# Using Autoregressive with Exogenous Input Models to Study Pulsatile Flows
## 4 ARX Model Results
### ARX Model Validation
For example, in the opening area of Figure 6, we can see at the beginning of the forward stage (\(t_{f}\)) how the leaflets began to open up to the highest position due to an inertial momentum before it went back to the value where it got stabilized by the flow. On the contrary, the pressure difference output was better estimated by our model because it did not contain a transitory behavior. This validation process showed that our model could be a promising support for the FSI simulations when dealing with different situations in a faster and still accurate way. Besides, it could eliminate the necessity of conducting several simulations before we can get a vast amount of information by \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{\(\Delta\)P} & \multicolumn{4}{c}{\(S_{v}\)} \\ \cline{2-10}
**Coef** & \multicolumn{2}{c}{**E = 1.5 MPa**} & \multicolumn{2}{c}{**E = 15 MPa**} & \multicolumn{2}{c}{**E = 1.5 MPa**} & \multicolumn{2}{c}{**E = 15 MPa**} \\  & **T = 0.9 s** & **T = 1.2 s** & **T =0.9 s** & **T = 1.2 s** & **Coef** & **T = 0.9 s** & **T = 12 s** & **T = 0.9 s** & **T = 1.2 s** \\ \hline  & - & - & - & - & \(b_{2}\) & \(2.15\times 10^{3}\) & \(5.78\times 10^{3}\) & \(4.14\times 10^{3}\) & \(4.13\times 10^{2}\) \\ \(d_{1}\) & \(-1.18\times 10^{8}\) & \(-1.03\times 10^{8}\) & \(-9.01\times 10^{7}\) & \(-7.46\times 10^{7}\) & \(b_{1}\) & \(-4.34\times 10^{3}\) & \(-1.16\times 10^{4}\) & \(-8.47\times 10^{3}\) & \(-1.00\times 10^{3}\) \\ \(d_{0}\) & \(1.18\times 10^{8}\) & \(1.03\times 10^{8}\) & \(9.00\times 10^{7}\) & \(7.45\times 10^{7}\) & \(b_{0}\) & \(2.19\times 10^{3}\) & \(5.84\times 10^{3}\) & \(4.33\times 10^{3}\) & \(5.94\times 10^{2}\) \\ \hline  & - & - & - & - & \(a_{3}\) & \(1.000\) & \(1.000\) & \(1.000\) & \(1.000\) \\ \(c_{2}\) & \(1.000\) & \(1.000\) & \(1.000\) & \(1.000\) & \(a_{2}\) & \(1.234\) & \(1.236\) & \(1.347\) & \(1.546\) \\ \(c_{1}\) & \(0.032\) & -0.070 & \(0.671\) & \(0.990\) & \(a_{1}\) & \(-0.147\) & \(-0.427\) & \(-0.517\) & \(-0.594\) \\ \(c_{0}\) & \(0.330\) & \(0.449\) & \(-0.131\) & \(-0.460\) & \(a_{0}\) & \(-0.098\) & \(0.173\) & \(0.103\) & \(-0.023\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Coefficients calculated for the validation process. Figure 6: Validation results obtained using the coefficients maps to interpolate and extrapolate the new cases. In (**a**) we have results for the pressure difference cases and in (**b**) the reconstructed signals for opening area. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Elastic Modulus (_E_) (MPa)**} \\ \cline{2-10}
**Output** & \multicolumn{6}{c}{**Training**} & \multicolumn{6}{c}{**Validation**} \\ \cline{2-10}  & **0.1** & **1** & **2.15** & **3.16** & **10** & \(R_{tr}\) & **1.5** & **15** & \(R_{val}\) \\ \hline \(\Delta\)P & 89.84 & 97.03 & 96.14 & 95.42 & 82.94 & 92.27 & 97.77 & 68.39 & 83.08 \\ \(S_{v}\) & 92.76 & 93.21 & 89.35 & 92.57 & 82.81 & 90.14 & 92.48 & 94.13 & 93.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Average goodness of fit obtained for the training and validation processes. making small changes in the input signal and wait for very long times to see how these new conditions behave. We remark that our model follows trends, and it may require complementary techniques such as non-linear models to fully estimate the outputs.