Estimation and Sequential Monitoring of Nonlinear Functional Responses Using Wavelet Shrinkage

Statistical process control (SPC) is widely used in industrial settings to monitor processes for shifts in their distributions. SPC is generally thought of in two distinct phases: Phase I, in which historical data is analyzed in order to establish an in-control process, and Phase II, in which new data is monitored for deviations from the in-control form. Traditionally, SPC had been used to monitor univariate (multivariate) processes for changes in a particular parameter (parameter vector). Recently however, technological advances have resulted in processes in which each observation is actually an n-dimensional functional response (referred to as a profile), where n can be quite large. Additionally, these profiles are often unable to be adequately represented parametrically, making traditional SPC techniques inapplicable. This dissertation starts out by addressing the problem of nonparametric function estimation, which would be used to analyze process data in a Phase-I setting. The translation invariant wavelet estimator (TI) is often used to estimate irregular functions, despite the drawback that it tends to oversmooth jumps. A trimmed translation invariant estimator (TTI) is proposed, of which the TI estimator is a special case. By reducing the point by point variability of the TI estimator, TTI is shown to retain the desirable qualities of TI while improving reconstructions of functions with jumps. Attention is then turned to the Phase-II problem of monitoring sequences of profiles for deviations from in-control. Two profile monitoring schemes are proposed; the first monitors for changes in the noise variance using a likelihood ratio test based on the highest detail level of wavelet coefficients of the observed profile. The second offers a semiparametric test to monitor for changes in both the functional form and noise variance. Both methods make use of wavelet shrinkage in order to distinguish relevant functional information from noise contamination. Different forms of each of these test statistics are proposed and results are compared via Monte Carlo simulation.