Are principal components with estimated parameters feasible in multivariate process control?

Abstract

AbstractPrincipal components have been used in conjunction with Hotelling's T2 chart to monitor multivariate processes. It is known that prohibitively large sample sizes are needed to estimate the process parameters with enough precision to deploy the chart. We investigate whether principal components can be used to reduce the dimensionality of a process so that multivariate process control can be performed using estimated parameters. The chart based on the first k principal components, which we will refer to as the chart, is investigated. Specifically, we explore three research questions in this paper: (1) can the chart with estimated parameters be applied with moderate preliminary sample sizes?, (2) are there situations where the charts are able to detect a shift in the mean vector quicker than the T2 charts?, and (3) is it possible to exploit assumptions about the covariance matrix, such as equal covariances, to improve the performance of the chart. Using simulation, we find that for high dimensions, the chart with estimated parameters can be used to detect shifts in the direction of the first principal component. Otherwise, the chart requires very large preliminary sample sizes. When it is reasonable to assume equal covariances, the number of parameters to be estimated is substantially reduced and the performance of the chart is improved.