Discovering the Nature of Variation in Nonlinear Profile Data

Profile data have received substantial attention in the quality control literature. Most of the prior work has focused on the profile monitoring problem of detecting sudden changes in the characteristics of the profiles, relative to an in-control sample set of profiles. In this paper we present an approach for exploratory analysis of a sample of profiles, the purpose of which is to discover the nature of any profile-to-profile variation that is present over the sample. This is especially challenging in big data environments in which the sample consists of a stream of high dimensional profiles, such as image or point cloud data. We use manifold learning methods to find a low-dimensional representation of the variation, followed by a supervised learning step to map the low-dimensional representation back into the profile space. The mapping can be used for graphical animation and visualization of the nature of the variation, in order to facilitate root cause diagnosis. Although this mapping is related to a nonlinear mixed model sometimes used in profile monitoring, rather than assuming some prespecified parametric profile model and monitoring for variation in those specific parameters, our focus is on discovering an appropriate characterization of the profile-to-profile variation. We illustrate with two examples and include an additional example in the online supplement to this article on the Technometrics website.