L

5. Compute the sum of squared deviations from T for each variable w(p,p) = E É - T(m,p)

6. Normalize the weight matrix W so that the sum of the weights is equal to the number of variables, p

7. For the first couple of iterations (two or three would be sufficient) use the same chosen reference trajectory R/. Set R/ = T for the subsequent iterations.

8. Go to step 3 for the next iteration.

The change in the weight matrix can also be monitored to see the convergence and to determine consistent variables.

Example Consider a set of unequal/unsynchronized trajectories of 13 variables collected from 55 normal operation runs of fed-batch penicillin fermentation (L — 55, P — 13) at 0.2 h of sampling interval. The durations of each batch trajectory set varies from 418.4 h (2092 observations) to 499.8 h (2499 observations). Batch number 36 of length 445.6 h (2228 observations) is chosen arbitrarily and it was found to be appropriate since its length is close to the median length of 451 h (2255 observations). As a result, the number of batches that will be expanded and the number of batches to be compressed will be close. Prior to DTW implementation, the necessary initial steps explained in Figure 6.31 were taken. Each variable was scaled using averaged means and standard deviations to remove the effects of different engineering units. A low-pass filter was also used to remove white noise from especially off-gas measurements such as CO2 concentration. Figure 6.32 shows profiles of substrate, biomass, and penicillin concentrations

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