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Figure 6.17. Equalized batch lengths of five batches based on indicator variable technique in fed-batch operation.

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Figure 6.17. Equalized batch lengths of five batches based on indicator variable technique in fed-batch operation.

6.3.2 Dynamic Time Warping

When an appropriate indicator variable does not exist, other techniques can be implemented to synchronize data and equalize batch lengths. Dynamic Time Warping (DTW) technique is one of them. It has its origins in speech recognition. Unsynchronized feature vectors (trajectories) are a common problem in speech recognition [123, 406, 446, 547] since the same word can be uttered in varying intensities and durations by different speakers. Speech recognition systems should have the ability to interpret words independent of speakers [484]. This is analogous to batch process trajectory synchronization problem since similar events that take place in each batch run are required to be matched. DTW is a flexible, deterministic, pattern matching scheme which works with pairs of patterns. The time-varying features within the data are brought into line (synchronized) by time normalization. This process is known as "time warping" since the data patterns are locally translated, compressed, and expanded until similar features in the patterns between reference data and the new data are matched resulting in the same data length as the reference data [252, 533]. Basic description of DTW and different algorithms for implementing it have been reported [252, 406, 533, 485].

One of the pioneering implementations of DTW to bioprocesses was suggested by Gollmer and Posten [197] on the detection of important process events including the onset of new phases during fermentations. They have provided a univariate scheme of DTW for recognition of phases in batch cultivation of S. cerevisiae and detection of faults in fed-batch E. coli cultivations. Another application of DTW (by Kassidas et al. [270]) has focused on batch trajectory synchronization/equalization. They have provided a multivariate DTW framework for both off-line and on-line time alignment and discussed a case study based on polymerization reactor data.

To introduce the DTW theory, consider two sets of multivariate observations, reference set R, with dimensions j x P, and test set T, with dimensions ix P (Eq. 6.48). These sets can be formed from any multivariate observations of fermentation processes (or batch processes in general) where j and i denote the number of observations in R and T, respectively, and P the number of measured variables in both sets as p — 1,2,..., P.

R(j,p) : Reference Set, j = 1,2,... ,M T(i,p) : Test Set, i = 1,2,..., N

R = rip, r2p, ■ ■ ■, rjp,..., r^p T = t\p,t2p, ■ ■ ■ ,tip,... ,t^p (6.48)

Data lengths N and M will not be equal most of the time because the operating time is usually adjusted by the operators to get the desired product quality and yield in response to variations in input properties for each batch run and the randomness caused by complex physiological phenomena inherent in biochemical reactions. This problem could be overcome using linear time alignment and normalization based on linear interpolation or extrapolation techniques. Let i and j be the time indices of the observations in T and R sets, respectively. In linear time normalization, the dissimilarity between T and R for any variable trajectory is simply defined as

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