## 2 Materials and Methods
### Modeling \(V\)  
#### 2.4.1 Procedure 4  
Mean extra travel time might explain extra travel time variability. The Pearson's linear correlation coefficient (\(r\)) between \(T\) values (in vector \(A_{sp_{1}}\), corresponding to \(p_{2}=10\%\) to \(80\%\)) and \(V\) values (in vector \(C_{sp_{1}}\)) is expressed as \(corr(A_{sp_{1}}\), \(C_{sp_{1}}\)). For the different \(sp_{1}\) cases, \(corr(A_{60},C_{60})=0.7054\), \(corr(A_{65},C_{65})=0.8334\), \(corr(A_{75},C_{75})=0.7361\), \(corr(A_{85},C_{85})=0.8268\), and \(corr(A_{90},C_{90})=0.7543\). There is a positive correlation between \(A_{sp_{1}}\) and \(C_{sp_{1}}\) for any \(sp_{1}\) case, thus if \(T\) increases, so does \(V\). Now, we use a polynomial equation to explain \(V\) with \(T\), i.e., the independent values in \(A_{sp_{1}}\) and the dependent values in \(C_{sp_{1}}\). We propose that the choice of the polynomial equation degree should satisfy two conditions: the mean error \(\overline{E}=mean([E_{60}\ E_{65}\ E_{75}\ E_{85}\ E_{90}])<0.1\), where \(E_{sp_{1}}\) is the error between \(C_{sp_{1}}\) and \(\mathcal{C}_{sp_{1}}\), and the correlation coefficient \(|r|>0.60\) for each vector's parameter values and \(X_{1}\). A polynomial equation of order \(n=3\) satisfies both conditions. Then, \(\mathcal{C}_{sp_{1}}(k)=b_{1}A_{sp_{1}}(k)^{3}+b_{2}A_{sp_{1}}(k)^{2}+b_{3}A_{ sp_{1}}(k)+b_{4}\); the parameters' values are shown in Table 14. Vector \(P_{b_{1}}=[0.0077\ \ldots\ 0.0168]\) contains the \(b_{1}\) parameter values for \(sp_{1}\) cases {60,65,75,85,90}, \(P_{b_{2}}=[-0.1473\ \ldots\ -0.2285]\), \(P_{b_{3}}=[0.8393\ \ldots\ 0.9467]\), and \(P_{b_{4}}=[0.1177\ \ldots\ 0.0084]\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & \multicolumn{2}{c}{**Root Mean**} & \multirow{2}{*}{**R-Squared**} & \multirow{2}{*}{**\(p\)-Value**} \\  & & **Squared Error** & & \\ \hline \(\widetilde{a}_{1}=-0.0001\ x+0.0016\) & \(6.58\times 10^{-5}\) & \(0.972\) & \(0.00199\) \\ \(\widetilde{a}_{2}=0.0033\ x-0.0189\) & \(0.00208\) & \(0.958\) & \(0.0037\) \\ \(\widetilde{a}_{3}=-0.0027\ x^{4}+0.0395\ x^{3}-0.1907\ x^{2}\) & \(0\) & \(1\) & — \\ \(+0.3565\ x-0.2353\) & \(0\) & \(1\) & — \\ \hline \hline \end{tabular}
\end{table}
Table 12: Parameters’ models and statistics (procedure 3).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & \(\widetilde{A}_{sp_{1}=60}\) & \(\widetilde{A}_{sp_{1}=65}\) & \(\widetilde{A}_{sp_{1}=75}\) & \(\widetilde{A}_{sp_{1}=85}\) & \(\widetilde{A}_{sp_{1}=90}\) \\ \hline \(a_{1}\) & \(0.0016\) & \(0.0013\) & \(0.0010\) & \(0.0009\) & \(0.0008\) \\ \hline \(a_{2}\) & \(-0.0159\) & \(-0.0127\) & \(-0.0029\) & \(-0.0010\) & \(0.0055\) \\ \hline \(a_{3}\) & \(-0.0328\) & \(-0.0129\) & \(-0.0322\) & \(0.0278\) & \(-0.0991\) \\ \hline \hline \end{tabular}
\end{table}
Table 11: Vectors \(\hat{A}_{sp_{1}}\) parameters’ values (procedure 3).  
Thus, we obtain vectors \(\hat{\mathcal{C}}_{sp_{1}}\) with parameters in Table 14. For modeling \(\widetilde{b}_{1}\), \(\widetilde{b}_{2}\), \(\widetilde{b}_{3}\), and \(\widetilde{b}_{4}\) we use a polynomial equation of order \(n=3\). For \(\widetilde{b}_{1}\), we have \(P_{b_{1}}\) (dependent values) and \(X_{1}\) (independent values), and so on for \(\widetilde{b}_{2}\), \(\widetilde{b}_{3}\), and \(\widetilde{b}_{4}\) considering \(P_{b_{2}}\), \(P_{b_{3}}\), and \(P_{b_{4}}\), respectively. The parameters' models and statistics are presented in Table 15, where the models explain well (R-squared in the range \(\sim\)87% to \(\sim\)99%) the dependent variable (a parameter). The parameters' values for estimating \(\hat{\mathcal{C}}_{sp_{1}}(k)=\widetilde{b}_{1}A_{sp_{1}}(k)^{3}+\widetilde{b} _{2}A_{sp_{1}}(k)^{2}+\widetilde{b}_{3}A_{sp_{1}}(k)+\widetilde{b}_{4}\) are presented in Table 16.