Monitoring nonlinear profiles using a wavelet-based distribution-free CUSUM chart

Abstract

WDFTC is a wavelet-based distribution-free CUSUM chart for detecting shifts in the mean of a profile with noisy components. Exploiting a discrete wavelet transform (DWT) of the mean in-control profile, WDFTC selects a reduced-dimension vector of the associated DWT components from which the mean in-control profile can be approximated with minimal weighted relative reconstruction error. Based on randomly sampled Phase I (in-control) profiles, the covariance matrix of the corresponding reduced-dimension DWT vectors is estimated using a matrix-regularisation method; then the DWT vectors are aggregated (batched) so that the non-overlapping batch means of the reduced-dimension DWT vectors have manageable covariances. To monitor shifts in the mean profile during Phase II operation, WDFTC computes a Hotelling's T 2-type statistic from successive non-overlapping batch means and applies a CUSUM procedure to those statistics, where the associated control limits are evaluated analytically from the Phase I data. Experimentation with several normal and non-normal test processes revealed that WDFTC was competitive with existing profile-monitoring schemes.