# MOIRA-UNIMORE Bearing Data Set for Independent Cart Systems
## 7 Preprocessing Methodology
### Feature Ranking Methods
#### 7.1.1 One-Way ANOVA  
The One-Way Analysis of Variance (ANOVA) [42; 43; 44] is a statistical method used to identify significant differences between the means of two or more groups. In the context of feature ranking, ANOVA evaluates the relationship between a feature and the target variable, ranking features based on their ability to distinguish between different classes or  
\begin{table}
\begin{tabular}{c c} \hline
**Statistical Parameter** & **Mathematical Expression** \\ \hline Mean & \(\mu=\frac{1}{N}\sum_{i=1}^{N}x_{i}\) \\ \hline RMS & \(x_{\text{rms}}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}|x_{i}|^{2}}\) \\ \hline Standard Deviation & \(x_{\text{std}}=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}|x_{i}-\mu|^{2}}\) \\ \hline Shape Factor & \(x_{\text{shape}}=\frac{x_{\text{rms}}}{\frac{1}{N}\sum_{i=1}^{N}|x_{i}|}\) \\ \hline Kurtosis & \(x_{\text{kurt}}=\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\mu)^{4}}{\left(\frac{1}{N} \sum_{i=1}^{N}(x_{i}-\mu)^{2}\right)^{2}}\) \\ \hline Skewness & \(x_{\text{skew}}=\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\mu)^{3}}{\left(\frac{1}{N} \sum_{i=1}^{N}(x_{i}-\mu)^{2}\right)^{\frac{3}{2}}}\) \\ \hline Peak Value & \(x_{p}=\max_{i}|x_{i}|\) \\ \hline Impulse Factor & \(x_{IF}=\frac{x_{p}}{\frac{1}{N}\sum_{i=1}^{N}|x_{i}|}\) \\ \hline Crest Factor & \(x_{\text{crest}}=\frac{x_{p}}{\sqrt{\frac{1}{N}\sum_{i=1}^{N}x_{i}^{2}}}\) \\ \hline Clearance Factor & \(x_{\text{clear}}=\frac{x_{p}}{\left(\frac{1}{N}\sum_{i=1}^{N}\sqrt{|x_{i}|} \right)^{2}}\) \\ \hline Signal-to-Noise Ratio & \(SNR=10\log_{10}\left(\frac{P_{\text{false}}}{P_{\text{false}}}\right)\) \\ \hline Total Harmonic Distortion & \(THD=10\log_{10}\left(\frac{\sum_{i=1}^{N}p_{i}}{P_{1}}\right)\) \\ \hline Signal-to-Noise And Distortion Ratio & \(SINAD=10\log_{10}\left(\frac{P_{\text{signal}}}{P_{\text{noise+distortion}}}\right)\) \\ \hline \end{tabular}
\end{table}
Table 6: Summary of statistical parameters.  
\begin{table}
\begin{tabular}{c c c} \hline
**Label** & **Defect** & **Dimension (mm)** \\ \hline H & No Fault & – \\ \hline FIR025 & Inner Race Fault & 0.25 \\ \hline FIR05 & Inner Race Fault & 0.5 \\ \hline FIR10 & Inner Race Fault & 1.0 \\ \hline FIR15 & Inner Race Fault & 1.5 \\ \hline FOR025 & Outer Race Fault & 0.25 \\ \hline FOR05 & Outer Race Fault & 0.5 \\ \hline FOR10 & Outer Race Fault & 1.0 \\ \hline FOR20 & Outer Race Fault & 2.0 \\ \hline \end{tabular}
\end{table}
Table 7: List of the bearing defect labels.  
groups. It assumes that the data within each group follow a normal distribution and that the variances are homogeneous between the groups. Mathematically, the test statistic for One-Way ANOVA is derived by partitioning the total variance of the data into variance between groups and variance within groups. For a feature \(X\) divided into \(k\) groups with \(n\) total samples, the statistic \(F\) is calculated as  
\[F=\frac{\text{Variance Between Groups}}{\text{Variance Within Groups}}=\frac{\frac{1}{k-1}\sum_{i=1}^{k}n_{i}(\tilde{X}_{i}-\tilde{X})^{2}}{\frac{1}{n-k} \sum_{i=1}^{k}\sum_{j=1}^{n_{i}}(X_{ij}-\tilde{X}_{i})^{2}} \tag{3}\]  
where \(\tilde{X}_{i}\) is the mean of group \(i\), \(\tilde{X}\) is the overall mean, \(n_{i}\) is the number of samples in group \(i\), \(k\) is the number of groups, and \(X_{ij}\) is the \(j\)-th sample in group \(i\). A higher \(F\)-statistic indicates that the feature effectively discriminates between groups, making it more relevant for classification or regression tasks. Features are ranked based on their corresponding \(F\)-statistics.