Technometrics.

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

[15]  Jyh-Jen Horng Shiau,et al.  A MULTIVARIATE CONTROL CHART FOR DETECTING INCREASES IN PROCESS DISPERSION , 2010 .

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

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

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

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

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

[21]  Stefan H. Steiner,et al.  An Overview of Phase I Analysis for Process Improvement and Monitoring , 2014 .

[22]  Victoria S. Jordan,et al.  Distribution-Free Phase I Control Charts for Subgroup Location , 2009 .

[23]  Nandini Das,et al.  A New Multivariate Non-Parametric Control Chart Based on Sign Test , 2009 .

[24]  Jyh-Jen Horng Shiau,et al.  Effective Control Charts for Monitoring Multivariate Process Dispersion , 2012, Qual. Reliab. Eng. Int..

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

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

[27]  Subhabrata Chakraborti,et al.  Two simple Shewhart-type multivariate nonparametric control charts , 2012 .

[28]  Xuemin Zi,et al.  An Empirical-Likelihood-Based Multivariate EWMA Control Scheme , 2013 .

[29]  Mehmet Mert,et al.  Nonparametric control charts based on Mahalanobis depth , 2006 .

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

[31]  P. Rousseeuw,et al.  A fast algorithm for the minimum covariance determinant estimator , 1999 .

[32]  Chien-Wei Wu,et al.  A multivariate EWMA control chart for monitoring process variability with individual observations , 2005 .

[33]  Arthur B. Yeh,et al.  Monitoring the covariance matrix via penalized likelihood estimation , 2013 .

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

[35]  Arthur B. Yeh,et al.  Monitoring multivariate process variability with individual observations via penalised likelihood estimation , 2012 .

[36]  Hang Zhang,et al.  Detecting outliers in complex profiles using a χ2 control chart method , 2009 .

[37]  Nedret Billor,et al.  A Distribution-Free Multivariate Phase I Location Control Chart for Subgrouped Data from Elliptical Distributions , 2014, Technometrics.

[38]  P. Rousseeuw,et al.  Breakdown Points of Affine Equivariant Estimators of Multivariate Location and Covariance Matrices , 1991 .

[39]  N. José Alberto Vargas,et al.  Robust Estimation in Multivariate Control Charts for Individual Observations , 2003 .

[40]  William H. Woodall,et al.  Distribution of Hotelling's T2 Statistic Based on the Successive Differences Estimator , 2006 .

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

[42]  Mahmoud A. Mahmoud,et al.  The Inertial Properties of Quality Control Charts , 2005, Technometrics.

[43]  Douglas M. Hawkins,et al.  Self-Starting Multivariate Control Charts for Location and Scale , 2011 .

[44]  Frank B. Alt.  Multivariate Quality Control , 1984 .

[45]  William H. Woodall,et al.  Phase I Analysis of Linear Profiles With Calibration Applications , 2004, Technometrics.

[46]  Charles W. Champ,et al.  A Distribution-Free Phase I Control Chart for Subgroup Scale , 2010 .

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

[48]  Edgard M. Maboudou-Tchao,et al.  Monitoring the covariance matrix with fewer observations than variables , 2013, Comput. Stat. Data Anal..

[49]  Paulo Cortez,et al.  Modeling wine preferences by data mining from physicochemical properties , 2009, Decis. Support Syst..

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

[51]  William H. Woodall,et al.  A Comparison of Multivariate Control Charts for Individual Observations , 1996 .

[52]  Sungzoon Cho,et al.  Multivariate Control Charts Based on Hybrid Novelty Scores , 2014, Commun. Stat. Simul. Comput..

[53]  Gyo-Young Cho,et al.  Multivariate Control Charts for Monitoring the Mean Vector and Covariance Matrix , 2006 .

[54]  Charles W. Champ,et al.  Designing Phase I ―X Charts with Small Sample Sizes , 2004 .

[55]  P. Rousseeuw.  Least Median of Squares Regression , 1984 .

[56]  Connie M. Borror,et al.  Robustness properties of multivariate EWMA control charts , 2003 .

[57]  Regina Y. Liu.  On a Notion of Data Depth Based on Random Simplices , 1990 .

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

[59]  Edgard M. Maboudou-Tchao,et al.  A LASSO Chart for Monitoring the Covariance Matrix , 2013 .

[60]  William H. Woodall,et al.  High breakdown estimation methods for Phase I multivariate control charts , 2007, Qual. Reliab. Eng. Int..

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