Overlay of a batch drying process, 36, 37 with the temperature profile (a) showing different durations (c) for its distinct phases (d) colored by date. A common challenge in manufacturing is that that the duration of each phase varies batch-to-batch. To aid the visualization (b), a simple alignment using the automation triggers (in this case, phase start and end) can be done (d,f). Notice that this simple alignment technique 46 allows the finding of anomalous batches by visual inspection (or density-based analysis, not shown).

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... drying process is a good illustration of how the duration can vary in batch manufacturing processes (see Fig. 2). In the dryer dataset, the main source of variability comes from the batch-to-batch variation in the amount of product loaded and its solvent ...

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... reference batch. The reader is referred to previous literature 40,41,45,47,50,101 for an in-depth explanations of the mathematics behind the DTW algorithm and The main obstacle in the application of DTW is the presence of singularities, which occur when a single point on a variable trajectory maps onto a large subsection of the other time series (Fig. 20). Multiple variants of the original DTW algorithm proposed by Kassidas et al. 47 have sprung ever since, many of them addressing the formation of singularities. In the following, the different variants will be explained and classified based on the modifications made. ...

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... time index Reference time index Query time index some intermediate P value with a desirable tradeoff between reduced time distortion and an acceptable alignment (Fig. 22). The same logic applies to the alignment of a single batch (Fig. 23). A possible way to calculate the optimal P value is presented in Spooner et al. ...

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... time index Reference time index Query time index some intermediate P value with a desirable tradeoff between reduced time distortion and an acceptable alignment (Fig. 22). The same logic applies to the alignment of a single batch (Fig. 23). A possible way to calculate the optimal P value is presented in Spooner et al. ...

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... P increases, it approaches a linear interpolation from the start to the end of the batch (linear warping paths). Figure 22: Sakoe-Chiba local constraint values (P) modify the results of DTW. Batch reference (red) is the one with median duration, that is also on-spec (colormap indicates batch date). ...

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... constraint The main functions of the global constraint are to avoid extreme warpings and to reduce the computational load by reducing the number Euclidean distance calculations. In the original DTW algorithm, arbitrary global constraints such as the ones shown in Fig. 25 were proposed. 47 Relaxed Greedy DTW (rgDTW), on the other hand, proposed setting the upper and lower bounds of the global constraint by taking the maximum and minimum grid cells reached by the optimal warping paths of a set of previous batch runs (Fig. 24). 44 Its main advantages are that the computational load is reduced for online ...

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... In the original DTW algorithm, arbitrary global constraints such as the ones shown in Fig. 25 were proposed. 47 Relaxed Greedy DTW (rgDTW), on the other hand, proposed setting the upper and lower bounds of the global constraint by taking the maximum and minimum grid cells reached by the optimal warping paths of a set of previous batch runs (Fig. 24). 44 Its main advantages are that the computational load is reduced for online use, and an improved alarm rate when used for fault detection, as it reduces the variability in the online warping function every time a new measurement is taken by performing DTW on a moving window. Yet, it does not allow a bad alignment to be revised. Based ...

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... a local constraint (P>0) is applied, the alignment results can become largely insensitive to the variable trajectory pretreatment 108 (Fig. 26). Univariate and multivariate DTW Regarding the number of variables or tags, a single, multiple or all variables can be used for warping. Univariate DTW overfits the variable used for alignment, at the expense of the other variables' alignment, as shown in Fig. 27. On the other hand, multivariate DTW tries to obtain the best alignment ...

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... results can become largely insensitive to the variable trajectory pretreatment 108 (Fig. 26). Univariate and multivariate DTW Regarding the number of variables or tags, a single, multiple or all variables can be used for warping. Univariate DTW overfits the variable used for alignment, at the expense of the other variables' alignment, as shown in Fig. 27. On the other hand, multivariate DTW tries to obtain the best alignment possible for all variables, making it more robust and less prone to singularities, as in the univariate case some automation triggers that hold valuable information for warping may not appear on the variable used for alignment. For the multivariate DTW, all ...

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... DTW Multivariate DTW Figure 27: Comparison of univariate DTW performed using the dryer temperature only and a multivariate DTW, including the tank level, dryer pressure and agitator speed. Colormap indicates batch date. ...

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... weighting 50 and optimal local constraint P 45 can yield the best results in terms of robustness. This involves a certain amount of hyper-parameter tuning in the first place. In case the obtained alignment is not satisfactory, then a multivariate dDTW variant may be combined with a local constraint. 40 DTW does not necessarily align by phase (Fig. 28); rather it warps to obtain a better alignment of the variable trajectories with respect to the reference batch. If information on automation triggers is available, DTW may be performed stage-wise to align both between and within stages. ...