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Article Excerpt
1. Introduction
Control charts are widely used for monitoring process and quality improvement (see, Montgomery (2004)). Most statistical process control techniques assume that consecutive observations from a process are independent and identically distributed (i.i.d) over time. However, with the development of high sampling frequency in the data collection, observations are more likely to be autocorrelated. The Run Length (RL) properties of traditional control charts, such as Shewhart, CUSUM and EWMA, are strongly degraded by data autocorrelation. Thus, there has been considerable interest in recent years in designing procedures for handling autocorrelation. Assuming that the underlying time series model is known, two main approaches have emerged. In the first, the original data are directly monitored using a standard control chart whose control limits are adjusted to account for the autocorrelation (Vasilopou-los and Stamboulis, 1978; Yashchin, 1993; Schimd and Schone 1997; Runger, 2002). The second approach consists in monitoring the forecast errors (residuals) to identify unusual observations. When the time-series model is correctly specified, the residuals are i.i.d with mean zero for an In-Control (IC) process. Consequently, it is possible to use traditional control schemes with well-understood RL properties (Alwan and Roberts, 1988; Harris and Ross, 1991; Montgomery and Mastrangelo, 1991; Wardell et al., 1994; Runger and Willemain, 1995; Runger et al., 1995; Apley and Shi, 1999; Lu and Reynolds, 1999a, 1999b; Shu et al., 2002).
As mentioned above, both approaches rely on accurate process model knowledge. In practice, the structure of dependence and/or the time series parameters have to be estimated on the basis of n observations from an IC process. The typical design procedure consists in controlling the false alarm rate. In the presence of modeling errors, the rate of incorrect signals is a random variable, being a function of the estimated model parameters. Thus, if the fitted model is inaccurate, the control limits of the modified and residual control schemes will fail to provide the desired RL properties. Indeed, much of the recent research that investigates the impact of estimation error, shows that even small errors in parameter estimates can significantly alter the RL characteristics (Adams and Tzeng, 1998; Boyles, 2000; Kramer and Schmid, 2000; Apley, 2002; Apley and Lee, 2003; Testik, 2005; Jensen et al., 2006). Furthermore, the identification of an appropriate time series model is sometimes difficult and requires skill obtained by experience.
Although the adverse impact of model uncertainty on the RL performance is well documented, only a few studies suggest practical guidelines to tackle this issue. A significant contribution to a robust design for dependent data is the pioneering work of Apley (2002) which provided a design method of the EWMA chart for ARMA processes. Since the proposed control limits are a function of the co-variance matrix of the ARMA parameter estimates, the resulting chart is robust to parameter modeling errors. Apley and Lee (2003) also derived an approximate upper onesided confidence interval for the standard deviation of the EWMA control statistic which can be used to widen the control limits by an aumont that depends on the level of model uncertainty and on how conservative is the design practitioner. Testik (2005), following an approach similar to Apley (2002), suggested another method to widen the residual EWMA control limits for a stationary first-order autoregressive process. As a result of incorporating parameter uncertainty, all these control limits are wider than the standard control limits used when models are assumed perfect. Hence, such approaches clearly give some protection against an unacceptably rate of false alarms together with a certain amount of decrease in the EWMA out-of-control performance, as is the case with the more conservative procedure proposed by Apley and Lee (2003). Two drawbacks characterize these methods. First, a key step of these approaches consists in writing the residual EWMA control statistic as the output of a linear filter applied to an ARMA process. Then, approximated closed-form expressions for the standard deviation of the EWMA chart statistic are used to derive the control charts limits. Hence, the design procedure strictly depends on the EWMA chart characteristics and it is not obvious how to extend it to other control charts. Second, only estimation errors are considered assuming a complete knowledge of model structure. However, in practical situations, the order of the model is often unknown and the combined effect of model misspecification and parameter estimation should be addressed in designing and setting up control charts (Jensen et al., 2006).
This paper explores a general design procedure for residual-based control charts in the presence of model uncertainty. This procedure is based on the very mild assumption that the true underlying process allows an autoregressive representation of order infinity with Gaussian innovations. A design approach based on the AR-sieve bootstrap algorithm (Buhlmann, 1998a, 1998b, 2002; Alonso et al., 2002, 2003) is used to take into account the effects of modeling errors. The control limits are computed via stochastic approximation (Ruppert, 1991; Kushner and Yin, 2003) so that a given constraint on the random false alarm rate is satisfied. The proposed design procedure is illustrated for three control charts: the Generalized Likelihood Ratio (GLR) (Willsky and Jones, 1976; Basseville and Nikiforov, 1993; Superville and Adams, 1994; Siegmund and Venkatraman, 1995; Apley and Shi, 1999; Lai, 2001) and the traditional CUSUM and EWMA. We also compare the bootstrap control limits to the control limits suggested by Apley (2002), Apley and Lee (2003) and Testik (2005) for a residual EWMA control chart.
2. Framework
Assume that, when a system is under control, observations are generated by a Gaussian stationary process, [x.sub.t], that allows an autoregressive representation of order infinity, AR([infinity])
[x.sub.t] - [mu] = [[infinity].summation over (j = 1)][[empty set].sub.j]([x.sub.t-j] - [mu]) + [[member of].sub.t], t [member of] Z,
where [mu] = E([x.sub.t]), [[empty set].sub.j] are parameters such that [[SIGMA].sub.j] [[empty set].sub.j.sup.2] < [infinity] and [[member of].sub.t] is an i.i.d. innovation sequence following a Gaussian distribution with E([[member of].sub.t]) = and E([[member of].sub.t.sup.2]) = [[sigma].sup.2]. This class of models includes stationary and invertible autoregressive moving average models. We will denote with [beta] the infinite dimensional parameter vector ([mu], [sigma], [[empty set].sub.1], [[empty set].sub.2], ...). Observe that [beta] completely determines the process probability distribution.
Suppose that a persistent shift in the mean occurs at some unknown time [tau]. Thus, the process data to be monitored are given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Other types of deviations, such as transient shifts or linear drifts, can also be considered. Let
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
be the best mean-squared predictor of [y.sub.t] based upon [y.sub.t-1], ... [y.sub.1] and
[[upsilon].sub.t.sup.2] = E[[([y.sub.t] - [[^.y].sub.t]).sup.2]],
the mean-squared prediction error computed under the hypothesis that t < [tau], i.e., assuming that the process is IC at time t. Since [y.sub.t] is Gaussian, the best predictor is linear and [^.y.sub.t]([beta]) and [[upsilon].sub.t.sup.2]([beta]) can be computed by either the Durbin-Levinson or the innovation algorithms and, when 1 - [[SIGMA].sub.j][[empty].sub.j][z.sup.j] is rational in z, by using a Kalman filter approach. For the computational details see Brockwell and Davies (1996).
Residual control charts are based on the standardized one-step prediction errors:
[a.sub.t]([beta]) = [[y.sub.t] - [[^.y].sub.t]([beta])/[[upsilon].sub.t]([beta])] = [[~.x].sub.t] + [delta][~.[Florin]].sub.[tau]](t), t = 1,2, ...,
where [[~.x].sub.t] and [~.[Florin].sub.[tau]](t) are the outputs of the linear filter defining [a.sub.t]([beta]), when the input is [x.sub.t] and [mu] + 1, respectively....
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