| Advances in measurement technology have led to an interest in methods for analyzing functional response data, also known as profiles. Profiles are response variables that, rather than taking on a single value, can be considered a function of one or more independent variables. In quality engineering, profiles present challenges for both statistical process monitoring and experimentation because they tend to be high dimensional. High dimensional responses can result in low power tests statistics and may preclude the use of conventional multivariate statistics. Moreover, profile responses can differ at any combination of locations along the independent variable axes, compared to a simple increase or decrease for a single-valued response. This leads to potentially ambiguous interpretation of results and may induce a disparity in the ability to detect differences that occur at only a few points (a local difference) compared to a systematic difference that impacts the entire length of the profile (a global difference). Wavelet-based methods show a strong potential for addressing these challenges. This dissertation presents an overview of wavelets, emphasizing the potential advantages of wavelets for statistical process monitoring applications. Next, the performances of wavelet-based, parametric, and residual control chart methods to quickly detect a range of local and global within-profile change types are compared and contrasted. Finally, four methods are proposed for testing hypotheses about profile differences between treatments. The performance of these methods are compared and an extension to one-way ANOVA is introduced. We conclude that for both profile monitoring and hypothesis testing applications, wavelet-based methods can out-perform other approaches. In addition, wavelet-based statistical methods tend be more robust than competing approaches when the local or global nature of process changes or profile differences are not known a priori. |
