نمودار حداقل فاصله اقلیدسی بر اساس رگرسیون بردار پشتیبانی برای نظارت به معنای تغییرات همبستگی خودکار فرایندها

Though traditional control charts have been widely used as effective tools in statistical process control (SPC), they are not applicable in many industrial applications where the process variables are highly auto-correlated. In this study, one new minimal Euclidean distance (MED) based monitoring approach is proposed for enhancing the monitoring mean shifts of auto-correlated processes. Support vector regression (SVR) is used to predict the values of a variable in time series. Through calculating minimal Euclidean distance (MED) values over time series, a novel MED chart is developed for monitoring mean shifts, and it can provide a comprehensive and quantitative assessment for the current process state. The performance of the proposed MED control chart is evaluated based on average run length (ARL). Simulation experiments are conducted and one industrial case is illustrated to validate the effectiveness of the developed MED control chart. The analysis results indicate that the developed MED control chart is more effective than other control charts for small process mean shifts in auto-correlated processes, and it can be used as a promising tool for SPC.

The ability to monitor and reduce process variation for cost reduction and quality improvement in industrial processes plays a critical role in the success of one enterprise in today's globally competitive marketplace (Montgomery, 2001, Wu et al., 2007, Du et al., 2008 and Lee et al., 2012). Control charts have been widely used as effective tools in statistical process control (SPC) for monitoring the process variation in industrial applications. In particular, control charts for monitoring independent observations have been extensively investigated and applied (Alt, 1984, Mason et al., 1995, Aparisi and Haro, 2003, Yang and Rahim, 2005, Torng et al., 2009, De Magalhãesa et al., 2009, Wu et al., 2009, Du and Xi, 2010, Ou et al., 2011 and Costa and Machado, 2011). With tremendous growth of advanced automatic data inspection and measurement techniques during the past few years, the process variables are being collected automatically at higher rates, therefore, for many industrial applications, a basic statistical assumption of independence is often violated, i.e. the data collected at regular time intervals from the processes is serially auto-correlated (Montgomery and Friedman, 1989 and Cook and Chiu, 1998). Several attempts have been made to extend traditional SPC techniques to deal with auto-correlated processes. One of the most interesting approaches to SPC for auto-correlated processes was proposed by Alwan and Roberts (1988). They introduced two charts, which they referred to as the common-cause control chart (CCC) and the special-cause control chart (SCC). CCC is a plot of forecasted values that are determined by fitting the correlated process with an autoregressive moving average model (ARMA), and SCC is a traditional Shewhart chart of the residuals. Their work attracted further investigation on time series modeling techniques application for monitoring correlated processes (Montgomery and Friedman, 1989, Montgomery and Mastrangelo, 1991, Wardell et al., 1992, Wardell et al., 1994, Schmid, 1997, Adams and Tseng, 1998, Timmer et al., 1998, Jiang et al., 2000, Wright et al., 2001, Orlando et al., 2002 and Kalgonda and Kulkarni, 2004). The time series based control charts approaches essentially involve fitting an adequate time series model to the correlated process data and applying a traditional control chart to the stream of residuals from the time series model. All these control chart approaches have been shown to improve the monitoring performance in the presence of auto-correlation. However, these time series modeling techniques require that a strict model has been identified for the time series of process observations before residuals can be obtained (Hwarng, 2004), and their performance is not very good for monitoring small shifts (Wardell et al., 1994), and they require one to have some skill in time series analysis (Box et al., 1994). Therefore, some other control charts based on residual have been developed for enhancing the performance of monitoring correlated processes (Testik, 2005 and Pan and Jarrett, 2007). Recently, some researchers tried to find alternative methods that allow less restrictive assumptions, more flexibility and adaptability to real data situations. Examples of such techniques are machine learning methods such as neural network (NN) and support vector machine (SVM). These techniques allow learning the specific structure directly from the data and can be applied without forcing any assumptions. Some authors have proposed NN approaches as effective tools for monitoring auto-correlated processes (Cook and Chiu, 1998, Cook et al., 2001, Zobel et al., 2004, Hwang, 2005, Pacella and Semeraro, 2007, Jamal et al., 2007 and Du and Xi, 2011). Support vector machine (SVM) has recently become a new generation learning system based on recent advances on statistical learning theory for solving a variety of learning, classification and prediction problems (Cortes and Vapnik, 1995, Gunn, 1998, Cristianini and Shawe-Taylor, 2000 and Deng and Yeh, 2011). SVMs calculate a separating hyperplane that maximizes the margin between data classes to produce good generalization abilities. The main difference between NNs and SVMs is in their risk minimization (Gunn, 1998). In case of SVMs, structural risk minimization principle is used to minimize an upper bound based on an expected risk, whereas in NNs, traditional empirical risk minimization is used to minimize the error in the training of data. The difference in risk minimization leads to a better generalization performance for SVMs than NNs (Gunn, 1998). Support vector regression (SVR) is an important extension of SVM and is a regression method by introduction of an alternative loss function (Vapnik, 1998). The applications of SVM to monitor the process variation are spare. Chinnam (2002) demonstrated that SVM can be extremely effective in minimizing both type I and type II errors for detecting shifts in the auto-correlated processes, and performed as well or better than traditional Shewhart control charts and other machine learning methods. Sun and Tsung (2003) and Kumar et al. (2006) developed one kernel-distance based K-chart using support vector for monitoring the independent observations. Issam and Mohamed (2008) presented SVR based cumulative sum (CUSUM) control chart for auto-correlated process. In their approach, SVR is firstly calculated and then CUSUM is applied to the stream of residuals from SVR. Therefore, two methods including SVR and CUSUM need to be calculated in their approach. In this paper, one new minimal Euclidean distance (MED) based control chart is developed as a promising tool for monitoring auto-correlated processes. SVR is used to predict the values of a variable in time series. By using MED, the quantization error is provided for quantifying the deviation degree of current process with in-control process state space. Depending on how far away the current process is deviating from the in-control process state, a quantitative assessment index can be obtained by calculation of the MED of the new measurement data to the SVR trained by in-control process datasets. The rest of this paper is organized as follows. The SVR theory is reviewed briefly in Section 2. The methodology using the proposed MED chart for monitoring auto-correlated processes is developed in Section 3. Experiments and performance analysis of the MED chart are conducted based on average run length (ARL) in Section 4. Further analysis of MED chart and one industrial application is illustrated to validate the effectiveness of the MED chart in 5 and 6, respectively. A procedure for applying the MED chart into real processes is presented in Section 7. Finally, the conclusions and future work are given in Section 8.

Monitoring the mean shifts in auto-correlated processes has been a challenging task for traditional SPC techniques. This study has proposed one new monitoring approach based on SVR for recognizing the mean shifts of auto-correlated processes. The monitoring approach is capable of providing a comprehensible and quantitative assessment value for current process state through calculating the MED values. Based on these MED values over time series, a novel MED control chart is developed to monitor the mean shifts of auto-correlated processes. Some important details of the construction of the MED chart are discussed and analyzed using the simulation experiments. The simulation results indicate that the MED chart shows the improved performance, which outperforms those of some statistics-based charts and the NN-based control scheme for small process mean shifts in auto-correlated processes. Moreover, the influences of some key parameters of SVR upon its performance are analyzed, which aims to find the suitable parameters for constructing the MED chart. The effectiveness of the MED chart is further validated through the datasets from one case, and a general procedure for using the MED chart in the industrial applications is also proposed. Some merits of the proposed MED chart are concluded as follows: the MED chart possesses high performance for detecting the mean shifts of auto-correlated processes immediately based on ARL. The best time to identify a shift in a process is immediately after the shift has occurred since it becomes more difficult to identify shifts as more observations are taken. Early detection makes the root cause of the signal easier to identify, whereas causes of shifts that occurred in the distant past are difficult to identify. Increasing the probability of early detection would result in the fastest rate of continuous quality improvement. Furthermore, just like CUSUM chart, the MED chart also provides some important process information including quantitative assessment values, the starting shift points and the whole tendency state of process state, which are very important for quality engineers to identify the root causes as soon as possible. There are several possible directions for future research. Firstly, further work can focus on finding out the better optimization algorithms to find an optimal subset of the parameters of SVR. Secondly, further investigation can be done to apply the proposed MED chart into monitoring the covariance changes, which is also important in SPC. Thirdly, in the future more industry cases need be collected in order to further validate and improve the MED chart.