## Mvx

1x1x1 1 xJxK

Figure 4.9. Batch process data arrangement and decomposition in three-way array [433].

1x1x1 1 xJxK

Figure 4.9. Batch process data arrangement and decomposition in three-way array [433].

variables and their time variation. A mathematically equivalent unfolding would be to take slices off the side of X and place them down the time axis, which also forms a (J x JK) dimensional matrix. The latter unfolding orders the matrix with the history of each variable kept together while the former orders the matrix with all the measurements taken at the same time kept together. After mean-centering and scaling the unfolded data matrix, PCA is applied. Each of the p, however, is really an unfolded version of the loadings matrix PQ. After vectors p are obtained, Pa can be obtained by reversing the unfolding procedure. Similarly, the three-way array E can be formed by folding the PCA residual matrix E. For the unfolded X:

MPCA explains variation of measured variables about their average trajectories. Subtracting the average trajectory from each variable (accomplished by mean centering the columns of the unfolded matrix X) removes most of the nonlinear behavior of the process (see Figure 4.11). Batch process models, developed based on historical data of batch runs yielding good products, using MPCA provide the foundation to develop statistical process monitoring and quality control systems in Section 6.4.

### 4.5.2 Multiway Partial Least Squares-MPLS

Traditional SPC methods in batch processes are usually limited to end-product quality measurements [598, 614] or to a single variable measured throughout the batch. Most batch processes operate in open loop with respect to product quality variables due to lack of on-line sensors for tracking these variables. Upon completion of the batch, off-line quality measurements are usually made in the laboratory. MPCA makes use of process