# Energy-Based Prognosis of the Remaining Useful Life of the Coating Segments in Hot Rolling Mill
## 5 Industrial Case Study and Approach Validation
### Discussion  
Nine Monte Carlo predictions were performed to validate the proposed approach. Monte Carlo experiments are used to provide an interval estimation for an examined random variable, given a specified rate of reliability. Accuracy was considered as the examined variable in this case. Two different types of trailing arms were processed during the periods captured by the available datasets. Within a 99% confidence level, and according to Monte Carlo simulation theory [48], the accuracy of the method is, in general, estimated to be inside the open interval (98.11% and 99.49%). Independence among the MAPE and the maximum allowed embodied energy capacity (\(W_{p}\)) was examined via Pearson's Correlation Coefficient. Covariance among them was measured equal to \(r_{p}\)\(\approx\) 0.3, which interprets that accuracy is independent of which wear level is selected as the segments' exchange point.  
In parallel, prediction confidence estimation via MLE (described in Section 3) was performed at the same time with the Monte Carlo simulations. Plotting the confidence values of the interval RUL prediction during the trailing arms' processing (see Figure 9 that corresponds to dataset 6, with \(x\)-axis indicating the number of produced products and the \(y\)-axis the prediction confidence), the evidence has shown that, in confidence levels between 30% and 50%, the success of the predictions was in a critical point in most cases. \"In critical point\" means that either the prediction interval fails but it is still close on the actual RUL value, or the prediction interval is marginally successful. Prediction confidence increases as the tools get closer to the end of their life. In extreme cases in which prediction confidence remains within the \"moderate\" status area for a very long time after the start of usage, it may be due to huge variations on production rhythm, or it may be implied that the data quality from the measurement system is not sufficient. Predictions with confidence higher than 50% can be taken seriously into account, while \"moderate-level\" confidence predictions should not pass indifferently, either.  
\begin{table}
\begin{tabular}{c} **Prediction** \\ **(100\%-MAPE)** \\ \end{tabular} \\ \hline
1 & 0.59\% & 99.41\% & 6 & 0.83\% & 99.17\% \\
2 & 0.65\% & 99.35\% & 7 & 1.20\% & 98.80\% \\
3 & 2.23\% & 97.77\% & 8 & 2.87\% & 97.13\% \\
4 & 0.90\% & 99.10\% & 9 & 0.90\% & 99.10\% \\
5 & 0.62\% & 99.38\% & & & \\ \hline \hline \end{tabular} MAPE = mean absolute percentage error.  
\end{table}
Table 4: Accuracy results.  
Prediction confidence also depends on the applied tolerance of the constructed forecast interval. A higher tolerance amplitude leads to a higher validity of the predictions, but selecting a very high tolerance value might not make sense, as it will result in too lenient forecasts. In the current case study, the selection of an interval tolerance amplitude of plus/minus 40 pieces (equally an 80 product-long interval) results in having a prediction with a maximum replacement time declination around 1 h, since the processing of each trailing arm lasts about 45 s. This prediction interval is austere enough, so the selection of a higher tolerance amplitude would also be acceptable in the current case.