of drivers, and thus parameterizes a families of velocity and flow rate curves, respectively. We construct these families of curves by generalizing the least-squares fit to a weighted least-squares fit, as follows.

Given a weight parameter $0<\beta<1$, we consider the minimization problem $$\label{eq:WLSQ}
\min_{\alpha,\lambda,p}\left\{
(1-\beta) \sum^{n}_{j=1}{\left((Q_{\alpha,\lambda,p}(\rho_{j})-Q_{j})_{+}\right)^2}+
\beta \sum^{n}_{j=1}{\left((Q_{\alpha,\lambda,p}(\rho_{j})-Q_{j})_{-}\right)^2}
\right\}\;,$$ where $$\begin{aligned}
\left(Q_{\alpha,\lambda,p}(\rho_{j})-Q_{j}\right)_{+}\
&= \max\left\{ Q_{\alpha,\lambda,p}(\rho_{j})-Q_{j},0\right\}\;, \\
\left(Q_{\alpha,\lambda,p}(\rho_{j})-Q_{j}\right)_{-}\
&= \max\left\{-Q_{\alpha,\lambda,p}(\rho_{j})+Q_{j},0\right\}\;.\end{aligned}$$ For $\beta=\frac{1}{2}$, problem reduces to , i.e., the LWR equilibrium curve is recovered. For $\beta<\frac{1}{2}$, data below the curve is penalized more, and consequently the resulting curve moves downwards. In turn, if $\beta>\frac{1}{2}$, curves above the equilibrium curve are obtained.

The weighted least-squares problem generates a one-parameter family of curves $Q_\beta(\rho) = Q_{\alpha(\beta),\lambda(\beta),p(\beta)}(\rho)$, parameterized by $\beta$; and consequently it also generates a family of velocity functions $$V_\beta(\rho) = \begin{cases}
Q_\beta(\rho)/\rho &\text{if~}\rho>0 \\
\frac{\partial Q_\beta}{\partial\rho}(0) &\text{if~}\rho=0 \end{cases}\;.$$ In this paper, we restrict to the case that the velocity curves in the family are non-intersecting, i.e., $$\label{eq:non_intersecting}
\text{If~}\beta_1<\beta_2\;,\text{~~then~~}
V_{\beta_1}(\rho)<V_{\beta_2}(\rho)
\text{~~for~~}\rho\in [0,\rho_{\text{max}})\;.$$ Note that in general (i.e., for a general family of flow rate functions, and for general data points), property is not necessarily satisfied; and furthermore, problem need not have a unique solution. However, in all cases studied in this paper, the minimization 