# On Training Data Selection in Condition Monitoring Applications--Case Azimuth Thrusters
## 2 Materials and Methods
### Training Subset Selection  
The algorithm for identifying CM models was originally presented in [6] and a flowchart of its stages is shown in Figure 1. The training subset selection process is highlighted there. It controls simultaneously the sample selection within the full training set and the selection of operating areas for monitoring. The process uses features extracted from the acceleration signal, the values of shaft rotational speed and user-defined parameters to guide the selection. The process checks the available samples in various operating areas, defined by the ranges of rotational speed values within the training subsets, and finally selects the subsets suitable for model training. To clarify, the process was named'speed range selection' in the original version [6], but the name is modified here to refer to the more general concept of training data selection. The main function of training data selection is to identify the subsets in which the predictions of linear regression models result in residuals that are normally distributed. The regression models utilize shaft speed values to predict the values of vibration features. Therefore, a residual (r\({}_{i}\)) can be defined as  
\[\text{r}_{i}=\text{y}_{i}-\beta_{0}-\beta_{1}\text{x}_{i}, \tag{1}\]  
where \(\text{y}_{i}\) is a calculated feature value, \(\text{x}_{i}\) is a shaft speed value and i is the sample number. Parameters \(\beta_{0}\) and \(\beta_{1}\) are the intercept and the regression coefficient, respectively. The selected features are described in Table 1 and their residuals were monitored together in suitable combinations. Features 1-3 were selected for the time-domain model, features 4-8 for the bearing model and features 9-11 for the gear model. The reasoning for the selected features, the system parameters for their calculation and the use of their residuals for condition monitoring were presented in [6]. The applied system parameters included Ball Pass Frequency Inner Race (BPFI), Ball Pass Frequency Outer Race (BPFO), Ball Spin Frequency (BSF), Fundamental Train Frequency (FTF) and Gear Mesh Frequency (GMF). As shown in Figure 1, the algorithm uses preset parameters that control its performance, and it gives model parameters that are identified during training. The preset parameters that guide the speed range selection are defined in Section 2.1.1. The studied model parameters given by the algorithm include the rotational speed ranges and the confidence intervals of regression coefficients. The applied quality control (QC) parameters are the same as in the previous study [6], namely, the moving window size is 100 acceleration values, the limit for the range of moving mean range is 0.5 g and the limit for the absolute mean of the signal is 2 g. However, the absolute mean check was unnecessary in the case study where the signal offset was removed by the system onboard. The algorithm uses variance inflation factor (VIF) values to assess the multicollinearity of residual sets. For the sake of brevity, the analysis of such values is omitted here, but to clarify, they were strictly \(<\)10 based on random sampling.