Improving the detection of unusual observations in high-dimensional settings

Multivariate control charts are used to monitor stochastic processes for changes and unusual observations. Hotelling's T² statistic is calculated for each new observation and an out-of-control signal is issued if it goes beyond the control limits. However, this classical approach becomes unreliable as the number of variables p approaches the number of observations n, and impossible when p exceeds n. In this paper, we devise an improvement to the monitoring procedure in high-dimensional settings. We regularise the covariance matrix to estimate the baseline parameter and incorporate a leave-one-out re-sampling approach to estimate the empirical distribution of future observations. An extensive simulation study demonstrates that the new method outperforms the classical Hotelling T² approach in power, and maintains appropriate false positive rates. We demonstrate the utility of the method using a set of quality control samples collected to monitor a gas chromatography–mass spectrometry apparatus over a period of 67 days.