## 2 Materials and Methods
### Modeling \(V\)
#### 2.4.2 Procedure 5  
The linear correlation between \(p=[10\ 20\ \dots\ 70\ 80]\) (\(p_{2}\) values) and \(C_{sp_{1}}\) (\(V\) values) for \(sp_{1}\) cases [60,65,75,85,90] is always positive and in the range of \(\sim 0.85\) to \(\sim 0.93\). As the percentage of slow vehicles increases, extra travel time variability increases. Now, with a polynomial equation, the dependent variable \(V\) is explained by the independent variable \(p_{2}\). For the polynomial degree (starting with \(n=1\)), we seek to meet two conditions (the same as in procedure 4): (1) \(\overline{E}<\)0.1 and (2) \(|\mathbf{r}|>0.60\) for each \(P_{b}\) vector. \(\overline{E}=mean([E_{60}\ E_{65}\ E_{75}\ E_{85}\ E_{90}\ ])\), where \(E_{sp_{1}}\) is the error between \(C_{sp_{1}}\) and \(\widetilde{C}_{sp_{1}}\), and \(r=corr(P_{b},X_{1})\) is the linear correlation coefficient between \(P_{b}\) (a vector containing the \(b\) parameter values corresponding with the \(sp_{1}\) cases) and \(X_{1}\). With \(n=5\), both conditions are met. The parameters' values for modeling \(\tilde{\mathcal{C}}_{sp_{1}}(k)=b_{1}p_{2}{}^{5}+b_{2}p_{4}{}^{4}+b_{3}p_{2}{} ^{3}+b_{4}p_{2}{}^{2}+b_{5}p_{2}+b_{6}\) are presented in Table 17. The \(b_{1}\) parameter values are in vector \(P_{b_{1}}\), the \(b_{2}\) parameter values are in vector \(P_{b_{2}}\), and so on.  
Each vector \(P_{b}\) was modeled with a linear equation, with \(X_{1}\) as the independent values and \(P_{b}\) as the dependent values. The models and statistical results are shown in Table 18.  
With the models in Table 18, we calculate each parameter for \(X_{2}\) values (corresponding to \(sp_{1}\) cases \(\{70,80\}\)). The parameters' values for describing \(\hat{\mathcal{C}}_{sp_{1}}(k)=\widetilde{b}_{1}p_{2}{}^{5}+\widetilde{b}_{2}p_ {2}{}^{4}+\widetilde{b}_{3}p_{2}{}^{3}+\widetilde{b}_{4}p_{2}{}^{2}+ \widetilde{b}_{5}p_{2}+\widetilde{b}_{6}\) are presented in Table 19.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**Root Mean**} & \multirow{2}{*}{**R-Squared**} & \multirow{2}{*}{**\(p\)-Value**} \\  & & **Squared Error** & & \\ \hline \(\widetilde{b}_{1}=-0.0006\ x^{3}+0.0086\ x^{2}-0.0305\ x+0.0317\) & 0.00624 & 0.873 & 0.445 \\ \hline \(\widetilde{b}_{2}=0.0062\ x^{3}-0.0839\ x^{2}+0.303\ x-0.3799\) & 0.0313 & 0.961 & 0.249 \\ \hline \(\widetilde{b}_{3}=-0.0161\ x^{3}+0.2199\ x^{2}-0.8209\ x+1.452\) & 0.0192 & 0.997 & 0.0692 \\ \hline \(\widetilde{b}_{4}=0.0038\ x^{3}-0.0511\ x^{2}+0.1695\ x-0.0015\) & 0.0128 & 0.991 & 0.119 \\ \hline \hline \end{tabular}
\end{table}
Table 15: Parameters’ models and statistics (procedure 4).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \(\tilde{\mathcal{C}}_{sp_{1}=60}\) & \(\tilde{\mathcal{C}}_{65}\) & \(\tilde{\mathcal{C}}_{75}\) & \(\tilde{\mathcal{C}}_{85}\) & \(\tilde{\mathcal{C}}_{90}\) \\ \hline \(b_{1}\) & 0.0077 & 0.0031 & 0.0022 & 0.0221 & 0.0168 \\ \hline \(b_{2}\) & \(-0.1473\) & \(-0.0763\) & \(-0.0943\) & \(-0.2551\) & \(-0.2285\) \\ \hline \(b_{3}\) & 0.8393 & 0.5507 & 0.6649 & 0.9454 & 0.9467 \\ \hline \(b_{4}\) & 0.1177 & 0.1706 & 0.0992 & 0.0200 & 0.0084 \\ \hline \hline \end{tabular}
\end{table}
Table 14: Vectors \(\hat{\mathcal{C}}_{sp_{1}}\) parameters’ values (procedure 4).