# Probabilistic Condition Monitoring of Azimuth Thrusters Based on Acceleration Measurements
## 2 Materials and Methods
### Condition Monitoring Algorithm
#### 2.2.2 Feature Extraction
The kurtosis is a shape indicator for the distribution of the signal and it is sometimes used in the detection of defect-related impulses in the signal [24]. On the other hand, the generalized norm is sensitive to both the signal amplitude changes and to shock-like effects [38]. An additional characteristic of interest is the ratio between the shock-like effects and the general amplitude level, such as the ratio between a high-order norm and a low-order norm. Therefore, three features were selected for the time domain model and they are presented in Table 3 as features no. 1-3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Feature** & **Details** \\ \hline
1 & Generalized norm (\(l_{10}\)) & Order of norm, \(p=10\) \\
2 & Ratio of norms (\(l_{20}/l_{2}\)) & Ratio of high-order norm (\(p=20\)) to low-order norm (\(p=2\)) \\
3 & Kurtosis & Indicator for the tails of probability distribution \\
4 & BPFI feature & Median amplitude of 1–10 BPFI harmonics \\
5 & BPFO feature & Median amplitude of 1–10 BPFO harmonics \\
6 & BSF feature & Median amplitude of [1, 2, 4, 6] BSF harmonics \\
7 & BPFI sideband feature & Median amplitude of the nearest sidebands on both sides of BPFI harmonics (20 sidebands \\  & & altogether, spaced at shaft rotational frequency) \\
8 & BSF sideband feature & Median amplitude of the nearest sidebands on both sides of BSF harmonics (8 sidebands \\  & & altogether, spaced at FTF) \\
9 & GMF feature 1 & Median amplitude of 1–4 GMF harmonics and two nearest sidebands on both sides (20 frequency components altogether) \\
10 & GMF feature 2 & Median amplitude of 1 \(\times\) GMF and two nearest sidebands on both sides (5 frequency components altogether) \\
11 & GMF feature 3 & Median amplitude of 2 \(\times\) GMF and two nearest sidebands on both sides (5 frequency components altogether) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Description of features used in this study. Features no. 1–3 are computed from time series, whereas features no. 4–11 are extracted from amplitude spectra based on system-specific definitions. The frequency domain information in the acceleration signal is influenced by the operation of various components in the thruster. Therefore, it is useful to focus on the amplitude changes of specific frequencies in order to identify the responses of specific rotating components from the overall vibration. However, the selection of the frequencies for a predictive algorithm is challenging, because it is not known beforehand which components reveal the changes inflicted by defects. The defects in rolling element bearings are often diagnosed based on the harmonics of defect frequencies and their sidebands [31]. Then again, the damage in gears is often diagnosed based on the gear rotating frequency, the meshing frequency and their harmonics, and sidebands [23]. Such frequency components could be monitored individually or together. In this study, several components were monitored together by computing a quantile value from them. The quantile selection is further reasoned in Section 2.2.3 because it influences the distribution of data samples, and therefore, the model performance. The envelope spectrum is commonly used in the bearing diagnosis conducted by an expert [39]. However, the automated selection of an appropriate frequency band for demodulation is complicated [33; 40], and therefore, the amplitude spectrum was used in this study instead. The computational bearing frequencies (see Table 1) exhibit some uncertainty due to the measurement precision, skidding and variations in the rotational speed. Therefore, each frequency component in this study was selected from the range \([f(1-\varepsilon),f(1+\varepsilon)]\). The parameter \(f\) is the computational frequency of the monitored component and \(\varepsilon\) is the error, which is set \(\varepsilon=0.01\). The component with the maximum amplitude was chosen to represent the component of interest from this range. The selected features for bearing monitoring are shown in Table 3 as features no.