In most statistical process control (SPC) applications, it is assumed that the quality of a process or product can be adequately represented by the distribution of a univariate quality characteristic or by the general multivariate distribution of a vector consisting of several correlated quality characteristics. In many practical situations, however, the quality of a process or product is better characterized and summarized by a relationship between a response variable and one or more explanatory variables. Thus, at each sampling stage, one observes a collection of data points that can be represented by a curve (or profile). In some calibration applications, the profile can be represented adequately by a simple straight-line model, while in other applications, more complicated models are needed. In this expository paper, we discuss some of the general issues involved in using control charts to monitor such process- and product-quality profiles and review the SPC literature on the topic. We relate this application to functional data analysis and review applications involving linear profiles, nonlinear profiles, and the use of splines and wavelets. We strongly encourage research in profile monitoring and provide some research ideas.
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