## 2 Materials and Methods
### The Methodology
#### 2.1.2 Feature Extraction and Selection  
Feature extraction is a critical aspect of fault diagnosis [37]. Condition monitoring data are weak, meaning that they are not directly correlated to the fault information. In addition, signals are collected from machinery at high frequencies to gain as much informative content as possible. Instead of using raw signals, the fault diagnosis process includes a signal processing step to extract synthetic and relevant fault features. Features are usually extracted in the time, frequency, or time-frequency domain from signal segments of a certain length. In this case, the length is set to 800 ms, equal to the cycle duration. In addition, only time-domain features are extracted, as they require low computational time for calculation and are suitable for streaming applications. In particular, ten typical time-domain features are extracted for each cycle, i.e., peak, peak-to-peak, mean, absolute mean, RMS, crest factor, skewness, kurtosis, shape factor, and impulse factor [38]. The trend of the extracted features in each condition is shown in Figure 5. During the nominal condition, an initial increasing trend can be observed for some features, e.g., kurtosis, which corresponds to the warm-up phase of the machine. Finally, fault 3 is very similar to the nominal condition. Indeed, the signal collected with this sensor is not able to capture the fault information. As the company already provided the susbsyem with another sensor, that collects a signal able to reveal the health condition of the machinery, further investigation using other kinds of features, e.g., extracted in the time-frequency domains, was not performed. In most ML methods, feature selection before fault diagnosis helps to improve the final classification accuracy [39]. Indeed, feature extraction leads to an increase in the feature vector dimension. However, not all the extract features can affect fault diagnosis. Therefore, the optimal features that have intrinsic information about the faults should be retained [40]. In other words, the information retained in the feature subset must demonstrate sensitivity to variation in events observed by a system [41]. Feature selection aims to reduce the number of variables in a dataset by selecting the more discriminative variables and eliminating the redundant ones [42]. In contrast to dimensionality reduction techniques, feature selection does not apply any transformation to the original feature set. Therefore, feature selection preserves the original physical meaning of each variable. For this reason, it is particularly suitable for fault diagnosis in industrial contexts [43]. Figure 4: Cycle differences between nominal and faulty conditions. Different evaluation measures and search techniques are used to produce a suitable feature subset in feature selection [44]. In this paper, the Pearson correlation analysis and the Recursive Feature Elimination (RFE) have been applied, as they are two widespread approaches used in the context of fault diagnosis. Pearson's correlation analysis is a filter-based method used for redundancy elimination. It assumes that if two variables have a high linear relationship, they contain redundant information [45]. A fixed threshold usually determines if features are highly correlated or not. The RFE is a supervised feature selection method, which assigns a weight to the features based on a particular classification model and recursively removes the features with the most negligible weight [46]. In the present paper, first, the Pearson correlation is conducted, and a threshold equal to the absolute value of 0.9 is chosen. Then, the correlation matrix is first built using the Pearson correlation coefficient computed among all features and expressed by Equation (1).