# Using Autoregressive with Exogenous Input Models to Study Pulsatile Flows
## 3 System Identification Method
#### 3.0.1 ARX Model  
The goal of this model is to obtain the \(\theta\) vector, which contains the parameters (coefficients) that represent the relation between the input and output of the system. \[\theta=[a_{1},\dots,a_{na};b_{1},\dots,b_{nb}]^{T} \tag{4}\]  
Hence, the process to calculate the optimal parameter vector can apply the hereditary identification principle by using the experimental expectation [47] instead of the mathematical one [48]. This principle keeps the criterion minimization task quadratic; as a result, it avoids the use of non-linear optimization techniques, such as the gradient-based or Gauss-Newton techniques to derivate and reduce the order of the coefficients. For this case, the distance (the difference) among the estimated trajectories (behavior of the signal after every iteration) is minimized. Hence, we could present:  
\[\mathbf{E}^{t}[\theta_{\bullet}|y_{\bullet}]=\frac{1}{t}\sum_{\tau=1}^{t} \theta_{\tau}^{T}y_{\tau} \tag{5}\]  
as the general form to obtain the coefficients \(\theta_{\tau}\), \(y_{\tau}\) as our main input/output signal using the experimental expectation over \([1,\dots,t]\), and \(\theta_{\bullet}=\{\theta_{\tau},\tau=1,\dots,t\}\). Now, in the case of the ARX \begin{table}
\begin{tabular}{c c c c} \hline \hline \(f\) & \(T\) & \(U_{max}\) & \(V\) \\ (Hz) & (s) & (m/s) & (m\({}^{3}\)) \\ \hline
1.00 & 1.000 & 0.1448 & 6.129 \(\times\)\(10^{-5}\) \\
0.90 & 1.111 & 0.1303 & 6.128 \(\times\)\(10^{-5}\) \\
0.80 & 1.250 & 0.1158 & 6.127 \(\times\)\(10^{-5}\) \\
0.70 & 1.429 & 0.1013 & 6.126 \(\times\)\(10^{-5}\) \\
0.66 & 1.502 & 0.0964 & 6.127 \(\times\)\(10^{-5}\) \\
0.60 & 1.667 & 0.0869 & 6.132 \(\times\)\(10^{-5}\) \\
0.50 & 2.000 & 0.0724 & 6.129 \(\times\)\(10^{-5}\) \\
0.40 & 2.500 & 0.0579 & 6.127 \(\times\)\(10^{-5}\) \\
0.33 & 3.003 & 0.0482 & 6.124 \(\times\)\(10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Velocities used for the parametric study. model, we considered \(y_{\tau}\) as the output, \(u_{\tau}\) as the exogenous input, and \(\hat{y}_{\tau}^{t}\) as the optimal past estimator of the behavior at instant \(t\) for data over \([1,t]\), and it is calculated as follows:  
\[\hat{y}_{\tau|\tau-1}^{t}=\sum_{i=1}^{n}a_{i,t}\hat{y}_{\tau-i|\tau-i-1}^{t-i}+ \sum_{i=0}^{n}b_{i,t}u_{t-i} \tag{6}\]  
Then, the regression vector that contains the coefficients to describe the behavior of the system is defined by:  
\[\varphi_{\tau}^{t-1}=[y_{\tau},\cdots,y_{\tau-n+1},\hat{y}_{\tau-1|\tau-2}^{t- 1},\cdots,\hat{y}_{\tau-n|\tau-n-1}^{t-n},u_{\tau},\cdots,u_{\tau-n+1}] \tag{7}\]  
Then, the parameter estimation is done by hereditary computation; therefore, it is required to know the experimental autocorrelation matrix of \(\varphi_{\tau}^{t-1}\) and the inter-correlation vector of \(\varphi_{\tau}^{t-1}\) and \(y_{\tau}\). Hence, to compute the system parameters \(\theta_{t}\) at instant \(t\), it is necessary to keep in memory all the past estimations \(\hat{y}_{\tau|\tau-1}^{t-1},\cdots,\hat{y}_{\tau|\tau-1}^{t-n}\). Therefore, if we were not using hereditary computation, the complexity and computational cost would be higher. The general algorithm used to estimate the \(\theta_{t}\) vector is:  
1. Initialization. As data before \(t=1\) are unknown, the projection over it is zero (division by a covariance a priori infinite). It is then possible to think of it as zero: \[\hat{y}_{\tau|\tau-1}^{1-i},\quad\forall\tau<1,\forall i=1,\cdots,n\] (8)
2. At instant \(t-1\), we already have data from the previous optimized estimations. \[\hat{y}_{\tau|\tau-1}^{t-i},\forall\tau=1,\cdots,t-1,\forall i=1,\cdots,n\]
3. At instant \(t\), new data \(y_{t},u_{t}\) are available in order to update the experimental auto-correlation matrix and the inter-correlation vector: \[\mathbf{E}^{t}[\varphi_{\bullet}^{t-1}(\varphi_{\bullet}^{t-1})^{T}]=\frac{1 }{t}\sum_{\tau=1}^{t}\varphi_{\bullet}^{t-1}(\varphi_{\bullet}^{t-1})^{T}\] (9) \[\mathbf{E}^{t}[y_{\bullet}\varphi_{\bullet}^{t-1}]=\frac{1}{t}\sum_{\tau=1}^{ t}y_{\bullet}\varphi_{\bullet}^{t-1}\] (10)
4.