There are vast amount of papers discussing multivariate process monitoring schemes for either location or dispersion of the process, but only some recent papers are re-viewed here. For more related references, see the papers described in the following and the references cited therein.
Among the multivariate monitoring schemes developed under the multivariate normal distribution, the most popular method could be the Hotelling’s T2 control chart. It is widely used for monitoring the location of the process in both Phase I and Phase II analysis. However, the Hotelling’s T2 chart is notorious for its poor power in detecting small location shifts. To get more detecting power for small shifts, Crosier (1988) and Lowry et al. (1992) proposed the multivariate CUSUM and multivariate EWMA control charts, respectively. Mason et al. (2003) pointed out that there would be some special systematic patterns rather than the random pattern in the Hotelling’s T2 chart if some specific conditions occur in the process.
Vargas (2003) and Jensen et al. (2007) proposed their T2 control charts based on the robust estimators of the location and scatter matrix for Phase I applications.
They claimed that the control chart using the minimum volume ellipsoid (MVE) or minimum covariance determinant (MCD) estimator is more powerful in detecting reasonable number of outliers than the regular Hotelling’s T2 control chart. The use of the scatter matrix estimated by successive difference was also discussed in their papers. Later, Williams et al. (2006) derived the asymptotic distribution of the T2 statistic based on the successive-difference estimator of the scatter matrix.
process mean in both Phase I and Phase II analysis. To avoid a long process of collecting in-control data, Hawkins and Maboudou-Tchao (2007) developed a self-starting EWMA procedure to monitor the process mean. Assuming only a few dimensions of the vector shift, Zou and Qiu (2009) proposed an EWMA control chart integrating the LASSO-based testing statistic.
The aforementioned methodologies focus mainly on monitoring the process mean. Nevertheless, the scatter matrix should also be monitored in real appli-cations. For Phase II applications, Yeh et al. (2004) proposed an EWMA control chart based on the likelihood ratio test statistic for comparing the sample covari-ance matrix of the incoming grouped data with that of the reference sample. Yeh et al. (2005) considered the EWMA of XtXt′, where Xt is the observed vector at time t, as the estimator of the scatter matrix, and proposed the control charts based on the entries of the estimated scatter matrix to monitor the variability of the process. Huwang et al. (2007) considered not only the EWMA of XtXt′, but also the EWMA of (Xt− ˆµt)(Xt − ˆµt)′, where ˆµt is the EWMA of Xt, as the estimators of the scatter matrix. The trace of each estimated scatter matrix was utilized to construct a Shewhart-type control chart. Hawkins and Maboudou-Tchao (2008) adopted the XtXt′ version of the scatter matrix estimator described above and applied the Alt’s likelihood ratio statistic (Alt, 1984) to monitor the process dispersion. To gain more power than the usual two-sided test on the scatter matrix, Yen and Shiau (2010) derived the likelihood ratio test statistic for testing one-sided alternative hypothesis of increasing process dispersion and developed a control chart accordingly. Yen et al. (2012) further developed an effective chart for detecting dispersion increase and decrease simultaneously by combining two one-sided charts.
Some authors developed multivariate control charts to monitor the location and dispersion of the process simultaneously. Reynolds and Cho (2006) constructed two T2-type control charts based on the EWMA of each component of Xt and Xt2, respectively (Xt2 refers to the vector of the square of each component of
Xt), then combined the two T2 charts to monitor the mean and scatter matrix simultaneously. Maboudou-Tchao and Hawkins (2011) combined the self-starting monitoring scheme for the mean in Hawkins and Maboudou-Tchao (2007) and the EWMA procedure for the scatter matrix in Hawkins and Maboudou-Tchao (2008) for the same purpose.
The methodologies described above were all developed based on the normality assumption of the observations. However, this assumption is often violated in prac-tice. Stoumbos and Sullivan (2002) and Testik et al. (2003) studied the robustness of the multivariate EWMA control chart. They pointed out that the multivariate EWMA chart is quite robust to normality if one choose a small weighting param-eter λ. But how small λ should be depends on the distribution of the data, which is often difficult to estimate in practical applications. Therefore, distribution-free schemes for process monitoring are definitely in need.
Qiu and Hawkins (2001, 2003) constructed the CUSUM control chart based on the so-called antiranks of vectors, in which the antiranks are the indices of the order statistics. Liu (1995) proposed three control charts, r, Q, and S charts, which can be viewed as the univariate X, ¯X, and CUSUM charts applying on the depth of the multivariate data. Liu et al. (2004) constructed a nonparametric moving average (MA)-chart derived from the notation of data depth for multivariate data. The simplical depth (Liu, 1990) was considered in their methodology. Hamurkaro˘glu et al. (2004) demonstrated the use of the r and Q charts under the Mahalanobis depth (Mahalanobis, 1936).
Qiu (2008) considered an approach involving the log-linear model to construct a CUSUM chart based on the Pearson’s χ2 statistic. Zou and Tsung (2011) pro-posed an EWMA monitoring procedure based on the spatial signs of vectors (in-troduced in Section 2.2.5). To incorporate the information in the multivariate data more than just the multivariate direction, Zou et al. (2012b) proposed a spa-tial rank-based multivariate EWMA control chart. In addition, they incorporated the self-starting procedure into the the proposed monitoring scheme. Boone and
Chakraborti (2012) considered the univariate sign and Wilcoxon sign-rank statis-tics for each component of the multivariate data and constructed the Hotelling’s T2-type control charts based on these distribution-free statistics to monitor the process location.