Monitoring the mean vector and the covariance ­matrix of multivariate processes with sample means and sample ranges

The joint X and R charts and the joint X and S 2 charts are the most common charts used for monitoring the process mean and dispersion. With the usual sample sizes of 4 and 5, the joint X and R charts are slightly inferior to the joint X and S 2 charts in terms of efficiency in detecting process shifts. In this article, we show that for the multivariate case, the charts based on the standardized sample means and sample ranges (MRMAX chart) or on the standardized sample means and sample variances (MVMAX chart) are similar in terms of efficiency in detecting shifts in the mean vector and/or in the covariance matrix. User’s familiarity with the computation of sample ranges is a point in favor of the MRMAX chart. An example is presented to illustrate the application of the proposed chart.

[1]  Antonio Fernando Branco Costa,et al.  A New Chart for Monitoring the Covariance Matrix of Bivariate Processes , 2008, Commun. Stat. Simul. Comput..

[2]  Antonio Fernando Branco Costa,et al.  Bivariate control charts with double sampling , 2008 .

[3]  Antonio Fernando Branco Costa,et al.  Synthetic control charts with two-stage sampling for monitoring bivariate processes , 2007 .

[4]  Marcela Aparecida Guerreiro Machado,et al.  Gráfico de controle de VMAX para o monitoramento da matriz de covariâncias , 2008 .

[5]  A.B. Dumont,et al.  Quality in Engineering , 1950, Proceedings of the IRE.

[6]  Antonio Fernando Branco Costa,et al.  The double sampling and the EWMA charts based on the sample variances , 2008 .

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

[8]  H. Hotelling,et al.  Multivariate Quality Control , 1947 .

[9]  Maria E. Calzada,et al.  THE ROBUSTNESS OF THE SYNTHETIC CONTROL CHART TO NON-NORMALITY , 2001 .

[10]  Antonio Fernando Branco Costa,et al.  A new chart based on sample variances for monitoring the covariance matrix of multivariate processes , 2009 .

[11]  J. Wolfowitz,et al.  Introduction to the Theory of Statistics. , 1951 .

[12]  Trevor A Spedding,et al.  A Synthetic Control Chart for Detecting Fraction Nonconforming Increases , 2001 .

[13]  Antonio Fernando Branco Costa,et al.  The synthetic control chart based on two sample variances for monitoring the covariance matrix , 2009, Qual. Reliab. Eng. Int..

[14]  Eugenio K. Epprecht,et al.  Monitoring the process mean and variance using a synthetic control chart with two-stage testing , 2009 .

[15]  Zhang Wu,et al.  Design of the sum-of-conforming-run-length control charts , 2001, Eur. J. Oper. Res..

[16]  F. Aparisi,et al.  Statistical properties of the lsi multivariate control chart , 1999 .

[17]  F. Aparisi,et al.  GENERALIZED VARIANCE CHART DESIGN WITH ADAPTIVE SAMPLE SIZES. THE BIVARIATE CASE , 2001 .

[18]  Chung-Ho Chen,et al.  Economic-Statistical Design of Multivariate Control Charts Using Quality Loss Function , 2002 .

[19]  H. Hotelling.  Multivariate Quality Control-illustrated by the air testing of sample bombsights , 1947 .

[20]  Anja Vogler,et al.  An Introduction to Multivariate Statistical Analysis , 2004 .

[21]  A. Costa,et al.  Monitoring bivariate processes , 2009 .

[22]  Marcela A. G. Machado,et al.  Control charts for monitoring the mean vector and the covariance matrix of bivariate processes , 2009 .

[23]  Y. Takemoto,et al.  A study of multivariate $(\bar{X},S)$ control chart based on Kullback–Leibler information , 2005 .

[24]  H. Moskowitz,et al.  Univariate X¯ control charts for individual characteristics in a multinormal model , 2000 .

[25]  Trevor A Spedding,et al.  A Synthetic Control Chart for Detecting Small Shifts in the Process Mean , 2000 .

[26]  Antonio Fernando Branco Costa,et al.  The use of principal components and univariate charts to control multivariate processes , 2008 .

[27]  T. W. Anderson.  An Introduction to Multivariate Statistical Analysis , 1959 .

[28]  Shing I. Chang,et al.  Multivariate EWMA control charts using individual observations for process mean and variance monitoring and diagnosis , 2008 .

[29]  Smiley W. Cheng,et al.  A New Multivariate Control Chart for Monitoring Both Location and Dispersion , 2005 .

[30]  R. Plackett.  An introduction to the theory of statistics , 1972 .

[31]  Trevor A Spedding,et al.  Implementing Synthetic Control Charts , 2000 .

[32]  David S. Stoller,et al.  Contrôle estatístico de qualidade , 1966 .

[33]  Eugenio K. Epprecht,et al.  Synthetic control chart for monitoring the pprocess mean and variance , 2006 .

[34]  Michael B. C. Khoo,et al.  A New Bivariate Control Chart to Monitor the Multivariate Process Mean and Variance Simultaneously , 2004 .

[35]  W. A. Wallis,et al.  Techniques of Statistical Analysis. , 1950 .

[36]  William H. Woodall,et al.  Evaluating and Improving the Synthetic Control Chart , 2002 .