## Multivariate Statistical Paradigms for Batch Process Modeling

Data collected from batch or fed-batch processes have a three-dimensional structure. This array is different than the two-dimensional structure (variables x time) resulting from continuous process data. The dimensions in batch data are batch runs, variables and time (Figure 4.9). Data are arranged into a three-dimensional array (I x J x K) where I is the number of batches, J is the number of variables and the K is the number of sampling times in a given batch. Consequently, the PCA and PLS based methods discussed in Sections 4.1 and 4.2.4 must be modified to handle three-dimensional data.

### 4.5.1 Multiway Principal Component Analysis-MPCA

MPCA is based on PCA . It is equivalent to performing ordinary PCA on a large two-dimensional matrix constructed by unfolding the three-way array. The use of MPCA for batch process monitoring was proposed in mid 1990s  and applied to monitor a polymerization reactor. It has been extended to nonlinear PCA  and wavelet decomposition based multiscale PCA techniques for analysis of batch process data .

Batch process data are arranged into a three-dimensional array X The underbar is used to denote a three dimensional matrix. MPCA decomposes the X array into a summation of the product of score vectors ta and loading matrices Pa, plus a residuals array E that is minimized in a least-squares sense, as where ® is the Kronecker product (X = t ® P is X(h 3, k) — t(i)P(j> k)) and A is the number of principal components (PC) retained .

This three-way array is unfolded and scaled properly prior to MPCA (Figure 4.10). Unfolding of three-way array X can be performed in six possible ways. For instance, X can be unfolded to put each of its vertical slices (I x J) side by side to the right, starting with the slice corresponding to the first time interval. The resulting two-dimensional matrix has dimensions (I x JK). This particular unfolding allows one to analyze variability among the batches in X by summarizing information in the data with respect to a a a=1