Multiscale SPC using wavelets: Theoretical analysis and properties

Most practical process data contain contributions at multiple scales in time and frequency, but most existing SPC methods are best for detecting events at only one scale. For example, Shewhart charts are best for detecting large, localized changes, while EWMA and CUSUM charts are best for detecting small changes at coarse scales. A multiscale approach for SPC, adaptable to the scale of relevant signal features and developed based on wavelet analysis, detects abnormal events at multiple scales as relatively large wavelet coefficients. Univariate and multivariate multiscale SPC (MSSPC) for detecting abnormal operation are theoretically analyzed, and their properties are compared with existing SPC methods based on their average run lengths. SPC methods are best for detecting features over a narrow range of scales. Their performance can deteriorate rapidly if abnormal features lie outside this limited range. Since in most industrial processes, the nature of abnormal features is not known a priori, MSSPC performs better on average due to its adaptability to the scale of the features and for monitoring autocorrelated measurements since dyadic wavelets decorrelate most stochastic processes. MSSPC with dyadic discretization is appropriate for SPC of highly autocorrelated or nonstationary stochastic processes. If normal measurements are uncorrelated or contain only mild autocorrelation, it is better to use MSSPC with integer or uniformly discretized wavelets. Many existing methods such as MA, EWMA, CUSUM, Shewhart, batch means charts, and their multivariate extensions are special cases of MSSPC.