Phase I analysis of multivariate profiles based on regression adjustment☆
Introduction
Because of recent progress in sensing and information technologies, automatic data acquisition has become the norm in various industries. Consequently, a large amount of quality-related data of certain processes have become available. Statistical process control (SPC) based on such data is an important component of process monitoring and control. In many applications, the quality of a process is characterized by the relationship between a response variable and one or more explanatory variables. A collection of data points of these variables can be observed at each sampling stage, which can be represented by a curve (i.e., profile). In some calibration applications, the profile can be described adequately by a linear regression model. In other applications, however, more flexible models are necessary in order to describe the profiles properly. An extensive discussion of the related research problems has been given by Woodall, Spitzner, Montgomery, and Gupta (2004).
Profile monitoring has been extensively studied in SPC and several methods have been developed for monitoring linear and nonlinear profile data. Some examples include the use of multivariate control charts for monitoring linear and nonlinear regression coefficients (Kang and Albin, 2000, Mahmoud and Woodall, 2004, Zou et al., 2007, Williams et al., 2007), monitoring methods based on mixed-effect models (Jensen et al., 2008, Paynabar et al., 2012), dimension-reduction techniques (Lada et al., 2002, Ding et al., 2006, Chicken et al., 2009, Paynabar and Jin, 2011, Viveros-Aguilera et al., 2014), and methods for monitoring roundness profiles (Colosimo and Pacella, 2007, Colosimo et al., 2014). Extensive discussion about various research problems on profile monitoring can be found in Woodall, 2007, Noorossana et al., 2011, Qiu, 2014.
Most recent studies concentrated on the situation with a univariate profile that only contains one response variable. Although such profiles can characterize various applications as described in the literature, multivariate functional profiles in which multiple response variables are involved simultaneously may be even more representative of most industrial applications in certain real world practices. When the correlation structure between quality characteristics is ignored and profiles are monitored separately, then misleading results may be expected (c.f., Lowry et al., 1992, Hawkins, 1991 for relevant discussions). However, research on the monitoring and diagnosis of multivariate general profiles is still scanty. Noorossana, Eyvazian, Amiri, and Mahmoud (2010) discussed multivariate linear profile monitoring in Phase I analysis, mainly based on the ordinary least square estimation. Another relevant work is Zou, Ning, and Tsung (2012) which focused on a study of the Phase II method for monitoring a general multivariate linear profile by using the LASSO-based multivariate SPC techniques. More recently, Chou, Chang, and Tsai (2014) developed a process monitoring strategy for monitoring multiple correlated nonlinear profiles. Nevertheless, how to apply conduct Phase I monitoring for general multivariate profiles including nonlinear profiles, still remains a challenge and has not been thoroughly investigated in the literature. In practice, Phase I process control is crucially important to check the stability of historical profile data and to obtain accurate estimates of the baseline model parameters used for Phase II monitoring (Zhang & Albin, 2009). The main objective of this paper is to develop a new nonparametric method for Phase I monitoring of multivariate profile data.
In the literature of nonparametric profile monitoring, the nonparametric profile model (c.f. Zou, Tsung, & Wang, 2008) is usually consideredwhere is the ith sample collected over time, is the jth design point in the ith profile, g is a smooth nonparametric profile, and s are i.i.d. normal random errors with mean 0 and variance . Zou et al. (2008) developed a procedure based on the combination of local linear smooth and traditional SPC charting techniques. Furthermore, to account for the within-profile correlations, Qiu, Zou, and Wang (2010) proposed to use the nonparametric mixed-effects model which allows flexible variance–covariance structures. Recently, Hung, Tsai, Yang, Chuang, and Tseng (2012) introduced a technique called Support Vector Regression to model the profile relationship between the response variable and explanatory variables, while the within-profile correlation is accommodated by using a resampling technique called block bootstrap. Closely related to this idea, Chuang, Hung, and Yang (2013) provided an easy-to-implement and computationally cheaper framework for monitoring nonparametric profiles by taking into account the within-profile correlation.
It is not straightforward to extend (1) to multivariate settings. One technical challenge is how to model such multivariate profiles by taking the correlations between curves into account. Recent and representative work is Soleimani and Noorossana (2014). However, such traditional methods developed for a linear regression may not fully characterize the information from a complex profile datastream where between-curves correlations may be highly related to the covariates (design points). Although certain efforts have been made on the within profile autocorrelation in simple linear profiles or parametric profiles (see Soleimani et al., 2009, Soleimani et al., 2013, Wang and Tamirat, 2014, Khedmati and Niaki, 2015), the complexity of simultaneously handling the between-curves and within-profile correlations still challenges us. Engineering applications that give rise to profile data often lead to correlated error terms. As demonstrated by Qiu et al. (2010), neglecting the within-profile correlation will result in adverse effects on both in-control (IC) and out-of-control (OC) properties of control schemes. The situation may be more serious for multivariate profile processes due to the intricacy of the model and the large number of responses. Moreover, how to integrate an appropriate regression function nonparametric test with classical SPC techniques is not quite straightforward.
In this paper, we try to deal with all the aforementioned challenges, and to resolve the latent issue of existing nonparametric modelling and monitoring methods that are unable to efficiently utilize full information from a multivariate profile process. We start by introducing a manufacturing example taken from an etching process.
Section snippets
A motivating example
We use an industrial etching process example taken from semiconductor manufacturing to illustrate the motivation for this research. The etching chamber is equipped with more than 50 sensors which record the values of several variables with time during a batch. For illustrative purposes, only 7 major variables will be considered. Variable is related to spectral analysis of chamber gas, while to relate to plasma operations. Variables and are the chamber temperature and
Regression-adjustment-based profile monitoring
Our proposed methodology is termed as regression-adjusted-based multivariate profile monitoring (RAMP) hereafter and is described in three parts. In Section 3.1, we introduce a regression-adjusted varying coefficient model to handle the multivariate nonparametric profile monitoring problem. Its model estimation is discussed in Section 3.2. In Section 3.3, we develop a Phase I profile monitoring method based on the proposed model.
Simulation study
First, we study the RAMP’s in-control performance under the empirical false alarm rate. The number and variety of models are too large to allow a comprehensive, all-encompassing comparison. Our goal is to show the effectiveness, robustness and sensitivity of the proposed RAMP method, and thus we only choose certain representative models for illustration. Specifically, is considered and the following five functions are used for :
Example revisited
Here, we revisit the etching profile monitoring case presented in Section 2 and use that example to demonstrate how to implement the proposed scheme step by step in practice. Recall that we have wafers and equally spaced time points of size in this example. Following the proposed procedure, we first set up the model (2) by taking the two environmental variables and as the parametric part and the other five variables as . Hence, five sperate regression-adjusted varying
Concluding remarks
Motivated by a real-data application in semiconductor industries, we develop a new Phase I modelling and monitoring framework based on the regression-adjustment technique and functional principal component analysis (FPCA) for multivariate profile data. The proposed method could effectively find the potential outlying profiles in a historical dataset as shown by numerical and real-data analysis.
There are a number of issues not addressed here that could be the topics of future research. First,
Acknowledgment
The authors would like to thank the three anonymous referees, Area Editor, and Editor for their many helpful comments that have resulted in significant improvements in the article. This research was supported by the NNSF of China Grants 11431006, 11131002, 11371202, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China 201232.
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This manuscript was processed by Area Editor Min Xie.