# New Approaches of Stochastic Models to Examine the Vibration Features in Roller Bearings
## 6 Stochastic Modelling of Vibration Features
### Stochastic Association amongst Inner-Race Faulty Features with Healthy Features  
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2. Impulse Factor \[\tau_{t}^{i}=4.0324+1.089\tau_{t}^{h}-0.882(\tau_{t-1}^{i}+\tau_{t-2}^{i})-0.99( \omega_{t-1}^{i}+\omega_{t-2}^{i}+\omega_{t-3}^{i})+\omega_{t}^{i}\] (4) 3. Crest Factor \[\lambda_{t}^{i}=0.0056+0.986\lambda_{t}^{h}-0.749(\lambda_{t-1}^{i}+\lambda_{t- 2}^{i})-0.978(\omega_{t-1}^{ci}+\omega_{t-2}^{ci})+\omega_{t}^{ci}\] (5) 4. Shape Factor \[\theta_{t}^{i}=1.096\theta_{t}^{h}-0.953\theta_{t-1}^{i}+0.887\omega_{t-1}^{ si}+\omega_{t}^{si}\] (6) 5. Margin Factor \[\xi_{t}^{i}=0.012-1.165\xi_{t}^{h}-0.767\xi_{t-1}^{h}-0.884(\omega_{t-1}^{mi}+ \omega_{t-2}^{mi})+\omega_{t}^{mi}\] (7) 6. Peak-Peak Factor \[\rho_{t}^{i}=0.416-3.296\rho_{t}^{h}+0.684\rho_{t-1}^{h}+0.821(\omega_{t-1}^{ pi}+\omega_{t-2}^{pi})+\omega_{t}^{pi}\] (8) 7. RMS Value \[\Phi_{t}^{i}=17.96-6.037\Phi_{t}^{h}+0.84\Phi_{t-1}^{h}+\omega_{t}^{ri}\] (9) 8. Kurtosis \[\text{K}_{t-1}^{i}=2.39\text{K}_{t}^{h}-0.994\text{K}_{t-1}^{h}-0.979(\omega_{ t-1}^{ki}+\omega_{t-2}^{ki})+\omega_{t}^{ki}\] (10)  
The stability and significance of stochastic models of inner-race faulty features compared with healthy features by using the \(p\) value test and good fitness test R\({}^{2}\) values are explained in Table 2.  
Figure 4: Variability in extracted features under investigated experimental setup conditions. (**a**) variation in skewness, (**b**) variation in kurtosis, (**c**) variation in crest factors, (**d**) variation in RMS, (**e**) variation in impulse factors, (**f**) variation in margin factors, (**g**) variation in shape factors, (**h**) variation in peak peak values.  
The stochastic regression model of the impulse factor and crest factor shows a significant level of 5% because the \(p\)-values of all the coefficients in the model are \(<\)0.05, indicating that the model fits within a 95% confidence interval. Additionally, the \(R^{2}\) test result also shows 75% and 73%, which are good significance levels according to the standard, where 70% to 90% are acceptable. Hence, the developed model efficiently forecasts faults in the signal data at time t. Similarly, the \(p\) values of the coefficients of the shape factor model are less than 0.01, which indicates the highest significance at a 99% level of significance, and the \(R^{2}\) value is 77%, which is a good indicator of the variance of the dependent variable. The \(p\)-values of the coefficients in the model of the margined factor are as follows: 0.005, 0.014, 0.056, and 0.075. The first three values indicate strong evidence against the null hypothesis at a 95% level, while the fourth value has low evidence against the null hypothesis at a level of 92.5%. The \(R^{2}\) value is 75%, which is a good indicator of the variance of the dependent variable. The \(p\)-values of the coefficients in the peak to peak value are as follows: 0.006, 0.087, 0.042, 0.099. These values are significant, and the \(R^{2}\) value is 68%. The \(p\)-values of the coefficients in the model of RMS are as follows: 0.001 \(\leq\) 0.005, 0.002 \(\leq\) 0.005, but the third coefficient (0.089) \(\geq\) 0.005 and \(<\) 0.1, indicating low significance at this coefficient. Overall, the \(R^{2}\) value is 91%, showing that the model is statistically significant. The statistically significant model of Kurtosis has high significance because the \(p\)-values of the coefficients in the model are all 0.001, showing the highest level of significance at 99%. The value of \(R^{2}\) is 70%, indicating the maximum variance in the dependent variable.