# A Novel Framework for Online Remaining Useful Life Prediction of an Industrial Slurry Pump
## 3 Results
### Error Analysis  
The weighted average accuracy of prediction was utilized to assess the efficiency of the proposed and existing methods for estimating the _RUL_ of the slurry pump. The weighted average accuracy of prediction can be calculated using Equation (6) [12].  
\[\text{{Weighted average of accuracy}}=\frac{\omega_{j}*\left(1-\frac{\left|\text{{RUL }}_{A}(x_{j})-\widehat{RUL}\left(x_{j}\right)\right|}{\widehat{RUL}_{A}(x_{j}) }\right)*100}{\sum_{j}\omega_{j}} \tag{6}\]  
where  
\(\omega_{j}=\text{weight of particular operation hours}\)  
\(x_{j}=\text{operation hours}\)  
\(RUL_{A}=\text{actual {RUL}}\) in terms of weights  
\(R\widehat{UL}=\text{predicted {RUL}}\) in terms of weights  
The weights of operation hours \(\omega_{j}\) are directly proportional to the number of operation hours \(x_{j}\)[12]. For estimating the weighted average of accuracy for channel 2 datasets, the weights of all the considered operating hours were calculated, as follows.  
\[w_{258}=258/2094\]  
\(w_{288}=288/2094\)  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Channel 4 Datasets**} \\ \hline  & **No. of Operating** & **Actual _RUL_ with respect to** & **Predicted _RUL_ (Days)** & **Accuracy (\%)** \\ \hline Proposed Method & & & 20.45 & 56.28 \\ \hline Curve-fitting method & 456 & 36.33 & 20.76 & 57.14 \\ \hline NARX & & & Not Applicable & - \\ \hline Proposed Method & & & 22.00 & 60.55 \\ \hline Curve-fitting method & 496 & 36.33 & 22.43 & 61.73 \\ \hline NARX & & & Not Applicable & - \\ \hline Proposed Method & & & 24.16 & 66.50 \\ \hline Curve-fitting method & 565 & 36.33 & 24.38 & 67.10 \\ \hline NARX & & & Not Applicable & - \\ \hline Proposed Method & & & 29.79 & 81.99 \\ \hline Curve-fitting method & 655 & 36.33 & 28.28 & 77.84 \\ \hline NARX & & & Not Applicable & - \\ \hline Proposed Method & & & 31.91 & 87.83 \\ \hline Curve-fitting method & 720 & 36.33 & 31.07 & 85.52 \\ \hline NARX & & & Not Applicable & - \\ \hline Proposed Method & & & 34.95 & 96.20 \\ \hline Curve-fitting method & 824 & 36.33 & 33.58 & 92.43 \\ \hline NARX & & & 34.66 & 95.40 \\ \hline Proposed Method & & & 35.20 & 96.88 \\ \hline Curve-fitting method & 834 & 36.33 & 34.14 & 93.97 \\ \hline NARX & & & 35.04 & 96.44 \\ \hline \hline \end{tabular}
\end{table}
Table 4: _RUL_ predicted by developed and other existing models for channel 4.  
where the denominator 2094 is the sum of all the operating hours.  
The calculated \(\omega_{j}\), \(RUL_{A}\), and \(R\hat{UL}\) were substituted into the equation (6) for obtaining the weighted average of accuracy. The same procedure was revised for channel 4. Table 5 presents the weighted actual and predicted \(RULs\), while the results for the weighted average accuracy of prediction are shown in Table 6.  
In the case of channel 2, it can be observed that when other methods are not even applicable to generate \(RUL\) results, the proposed method still yields its results with 42.15% weighted average accuracy. It should also be noticed that curve fitting and NARX methods are designated as \"Not Applicable\" in Table 6. This is due to the reason that they did not produce any \(RUL\) prediction results, as shown in Table 5. Similarly, in the case of channel 4, NARX is still not applicable. On the other hand, the weighted average accuracy of the curve-fitting method is still less than the proposed method, as depicted in Table 6.