## 2 IMS Bearing Dataset and Signal Analysis Techniques
### Signal Analysis Techniques  
In this Section, a list and a brief mathematical introduction of the diagnostic techniques used to analyze vibration signals are provided. Techniques are divided into the three main phases of the condition monitoring process, i.e., fault detection, fault diagnosis and fault prognosis. \begin{table}
\begin{tabular}{l l} \hline Model & \multicolumn{1}{c}{Rexnord ZA-2115} \\ \hline Pitch diameter \(D\) & 71.5 mm (2.815 inch) \\ \hline Rolling element diameter \(d\) & 8.4 mm (0.331 inch) \\ \hline Number of rolling elements per row \(n\) & 16 \\ \hline Load contact angle \(\phi\) & 15.17\({}^{\circ}\) \\ \hline Static load \(Q\) & 26,690 N (6000 lbs.) \\ \hline Shaft angular velocity \(\omega\) & 2000 rpm \\ \hline \end{tabular}
\end{table} 
Table 2: Rolling bearing mechanical characteristics and working conditions (from [33]). Figure 3: Rolling element bearing (from [3]). \begin{table}
\begin{tabular}{l l} \hline Nominal rotation frequency of the shaft & 33.3 Hz \\ \hline Ball Pass Frequency Outer Race (BPFO) & 236 Hz \\ \hline Ball Pass Frequency Inner Race (BPFI) & 297 Hz \\ \hline Ball Spin Frequency (BSF) & 278 Hz (2 \(\times\) 139 Hz) \\ \hline Fundamental Train Frequency (FTF) & 15 Hz \\ \hline \end{tabular}
\end{table} 
Table 3: Characteristic frequencies of the IMS test rig (from [33]). Moreover, within the fault diagnosis techniques, signal denoising techniques, filters and cyclostationary techniques are considered. Fault detection techniques:  
Mean;  
Variance;  
Standard Deviation;  
Pea;  
Root Mean Squared value (RMS);  
Skewness;  
Crest Facto;  
Clearance Factor;  
Shape Factor;  
Impulse Factor;  
Peak-to-Peak;  
Hjorth's Parameters;  
Detectivity. Fault diagnosis techniques:  
Short-Time Fourier Transform (STFT);  
Power Spectral Density (PSD);  
Squared Envelope Spectrum (SES);  
Autoregressive Linear Prediction (ALP);  
Time-Synchronous Averaging (TSA);  
Kurtosis (time domain);  
Kurtogram. Signal denoising techniques:  
Daubechies' Wavelets;  
Cepstrum Pre-Whitening (CPW). Filters:  
FIR;  
Adaptive. Cyclostationary techniques:  
Spectral Correlation (SC);  
Cyclic Spectral Coherence (CSC);  
Improved Envelope Spectrum (IES);  
Wigner-Ville Distribution (WVD). Fault prognosis techniques:  
Correlation;  
Monotonicity;  
Robustness. The use of numerous statistical parameters is due to their limited analytical and computational complexity and their general and intuitive meaning for detection purposes, although they just play the role of high-level indicators and need to be followed by more in-depth, specific and elaborate techniques. Table 4 shows the mathematical definition of the most common statistical parameters. Hjorth's parameters and Detectivity are statistical time-domain quantities, but their application in the context of signal diagnosis is still unusual. The formers (Activity, Mobility, Complexity) were introduced by [PERSON] and [PERSON] in 1970 and are commonly used in the analysis of electroencephalography (EEG) signals for feature extraction [35] and in the tactile signal analysis in the robotic area [36]. They are related to the variance of the signal and of its subsequent derivatives, and can be computed with minimum effort and resources, suggesting their use in real-time application for the condition monitoring of ball bearings or as a feature array for Machine Learning techniques. The \"Activity\" of a vibration signal (Act\((x)\)) is defined as the variance of its amplitude, and therefore it has the same dimension of the signal (the one of the accelerations in the current case) and it is directly related to its power. \"Mobility\" (Mob\((x)\)) is related to the variance of the vibration signal velocity, so it is associated to the jerk (first time derivative of the signal) and its dimension is expressed as a ratio per time unit. \"Complexity\" (Com\((x)\)) is a dimensionless parameter, connected to the variance of the acceleration of the signal, hence to the snap (second time derivative of the signal and four-time derivative of space).