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

A wavelet-based distribution-free tabular CUSUM chart based on adaptive thresholding, WDFTC_a is designed for rapidly detecting shifts in the mean of a high-dimensional profilewhose 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, 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-type statistics are computed to set the parameters of a CUSUM procedure. To monitor shifts in the mean profile during Phase II (regular) operation, WDFTC_a computes a similar Hotelling's T²-type statistic from successive thresholded batch means of the wavelet-transformed noise vectors using the in-control thresholds; thenWDFTC_a applies the CUSUM procedure to the resulting T²-type statistics. Experimentation with several normal and non-normal test processes revealed that WDFTC_a outperformed existing non-adaptive profile-monitoring schemes.