Changliang Zou

MONITORING GENERAL LINEAR PROFILES USING MULTIVARIATE EXPONENTIALLY WEIGHTED MOVING AVERAGE SCHEMES

We propose a statistical process control scheme that can be implemented in industrial practice, in which the quality of a process can be characterized by a general linear profile. We start by reviewing the general linear profile model and the existing monitoring methods. Based on this, we propose a novel multivariate exponentially weighted moving average monitoring scheme for such a profile. We introduce two other enhancement features, the variable sampling interval and the parametric diagnostic approach, to further improve the performance of the proposed scheme. Throughout the article, we use a deep reactive ion etching example from semiconductor manufacturing, which has a profile that fits a quadratic polynomial regression model well, to illustrate the implementation of the proposed approach.

CONTROL CHART BASED ON CHANGE-POINT MODEL FOR MONITORING LINEAR PROFILES

A control chart based on the change-point model is proposed that is able to monitor linear profiles whose parameters are unknown but can be estimated from historical data. This chart can detect a shift in either the intercept, slope or standard deviation. Simulation results show that the proposed approach performs well across a range of possible shifts, and that it can be used during the start-up stages of a process. Simple diagnostic aids are also given to estimate the location of the change and to determine which of the parameters has changed.

NONPARAMETRIC PROFILE MONITORING BY MIXED EFFECTS MODELING

In some applications, the quality of a process is characterized by the functional relationship between a response variable and one or more explanatory variables. Profile monitoring is for checking the stability of this relationship over time. Control charts for monitoring nonparametric profiles are useful when the relationship is too complicated to be described parametrically. Most existing control charts in the literature are for monitoring parametric profiles. They require the assumption that within-profile measurements are independent of each other, which is often invalid in practice. This article focuses on nonparametric profile monitoring when within-profile data are correlated. A novel control chart is suggested, which incorporates local linear kernel smoothing into the exponentially weighted moving average (EWMA) control scheme. In this method, within-profile correlation is described by a nonparametric mixed-effects model. Our proposed control chart is fast to compute and convenient to use. Numerical examples show that it works well in various cases. Some technical details are provided in an Appendix available online as supplemental materials.

Multivariate Statistical Process Control Using LASSO

This article develops a new multivariate statistical process control (SPC) methodology based on adapting the LASSO variable selection method to the SPC problem. The LASSO method has the sparsity property of being able to select exactly the set of nonzero regression coefficients in multivariate regression modeling, which is especially useful in cases where the number of nonzero coefficients is small. In multivariate SPC applications, process mean vectors often shift in a small number of components. Our primary goals are to detect such a shift as soon as it occurs and to identify the shifted mean components. Using this connection between the two problems, we propose a LASSO-based multivariate test statistic, and then integrate this statistic into the multivariate EWMA charting scheme for Phase II multivariate process monitoring. We show that this approach balances protection against various shift levels and shift directions, and thus provides an effective tool for multivariate SPC applications. This article has supplementary material online.

Monitoring Profiles Based on Nonparametric Regression Methods

The use statistical process control (SPC) in monitoring and diagnosis of process and product quality profiles remains an important problem in various manufacturing industries. The SPC problem with a nonlinear profile is particularly challenging. This article proposes a novel scheme to monitor changes in both the regression relationship and the variation of the profile online. It integrates the multivariate exponentially weighted moving average procedure with the generalized likelihood ratio test based on nonparametric regression. The proposed scheme not only provides an effective SPC solution to handle nonlinear profiles, which are common in industrial practice, but it also resolves the latent problem in popular parametric monitoring methods of being unable to detect certain types of changes due to a misspecified, out-of-control model. Our simulation results demonstrate the effectiveness and efficiency of the proposed monitoring scheme. In addition, a systematic diagnostic approach is provided to locate the change point of the process and identify the type of change in the profile. Finally, a deep reactive ion-etching example from semiconductor manufacturing is used to illustrate the implementation of the proposed monitoring and diagnostic approach.

Likelihood Ratio-Based Distribution-Free EWMA Control Charts

Nonparametric or distribution-free charts are useful in statistical process control when there is a lack of or limited knowledge about the underlying process distribution. Most existing approaches in the literature are for monitoring location parameters. They may not be effective with a change of distribution over time in many applications. This paper develops a new distribution-free control chart based on the integration of a powerful nonparametric goodness-of-fit test and the exponentially weighted moving-average (EWMA) control scheme. Benefiting from certain desirable properties of the test and the proposed charting statistic, our proposed control chart is computationally fast, convenient to use, and efficient in detecting potential shifts in location, scale, and shape. Thus, it offers robust protection against variation in various underlying distributions. Numerical studies and a real-data example show that the proposed approaches are quite effective in industrial applications, particularly in start-up and short-run situations.

A Multivariate Sign EWMA Control Chart

Nonparametric control charts are useful in statistical process control (SPC) when there is a lack of or limited knowledge about the underlying process distribution, especially when the process measurement is multivariate. This article develops a new multivariate SPC methodology for monitoring location parameters. It is based on adapting a powerful multivariate sign test to online sequential monitoring. The weighted version of the sign test is used to formulate the charting statistic by incorporating the exponentially weighted moving average control (EWMA) scheme, which results in a nonparametric counterpart of the classical multivariate EWMA (MEWMA). It is affine-invariant and has a strictly distribution-free property over a broad class of population models. That is, the in-control (IC) run length distribution can attain (or is always very close to) the nominal one when using the same control limit designed for a multivariate normal distribution. Moreover, when the process distribution comes from the elliptical direction class, the IC average run length can be calculated via a one-dimensional Markov chain model. This control chart possesses some other favorable features: it is fast to compute with a similar computational effort to the MEWMA chart; it is easy to implement because only the multivariate median and the associated transformation matrix need to be specified (estimated) from the historical data before monitoring; it is also very efficient in detecting process shifts, particularly small or moderate shifts when the process distribution is heavy tailed or skewed. Two real-data examples from manufacturing show that it performs quite well in applications. This article has supplementary material online.

A self-starting control chart for linear profiles

A self-starting control chart based on recursive residuals is proposed for monitoring linear profiles when the nominal values of the process parameters are unknown. This chart can detect a shift in the intercept, the slope, or the standard deviation. Because of the good properties of the plot statistics, the proposed chart can be easily designed to match any desired in-control average run length. Simulated results show that our approach has good charting performance across a range of possible shifts when the process parameters are unknown and that it is particularly useful during the start-up stage of a process.

Coordinate-independent sparse sufficient dimension reduction and variable selection

Sufficient dimension reduction (SDR) in regression, which reduces the dimension by replacing original predictors with a minimal set of their linear combinations without loss of information, is very helpful when the number of predictors is large. The standard SDR methods suffer because the estimated linear combinations usually consist of all original predictors, making it difficult to interpret. In this paper, we propose a unified method— coordinate-independent sparse estimation (CISE)-that can simultaneously achieve sparse sufficient dimension reduction and screen out irrelevant and redundant variables efficiently. CISE is subspace oriented in the sense that it incorporates a coordinate-independent penalty term with a broad series of model-based and model-free SDR approaches. This results in a Grassmann manifold optimization problem and a fast algorithm is suggested. Under mild conditions, based on manifold theories and techniques, it can be shown that CISE would perform asymptotically as well as if the true irrelevant predictors were known, which is referred to as the oracle property. Simulation studies and a real-data example demonstrate the effectiveness and efficiency of the proposed approach.

LASSO-based multivariate linear profile monitoring

In many applications of manufacturing and service industries, the quality of a process is characterized by the functional relationship between a response variable and one or more explanatory variables. Profile monitoring is for checking the stability of this relationship over time. In some situations, multiple profiles are required in order to model the quality of a product or process effectively. General multivariate linear profile monitoring is particularly useful in practice due to its simplicity and flexibility. However, in such situations, the existing parametric profile monitoring methods suffer from a drawback in that when the profile parameter dimensionality is large, the detection ability of the procedures commonly used T2-type charting statistics is likely to decline substantially. Moreover, it is also challenging to isolate the type of profile parameter change in such high-dimensional circumstances. These issues actually inherit from those of the conventional multivariate control charts. To resolve these issues, this paper develops a new methodology for monitoring general multivariate linear profiles, including the regression coefficients and profile variation. After examining the connection between the parametric profile monitoring and multivariate statistical process control, we propose to apply a variable-selection-based multivariate control scheme to the transformations of estimated profile parameters. Our proposed control chart is capable of determining the shift direction automatically based on observed profile data. Thus, it offers a balanced protection against various profile shifts. Moreover, the proposed control chart provides an easy but quite effective diagnostic aid. A real-data example from the logistics service shows that it performs quite well in the application.