Control Charts Monitoring For General Changes

Control Charts have been widely used to monitor one or some variables in various industrial or service processes by using statistical methods and computing techniques, to improve the quality of production and service. Most control charts are used for mon-itoring location or scale parameters. Although the problem of monitoring the location or scale of a process is important in many applications, on-line monitoring of general changes in an entire distribution are highly desirable, because other distribution char-acteristics, such as the shape, is also important quality indicator. Most control charts assume that the quality of a process can be adequately represented by the distribution of a quality index. While parametric control charts are only useful in certain appli-cations, there is often a lack of enough knowledge about the process distribution. In many cases, there is also a need to monitor variables in start-up or short-run situations, so self-starting control charts are needed.With these situations, we aim to consider the following topics:control charts mon-itoring for process dispersion, control charts monitoring for any distributional changes, multivariate control charts monitoring for general changes using multivariate Smirnov test, and multivariate control charts monitoring for general changes using minimal s-panning tree (MST). We develop some new control charts, with the help of statistical simulation and computation, to tackle these challenging problems. Then we give an introduction, respectively.Most control charts assume that the quality of a process can be adequately rep-resented by the distribution of a quality characteristic and the in-control (IC) and out-of-control (OC) distributions are the same with only differing parameters. While para-metric control charts are only useful in certain applications, there is often a lack of enough knowledge about the process distribution. For example, univariate process data are often assumed to have normal distributions, although it is well recognized that, in many applications, particularly in start-up situations, the underlying process distribu-tion is unknown and not normal, so that statistical properties of commonly used charts, designed to perform best under the normal distribution, could potentially be (high-1y) affected. So nonparametric control charts are needed in such situations. Most of nonparametric charts focus on monitoring process median, but monitoring the process dispersion is also highly desirable. However there are far few nonparametric control charts which can monitor process dispersion. In Chapter1, we develop a new nonpara-metric control chart by integrating a two-sample nonparametric test (Mood1954) into the effective change-point model. Simulation studies show that the proposed method is superior to other nonparametric schemes in monitoring dispersion. As it avoids the need for a lengthy data-gathering step before charting (although it is generally neces-sary and advisable to have about at least20warm-up samples) and it does not require knowledge of the underlying distribution, so the proposed chart is particularly useful in start-up or short-run situations.Most of the nonparametric charts focus on detecting shifts in location parameters. However, monitoring for any change such as the scale and shape in an entire distribution is highly desirable. For example, a distributional change only involving a decrease in s-cale stand for the improvement of process quality, which needs timely learning and gen-eralizing by practitioners. However there are far fewer distribution-free control charts which can detect any distributional change. Zou and Tsung (2010) proposed a chart which incorporates a new goodness-of-fit (GOF) test using the nonparametric likeli-hood ratio into a EWMA chart. Their method is very easy in computation, but leaves a tuning parameter λ to choose. Ross and Adams (2012) proposed a chart which inte-grates the omnibus Kolmogorov-Smirnov and Cramer-von-Mises tests into the change-point model. Both of them can detect more general changes than location shifts. As we known, GOF test can detect any change in distribution. The well-known tests include Kolmogorov-Smirnov, Cramer-von-Mises, and Anderson-Darling tests. Zhang (2002,2006) proposed a new approach of parameterization to construct a general GOF test based on the nonparametric likelihood ratio. It not only generates the foregoing tradi-tional tests but also produces new types of omnibus tests that are generally much more powerful than the old ones. In Chapter2, we develop a new distribution-free control chart by integrating the powerful two-sample nonparametric likelihood ratio GOF test (Zhang (2006)) into the effective change-point model. Simulation studies show that the proposed method is superior to other nonparametric schemes in monitoring any distributional change at most cases. Especially it is much better in monitoring any dis-tributional change involving a decrease in scale. As it avoids the need for a lengthy data-gathering step before charting (although it is generally necessary and advisable to have about19warm-up samples) and it does not require knowledge of the underlying distribution, the proposed chart is particularly useful in start-up or short-run situations. A real-data example shows that it performs quite well in applications.Multivariate statistical process control (MSPC) charts are particularly useful, when there is need to monitor several quality characteristics of a process simultaneously. Most MSPC methods are based on a fundamental assumption that the process data have multinormal distributions. However, it is well recognized that in many applications, the underlying process distribution is unknown and not multinormal, so that the statistical properties of commonly used charts, which were designed to perform best under the normal distribution, could potentially be highly affected. Distribution-free or robust charts may be useful in such situations. The good performance of most MSPC charts is generally based on a large number of historical observations so as to have enough information on the unknown distribution. In many applications, however, we have no many historical observations. The number of IC historical observations used for cali-brating the necessary parameters are often rather small. In such situations, there would be considerable uncertainty in the parameter estimation, which in turn would distort the IC run-length distribution. Self-starting methods, which can handle sequential moni-toring and estimating simultaneously, were developed accordingly. Recently, Zou et al.(2012) developed a multivariate self-starting method based on spatial rank EWMA (S-REWMA) for monitoring location parameters. It has distribution-free properties over a broad class of population models in the sense that the IC run-length distribution is (or is always very close to) the nominal one when the same control limit designed for a multi-normal distribution is used. But their method leaves a tuning parameter λ to choose, and when λ is not small, say λ≥0.05, its IC ARL performance will be unsatisfactory. So their procedure may be only efficient to small or moderate shifts. In Chapter3, we develop a new robust and self-starting multivariate control chart by integrating a two-sample multivariate smirnov test based on multivariate empirical distribution function (MEDF)([?]) into the effective change-point model. Simulation studies show that the proposed method is superior to SREWMA scheme of Zou et al.(2012) in monitoring large process shifts.Distribution-free control charts are useful in statistical process control (SPC) when there is limited knowledge about the underlying process distribution, especially for multivariate observations. As we know, two-sample multivariate smirnov test based on MEDF is not very effective. In Chapter4, we develop a new distribution-free and self-starting multivariate procedure based on minimal spanning tree (MST) in graph theory (Friedman and Rafsky (1979)), which integrates a multivariate two-sample goodness-of-fit (GOF) test based on MST and the change-point model. As expected, simulation results show that our proposed control chart is quite robust to non-normally distribut-ed data, and moreover, it is efficient in detecting process shifts, especially moderate to large shifts, which is one of the main drawbacks of most distribution-free control charts in the literature. As our proposed control chart avoids the need for a lengthy data-gathering step before charting (although it is generally necessary and advisable to collect several warm-up samples) and it does not require knowledge of the underlying distribution, the proposed chart is particularly useful in start-up or short-run situation-s. Comparison results and a real data example show that our proposed chart has great potential for application.In Chapter5, we give the conclusions and related considerations.