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200 400 600 800 Number of samples

Figure 6.44. Trajectories of process variables of normal (solid curve), a batch with 10% step decrease (Variable 2) (dotted curve), with a small drift (Variable 3) (dashed curve) and with a large drift (Variable 3) (dashed-dotted curve).

200 400 600 800 Number of samples

200 400 600 800 Number of samples

Figure 6.44. Trajectories of process variables of normal (solid curve), a batch with 10% step decrease (Variable 2) (dotted curve), with a small drift (Variable 3) (dashed curve) and with a large drift (Variable 3) (dashed-dotted curve).

Hotelling's T2 [244] and follows the beta distribution. It can also be calculated for each batch as [254]

where the PCA scores t in dimension a have variance Xa (or estimated variance s2 from the scores of the reference set), which is the a-th largest eigenvalue of the scores covariance matrix S. If tables for the beta distribution are not readily available, this distribution can be approximated using Eq. 6.99 [594].

T2 values are calculated throughout the batch. As soon as the batch is complete Eq. 6.100 is applied for each observation at time interval k [435].

T2 values for each time interval k for a new batch can also be calculated similar to Eq. 6.98 as

Dk values follow F-distribution [594]

where A denotes the number of PCs and I the number of batches in the reference set.

Squared Prediction Error (SPE) plot shows large variations and deviations from the normal operation that are not defined by the model. The ith elements of the t-score vectors correspond to the ith batch with respect to the other batches in the database over the entire history of the batch. The P loadings matrices summarize the time variation of the measured variables about their average trajectories. If a new batch is good and consistent with the normal batches (used to develop MPCA model), its scores should fall within the normal range and the 'sum of the squared residuals' (Q-statistic) should be small. The Q statistic for end-of-batch SPM for batch i is written as

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