## 4 The Proposed Approach
### Proposed Approach 1  
This approach evaluates condition indicators extracted from the vibration signal in time domain, frequency domain, and a combination of both called time + frequency domain for crack detection in railway axles. In this approach, we analyze the two accelerometers located along the longitudinal direction: on the right side (RL) and another on the left side (LL). According to the work developed by [PERSON] et al. [12], these accelerometers and their orientation offer the best information. Figure 2(a) shows the workflow of the first approach. It has four stages: (1) Data Acquisition, (2) Feature Extraction, (3) Feature Selection, and (4) Classification. Each stage includes the following tasks:  
1. Data acquisition: two sets of signals are acquired, from the left accelerometer (LL) along the longitudinal direction and from the right accelerometer (RL) along the same direction under different fault, load, and speed conditions, as explained in Section 3.2. 2. Feature extraction: the feature extraction was applied as explained in Section 3.2. Figure 4 shows examples of vibration signals in time domain, and signals in the frequency domain using FFT and PSD from both the left and right accelerometers, for normal (healthy) condition and fault level 1. For each signal, different CIs were obtained according to three domains: (i) 30 CIs for time domain (CI_T), (ii) 24 CIs for frequency domain (CI_F), and (iii) 54 CIs from time + frequency domain (CI_TF). Therefore, six data sets were completed. Three data sets were obtained with the left-side accelerometer along the longitudinal direction (LL): the first CI_TL contains CIs extracted from the left side accelerometer and in time domain, the second CI_FL contains the CIs in frequency domain for the same accelerometer, and finally, the third CI_TTL has CIs in the time + frequency domain, also with the same accelerometer. Similarly, with the right-side accelerometer along the longitudinal direction (RL), we obtained other three data sets: CI_TR, CI_FR, and CI_TFR on time, frequency and time + frequency domain, respectively. Figure 2: Location of accelerometers and crack fault. (**a**) Location of the crack in the railway axle. (**b**) Location of accelerometers and crack fault. 3. Feature selection: this process starts by removing correlated features. Correlation coefficient value on 0.8-1 between two CIs suggests these two features are highly correlated [30]. In this study, a threshold value equal to or greater than 0.95 (95%) was selected to identify two highly-correlated features, in all data sets. This threshold was stated after obtaining good performance in classification. Next, a normalization process was applied by scaling characteristics between \(-1\) and \(1\). Normalization can be applied to Normal distributed data, as well as to data with another type of distribution. This process helps KNN to give equal importance to all CIs. After the preprocessing step, a Random Forest model was implemented with 40 trees as the main parameter. MDA metric was used to select the ten most important CIs, and then, the selection of the ten best-ranked features from each dataset. 4. Classification: each dataset is organized in a matrix of samples in the rows and CIs in the columns. Each pre-selected dataset (built from 10 ranked CIs) in stage 3 is classified using KNN (a value of k = 3 was chosen as the main parameter after testing different number of neighbors) and the cosine distance metric was used. Usually, when using k-folds cross-validation, values of k = 5 or k = 10 are chosen due to the good results obtained empirically [31]. In our work, the best results were obtained for k = 5. Therefore, a five-fold cross-validation strategy was carried out; the average accuracy and standard deviation (_std_) on the cross-validation process were calculated from 5 runs.