# A Novel Framework for Online Remaining Useful Life Prediction of an Industrial Slurry Pump
## 2 Methodology
### Development and Incorporation of the Smart Learning Rate Mechanism  
After the automatic selection of the different data points of HDTs, the required iteration process of prediction was started. The iteration process produced one prediction point in return to one complete iteration run, as discussed above. If the produced prediction point was found to be greater than its preceding data point, i.e., \(\hat{y}_{p+m}\)\(>\)\(\hat{y}_{p+m-1}\), then it was accepted to append into the input vector for the next iteration. The iteration process remained in progress until it produced the acceptable prediction points, i.e., \(\hat{y}_{p+m}\)\(>\)\(\hat{y}_{p+m-1}\). After a few acceptable prediction points, the iteration system produced a prediction point that was less than its preceding data point, i.e., \(\hat{y}_{p+m}\)\(<\)\(\hat{y}_{p+m-1}\). The simulation program considered a \"descending\" prediction point as an outlier or a rejected prediction point and re-iterated the prediction process for that particular time step. The system continued the re-iteration process repeatedly until the resulting prediction point did not arrive to be greater than the earlier prediction point, i.e., \(\hat{y}_{p+m}\)\(>\)\(\hat{y}_{p+m-1}\).  
Since the iteration system was only iterating a maximum of 458 and 834 data points of channel 2 and channel 4 HDTs, respectively, the system was producing too many outliers or rejected values. In some cases, the situation was such that even after 1000 iterations for a  
Figure 6: The architecture of the developed hybrid deep LSTM model.  
particular time step, the condition was still \(\hat{y}_{p+m}<\hat{y}_{p+m-1}\). An investigation was made to dig out the reason for this unwanted phenomenon. After a deep analysis, it came to the authors' knowledge that the base learning rate value helped the developed hybrid LSTM model for a few iterations for obtaining acceptable prediction points. However, later on, it leads the training process to suffer from an under- / overfitting problem for converging towards the global minima of the gradient descent. The concept of learning rate and its role in modeling for reaching towards global minima can be seen in [18,19].  
In order to tackle the frequent situation of \(\hat{y}_{p+m}<\hat{y}_{p+m-1}\), a novel smart learning rate mechanism was developed and incorporated into the iteration system. The developed mechanism was overviewing the whole iteration process. As it observed consecutive 20 rejected values, i.e., \(\hat{y}_{p+m}<\hat{y}_{p+m-1}\), it automatically changed the base learning rate by a multiple factor of standard deviation value of the considered HDT. The proposed learning rate mechanism worked as the \"catalyst\" for the whole prediction process since its triggering was setting the system to again begin to provide acceptable prediction points, i.e., \(\hat{y}_{p+m}>\hat{y}_{p+m-1}\). The concept of accepted and rejected prediction points is depicted in Figure 7. The iteration system can also work without the developed learning rate mechanism, but it is quite possible that it would take many days to produce the required number of acceptable prediction points.  
Figure 7: Accepted or rejected prediction point for becoming the part of the input vector for the next iteration in (**a**) channel 2 and in (**b**) channel 4.