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Figure 6.39. Landmark registration results for some of the concentration profiles (S: glucose concentration, X: biomass concentration, P: penicillin concentration). Dashed lines represent the locations of mean landmarks.

profiles in the reference batch set are aligned similarly so that the same set of nonlinear warping functions (Figure 6.38(c)) are used to align rest of the variable profiles. Since the same physiological events affect most variables such as concentration profiles, their landmark locations overlap in time (Figure 6.39).

Alignment of variable profiles online in real-time of a new batch using landmark registration

After aligning the reference batch profiles, the necessary information is available for implementing the alignment procedure for a new batch online in real-time. The iterative online landmark estimation procedure described earlier is used in this example. The necessary information from reference batch alignment includes reference landmark locations matrix £m and its mean vector, and the aligned reference set to calculate average profiles. Since a combined landmark location vector is used from different process variables, the corresponding reference profile is used as a comparative template while implementing the online procedure. For instance, the inflection point in biomass concentration profile determines the location of the first

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Figure 6.40. Online landmark estimation results. Dashed lines represent estimated landmark locations.

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Figure 6.40. Online landmark estimation results. Dashed lines represent estimated landmark locations.

landmark in this example. Therefore, the mean biomass concentration profile that is calculated from the aligned reference set is used while searching for an estimated first landmark. Another decision should be made about the landmark search space. For practical purposes, it is chosen as the range of reference landmark locations. The search space is depicted as a band for each landmark in Figure 6.40. The plots on right side of Figure 6.40 represent sum of squared errors (SSE) between the mean and replicate profiles with potential landmarks in search space. A minimum of the SSE indicates the landmark point in the search space that gives the closest match between new batch profile and the template profile. Once the locations of the landmarks are estimated, the rest of the profiles are aligned with respect to estimated landmark(s). The results in Figure 6.40 show that the iterative technique successfully detects landmark locations. □

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