Distribution-free multivariate time-series monitoring with analytically determined control limits

Di Liu, Heeseon Kim, Seong-Hee Kim, Taeheung Kim, Dongki Lee and Yao Xie

International Journal of Production Research, 2023, vol. 61, issue 20, 6960-6977

Abstract: We consider the problem of detecting a shift in the mean of a multivariate time-series process with general marginal distributions and general cross- and auto-correlation structures. We propose a distribution-free monitoring procedure that does not need model fitting or trial-and-error calibration for control limits, which makes the procedure convenient to be implemented when a facility consists of many processes to be monitored. The main idea is to convert each observation vector into a one-dimensional $ T^2 $ T2 quantity that captures cross-correlation. The $ T^2 $ T2 quantities form a univariate auto-correlated process, and CUSUM statistics are constructed on the $ T^2 $ T2 quantities. Then using the fact that the CUSUM statistics on the auto-correlated process behave as a reflected Brownian motion asymptotically under some conditions, the control limits of the CUSUM procedure are analytically determined by setting the first-passage time of the Brownian motion equal to a target in-control average run length. We compare the performance of our procedure with three competing procedures on simulated data with various cross- and auto-correlation and real data from a wafer etching process. The proposed procedure delivers actual in-control average run lengths close to the target and shows comparable or better performance in detecting a shift in mean than the competitors.

Date: 2023
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DOI: 10.1080/00207543.2022.2140364

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