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Profile monitoring is a relatively new technique in quality control best used where the process data follows a profile (or curve) at each time period. Little work has been done on the monitoring on nonlinear profiles. Previous work has assumed that the measurements within a profile are uncorrelated. To relax this restriction we propose the use of nonlinear mixed models to monitor the nonlinear profiles in order to account for the correlation structure. We evaluate the effectiveness of fitting separate nonlinear regression models to each profile in Phase I control chart applications for data with uncorrelated errors and no random effects. For data with random effects, we compare the effectiveness of charts based on a separate nonlinear regression approach versus those based on a nonlinear mixed model approach. Our proposed approach uses the separate nonlinear regression model fits to obtain a nonlinear mixed model fit. The nonlinear mixed model approach results in charts with good abilities to detect changes in Phase I data and has a simple to calculate control limit.

In some statistical process control applications, the quality of a process or product can be characterized by a relationship between a response variable and one explanatory variable, which is referred to as profile. We give an example here of a profile that can be described using a polynomial model. This example comes from the automotive industry, where one of the most important quality characteristics of an automobile engine is the relationship between the torque produced by an engine and the engine speed in revolutions per minute. We find for this data set that a second-order polynomial works well. In addition, we show that there is autocorrelation within each profile, thus an ordinary least-square method that ignores the autocorrelation is inappropriate. We propose a linear mixed model method as an alternative approach. After the reduction of the data to a series of parameter estimates, we then conduct a step-by-step Phase I analysis of the polynomial profiles monitoring using a T 2 -based procedure to check the stability of the process and whether or not there are outlying profiles. The remaining profiles are used to form the estimated mean vector and variance-covariance matrix to be used in Phase II studies. Finally, a brief discussion is presented to show how one can use these parameters in Phase II.

A goal of Phase I analysis of multivariate data is to identify multivariate outliers and step changes so that the Phase II estimated control limits are sufficiently accurate. High breakdown estimation methods based on the minimum volume ellipsoid (MVE) or the minimum covariance determinant (MCD) are well suited for detecting multivariate outliers in data. As a result of the inherent difficulties in their computation, many algorithms have been proposed to detect multivariate outliers. Due to their availability in standard software packages, we consider the subsampling algorithm to obtain the MVE estimators and the FAST-MCD algorithm to obtain the MCD estimators. Previous studies have not clearly determined which of these two available estimation methods is best for control chart applications. The comprehensive simulation study presented in this paper gives guidance for the correct use of each estimator. Control limits are provided. High breakdown estimation methods based on the MCD and MVE approaches can be applied to a wide variety of multivariate quality control data.