# New Approaches of Stochastic Models to Examine the Vibration Features in Roller Bearings
## 7 Optimization Result and Discussion
Figure 6: Reponses to One S.D innovation of the impulse factor. (**a**) Inner race fault, (**b**) outer race fault. Figure 7: Reponses to One S.D innovation of the crest factor. (**a**) Inner race fault (**b**) outer race fault. The statistical features of a healthy bearing exhibit stability over time, in stark contrast to the variations observed in extracted features when inner or outer race faults are present. The magnitude of these statistical features intensifies with the rate of inner or outer race defect propagation and damage growth, showcasing both linear and non-linear tendencies. Specifically, the impulse factor for the outer race shows a more pronounced deviation from a healthy bearing compared to the inner race, as evidenced by the estimated coefficients. Figure 11: Reponses to One S.D innovation of the shape factor. (**a**) Inner race fault, (**b**) outer race fault. Figure 9: Reponses to One S.D innovation of kurtosis. (**a**) Inner race fault, (**b**) outer race fault. This discrepancy arises from variations in crest factor information for healthy and faulty bearings, extracted from both vibration and acoustic emission signals. A low crest factor, typical for a healthy bearing, signifies a smooth and steady vibration signal. Conversely, in the presence of a fault, the vibration signal becomes more complex, leading to an increased crest factor. Consequently, the coefficients of the crest factor exhibit more significant variations in the case of outer race faults than in inner race faults. Examining the shape factor reveals that a healthy bearing typically has a shape factor close to unity, indicating a smooth and steady vibration signal with minimal amplitude variation. The presence of a fault, however, induces changes in the vibration signal's complexity, resulting in a shift in the shape factor. The estimated shape factor equation proves valuable for enhancing the accuracy and robustness of fault diagnostics in condition monitoring of rotating machinery. The margin factor, representing the difference between the peak and RMS value of vibration signals, is typically high for a healthy bearing. However, the development of inner or outer race defects alters these signals, leading to variations in the margin factor and adjusted coefficients in the estimated models. Conversely, the peak-to-peak factor is generally low in a healthy bearing, indicating a continuous and smooth vibration signal. In the presence of inner or outer race faults, this factor escalates, resulting in fluctuating computed quantities. The RMS value, a widely-used statistical measure of vibration signal amplitude, remains relatively stable over time for a healthy bearing, serving as an indicator of the overall vibration level generated by the bearing. However, in the presence of a fault, the vibration signal complexity increases, causing changes in the RMS value and deviations in the predicted coefficients. Lastly, kurtosis, a statistical measure of the shape of a probability distribution, plays a crucial role in vibration analysis and condition monitoring of rotating machinery, including bearings. A healthy bearing typically exhibits a low kurtosis value due to the smooth functioning of the roller bearing. In Contrast, inner and outer race faulty bearings show fluctuations in kurtosis due to impulsive vibration signals, prompting corresponding adjustments in the coefficients.