## 4 Results
### Fault Diagnosis with ANOVA  
In this section, we provide the results obtained applying the functional ANOVA procedures to the set of curves obtained from the time-frequency distributions given by the Dragon Transform from the stator currents of all laboratory experiments on three different situations. This procedure will serve to analyze whether the mean curves for the three categories in which fault severities are classified (healthy motor, medium fault, and full fault) can be considered the same or if significant differences among mean curves can be detected. In the first scenario, motors under high load are considered, using the USH values provided by the Dragon Transform as input values for the functional ANOVA. Notice that as only the USH values are used and not the LSH values, the procedure is not  
Figure 5: Time-frequency distributions of the stator current for the following cases: (**a**) Healthy motor under low load; (**b**) Healthy motor under high load; (**c**) Motor with one broken rotor bar under low load; (**d**) Motor with one broken rotor bar under high load.  
Figure 6: Fault severity quantification curves from the time-frequency distribution of Figure 5 for the following: (**a**) Healthy motor under low load; (**b**) Healthy motor under high load; (**c**) Motor with one broken rotor bar under low load; (**d**) Motor with one broken rotor bar under high load.  
considering all the available information and thus even better results would be obtained if both sidebands are used as input. The results for this first scenario are given in Figure 7. The left panel shows the mean curves for each of the three states considered (solid lines, red for healthy motor state R1, green for medium fault state R2 and blue for full fault state R3) together with all curves for each state (dotted color lines) and the global mean (solid black line). In the middle panel the mean curves are pictured together with dotted grey lines simulating the type of variability expected for the mean curves if the equality of means condition \(H_{0}\) holds. It can be seen that the mean colored curves, in all cases monitored, are placed apart from the grey curves suggesting that the null hypothesis \(H_{0}\) does not hold. This appreciation is confirmed in the right panel by seeing the bootstrap simulated values for the functional ANOVA test statistic (dashed red vertical line) and that show how the value of the test statistics are highly unlikely to be generated by the distribution of the test statistic under the null hypothesis. The \(p\)-value is close to 0, which clearly reinforces the rejection decision of the equality of mean curves stated under the null hypothesis \(H_{0}\).  
The second scenario is that of low-load motors, which is a condition that makes diagnosis more difficult. As in the previous case, only the functional ANOVA procedure with the USH values is provided. The results obtained are shown in Figure 8. The third panel of the figure shows that the functional ANOVA also clearly detects the differences among the three mean curves.  
Finally, to check what would happen for the general case when the motor can be working under different load levels, a third scenario is considered where all USH values are provided independently of the motor load. The first panel of Figure 9 shows that this should be a more difficult situation for the functional ANOVA procedure since there is more confusion among the curves. However, the third panel of Figure 9 shows that even in this more general scenario, the functional ANOVA detects the difference among the mean curves of the three different motor states with a 0 \(p\)-value.  
These results open the door to consider the Dragon Transform curves