  Monitoring multichannel profiles has important applications in manufacturing
systems improvement, but it is non-trivial to develop efficient statistical
methods due to two main challenges. First, profiles are high-dimensional
functional data with intrinsic inner- and inter-channel correlations, and one
needs to develop a dimension reduction method that can deal with such intricate
correlations for the purpose of effective monitoring. The second, and probably
more fundamental, challenge is that the functional structure of multi-channel
profiles might change over time, and thus the dimension reduction method should
be able to automatically take into account the potential unknown change. To
tackle these two challenges, we propose a novel thresholded multivariate
principal component analysis (PCA) method for multi-channel profile monitoring.
Our proposed method consists of two steps of dimension reduction: It first
applies the functional PCA to extract a reasonable large number of features
under the normal operational (in-control) state, and then use the
soft-thresholding techniques to further select significant features capturing
profile information in the out-of-control state. The choice of tuning parameter
for soft-thresholding is provided based on asymptotic analysis, and extensive
simulation studies are conducted to illustrate the efficacy of our proposed
thresholded PCA methodology.
