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... X(lxKJ)

Figure 6.79. (a) Partitioning of process measurements space and (b) restructuring for online quality prediction framework.

estimate future portions of variable trajectories. However, this technique lacks online prediction of end-of-batch quality in real-time. A two-step integrated modeling approach is proposed to account for online quality prediction [606]. The first step is similar to MPLSV modeling discussed earlier. After reference batch data are aligned using IV technique, batch progress is determined according to percent increments on local batch time (or another IV) so that batches in the reference set are partitioned based on these increments that are chosen arbitrarily such as 10%, 20% of zprecĀ” (Figure 6.79(a)). Each partition of X {IK x J) is rearranged and inserted into matrix X (I x KJ) as shown in Figure 6.79(b). This is similar to transition between MPLSV and MPLSB modeling. The difference is that whenever a partition is rearranged, i.e. some percent of the batch is completed, an MPLSB model is developed between this partial data and the final product quality matrix Y. This gives an opportunity to predict end-of-batch quality on percent progress points reflected by partitions. The number of quality predictions will be equal to the number of partitions in this case.

Example. MPLS-based SPM and quality framework is illustrated using the simulated data set of fed-batch penicillin production presented in Section 6.4.1. Because of the modeling concerns about discontinuity, the data set is preprocessed for partitioning according to process phases. First, the batch/fed-batch switching point is found for each batch and data are divided into two sets as phase 1 and phase 2. Because the third variable (substrate feed) is zero in batch phase, only 13 variables are left in this first set. Data alignment is performed by using the IV technique. Since an indicator variable is not available for the entire batch run, separate indicator variables are selected for each phase. Variable 9, the culture volume decrease, is a good candidate to be chosen as an indicator variable for phase 1. A new variable called 'percent substrate fed' is calculated from variable 3 (substrate feed) and used as an indicator variable for phase 2 data set. This variable is added as the 15th variable to the data set of phase 2. It is assumed that fed-batch phase is completed when 25 L of substrate is added to the fermenter. Data are re-sampled by linear interpolation at each 1 percent completion of volume decrease for phase 1 and at each 0.2 percent of total substrate amount added for phase 2. Data alignment is achieved yielding in equal number of data points in each phase such that the data lengths are K1 = 101 and K'2 = 501, respectively.

MPLS model development stage: Model development includes two stages. In the first stage, an MPLSV model is developed between process variables matrix (unfolded in variable direction) and an indicator variable. This model is used for online SPM purposes. The second stage involves developing predictive MPLSB models between available data partitions matrix (rearranged process variables matrix in batch direction) and end-of-batch quality matrix.

An MPLSV model is developed for phase 1 between autoscaled XI (IK 1 x J1) and the IV vector zl (IK 1 x 1) by using 5 latent variables. The number of latent variables should be chosen large enough to explain most of the information in zl block because the MPLSV model is used to predict batch evolution. Cross validation is used to determine the number of latent variables. XI (IK 1 x J1) can be rearranged into matrix XI (I x A'lJl) to develop an MPLSB model to obtain an estimate of end-of-batch quality at the end of phase 1. Since all K1 measurements of the first phase have been recorded by the beginning of the second phase, there would be no estimation of variable trajectories required and I x KJ partitioning can be used for modeling. Autoscaled XI (IxKlJl) and product quality matrix Y (IxM) are used as predictor and predicted blocks, respectively. Similarly, another MPLSV model is developed for phase 2 between autoscaled X2 (I x K2J2) and IV vector z2 (IK2 x 1).

In the second modeling stage, quality prediction models are developed. To develop the first MPLSB model, data are collected in 50% increment of phase 1 resulting in two data partitions Xjj and Xi,2 (Figure 6.79b). A similar approach is followed in phase 2 for every 20% increase in phase 2 evolution resulting in five data partitions (X2,m n = 1,..., 5). MPLSB

Table 6.14. Explained variance of MPLSB models for online quality prediction

Model no.

X-block

Y-block

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