## 3 Fault Diagnosis with Original Data
### Statistical Analysis  
The third approach is that of statistical analysis, which uses the extracted features from the FFT of the signal. The FFT of the signal is analyzed based on the statistical features that have eight parameters: mean, median, min, max, kurtosis, skewness, standard deviation, and range. These features are obtained along all four axes: \(g\_x\), \(g\_y\), \(g\_z\), and \(g\_t\). Each data sample contains 32 elements. Due to low input data dimensions, these approaches are suitable with ANN, K-mean clustering, and SVM. The statistical approach not only provides another efficient method for fault classification but also plays an important role in evaluating the generated data. Using statistical analysis, we extract the overall characteristics of the vibration signal. Based on these characteristics, we can compare the generated data with the original data. If the generated data has high accuracy in the statistical analysis, we can conclude that it has a high similarity level with the original data and can be used for the training process. Before feeding data into the AI models, we should understand the variations in the statistical characteristics of the FFT of the signal.  
The statistical relationship between the normal and broken machine signals can be considered a reliable classification method, as shown in Figure 7. The statistical elements  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Full FFT** & **Reduced FFT** \\ \hline ANN & 100\% & 100\% \\ \hline K-mean clustering & 100\% & 100\% \\ \hline SVM & 100\% & 100\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Performance of artificial neural network (ANN), K-mean clustering, and support vector machine (SVM) with full FFT and reduced FFT of the signal.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Data Augmentation** & **AI Model** & **Full FFT** & **Reduced FFT** \\ \hline \multirow{3}{*}{Jittering} & ANN & 79.28\% & 88.24\% \\ \cline{2-3}  & K-mean clustering & 75.71\% & 80.35\% \\ \cline{2-3}  & SVM & 85.88\% & 86.74\% \\ \hline \multirow{3}{*}{Scaling} & ANN & 80.21\% & 87.25\% \\ \cline{2-3}  & K-mean clustering & 75.36\% & 82.35\% \\ \cline{2-3}  & SVM & 83.91\% & 85.74\% \\ \hline \multirow{3}{*}{Slicing} & ANN & 82.57\% & 89.41\% \\ \cline{2-3}  & K-mean clustering & 76.71\% & 81.85\% \\ \cline{2-3}  & SVM & 91.76\% & 87.74\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Performance of ANN, K-mean clustering, and SVM with traditional data augmentation using full FFT and reduced FFT of the signal.  
of the normal and broken signal are similar in the \(g\_y,g\_z\), and \(g\_t\) directions, but they are different in the \(g\_x\) direction. The most difference is seen in the maximum value (4) and kurtosis value (5) of the \(g\_x\) signal. In Figure 8, if we break down the relationship between them, we can easily classify the normal and broken machine signals.  
The performance of ANN, K-mean clustering, and SVM based on statistical analysis are shown in Table 4. The ANN algorithm reaches over 95% accuracy in three scenarios, which proves that robust classification features were adopted. In contrast, K-mean clustering and SVM have the worst performance as compared with full FFT analysis. In the classification using real data, K-mean clustering and SVM showed poor performance as compared with a full analysis approach. However, this characteristic can be used to analyze the performance of the generated data for an unstable classification method. This approach needs to be considered when the generated data is used.  
Figure 8: Relationship between the maximum and kurtosis values of the normal and broken FFT of the signals. (Reprinted with permission from Ref. [32]. Copyright 2020 IEEE).  
Figure 7: Statistical elements of normal and broken FFT of the signal. (Reprinted with permission from Ref. [32]. Copyright 2020 IEEE).