Figure 4.10. Batch data representation and unfolding process. The rows are batches and columns are the variables, Vj, sampled at each time [433], variable trajectory measurements (X) taken throughout the duration of the batch to help classify a batch as 'good' or 'bad'. MPLS [661] is an extension of PLS that is performed using both process data (X) and the product quality data (Y) to predict final product quality during the batch [434], The unfolding process, mean-centering and scaling issues apply to MPLS technique as well (Section 4.5.1). There is also a Y matrix of quality variables in addition to three-way data matrix, as shown in Figure 4.10. After unfolding this three-way array into two dimensions, the algorithm explained for PLS in Section 4.2.4 is applied to this unfolded three-way array [298, 434],

For batch data, MPLS decomposes the X(J x JK) and Y(I x M) matrices into a summation of A score vectors [t (I x 1), u(I x 1)], loading vectors [p(JK x 1), q(M x 1)], weights w(JK x 1) and model residual matrices E(I x JK), F(I x M). t, u, p, q and w can be combined into T(I x A), U (I x A), P (JK x A), Q (M x A) and W (JK x A) matrices to build matrix form of equations Eq. 4.19 and 4.20 in Section 4.2.4. A denotes the number of latent variables included in the MPLS model.

where T is given by

Transformed (autoscaled) Concentration Profiles of 20 Batches

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