A Multivariate Process Variability Monitoring Based on Individual Observations

In order to have a better understanding whether or not an additional observation has changed the covariance structure, a new statistic will be introduced. This statistic will be defined as the scatter matrix issued from augmented data set subtracted by that from historical data set. Under normality assumption, the distribution of its Frobenius norm will be derived and, for practical purpose, a chi-square approximation will be presented. This statistic and Wilks' will be used to construct a new procedure for monitoring process variability based on individual observations. The performance of this procedure in providing information about the effect of an additional observation on covariance structure is promising. An industrial application will be presented to illustrate its advantage. . The idea behind the present paper was inspired by the use of Wilks' statistic (1963) for the second scenario. This monitoring procedure was originally introduced by Mason, Chou and Young (2009) and developed in Mason, Chou and Young (2010) in order to identify the quality characteristics that contribute to the out-of-control signal. What makes Wilks's statistic important in this area of industrial application is that it has direct and simple geometrical interpretation and it is easy to implement in practice especially when p is not too large. Based on Wilks' statistic, the effect of an additional observation on covariance structure is measured as the ratio of the scatter matrix determinant issued from a historical data set (HDS) and that issued from the augmented data set (ADS). The latter data set consists of HDS and an additional observation. It is thus proportional to the ratio of the generalized variance (GV) of HDS and that of ADS. Geometrically, see Anderson (2003), it is the ratio of the volume of the p-dimensional parallelotope related to HDS and that related to ADS. Since the covariance structure is absolutely determined by the eigenvalues and eigenvectors of covariance matrix, then the use of Wilks' statistic to detect the effect of an additional observation on covariance structure might be misleading. This is caused by the fact that GV is only the product of all eigenvalues. It might happen then that Wilks' statistic fails to detect that effect whereas actually the covariance structure has changed. To illustrate the situation, it is sufficient to consider two different covariance matrices having the same GV. Let us consider the following two hypothetical covariance matrices 1