# Probabilistic Condition Monitoring of Azimuth Thrusters Based on Acceleration Measurements
## 2 Materials and Methods
### Condition Monitoring Algorithm
#### 2.2.3 Residual Calculation  
The use of residuals of linear regression improves the applicability of the proposed method in varying shaft speeds because the strong correlation between the shaft speed and feature values is removed. This is demonstrated in Table 4 with correlation coefficients [37] calculated from the data of thruster 1. The residuals (\(r_{i}\)) of linear regression can be defined by  
\[r_{i}=y_{i}-(\beta_{0}+\beta_{1}x_{i}), \tag{3}\]  
Figure 4: Amplitude spectra of thruster 1 with shaft speed 346.96 rpm. The upper plot shows a case without reported damage and the lower plot shows a case where the inner race of pinion shaft support bearing was damaged. The median and upper quartile of the amplitudes of 1–10 harmonics of BPFI are shown.  
Figure 5: Amplitude spectra of thruster 2 with shaft rotational speed 394.13 rpm. The upper plot shows a case without reported damage and the lower plot shows a case where wear on pinion and wheel gear surfaces was confirmed. The median and upper quartile of 20 amplitudes including the values of 1–4 harmonics and sidebands of gear mesh frequency are shown.  
where \(y_{i}\) is the calculated feature value, and \(\beta_{0}\) and \(\beta_{1}\) are the intercept and the regression coefficient of the linear regression model, identified using the least squares fitting. The predictor \(x_{i}\) is the shaft speed and \(i\) is the sample number. The residual calculation is done in the \"speed range selection\" and \"inference\" processes in the flowcharts shown in Figure 2.  
As was mentioned in Section 2.2.2, the nature of the selected feature has a significant impact on the distribution of its residual when the linear model is applied. Figure 6 demonstrates the effect of selected quantile on the distribution of data when different quantiles were calculated from 1-10 BPFI harmonics. The graphs show that the tails of the distribution moved further away from the line indicating normal distribution when high quantiles such as upper quartile or maximum were used. Additionally, the \(p\)-values of Kolmogorov-Smirnov test [41] decreased when the higher quantiles were used.  
In this study, the median was selected as the quantile for the features computed from the amplitude spectrum (features no. 4-11) because the associated residual sometimes follows the normal distribution better than the residuals associated with the higher quantiles. However, this is not a general rule, and therefore, the check for normal distribution is included in the selection of the operational area, which is introduced in the following section.  
The application of non-linear models for regression is beyond the scope of this study. Such models could improve the fit of the model, but there is a risk that the identified model becomes complicated and overfitted reducing its robustness in later use. The training of non-linear models would benefit from large data sets, which are often unavailable.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**No.** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** \\ \hline Features & 0.75 & 0.08 & \(-\)0.14 & 0.69 & 0.32 & 0.67 & 0.83 & 0.83 & 0.71 & 0.61 & 0.36 \\ Residuals & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Correlation coefficients of shaft speed with feature values and residuals computed from the identification data of thruster 1.  
Figure 6: Normal probability plots demonstrating the effect of quantiles in the distribution of residual values. The plot on the left shows the residual of feature no. 4 (median), whereas the upper quartile is shown in the middle and maximum is shown on the right. The parameter H (0 or 1) is the hypothesis result from Kolmogorov-Smirnov test at the 5% significance level.