## 4 Results and Discussion
### Random Forest Regression Model for Reliability Prediction  
The linear regression model has good predictive ability but uses hourly average data from operating time periods since traditional linear regression models have limitations in the use of large databases. To predict the reliability and remaining useful life in real time, it is necessary to create a model that uses instantaneous data from sensor readings. The node layout for the entire model creation process is presented in Figure 14. The first node, identified as Node 18, is responsible for reading the spreadsheet containing the operational data table; then, Node 19 partitions the data into the training set and validation set with a 30%/70% split. The model is created at the node labeled Node 23 using all 12 sensor readings as predictors. Then, on Node 24, using the random partition of 70% of the data as inputs, the reliability prediction is obtained. The evaluation of the model is performed by the statistics obtained from Node 22, and the export of the predicted values is carried out for further analysis at Node 25. The model obtained was evaluated in relation to the metrics presented in Table 6, in which a reasonable R\({}^{2}\) value was obtained. The model can explain 63.7% of the variation in reliability from the sensor readings used as predictors. The error values are also reasonable for use in prediction. The model is also evaluated in terms of residual distribution, as presented in Figure 15, where an error histogram is displayed. As expected, Figure 16 shows a strong relationship \begin{table}
\begin{tabular}{c c} \hline \hline
**Metric** & **Result** \\ \hline R\({}^{2}\) & 0.637 \\ Mean Absolute Error & 0.042 \\ Mean Square Error & 0.003 \\ Square Root of Mean Square Error & 0.056 \\ Average Difference & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Random forest regression model evaluation. Figure 14: Node layout for creating a random forest regression model. Arrows indicate the data flow from nodes; each node ha its inputs on the left and outputs on the right. The square end line indicates that the model created at Node 23 is used as an input at Node 24 for random forest prediction. between the actual reliability values and those predicted by the model, with a correlation coefficient of 0.822. The graph presented in Figure 17 shows a comparison of the estimated reliability and the fitted values. As indicated in Figure 13, although some confidence intervals contain values higher than 100%, they have no meaning in real-world contexts. Figure 16: Scatter plot of reliability versus predicted reliability. Figure 17: Comparison of reliability versus values predicted by the random forest regression model. Y-axis ranges from 30% to 120% for improved visualization only. Figure 15: Histogram of the model residuals in relation to the estimated reliability. From the reliability estimated by the random forest regression model, a new expression is obtained for the remaining useful life of the ceramic plates. Figure 18 shows a comparison of the results of the analytical reliability model with the results of the fitted model. For the interval of hours evaluated, it is reasonable to adopt linear models to represent the behavior of the remaining useful life, according to the models and their respective R\({}^{2}\) values presented in Figure 18. It should be noted, however, that the models do not have the same slope or the same starting point of reliability at 0 hours operated. The linear models of the remaining useful life as a function of time, RUL(t), allow, however, a comparison of the regressions and to stablish an adjustment to the e analytical expression. An analysis of the best combination of both the coefficient multiplied by the constant and the slope coefficient is performed to minimize the square root of the mean squared error (RQSME). A contour plot of this analysis is presented in Figure 19, in which the best combination of values is identified as a constant multiplier index of 1.75 and a linear coefficient multiplier index of 2.54.