Control chart based on likelihood ratio for monitoring linear profiles

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Abstract

A control chart based on the likelihood ratio is proposed for monitoring the linear profiles. The new chart which integrates the EWMA procedure can detect shifts in either the intercept or the slope or the standard deviation, or simultaneously by a single chart which is different from other control charts in literature for linear profiles. The results by Monte Carlo simulation show that our approach has good performance across a wide range of possible shifts. We show that the new method has competitive performance relative to other methods in literature in terms of ARL, and another feature of the new chart is that it can be easily designed. The application of our proposed method is illustrated by a real data example from an optical imaging system.

Introduction

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 quality characteristics. In many practical situations, however, the quality of process or product is characterized and summarized better by a relationship between a response variable and one or more explanatory variables. In particular, there has been recent interest in monitoring processes characterized by simple linear regression profiles. Most of the studies conducted in the monitoring of such linear profiles have been motivated by calibration applications. Mestek et al. (1994), Stover and Brill (1998), Lawless et al. (1999) and Kang and Albin (2000) presented some practical applications in industrial engineering.

Process monitoring mainly using control charts can be seen as a two stage process-Phase I and Phase II (Woodall, 2000). The goal in Phase I is to evaluate the stability of the process and, after dealing with any assignable causes, to estimate the in-control values of the process parameters. In contrast, the main concern in the analysis of Phase II is to quickly detect shifts in the process from the in-control parameter values estimated in Phase I. Different types of statistical methods are appropriate for the two phases, with each type requiring different measures of statistical performance. In Phase I it is important to assess the rate of false signals of a control chart with a given type one error determined by practitioners. In Phase II, the emphasis is on detecting process changes as quickly as possible. That is usually measured by parameters of the run length distribution, where the run length is the number of samples taken before an out-of-control signal is given.

Most of the literature concerned with profile monitoring deals with the Phase II analysis of linear profiles when the underlying in-control model parameters are assumed to be known. Kang and Albin (2000) proposed two control charts for Phase II monitoring of linear profiles. One of these is a multivariate T2 chart and the other is a combination of an exponentially weighted moving average (EWMA) chart and a range (R) chart. Kim et al. (2003) proposed transforming the x values to achieve an average coded value of zero, and a method based on the combination of three EWMA charts was proposed for detecting a shift in the intercept, the slope and the standard deviation. Gupta et al. (2006) compared the performance of two phase II monitoring schemes for linear profiles, the control charting schemes proposed by Croarkin and Varner (1982) and Kim et al. (2003). The simulation study shows that the Croarkin and Varner (1982) method performed poorly compared to the combined control charting scheme of Kim et al. (2003). Recently, Zou et al. (2007) proposed a novel multivariate exponentially weighted moving average scheme for monitoring general linear profiles. They showed that their approach performed better than Kim et al. (2003) for small and moderate shifts.

For Phase I analysis, Kim et al. (2003) suggested replacing the Phase II EWMA charts with Shewhart charts. Mahmoud and Woodall (2004) studied the Phase I method for monitoring the linear profiles. Mahmoud et al. (2007) proposed a change-point method, based on the likelihood ratio statistics, to detect sustained changes in a linear profile data set in Phase I. They concluded that to protect against both kinds of changes, sustained and randomly occurring unsustained shifts, one could employ the change-point method in conjunction with the methods proposed by Mahmoud and Woodall (2004). A discussion about the problems in monitoring linear profiles is given in Woodall et al. (2004). Recently, Jensen et al. (2008) proposed the use of linear mixed models to monitor the linear profiles in order to account for any correlation structure within a profile and Williams et al. (2007) extended the use of the T2 control chart to monitor the coefficients resulting from a parametric nonlinear regression model fit to profile data.

Based on the generalized likelihood ratio test, we propose a new method to detect shifts in the linear profile. Moreover, the comparisons among our proposed method, the multivariate exponentially weighted moving average scheme of Zou et al. (2007) (henceforth referred to as MEWMA) and the combined control chart of Kim et al. (2003) (henceforth referred to as KMW) are carried out in this paper. We compare these three methods in terms of ARL performance under sustained shifts of different magnitudes in the intercept, slope and the error variance.

The rest of this paper is organized as follows. In Section 2, we review the existing two competitive monitoring methods, the MEWMA and KMW charts and present our proposed scheme. We present the proposed chart with VSI feature in Section 3 and compare the monitoring performance of the proposed scheme with those two methods in Section 4. In Section 5, the application of our proposed method is illustrated by a real data example from an optical imaging system. We summarize this paper in Section 6 with some conclusions.

Section snippets

Control chart for linear profiles

Denote by {(xi,yij),i=1,2,,n} the jth random sample collected over time. When the process is in control, the relationship between the response and explanatory variables is assumed to be yij=A0+A1xi+εij,i=1,2,,n, where the εij/σ are independently identically distributed (i.i.d) as a standard normal random variable, and the explanatory variable x is assumed to be fixed at n values. This is usually the case in the practical applications and is consistent with Kang and Albin (2000), Kim et al.

Adding the VSI performance to the proposed ELR chart

The variable sampling interval (VSI) scheme is a known approach to enhance the efficiency of SPC monitoring schemes. In recent years, several modifications have been suggested to improve traditional fixed sampling rate (FSR) policies that provide better performance than conventional charts in the sense of quicker responses to a process change. Among these, adding VSI in a control chart instead of a fixed sampling interval (FSI) is one of the most popular and useful approaches to improve the

Performance comparisons

When evaluating and comparing the performances of static control charts, the ARL performance is considered. This ARL performance is referred to as the zero-state ARL performance. In practice, it may be reasonable to assume that the process starts in control and then shifts at some random time t in the future. For an arbitrarily t>0, the ARL performance of a control chart is called steady-state ARL performance. In this paper, we only tabulate the zero-state ARLs in order to be consistent with 

A real data example

In this section, the application of our proposed ELR chart for monitoring linear profiles is illustrated by a real data example Gupta et al. (2006) used to compare the performance of two phase II monitoring schemes for linear profiles, the control charting schemes proposed by Croarkin and Varner (1982) and Kim et al. (2003). The data set consists of line widths of photo masks reference standards on 10 units (40 measurements) used for monitoring linear calibration profiles of an optical imaging

Conclusions and considerations

In this paper, we propose a new method for detecting shifts in intercept, slope and standard deviation for the linear profiles by using a single chart. The proposed scheme integrates the EWMA procedure with the generalized likelihood ratio statistics. The new chart can be easily designed and constructed. By the simulations, we show that the ELR chart performs similarly to the existing charts in terms of OC ARL. For detecting the standard deviation, our proposed new chart works significantly

Acknowledgments

The authors gratefully acknowledge the constructive comments of a co-editor and two anonymous referees that are very helpful for the improvement of this paper. This research is supported by the NSF of Tianjin Grant 07JCYBJC04300, the NNSF of China Grant 10771107 and 10711120448.

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