# Comparison of Traffic Flow Models with Real Traffic Data Based on a Quantitative Assessment
## 4 Results
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Criterion** & **Equation** & **Explanation** \\ \hline \multirow{3}{*}{2 Simplicity} & np—number of parameters & ne—number of equations \\  & \(\text{ca}_{2}=\text{w}_{21}\Big{(}\frac{\text{max}(\text{np})\text{-mp}_{1}}{ \text{max}(\text{np})\text{-min}(\text{np})}\Big{)}+\text{w}_{22}\text{ne}_{1}^ {-1}\) & i—the analyzed model \\  & \(\text{w}_{21},\text{w}_{22}\)—weights (sum up to 1) \\
3 Empirical accuracy & \(\text{ca}_{3}=\frac{\text{max}(\text{MAPE})\text{-MAPE}_{1}}{\text{max}( \text{MAPE})}\) & i—the analyzed model \\  & \(\text{ca}_{3}=\frac{\text{max}(\text{RMSE})\text{-RMSE}}{\text{max}(\text{ RMSE})\text{-min}(\text{RMSE})}\) & \\
4 Traffic flow parameters & \(\text{ca}_{4}=\frac{\text{w}_{5}}{5}\) & \(\text{nve}_{1}\)—number of the parameters of model i, \\  & which are estimated correctly \\ \hline Final assessment & \(\text{ca}=\text{w}_{2}\text{ca}_{2}+\text{w}_{3}\text{ca}_{3}+\text{w}_{4}\text{ ca}_{4}\) & \(\text{w}_{1},\text{w}_{2},\text{w}_{3}\)—weights (sum up to 1) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Formulas for model assessment and comparison—criteria 2, 3 and 4. \begin{table}
\begin{tabular}{l l c c c c c c c c c c} \hline  & \multicolumn{3}{c}{Assessment Criteria} & \multicolumn{3}{c}{Single-Regime Models—Group 1} & \multicolumn{3}{c}{Single-Regime Models—Group 2} & \multicolumn{3}{c}{Two-Regime Models} \\ \cline{3-11}  & Assessment Criteria & Greenshields & Drew & Pipes-Murjal & Newell & [PERSON] & [PERSON] & [PERSON] & [PERSON] & [PERSON] \\ \hline
1 & \multicolumn{3}{c}{_Statistical boundary conditions_} & & & & & & & & & \\  & BC1 & + & + & + & + & + & + & + & + & + \\  & BC2 & + & + & + & + & + & + & + & + & + \\  & BC3 & n.a. & n.a. & n.a. & n.a. & n.a. & n.a. & n.a. & + & + & + \\ \hline
2 & \multicolumn{3}{c}{_Simplicity_} & & & & & & & & & & \\  & No. of parameters & 2 & 3 & 3 & 3 & 3 & 4 & 3 & 3 & 7 \\  & No. of equations & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 \\  & \(\rm c_{2}\) & 1.00 & 0.90 & 0.90 & 0.90 & 0.90 & 0.80 & 0.65 & 0.65 & 0.25 \\ \hline
3 & \multicolumn{3}{c}{_Empirical accuracy_} & & & & & & & & & \\  & RMSE (nm/s) & 7.95 & 7.86 & 5.64 & 6.29 & 6.29 & 6.04 & 6.00 & 22.58 & 6.85 \\  & MAPE (\(\rm c_{3}\)) & 13.91 & 14.12 & 14.12 & 10.56 & 10.56 & 9.93 & 10.06 & 35.27 & 11.72 \\  & \(\rm c_{3}\) & 0.84 & 0.83 & 0.83 & 0.98 & 0.98 & 1.00 & 0.99 & 0.00 & 0.93 \\ \hline
4 & \multicolumn{3}{c}{_Parameter value (reported value)_} & & & & & & & & \\  & \(\rm v_{w}\) (km/h) & 121 & 109 & 118 & 109 & 109 & 110 & 114 & 100 & 120 \\  & \(\rm v_{w}\) (km/h) & 60 & 69 & 66 & 67 & 65 & 69 & 90 & 99 & 86 \\  & \(\rm k_{w}\) (pc/km) & 138 & - & 134 & 160 & 160 & 370 & 221 & 383 & 182 \\  & \(\rm k_{w}\) (pc/km) & 69 & 49 & 67 & 56 & 62 & 58 & 45 & 47 & 49 \\  & \(\rm q_{max}\) (pc/h) & 4153 & 4010 & 4198 & 4001 & 4005 & 4010 & 4080 & 3936 & 4214 \\  & \(\rm c_{4}\) & 0.00 & 0.20 & 0.20 & 0.40 & 0.20 & 0.40 & 0.20 & 0.20 & 0.40 \\ \hline Final assessment & 0.61 & 0.64 & 0.64 & 0.76 & 0.69 & 0.73 & 0.61 & 0.28 & 0.53 \\ \hline \end{tabular}  
* The highlighted cells (criterion 4) mean that the estimated values of the parameters are consistent with the expected value ranges (Table 3). **–** Weights for the criteria assessment were adopted as follows: \(w_{1}=w_{2}=0.55,w_{1}=w_{3}=w_{4}=0.33\).  
\end{table} 
Table 5: Assessment and comparison of traffic flow models for the field data. Figure 6: Fundamental diagram represented by single-regime models—group 1. Figure 7: Fundamental diagram represented by single-regime models—group 2. If we accept the failure to satisfy BC2, [PERSON]'s, [PERSON]'s, and [PERSON]'s models are worth considering. The first two perform relatively well on \(q_{max}\), traffic parameter representation, which is a feature other BC2 models did not meet (Table 6). However, all come with some limitations. [PERSON] and [PERSON]'s model carries a relatively big estimation error.