# Using Autoregressive with Exogenous Input Models to Study Pulsatile Flows
## 4 ARX Model Results
### ARX Model Validation  
One of the main goals of using an ARX model for this purpose is to have the possibility to use the parameter vectors found in the training part for estimating/simulating external conditions to the ones employed in the parametric study and verify the quality of our model without spending more time on FSI simulations. The simulation parameters considered for these changes are the same as the ones in the parametric study except for \(E\) and \(T\). These parameters now consider the following options: \(E~{}=~{}[1.5,15]\) MPa and \(T=[0.90,1.20]\)s. The selected ranges are close to the elastic conditions proposed in Section 2.1, and this allowed us to verify the behavior of our model in four different situations:  
1. [label=(0)]
2. E = 1.5 MPa, T = 0.9 s: the elastic modulus lies inside, and the period is outside the training dataset. 3. E = 1.5 MPa, T = 1.2 s: both conditions are located inside the training dataset. 4. E = 15 MPa, T = 0.9 s: both conditions lie outside the training dataset, whereas the period is located inside it. The validation process forms the transfer functions of the aforementioned desired values by using a bilinear interpolation with the coefficients found in Figure 4. Finally, Table 4 presents the interpolated values for these new cases. Figure 5: Signals reconstructed using the ARX model for \(E=2.15\) MPa during the training process. In (**a**), the pressure difference (\(\Delta\)P) and, in (**b**), the normalized opening area (\(S_{v}\)). The average accuracy obtained for reconstructing the nine periods was \(R=96.14\%\) and \(R=89.35\%\) for \(\Delta\)P and \(S_{v}\) respectively. The average goodness of fit of the five \(E^{\prime}s\) is listed in Table 5. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**E (MPa)** & **E = 0.1** & **E = 1** & **E = 2.15** & **E = 3.16** & **E = 10** \\ \hline
**T (s)** & \(R_{\Delta P}\) & \(R_{\Delta P}\) & \(R_{\Delta P}\) & \(R_{\Delta P}\) & \(R_{\Delta P}\) & \(R_{\Delta P}\) & \(R_{\Delta P}\) & \(R_{\Delta P}\) & \(R_{\Delta P}\) \\ \hline
1.000 & 99.473 & 98.187 & 97.966 & 93.910 & 99.342 & 87.863 & 99.134 & 91.972 & 93.703 & 60.463 \\
1.111 & 99.640 & 98.609 & 91.854 & 95.092 & 98.671 & 91.895 & 97.609 & 91.682 & 96.244 & 90.216 \\
1.250 & 99.471 & 98.814 & 98.664 & 91.902 & 98.828 & 90.882 & 91.800 & 86.692 & 91.735 & 78.225 \\
1.429 & 99.597 & 98.082 & 99.087 & 92.002 & 99.013 & 91.321 & 96.785 & 88.995 & 93.557 & 60.501 \\
1.502 & 99.437 & 98.526 & 99.344 & 92.971 & 98.464 & 91.646 & 97.298 & 92.869 & 90.121 & 94.031 \\
1.667 & 93.925 & 97.950 & 98.945 & 99.435 & 91.389 & 93.987 & 96.240 & 93.194 & 78.775 & 84.260 \\
2.000 & 78.528 & 97.250 & 98.449 & 91.927 & 97.914 & 92.525 & 97.247 & 96.177 & 98.154 & 92.525 \\
2.500 & 39.411 & 93.765 & 91.064 & 94.686 & 96.930 & 95.992 & 94.318 & 94.655 & 52.917 & 98.864 \\
3.003 & 99.154 & 98.233 & 97.968 & 97.003 & 84.707 & 68.096 & 88.427 & 96.912 & 51.302 & 86.288 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: R values obtained by the AutoRegressive with eXogenous (ARX) input model for both outputs. In order to compare the signals obtained by our ARX model, we previously conducted the FSI simulations of these new conditions used for validation. The results seen in Figure 6 show that some simulations were not able to finish the calculation process; this was because they were unable to find equilibrium or convergence. For these reasons, estimation of the outputs was merely conducted until the moment before the simulation stopped. The accuracy of the validation results is shown Table 5, where the average value of accuracy was \(R_{val}=93.31\%\) and \(R_{val}=83.08\%\) for \(S_{v}\) and \(\Delta\)P, respectively. One limitation of this method is that it only works for stationary signals; hence, when dealing with transitory behaviors, it was not able to follow the trend expected efficiently.