Monitoring Variation in a Multivariate Process When the Dimension is Large Relative to the Sample Size

A control procedure is presented for monitoring changes in variation for a multivariate normal process in a Phase II operation where the subgroup size, m, is less than p, the number of variates. The methodology is based on a form of Wilk' statistic, which can be expressed as a function of the ratio of the determinants of two separate estimates of the covariance matrix. One estimate is based on the historical data set from Phase I and the other is based on an augmented data set including new data obtained in Phase II. The proposed statistic is shown to be distributed as the product of independent beta distributions that can be approximated using either a chi-square or F-distribution. An ARL study of the statistic is presented for a range of conditions for the population covariance matrix. Cases are considered where a p-variate process is being monitored using a sample of m observations per subgroup and m < p. Data from an industrial multivariate process is used to illustrate the proposed technique.