## 2 Materials and Methods
### Machine Learning Condition Indicators  
From a mechanical vibration point of view, each drive train component excites the entire system at its corresponding characteristic frequency in a wind turbine. When an accelerometer is placed at a certain point in the machine, the signal obtained is a sum of the convolutions of the excitations of the different components and the transfer path of these excitations to the measuring element. The latter can be modelled mathematically as a specific impulse response function, as shown by [PERSON] in [15]. REBs are components that transmit the loads coming from the supported shaft. They consist of an outer race, an inner race, the rolling elements and a cage to maintain their relative position. When a spall appears on the surface of one of these parts due to fatigue, the rolling elements repetitively impact it. These impacts excite the system in the form of repetitive impulses at the corresponding fault's characteristic frequency, a function of the shaft rotation frequency, the REB geometry, the number of rolling elements and the load angle, as shown in Table 2.  
By computing the fast Fourier transform [16] of a time-domain signal captured by an accelerometer, the harmonics of such excitation will be observed in a specific bandwidth, where the impulse's resonant frequency will be the fault's characteristic frequency. Usually, the fault information will be masked in a complex signal such as that of a wind turbine's drive train. The envelope spectrum is a widely used tool for diagnosing REB faults in an early stage of development and limited size. Information about the different excitation sources is extracted from the spacing between impulse responses and not from excited frequencies. Therefore, the envelope spectrum consists of a series of impulse responses in the frequency domain spaced at the characteristic frequency of the different excitation sources, as shown in Figure 2. When obtaining the envelope spectrum of a time-domain signal for diagnosing REB faults, it is essential to demodulate the signal in the bandwidth where the fault information is present. For this purpose, a tool called the Kurtogram can be used. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Health Condition** & **Total Dataset** & **Class Labels** \\ \hline Normal & 8 & 0 \\ Inner race fault & 76 & 1 \\ Ball fault & 76 & 2 \\ Outer race fault & 147 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: 12k-sampled REBs signals. Health conditions and class labels. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Fault Description** & **Characteristic Frequency** & **Fault Location** \\ \hline BPFO & \(f_{r}\cdot\frac{N_{r}}{2}\cdot\left(1-\frac{d_{r}\cos\theta}{d_{p}}\right)\) & Outer race \\ BPFI & \(f_{r}\cdot\frac{N_{r}}{2}\cdot\left(1+\frac{d_{p}\cos\theta}{d_{p}}\right)\) & Inner race \\ BSF & \(f_{r}\cdot\frac{d_{p}}{d_{p}}\cdot\left(1-\frac{d_{p}\cos^{2}\theta}{d_{p}^{2}}\right)\) & Rolling element \\ \hline \hline \end{tabular} Where, \(f_{r}\) is Rotational frequency of the shaft [Hz]. \(d_{B}\) is Rolling element diameter [m]. \(d_{p}\) is Fitch diameter [m]. \(N_{r}\) is Number of rolling elements (for a single row) [-]. \(\theta\) is Load angle (contact angle from radial) [rad].  
\end{table} 
Table 2: Characteristic frequencies of REB faultsAs mentioned above, the rolling elements' impacts on a spall excite the mechanical system in the form of repetitive impulses at the corresponding fault's characteristic frequency. These impulses take the form of peaks in a time signal. Spectral Kurtosis (SK) is a remarkably sensitive indicator of a signal's peakedness. SK is a statistical method to detect non-Gaussian components in a signal, i.e., very impulsive components. [PERSON] and [PERSON] demonstrated the usefulness of SK in detecting faults in rotating machines [17]. The practical application of SK is the Kurtogram proposed by [PERSON] and [PERSON] in the same paper, a 2D colour map representing the SK at different levels and bandwidths of a signal.