## 5 Experimental Setup and Discussion  
In this section, we test the performance of the proposed framework to estimate traffic dynamics in a traffic network based on the LWR model. Traffic data are obtained from the circular traffic network shown in Figure 3 using PTV VISSIM. The length of each link is set to 15,015 m.  
In this study, we use synthetic demand to simulate the traffic in VISSIM. The traffic input flows are randomly assigned, for the initial 900 s of a simulation run, to the starting points of Links L1, L2 and L3 are 3000, 1000, and 2000 vehicles per hour, respectively, as presented in Table 1. In addition, to generate more dynamics in traffic behavior, speed reduction areas are activated for some time during the simulation run as well as activating a stochastic behavior of the vehicles. After the initial 900 s of a simulation run, the traffic is allowed to stabilize in the network for another 300 s, and trajectory data of the following 800 s are used in this study. We considered a closed traffic network in this study to observe the complete shockwave and congestion spillover in the network. To estimate the macroscopic traffic flow parameters, the cell size in the time-space diagram is set to 5 s along the time dimension, and 5 m along the space dimension. Thus, the size of the obtained uniform grid is 160 \(\times\) 3003. Five seconds is a common time step length for realistic speed data. For instance, we have real world speed data every five seconds for the major freeways from Alabama Department of Transportation. Some widely used datasets such as PeMS in California also provide speed data every five second. Therefore, we choose 5 s as \(\Delta\)T. The segment length is on trial basis. It is not too short to have the frequent empty cells with no detected vehicles. It is not too long to have significant heterogeneity on traffic states within each road segment. In the obtained data, there are cells, named empty cells, where the vehicles are not detected. The flow and density in the empty cells are zero, but the speed should be either close to free flow or the average speed of nearby cells. For simplicity, the speed in the empty cells is set to constant between two cells. The speed contour plot of all three links before and after filling the empty cells is presented in Figure 4a,b, respectively. The speed, density, and flow data of the network are explored in Figure 5. The figure shows that flow-density-speed follows a similar pattern; however, the jam density and peak flows are different in various links of the network. Therefore, we left this part on the framework to estimate the best-fit parameters. The LWR model is given by \(\partial_{t}\rho+\partial_{x}V[\rho]=0\). The flux function \(V=\rho v\) is described as a speed-density function, and density can also be described as a function of velocity. Figure 5 shows a nonlinear relationship between speed and density. However, we consider a linear relationship following [PERSON]'s Model, i.e., \(\rho=\lambda_{1}\,v+\lambda_{2}\). The flux function is given by \(V(v)=v(\lambda_{1}\,v+\lambda_{2})\), and the resulting LWR \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Link** & **Interval** & **Traffic Volume** & **Volume Type** \\ \hline \multirow{3}{*}{L1} & 0–900 & 3000 & Stochastic \\  & 900–1800 & 0 & Stochastic \\  & 2700–3600 & 0 & Stochastic \\ \hline \multirow{3}{*}{L2} & 0–900 & 1000 & Stochastic \\  & 900–1800 & 0 & Stochastic \\  & 1800–2700 & 0 & Stochastic \\  & 2700–3600 & 0 & Stochastic \\ \hline \multirow{3}{*}{L3} & 0–900 & 2000 & Stochastic \\  & 900–1800 & 0 & Stochastic \\  & 1800–2700 & 0 & Stochastic \\  & 2700–3600 & 0 & Stochastic \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Synthetic traffic demand assigned to links in the network.