Figure 3.5. Multivariate charts for detecting and diagnosing outliers. Variable 3 is glucose feed rate and variable 6 is dissolved oxygen concentration.

to handle missing data is used for estimation of these by restricting the estimates with observed values up to time interval k and the correlation structure of the reference set variables as defined by the loading matrix P of the MPCA model. The procedure followed in this example is based on locating the first outlier, replacing it with a PCA-based estimate and repeating this procedure with the next outlier. Projection of the already known observations made on J variables [Xk(kJx 1)] into the reduced space is performed by calculating the t^ scores (Eq. 3.20). This scores vector tjjfc is then used to predict the next observation set [xfc+i ((k + 1) J x 1)] with the outlier becoming the missing value (old value replaced by a 0) as shown in Eq. 3.21:

Note that P^ is a (kJ x R) matrix having as columns the elements of p-

loading vectors (pr) from all R principal components up to time k just before the outlier is observed. The matrix (P^Pfe) is well conditioned because all pr vectors are orthogonal to each other [435].

All these techniques use data sets that have already been collected. This is acceptable for model development and analysis of completed batches, but it is not satisfactory for real time process monitoring and control activities. Outlier detection during the progress of the batch is more challenging and a compromise must be made between speed and sophistication of the outlier detection technique used. As a first step, simple upper and lower limits for each measured variable may be used to identify major outliers. More sophisticated tools based on multivariate process monitoring techniques can also be used, noting the need to discriminate between real process faults (that may persist for a number of sampling times) and outliers that usually have shorter durations. Signal processing tools such as change detection techniques based on maximum likelihood ratios can be considered for critical variables [45]. Outlier detection and data reconciliation during the

Variable no. 3

Variable no. 6

Variable no. 3

500 1000 1500 Sample No.

Variable no. 3

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