# Quaternion Signal Analysis for Detection of Broken Rotor Fault Degrees in Induction Motors
## 3 Results  
In the experimental tests, a random test file from each rotor state was selected to create a sub-workbench for calculating \(f(\mu_{w})\). Our method then adapts one sample from each of the four vibration signals to construct the current quaternion \(q(t)\) and determines the rotation \(q_{r}(t)\) based on the previous quaternion sample \(q_{d}(t)\), as shown in (12). By  
Figure 5: The position of the sensors and the orientation of the signals. Figure 6: Provoked BRB drilling [27]. calculating the modulus of \(q_{r}(t)\) for each delayed quaternion sample, we obtain a total of 29,999 samples. Multiple shifted windows _w[n]_--utilizing 4000, 6000, 8000, 10,000, 12,000, 14,000, and 20,000 samples--are employed to calculate the mean (\(\mu\)) for each window, forming a function \(f(\mu)\) that characterizes fault behavior. This model function is then evaluated on the remaining test files (excluding those used in model formation). The dispersion of data points around the fitted function can be measured by root mean square error (RMSE), allowing us to evaluate how closely the data are concentrated around the fitted function [28]. We consider this measurement to be a metric to record the joint accuracy of our method across different rotor states. The process is repeated forty times using the original dataset with polynomial regression and Fourier series. The mean RMSE values from these forty trials are analyzed across polynomial orders from second to sixth. Figure 7 shows the average of the RMSE values obtained from the forty tests, with the different window samples tested along with multiple orders of the predictive wear functions. Analyzing the graph mentioned above, it is observed that as window size increases, the mean RMSE values remain relatively stable for each polynomial order, indicating that sample size has minimal impact on the efficiency of our method at a given order. Table 1 presents the average of the RMSE for each window size and polynomial order, along with the minimum and maximum RMSE values across repetitions. The lowest average of the RMSE observed is 0.256, achieved with 20,000 window samples in the second order, though a comparable result is obtained with 8000 samples, yielding a mean RMSE of 0.259 in the same order. In the sixth order, mean RMSE increases in all cases due to a broader spread in minimum and maximum values. Polynomial regression demonstrates consistent behavior, largely unaffected by sample size or polynomial order. Similarly, we calculated the mean of forty RMSE results to examine the behavior of the Fourier series as a regression method, ranging from the second to the sixth order. Figure 8 illustrates these results, showing that the method's efficiency is impacted by sample size, particularly from the fourth to the sixth order. While the Fourier series modeling displays consistent behavior at lower orders, its performance varies as the order increases. Figure 7: Mean RMSE behavior with polynomial modeling. Table 2 presents the mean RMSE for each window size and modeling order, along with the minimum and maximum RMSE values across forty repetitions. The lowest mean RMSE observed in the table is 0.268, achieved with 6000 window samples in the second order. It has been observed that the difference between the minimum and maximum values of orders is greater than the fourth order increases when using Fourier series modeling. Additionally, the average values of the RMSE for these orders change differently compared to those using polynomial modeling. This indicates that the functions derived from the Fourier series have less adaptation to the verification values. Comparing the average of the RMSE values in Tables 1 and 2 reveals that increasing the number of samples minimally impacts the mean RMSE across each order. However, polynomial modeling consistently yields a lower mean RMSE, indicating that the function \(f(\mu)\) achieves a closer fit to the original values and exhibits stable results as the order and window sample parameters increase.