Nonparametric multivariate covariance chart for monitoring individual observations

Parametric and nonparametric multivariate control charts that are proven very useful in monitoring the covariance matrix of multivariate normally or "nearly" normally distributed continuous datasets have been proposed in statistical process control (SPC) literature. However, in many recent practical applications of SPC, the underlying systems or processes are characterised by discrete or a mixture of discrete and continuous multivariate random variables. In such cases, the available multivariate control charts for monitoring the covariance matrix of continuous processes are inadequate. We propose a multivariate nonparametric Shewhart-type chart for monitoring shifts in the covariance matrix of multivariate discrete or mixture of discrete and continuous random variables. The proposed chart first projects the multivariate dataset into Euclidean space. It then uses the Alt's likelihood ratio obtained from the least absolute shrinkage and selection operator estimator that guarantees a well-conditioned estimate of the covariance matrix as the monitoring statistic. The proposed scheme does not require any parametric model assumptions and can be based on any distance measure of choice. It has the advantage of handling multivariate datasets of any type, including discrete, continuous or a mixture of discrete and continuous random variables. It uses the relationships among the process variables to build new variables that capture the dominant structure among the original variables. A bootstrap procedure is employed to obtain the control limit of the proposed chart for a suitable distance-based model through time. Simulation results show the advantage of the proposed chart in monitoring shifts in the covariance matrix. An illustrative example involving monitoring covariance structures of the lapping process in wafer semiconductor manufacturing and diagnosis single-proton emission computed tomography are provided to show the applications of the proposed chart.