# New Approaches of Stochastic Models to Examine the Vibration Features in Roller Bearings
## 6 Stochastic Modelling of Vibration Features
### Stochastic Association amongst Outer-Race Faulty Features with Healthy Features  
1. [label=()]
2. Impulse Factor \[\tau_{t}^{o}=1.731\tau_{t}^{h}+0.635\ \tau_{t-1}^{i}+0.99(\omega_{t-1}^{i}+ \omega_{t-2}^{i}+\omega_{t-3}^{i})+\omega_{t}^{i}\] (11) 3. Crest Factor \[\lambda_{t}^{0}=1.639\lambda_{t}^{h}\ +0.599\lambda_{t-1}^{i}+0.87(\omega_{t-1}^{ ci}+\omega_{t-2}^{ci}+\omega_{t-3}^{ci})+\omega_{t}^{ci}\] (12) 4. Shape Factor \[\theta_{t-1}^{o}=1.0426\theta_{t}^{h}\ +0.951(\omega_{t-1}^{si}+\omega_{t-2}^{si})+ \omega_{t}^{si}\] (13) 5. Margin Factor \[\xi_{t-1}^{o}=0.024-1.51\xi_{t}^{h}\ -0.045(\xi_{t-1}^{h}+\xi_{t-2}^{h})+0.7 12\omega_{t-1}^{mi}+\omega_{t}^{mi}\] (14) 6. Peak-Peak Factor \[\rho_{t}^{o}=1.636\rho_{t}^{h}\ +0.817(\rho_{t-1}^{h}+\rho_{t-2}^{h})+0.8 73(\omega_{t-1}^{pi}+\omega_{t-2}^{pi}+\omega_{t-3}^{pi})+\omega_{t}^{pi}\] (15) 6. RMS Value \[\Phi_{t}^{o}=7.38-2.987\Phi_{t}^{h}\ -0.524(\Phi_{t-1}^{h}+\Phi_{t-2}^{h})+ \omega_{t}^{ri}\] (16) \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Models** & \multicolumn{3}{c}{_p_**-Values**} & \(R^{2}\) \\ \hline Stochastic model of impulse factor & 0.042 & 0.015 & 0.026 & 0.047 & 0.75 \\ \hline Stochastic model of crest factor & 0.021 & 0.0151 & 0.0136 & 0.057 & 0.73 \\ \hline Stochastic model of shape factor & 0.001 & 0.001 & 0.001 & & 0.77 \\ \hline Stochastic model of margined factor & 0.005 & 0.014 & 0.056 & 0.075 & 0.75 \\ \hline Stochastic model of peak to peak value & 0.006 & 0.087 & 0.042 & 0.099 & 0.90 \\ \hline Stochastic model of root mean square & 0.012 & 0.091 & 0.076 & & 0.91 \\ \hline Stochastic model of kurtosis & 0.001 & 0.001 & 0.001 & & 0.70 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Stochastic model of inner-race faulty features with healthy features. * [(g)] Kurtosis \[\text{K}_{t}^{i}\,=20.17-4.586\text{K}_{t}^{h}\,+0.718\text{K}_{t-1}^{h}+0.917(\omega_ {t-1}^{h}+\omega_{t-2}^{h}+\omega_{t-3}^{h})+\omega_{t}^{ki}\] (17)  
The statistical stability of the extracted features models relating to outer-race faulty features and healthy features were checked by using the \(p\)-value test and \(R^{2}\) test. In Table 3, the \(p\)-values of the coefficient of the impulse factor model are (0.001, 0.057, 0.056) and crest factor model are (0.001, 0.063,0.044).This shows that the models are fit in the 94% to 95% confidence interval, and the \(R^{2}\) test result also shows 70% variance between the outer race fault at time t and the independent variables. Hence, the developed model is efficient at forecasting faults in the signal data at time t.  
The \(p\) values of the coefficient in the shape factor model are (0.01, 0.81), the first value indicates excellent stability at a 99% confidence level and the second value shows good stability at a 92% level, with \(R^{2}\) value of 68%. The values in the margined factor model are (0.046, 0.49, 0.089, 0.091). The first two values are less than 0.05, which indicates strong stability of the model at a 95% level, and the next two values have low stability, at a level of 92% with \(R^{2}\) value of 66%. The peak to peak value model has \(p\) values at (0.05, 0.069,0.092). The first value indicates high stability, at a 95% level, the second value has good stability at a level of 93%, and the third coefficient value indicates 91%, with an \(R^{2}\) value of 68%,  
The \(p\) values of the coefficient in the root mean square model are (0.001, 0.002, 0.89). The first two values are less than or equal to 0.02, which indicates the strong stability of the model at a 98% level, and the third value has low stability at a level of 91%, with an \(R^{2}\) value of 84%. This is a good indicator for the variance of the dependent variable. The kurtosis model has \(p\)-values at (0.071, 0.022, 0.078, 0.088). These values indicate a stability between 92% to 98%, with \(R^{2}\) value of 76%, which is a good indicator for the variance of the dependent variable.