## 6 Incremental Current Prediction Model
### Evaluation  
This last experiment serves to quantify the added value of the incremental grey-box model in a realistic setting. As such, we predict all welds in chronological order, as would be done in a production setting. We assume all welds in the first maintenance period (4679 welds) are available for training purposes and perform a warm-up run for all models by predicting these welds. We only report errors for the welds of periods 2 (9942 welds) and 3 (5289 welds). All model parameters (except \(\alpha\)) of the incremental grey-box model were determined during training. After predicting a weld, \(R_{mach}\) was updated using the actual measured weld current. We compare against two incremental versions of the average-based model and an oracle model. The first average-based model (Avg) predicts the welding current as the average of the last \(N\) measured welds in the same welding program. The second model (Avg ES) works similarly to the first but uses exponential smoothing to update the prediction value once the model has \(N\) historic values for the welding program. We only use exponential smoothing after \(N\) values to minimize the effect of noise on the first predictions. Finally, the oracle model is a non-updating version of the physics model that was trained on all of the data of the three periods and tracked a different resistance for each period. This means that the oracle model is evaluated on the same welds it was trained on. For all models, we calculate the MAP error for all welding currents in periods 2 and 3. To estimate how quickly each model adapts, we also calculate the MAP error when ignoring a predefined number of welds after either transition. Table 3 lists the prediction errors and gives several interesting insights. Looking at both variants of the average-based model, we see better performance when the model can adapt more quickly. This is demonstrated in the first variant (Avg) for the lower window size \(N\) and in the second variant (Avg ES) for the higher exponential smoothing factor \(\alpha\). Furthermore, we see that all average-based models perform better if we exclude more welds after a transition. This shows that the statistical models would need several days to adjust to the changed behavior. Looking at the grey-box model, we see better predictions for faster reactivity (higher \(\alpha\)) but do not see a major improvement beyond skipping 50 to 250 welds. This demonstrates the faster update speed of the incremental model over the average model. The grey-box model clearly outperforms both variants of the average-based model and even the oraclemodel. The former can be attributed to the shared parameters in the grey-box model, which adjusts the model for all welding programs, whereas the average model has to update each program independently. The latter can be attributed to the fact that the oracle assumes the absence of local trends. To visualize performance over time, we again define rejected welds as those with a prediction error greater than 3% and plot the total amount of rejected welds over time in Figure 4. We see that despite major differences in the slopes of all models, some general patterns emerge across models, meaning there is a degree of consensus. Overall, the average-based models have more rejected welds. Noticeably, both transitions are followed by a sudden increase in rejections as the average-based models have to update all welding programs. In contrast, the incremental models exhibit only minor jumps after either transition that further diminish for the more adaptive models. Note that small jump artifacts actually may be desired by the operator as a way to notice a behavioral change of the system. In this way, a series of rejected welds over a short period could instigate a further investigation. All findings point to the presence of both minor and major variations over time of the welding current that is related to the welding machine. The incremental grey-box model is significantly better at keeping up with these changes.