Multivariate Control Charts for Individual Observations

TLDR

Multivariate control charts are commonly built using Hotelling’s T² statistic, but when only individual observations are available during startup, conservative F and chi‑square approximations are typically used to set control limits. The study proposes an exact beta‑distribution method for constructing multivariate control limits during startup. The method derives control limits directly from the beta distribution rather than relying on conservative F and chi‑square approximations. An example from the chemical industry shows the beta‑distribution approach outperforms approximate techniques, especially with few subgroups.

Abstract

When p correlated process characteristics are being measured simultaneously, often individual observations are initially collected. The process data are monitored and special causes of variation are identified in order to establish control and to obtain a "clean" reference sample to use as a basis in determining the control limits for future observations. One common method of constructing multivariate control charts is based on Hotelling's T2 statistic. Currently, when a process is in the start-up stage and only individual observations are available, approximate F and chi-square distributions are used to construct the necessary multivariate control limits. These approximations are conservative in this situation. This article presents an exact method, based on the beta distribution, for constructing multivariate control limits at the start-up stage. An example from the chemical industry illustrates that this procedure is an improvement over the approximate techniques, especially when the number of subgroups is small.