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

Abstract

The joint <img src="/img/revistas/prod/2011nahead/aop_t6_0002_0329.jpg" /> and R charts and the joint <img src="/img/revistas/prod/2011nahead/aop_t6_0002_0329.jpg" /> and S² 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 <img src="/img/revistas/prod/2011nahead/aop_t6_0002_0329.jpg" /> and R charts are slightly inferior to the joint <img src="/img/revistas/prod/2011nahead/aop_t6_0002_0329.jpg" /> and S² 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.

Statistical Monitoring of the Covariance Matrix in Multivariate Processes: A Literature Review

“…These ideas were extended to the general multivariate case with p ≥ 2. See Costa and Machado [2011b] and Costa and Machado [2013].…”

Misleading signals in joint schemes for the mean vector and covariance matrix

“…There are fundamentally two types of joint schemes for and . On the one hand, the schemes make use of a single control chart to monitor both parameters; see, for example, Spiring and Cheng, Chen et al, Khoo, Chen and Thaga, Yeh and Lin, Zhang et al On the other hand, the most popular joint schemes that result from running simultaneously two individual charts, one for and another one for , such as the ones discussed by Chen and Thaga, Reynolds and Cho, Hawkins and Maboudou‐Tchao, Machado and Costa, Reynolds and Stoumbos, Zhang and Chang, Costa and Machado, Reynolds and Cho, Ramos et al, Ramos, Ramos et al, Morais et al…”

“…There are fundamentally two types of joint schemes for and . On the one hand, the schemes make use of a single control chart to monitor both parameters; see, for example, Spiring and Cheng, 20 Chen et al, 21 Khoo, 22 Chen and Thaga, 23 Yeh and Lin, 24 Zhang et al 25 On the other hand, the most popular joint schemes that result from running simultaneously two individual charts, one for and another one for , such as the ones discussed by Chen and Thaga, 23 Reynolds and Cho, 26 Hawkins and Maboudou-Tchao, 19 Machado and Costa, 27 Reynolds and Stoumbos, 28 Zhang and Chang, 29 Costa and Machado, 30 Reynolds and Cho, 31 Ramos et al, 32,33 Ramos,34,35 Ramos et al, 36 Morais et al 6 When we use any of these joint schemes, the multivariate quality characteristic is deemed to be out of control whenever a signal is triggered by either individual chart. Thus, a shift in the mean vector can be misinterpreted as a shift in the covariance matrix and vice versa.…”

Cumulative sum control charts for monitoring process mean and/or variance

“…Though a lot of works have been done in this field (Khoo 2004, Costa and Machado 2011, Yang and Rahim 2005, little attention has been received for developing charts that are highly effective for detecting small and moderate shifts in mean and variance for multivariate process (Zhang et al 2010). Thus this should be desirable to develop a single multivariate CUSUM chart to detect process shifts in mean and variance.…”