# Using Autoregressive with Exogenous Input Models to Study Pulsatile Flows
## 4 ARX Model Results  
In order to evaluate the quality of the algorithm addressed in this work, we used a multiple correlation coefficient \(R\). This percentage indicates how well \(\hat{y}_{\tau}^{t}\) explains \(y_{\tau}\)[48]:  
\[R=\left(1-\frac{\sum_{\tau=1}^{t}(y_{\tau}-\hat{y}_{\tau}^{t})^{2}}{\sum_{\tau =1}^{t}y_{\tau}^{2}}\right)\times 100 \tag{12}\]  
After the training part, we end up having forty-five \(\theta_{E,T}\) vectors per output, containing the coefficients' information. In order to construct the transfer functions, these coefficients follow the next expression (presented in Fourier transform structure) defined in [48]. \[ARX(s)=\frac{B(s)}{1-A(s)} \tag{13}\]  
where \(B(s)\) is the set of coefficients for the numerator and \(A(s)\) is the set of coefficients for the denominator. Each output has its own model: for \(S_{v}\), we employed \(A(s)\) and \(B(s)\); whereas for \(\Delta\)P, we used \(C(s)\) and \(D(s)\) to represent their respective coefficients. The models have a system order \(n_{a}\) and \(n_{c}\) for the \(a^{\prime}s\) and \(c^{\prime}s\) of the denominator and \(n_{b}\) and \(n_{d}\) for the \(b^{\prime}s\) and \(d^{\prime}s\) of the numerator of each equation. These equations were tested for \(n=[1,2,3,4]\), and we found the best goodness of fit in \(\Delta\)P when \(n_{c}=2\) and \(n_{d}=1\); whereas for \(S_{v}\), we obtained the best results when \(n_{a}=3\) and \(n_{b}=2\). Hence,  
Figure 3: (**a**) The parametric study is summarized as a for loop of five variations in \(E\) and nine in \(T\). (**b**) The system identification process uses the input/outputs signals to obtain forty-five \(\theta\) vectors. (**c**) The validation process simulates the output from new input signals for \(E^{*}\) and \(T^{*}\) representing the elastic modulus and periods that were not initially considered in the parametric study. according to Equation (13), we have the next structure for the transfer functions of each output using the hereditary ARX computation:  
\[(a)\ ARX(s)_{\Delta P}=\frac{d_{1}s+d_{0}}{s^{2}-c_{1}s-c_{0}} \tag{14}\]  
\[(b)\ ARX(s)_{S_{0}}=\frac{b_{2}s^{2}+b_{1}s+b_{0}}{s^{3}-a_{2}s^{2}-a_{1}s-a_{0}} \tag{15}\]  
The coefficients found during the identification part for all the elastic moduli proposed are presented as field maps in Figure 4a,b for \(\Delta\)P and \(S_{v}\), respectively. In both outputs, we observe that the zone around \(E=1\) MPa and \(T=2.5\) s presents the highest or the lowest values, depending on the particular behavior of each coefficient. In the case of \(\Delta\)P, we see a change close to a local extrema (\(E=10\) MPa and \(T=2.50\) s) for the \(c^{\prime}s\) coefficients, whereas for the \(d^{\prime}s\), we appreciate this change in the area close to \(E=1\) MPa and \(T=1\) s. On the other hand, the coefficients for \(S_{v}\) seem to have a constant behavior in all \(d^{\prime}s\) and \(b^{\prime}s\) coefficients surrounding the zone of \(E=1\) MPa and \(T=2.5\) s. As mentioned before, the quality of the algorithm was measured using the multiple correlation function \(R\); for that, \(R\) values of all combinations are shown in Table 3 (for more details of the entire set of reconstructed signals and the coefficients obtained for the SI model, see the Supplementary Material). In Figure 5a, the curves correspond to the reconstruction of \(\Delta\)P following the structure of Equation (14), and by comparing them with the experimental curves, their behavior follows the same trend. Conversely, Figure 5b shows that \(S_{v}\) is less accurate, since only the opening and closing moments were recovered; the reconstruction follows Equation (15). Regarding the computational time, the training process for each period took 13 s approximately; hence, obtaining the results of the nine periods for both outputs, for one elastic modulus, took less than 4 min to finish the process. Considering this time, our ARX model calculated all 45 \(\theta_{E,T}\) vectors for the training part of both outputs in about 20 min.