The use of principal components and univariate charts to control multivariate processes

Abstract

In this article, we evaluate the performance of the 2 T chart based on the principal components (PC chart) and the simultaneous univariate control charts based on the original variables (SU X charts) or based on the principal components (SUPC charts). The main reason to consider the PC chart lies on the dimensionality reduction. However, depending on the disturbance and on the way the original variables are related, the chart is very slow in signaling, except when all variables are negatively correlated and the principal component is wisely selected. Comparing the SU X , the SUPC and the 2 T charts we conclude that the SU X charts (SUPC charts) have a better overall performance when the variables are positively (negatively) correlated. We also develop the expression to obtain the power of two 2 S charts designed for monitoring the covariance matrix. These joint 2 S charts are, in the majority of the cases, more efficient than the generalized variance S chart.Keywords: principal component; simultaneous univariate control charts; multivariate process control. ResumoNeste artigo, avaliamos o desempenho do gráfico de 2 T baseado em componentes principais (gráfico PC) e dos gráficos de controle simultâneos univariados baseados nas variáveis originais (gráfico SU X ) ou baseados em componentes principais (gráfico SUPC). A principal razão para o uso do gráfico PC é a redução de dimensionalidade. Entretanto, dependendo da perturbação e da correlação entre as variáveis originais, o gráfico é lento em sinalizar, exceto quando todas as variáveis são negativamente correlacionadas e a componente principal é adequadamente escolhida. Comparando os gráficos SU X , SUPC e 2 T concluímos que o gráfico SU X (gráfico SUPC) tem um melhor desempenho global quando as variáveis são positivamente (negativamente) correlacionadas. Desenvolvemos também uma expressão para obter o poder de detecção de dois gráficos de 2 S projetados para controlar a matriz de covariâncias. Os gráficos conjuntos de 2 S são, na maioria dos casos, mais eficientes que o gráfico da variância generalizada S .

Advanced multivariate cumulative sum control charts based on principal component method with application

“…The reduction is achieved by transforming the original variables to a new set of variables called principal components (PC). For further importance of the PCM in control charts can be seen in the studies of Jackson, [23][24][25][26] Scranton et al, 27 Machado and Costa, 28 Yue and Liu, 29 Zaman et al, 30 and references therein.…”

Synthetic charts to control bivariate processes with autocorrelated data

“…If the statistical process control demands the monitoring of only two quality characteristics, the practitioner might prefer to work with two X charts, even knowing that the single T 2 chart was designed to control more than one quality characteristic. In comparison with the bivariate T 2 chart, the joint X charts have a better overall performance in signaling changes in the mean vector of correlated variables (Machado & Costa, 2008).…”

Simultaneous Univariate X_bar Charts to Control Bivariate Processes with Autocorrelated Data

“…The approach to overcome this drawback is the simultaneous use of several charts (simultaneous univariate (SU) charts); see Runger and Montgomery and Serel et al . Machado and Costa compared the performance of the T 2 chart with the SU charts in signaling changes in the mean vector of bivariate processes. When the variables are positively correlated, the SU charts have a better overall performance.…”