# New Approaches of Stochastic Models to Examine the Vibration Features in Roller Bearings
## 7 Optimization Result and Discussion  
The vibration signatures of roller element bearings exhibit intricate patterns in the presence of inner-race and outer-race faults when compared to a healthy state. To establish a robust relationship between the signals of faulty and healthy bearings, ARMA modelling techniques were employed to determine optimized parameters. Initially, Equations (1) and (2) were employed to ascertain the appropriate model orders. In Table 4 the estimated statistics, including the Akaike Information Criterion (AIC) and Schwarz Criterion (SC), are presented alongside their corresponding probability values for suitable ARMA (p, q) orders. These computations span a range from one to twelve. Subsequently, the stability of the constructed models is ensured through standard statistical \begin{table}
\begin{tabular}{l c c c c} \hline \multicolumn{1}{c}{**Models**} & \multicolumn{2}{c}{\(p\)**-Values**} & \multicolumn{1}{c}{\(R^{2}\)} \\ \hline Stochastic model of impulse factor & 0.001 & 0.057 & 0.056 & 0.701 \\ \hline Stochastic model of crest factor & 0.001 & 0.063 & 0.044 & 0.699 \\ \hline Stochastic model of shape factor & 0.001 & 0.081 & & 0.68 \\ \hline Stochastic model of margined factor & 0.046 & 0.049 & 0.089 & 0.091 & 0.66 \\ \hline Stochastic model of peak to peak value & 0.050 & 0.069 & 0.092 & & 0.68 \\ \hline Stochastic model of root mean square & 0.001 & 0.002 & 0.089 & & 0.84 \\ \hline Stochastic model of kurtosis & 0.071 & 0.022 & 0.078 & 0.088 & 0.76 \\ \hline \end{tabular}
\end{table} 
Table 3: Stochastic model of outer-race faulty features with healthy features. techniques, as evidenced by the necessary statistics provided in the parentheses of the estimated equations. The corresponding impulse response functions (IRFs) for various features of faulty bearings are displayed in Figures 5-11. The features of outer race faults show impacts at the ball pass frequency and its harmonics, as evidenced by spikes in the IRFs at these frequencies. The presence of sidebands further confirms the occurrence of outer faults. In contrast, for inner race faults, the impulse response produces impulses at the shaft rotation frequency and its harmonics. The IRFs exhibit synchronized irregularities with the shaft rotation, and multiple peaks of the shaft rotation frequency and sidebands are visible. These observations highlight distinctive patterns in the impulse response functions associated with different bearing faults, providing valuable insights for the diagnosis and analysis of bearing health. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Models** & **Order** & **AIC** & **BIC** \\ \hline Impulse Factor Inner Race Fault & ARMA (2,3) & 2.6516 & 2.8029 \\ \hline Impulse Factor Outer Race Fault & ARMA (1,3) & 4.3337 & 4.4547 \\ \hline Crest Factor Inner Race Fault & ARMA (2,2) & \(-\)10.6074 & \(-\)10.4561 \\ \hline Crest Factor Outer Race Fault & ARMA (1,3) & \(-\)8.8708 & \(-\)8.7498 \\ \hline Shape Factor Inner Race Fault & ARMA (1,1) & 5.6182 & 5.7392 \\ \hline Shape Factor Outer Race Fault & ARMA (0,2) & 7.9342 & 7.9987 \\ \hline Margin Factor Inner Race Fault & ARMA (1,2) & \(-\)12.4488 & \(-\)12.2974 \\ \hline Margin Factor Outer Race Fault & ARMA (2,1) & \(-\)10.4902 & \(-\)10.3806 \\ \hline Peak–Peak Factor Inner Race Fault & ARMA (2,3) & 2.6516 & 2.4857 \\ \hline Peak–Peak Factor Outer Race Fault & ARMA (1,3) & 4.3337 & 4.4547 \\ \hline RMS Inner Race Fault & ARMA (1,0) & 1.7768 & 1.8978 \\ \hline RMS Outer Race Fault & ARMA (2,0) & \(-\)0.4051 & \(-\)0.2341 \\ \hline Kurtosis Inner Race Fault & ARMA (1,2) & 1.1191 & 1.2067 \\ \hline Kurtosis Outer Race Fault & ARMA (1,3) & 4.2627 & 4.4141 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Order of suitable stochastic models with assessed statistics of selection criteria. Figure 5: Reponses to One S.D innovation of peak-to-peak values. (**a**) Inner race fault, (**b**) outer race fault. Figure 8: Reponses to One S.D innovation of RMS. (**a**) Inner race fault, (**b**) outer race fault.