# Probabilistic Condition Monitoring of Azimuth Thrusters Based on Acceleration Measurements
## 2 Materials and Methods
### Condition Monitoring Algorithm
#### 2.2.2 Feature Extraction  
The number of feature extraction methods proposed for the bearing diagnosis [21; 22], gear diagnosis [23] and other condition monitoring applications [24] is immense. The methods are traditionally based on the time domain, frequency domain or time-frequency domain processing [25], and more recently the techniques of feature learning [9; 26] and cyclostationary analysis [27; 28] have become popular in the scientific community. The frequency domain methods are useful in the identification of the symptoms that appear as regular impulses in the signals. On the other hand, the practical applications may have irregular symptoms [29], which may be revealed only in the time domain analysis. Furthermore, the time-frequency domain methods could be useful in the acceleration and deceleration stages of the operation [30]. The features should have the capability to indicate the symptoms of defects individually or together. The symptoms can sometimes be enhanced with various signal processing techniques [21; 31] and the features that reveal changes together could be selected based on some search algorithm [32] that optimizes the prediction accuracy. However, the computationally optimized signal processing parameters [33] and features [34] are prone to become case-dependent to the data sets analyzed, and then, the selections may not be useful for other data sets. In practical applications, the new data may reflect new symptoms which are not known by the models trained [10; 35]. The effects of the features on the selected model must be considered as well. For example, features that correlate strongly with other selected features may bring uncertainty to parameter estimation [36; 37]. Such feature correlations are sometimes reduced based on dimensionality reduction techniques, such as Principal Component Analysis [13; 15],  
Figure 3: Examples of three signals in data quality control: The signal on top is accepted by all checks, the signal in the middle is rejected in the check for moving mean range, and the signal below is rejected based on the moving mean range and absolute mean. but the use of unnecessary features in the original input domain magnifies the uncertainty of inference. Moreover, many of the features are sensitive not only to damage but also to environmental and operational variations [19]. The separation of changes caused by defects from other changes requires consideration. Therefore, the use of expert information in the selection of input data for a condition monitoring algorithm is essential. A good practice is to first select the features which famously have the diagnostic relevance, and secondly, to use additional tests which confirm their suitability to the selected modeling method (see Sections 2.2.4 and 2.2.5). In this study, the monitoring was restricted to momentarily fixed rotational speed, i.e., the rotational speed changes but is constant during the recording of one sample. Therefore, only the time domain and frequency domain methods were considered. Based on the success in previous practical applications [24], the generalized norms, their ratios and other statistical features could provide an efficient solution to the feature extraction in time domain. Therefore, the generalized norm [38], also named as \(l_{p}\) norm  
\[||x^{(\alpha)}||_{p}=\left(\frac{1}{N}\sum_{i=1}^{N}\bigl{|}x_{i}^{(\alpha)} \bigr{|}^{p}\right)^{\frac{1}{p}}, \tag{1}\]  
was applied in this study. The real number \(\alpha\) is the order of derivative, \(x\) is displacement, \(N\) is the number of data points, and the real number \(p\) is the order of norm. Only the acceleration signals (\(\alpha=2\)) were considered in this study. In addition, the kurtosis, given by  
\[Kurtosis=\frac{1}{N}\sum_{i=1}^{N}\left(\frac{x_{i}^{(\alpha)}-\mu}{\sigma} \right)^{4}, \tag{2}\]  
was selected. The parameter \(\mu\) is the mean and \(\sigma\) is the standard deviation.