Monitoring nonlinear profiles adaptively with a wavelet-based distribution-free CUSUM chart

A wavelet-based distribution-free tabular CUSUM chart based on adaptive thresholding, WDFTCa$ \text{ WDFTC}_a $ is designed for rapidly detecting shifts in the mean of a high-dimensional profile whose noise components have a continuous nonsingular multivariate distribution. First computing a discrete wavelet transform of the noise vectors for randomly sampled Phase I (in-control) profiles, WDFTCa$ \text{ WDFTC}_a $ uses a matrix-regularization method to estimate the covariance matrix of the wavelet-transformed noise vectors; then, those vectors are aggregated (batched) so that the non-overlapping batch means of the wavelet-transformed noise vectors have manageable covariances. Lower and upper in-control thresholds are computed for the resulting batch means of the wavelet-transformed noise vectors using the associated marginal Cornish–Fisher expansions that have been suitably adjusted for between-component correlations. From the thresholded batch means of the wavelet-transformed noise vectors, Hotelling’s T2$ T^2 $-type statistics are computed to set the parameters of a CUSUM procedure. To monitor shifts in the mean profile during Phase II (regular) operation, WDFTCa$ \text{ WDFTC}_a $ computes a similar Hotelling’s T2$ T^2 $-type statistic from successive thresholded batch means of the wavelet-transformed noise vectors using the in-control thresholds; then WDFTCa$ \text{ WDFTC}_a $ applies the CUSUM procedure to the resulting T2$ T^2 $-type statistics. Experimentation with several normal and non-normal test processes revealed that WDFTCa$ \text{ WDFTC}_a $ outperformed existing non-adaptive profile-monitoring schemes.