# Relaxed Rule-Based Learning for Automated Predictive Maintenance: Proof of Concept
## 2 Relaxed Separate-and-Conquer Rule Learning approach
\begin{table}
\begin{tabular}{|c|l|c|} \hline
**Alarm** & **Rules** & **Precision** \\ \hline \multirow{3}{*}{1017} & \(\frac{(\%ZP=0)\wedge(\mathit{Ads.HmiVis.EXTR.PRESS.I}\&6.3,170.95)}{(\% ZP=0)\wedge(\mathit{TEMP.E_{1}}\&18573,194.97)}\) & 98\% \\ \cline{2-3}  & \((\mathit{TEMP.E_{1}}\&49.6,50.1))(\(\mathit{TEMP.E_{10}}\&1646,769.1991))\wedge( \mathit{TEMP.ZPC}\&[214,184.8]) & 96\% \\ \cline{2-3}  & \((\mathit{TEMP.Cle}\&49.6,50.1))(\(\mathit{PRESE.Ple}\&[0.08])\wedge(\% E_{1}\&[-33.34,15.1])\) & 97\% \\ \hline \multirow{3}{*}{3167} & \((\mathit{Ads.HmiVis.CENTR.POT.M1B}\&[575,78.3])\wedge(\mathit{TEMP.Cle}\&[217, 42])\wedge(\mathit{TEMP.Ple}\&[26,31])\) & 96\% \\ \cline{2-3}  & \((\mathit{\%E_{21}}\&54.07))(\(\mathit{Ads.HmiVis.EXTR.PRESS.BC}\&[0.26,69.8])\wedge( \mathit{TEMP.Ple}\&[129,20.6])\) & 97\% \\ \cline{2-3}  & \((\mathit{TEMP.E_{1}}\&[170.53,195.02])(\mathit{TEMP.Ple}\&[26,13])\wedge( \mathit{TEMP.E_{1}}\&[180,197.1])\) & 94\% \\ \hline \multirow{3}{*}{3197} & \((\mathit{TEMP.PS_{5}}\&[36.88,37.3])\wedge(\mathit{Ads.HmiVis.EXTR.PRESS.I}\&78,170.8]) & 96\% \\ \cline{2-3}  & \((\mathit{TEMP.PS_{5}}\&[36.8,937.3])\wedge(\mathit{Ads.HmiVis.EXTR.PRESS.I}\&[78,1 70.8])\) & 96\% \\ \hline \multirow{3}{*}{3099} & \((\mathit{\%PS}\&[40.55,100])\wedge(\mathit{Ads.HmiVis.EXTR.PRESS.Je}\&[17, 58.16,19.3])\) & 97\% \\ \cline{2-3}  & \((\mathit{Ads.HmiVis.CENTR.POT.M1B}\&[575,78.3])\wedge(\mathit{\%E_{2}}\&[-81 8.64])\wedge(\% E_{1}\&[-33.34,29.4])\) & 93\% \\ \cline{2-3}  & \((\mathit{Ads.HmiVis.CENTR.POT.M1B}\&[58,78.8])\wedge(\mathit{TEMP.Cle}\&[217, 28.5])\wedge(\% TB\&[0,15.4])\) & 96\% \\ \hline \end{tabular}
\end{table} 
Table 1: Relaxed Separate-and-Conquer (RSC) (RUL = 0 s.)

### RUL prediction. We also test the ability of algorithms to predict the _remaining time to failure (or RUL - remaining useful life)_. In particular, for a time \(t\) seconds, we modify tables \(D_{1},\ldots,D_{5}\) to obtain the respective tables \(D_{1}^{t},\ldots,D_{5}^{t}\) as follows. Take table \(D_{i}\) and set the \(out\) column to 1 in those rows whose timestamp is at most \(t\) seconds before the timestamp of a row having \(out=1\) in \(D_{i}\). The resulting table is \(D_{i}^{t}\). We report experiments with two values of \(t\): 60 and 120, chosen for the sake of demonstration. The resulting rules and the testing results of the respective RSC, DT based and SeCo algorithms are reported in Tables 4-6 for \(RUL=60\) s and in Tables 7-9 for \(RUL=



Then \(FP=f/n\), where \(f\) is the number of such rows of \(N\) which are covered by at least one rule. We obtain \(FP=0.01\%\) for RSC, \(FP=0.2\%\) for DT based rule learner and \(FP=0.1\%\) for SeCo. Also, we perform \(TP\) and \(FP\) calculation on datasets \(D^{60}=D_{1}^{60}\vee\cdots\lor D_{5}^{60}\) and \(D^{120}=D_{1}^{120}\vee\cdots\lor D_{5}^{120}\), and obtain similar results. \(TP\) and \(FP\) calculation are provided in Table 10.

The proposed algorithm outputs rules predicting alarms (outcome 1). There are no rules making negative predictions (absence of the alarm). As a result, there are no false negative predictions. This, in turn, means that those measures that involve false negatives (TN, FN) do not make sense: for example, the accuracy coincides with the precision and the recall becomes equal \(1\).