## SPM of Continuous Processes with Multivariate Statistical Techniques

In traditional quality control of multivariable processes, a number of quality variables are monitored using Shewhart charts [542]. But because of inter-

actions among the variables that cause crosscorrelation, autocorrelation and colinearity, monitoring one variable at a time approach may become misleading and time consuming if the number of variables to be monitored is high. The potential for erroneous interpretations is illustrated in Figure 6.8 such that univariate charts of two quality variables (xi and X2) are constructed separately and depicted as a biplot by aligning one chart perpendicular to the other. The control limits of the two individual Shewhart charts (99 % upper (UCL) and lower (LCL) confidence limits) are now shown as a rectangle. All of the observations are inside the limits, indicating an in-control situation and consequently acceptable product quality. The ellipse represents the control limits for the in-control multivariable process behavior with 99 % confidence. When a customer complains about low-quality product for the batch corresponding to the sample indicated by ® in Figure 6.8, Shewhart charts do not indicate poor product quality but the multivariate limit does. Furthermore, if there are any samples outside the upper left or lower right corners of the Shewhart confidence region, but inside the ellipse, the consumer would not report them as poor quality products. The situation can be explained by inspecting the multivariate plot

of these variables. Given the joint-confidence region defined by the ellipse, any observation that falls out of this region is considered as out-of-control.

Traditional univariate techniques based on a single variable have been reviewed in the previous section. Despite their misleading nature, univariate charts are still used in industry for monitoring multivariable processes. Several multivariate extensions of Shewhart, CUSUM and EWMA have been proposed in the literature [351, 352, 594, 672]. The multivariate perspective helps one to unveil hidden relations that reside in process data and reach correct conclusions about product quality. There is significant motivation to develop a multivariable statistical process monitoring (SPM) framework to detect the existence, magnitude, and time of occurrence of changes that cause the process to deviate from its desired operation.

Biplots are useful when only a few variables are monitored. When the process has a large number of variables, monitoring tools based on projection techniques are more effective. These techniques rely on principal components analysis (PCA) and partial least squares (PLS) introduced in Sections 4.1 and 4.2.4.

Figure 6.9. The multivariate monitoring space, (a) Three dimensional representation, (b) Two dimensional representation.

Figure 6.9. The multivariate monitoring space, (a) Three dimensional representation, (b) Two dimensional representation.

### 6.2.1 SPM of Continuous Processes with PCA

SPM with PCA can be implemented by graphical and numerical tools. Two types of statistics, the statistical distance T2 and the principal components (PC) model residuals (I — PPT)X or squared prediction error (SPE) must be monitored. If a few PCs can describe the data, biplots of PC scores can be used as easy to interpret visual aids. Such biplots can be generated by projecting the data in Figure 6.9 to two dimensional surfaces PC1-PC2, PCi-Error and PC2-Error. Data representing normal operation (NO) and various faults are clustered in different regions, providing the opportunity to diagnose source causes as well [304].

Inspection of many biplots becomes inefficient and difficult to interpret when a large number of PCs are needed to describe the process. Monitoring charts based on squared residuals (SPE) and T2 become more useful. By appending the confidence interval (UCL) to such plots, a multivariate SPM chart as easy to interpret as a Shewhart chart is obtained.

PCA techniques have been used to monitor an LDPE reactor operation [297], high speed polyester film production [635], Tennessee Eastman simulated process [488] and sheet forming processes [508]. Multiscale PCA by using wavelet decomposition has been proposed [38].

6.2.2 SPM of Continuous Processes with PLS

Modern process data acquisition systems generate large amounts of process data, such as temperatures and flow rates. Measurements of process out-

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