Figure 4.10. Batch data representation and unfolding process. The rows are batches and columns are the variables, Vj, sampled at each time , variable trajectory measurements (X) taken throughout the duration of the batch to help classify a batch as 'good' or 'bad'. MPLS  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 , 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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