Profile Monitoring via Eigenvector Perturbation

Control charts are often used to monitor the quality characteristics of a process over time to ensure undesirable behavior is quickly detected. The escalating complexity of processes we wish to monitor spurs the need for more flexible control charts such as those used in profile monitoring. Additionally, designing a control chart that has an acceptable false alarm rate for a practitioner is a common challenge. Alarm fatigue can occur if the sampling rate is high (say, once a millisecond) and the control chart is calibrated to an average in-control run length ($ARL_0$) of 200 or 370 which is often done in the literature. As alarm fatigue may not just be annoyance but result in detrimental effects to the quality of the product, control chart designers should seek to minimize the false alarm rate. Unfortunately, reducing the false alarm rate typically comes at the cost of detection delay or average out-of-control run length ($ARL_1$). Motivated by recent work on eigenvector perturbation theory, we develop a computationally fast control chart called the Eigenvector Perturbation Control Chart for nonparametric profile monitoring. The control chart monitors the $l_2$ perturbation of the leading eigenvector of a correlation matrix and requires only a sample of known in-control profiles to determine control limits. Through a simulation study we demonstrate that it is able to outperform its competition by achieving an $ARL_1$ close to or equal to 1 even when the control limits result in a large $ARL_0$ on the order of $10^6$. Additionally, non-zero false alarm rates with a change point after $10^4$ in-control observations were only observed in scenarios that are either pathological or truly difficult for a correlation based monitoring scheme.