Monitoring charts multivariate

If the process is out-of-control, the next step is to find the source cause of the deviation (fault diagnosis) and then to remedy the situation. Fault diagnosis can be conducted by associating process behavior patterns to specific faults or by relating the process variables that have significant deviations from their expected values to various equipment that can cause such deviations as discussed in Chapter 7. If the latter approach is used, univariate charts provide readily the information about process variables with significant deviation. Since multivariate monitoring charts summarize the information from many process variables, the variables that inflate... 

Multivariate SPM methods with PCs can employ various types of monitoring charts. If only a few PCs can describe the process behavior in a satisfactory manner, biplots could be used as visual aids that are easy to interpret. Such biplots can be generated by projecting the data to two dimensional surfaces as PC versus PC2, PC versus SPE, and PC2-SPE as illustrated in Figure 5.1. 

Multivariate monitoring charts based on Hotelling s statistic (T ) and squared prediction errors SPEx and SPEy) are constructed using the PLS models. Hotelling s statistic for a new independent t vector is [298]... 

P Nomikos and JF MacGregor. Multivariate SPC charts for monitoring batch processes. Technometrics, 37 41-59, 1995. 

Tates AA, Louwerse DJ, Smilde AK, Koot GLM, Berndt H, Monitoring a PVC batch process with multivariate statistical process control charts, Industrial and Engineering Chemistry Research, 1999, 38, 4769 1776. 

Data generated from process analysis techniques are commonly displayed on control charts and the term statistical process control (SPC) is often used to describe the use of such data visualisation [5], As the amount of data available increases, due both to different measurements and greater frequency of measurements, then combinations of different data provide improved methods of monitoring the processes concerned. The procedures and concepts of multivariate SPC incorporating principal component analysis (PCA) and partial least squares (PLS) analysis then become important [6]. The different SPC approaches are all aimed at providing better process control and improved process understanding. 

Univariate and Multivariate Control Charts Univariate charts monitor a single process output, while multivariate charts are developed to monitor simultaneously dependent or independent multiple process outputs. Examples of multivariate control charts include the control chart, for known process variances, and the T control chart, for unknown process variances. 

Normality assumption Most traditional SPC tools are based on the assumption that the process output characteristic is normally distributed, among which Shewhart control charts and multivariate control charts. In some cases, the central limit theorem can be used to justify approximate normality when monitoring means, but in numerous cases normality is an untenable assumption, and one is unwilling to use another parametric model. A number of nonparametric methods are available in these cases. As data availability increases, nonparametric methods seem especially useful in multivariate applications where most methods proposed thus far rely on normality. 

If dependency exists. Hotelling 7 control chart is commonly used procedure for a multivariate process monitoring and controlling. For individual observations, suppose that there are m samples, each of size n=, and p is the number of quality characteristics observed in each sample. Let x be the sample mean vector and S be covariance matrix of these observations. The Hotelling 7 statistic is calculated as = x — (x — x). The phase I limits should be based on a beta distribution. 

Figure 21.9 provides a general comparison of univariate and multivariate SPC techniques (Alt et al., 1998). When two variables, xi and X2, are monitored individually, the two sets of control limits define a rectangular region, as shown in Fig. 21.9. In analogy with Example 21.5, the multivariate control limits define the dark, ellipsoidal region that represents in-control behavior. Figure 21.9 demonstrates that the application of univariate SPC techniques to correlated multivariate data can result in two types of misclassification false alarms and out-of-control conditions that are not detected. The latter type of misclassification occurred at sample 8 for the two Shewhart charts in Fig. 21.8.

Suppose that it is desired to use SPC techniques to monitor p variables, which are correlated and normally distributed. Let x denote the column vector of these p variables, x = col x, X2,..., Jc ]. At each samphng instant, a subgroup of n measurements is made for each variable. The subgroup sample means for the Ath sampling instant can be expressed as a column vector x(k) = col k), X2 k), Xp k) Multivariate control charts are traditionally based on Hotelling s statistic (Montgomery, 2009). 

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