Spacing-speed dependency on relative speeds to the adjacent lanes: a statistical test

Abstract

The spacing of a vehicle is the spatial separation between this vehicle and the vehicle ahead. It forms the basis of the fundamental diagram of traffic flow. Traditional car-following models assume that the spacing of a following vehicle is strictly a function of the speed of the lead vehicle in the same lane. The recent literature, however, reports a phenomenon of spacing-speed dependency on the relative speeds to the adjacent lanes, termed speed differential effect. The aim of this paper is to statistically test the speed differential effect. To quantitatively measure the speed differential effect, we develop a mixture spacing-speed model to capture the probabilistic nature of the spacing and speed relationship, and then relate this spacing-speed relation to the relative speeds to the adjacent lanes. We apply the developed model to test the speed differential effect. Our empirical analysis provides strong statistical evidence to support the speed differential effect.

KEYWORDS:

• Spacing-speed relation
• speed differential effect
• traffic flow theory

1. Introduction

The spacing of a vehicle is the spatial separation between this vehicle and the vehicle ahead. In practice, vehicle spacing plays a key role in road safety for avoiding rear-end collision. One example of early research on inter-vehicle spacing can be traced back to the study in Sun and Ioannou (1995). It concerns with the minimum safety spacing for collision-free vehicle following. In the recent years, due to the rapid development in automated/autonomous driving, quantifying vehicle spacings has become crucial for the design of automated/autonomous driving algorithms to enhance safety. As a result, a lot of attention has been paid to the research on vehicle spacing in the recent years. For example, Zhao, Oya, and El Kamel (2009) propose a safety spacing policy that ensures safe operation while at the same time improves traffic flow for adaptive cruise control. Campolo et al. (2017) investigate platooning control toward autonomous driving in which spacing and speed in a platoon of vehicles are examined. Xu et al. (2019) investigate dynamic cooperative automated lane-change manoeuvre based on a minimum safety spacing model.

From a theoretical perspective, the spacing-speed relation forms the basis of the fundamental diagram of traffic flow (Gartner, Messer, and Rathi 2001; Coifman 2014). It helps us better understand the inter-dependence among several important traffic variables such as traffic density, vehicle speed, and traffic volume. Except for lane-change models, however, almost all traffic flow models assume that the driving behaviour in one lane is independent of conditions in the adjacent lanes (Coifman and Ponnu 2020).

Ponnu and Coifman (2015) discover that the spacings drivers adopt are influenced by the differentials in speed between lanes, termed the speed differential effect. More specifically, they empirically demonstrate that the fundamental relationship in a high-occupancy-vehicle (HOV) lane can change shape in a systematic response to conditions in the adjacent general purpose (GP) lane. This finding suggests that the spacing-speed relation in an HOV lane is partially influenced by the speed relative to its adjacent GP lane; this is contrary to the existing traffic theory that assumes the longitudinal acceleration of a following vehicle is strictly a function of the relationship to the lead vehicle in the same lane (see, e.g. Brackstone and McDonald 1999; Gartner, Messer, and Rathi 2001; Rakha and Gao 2010).

As pointed out by Ponnu and Coifman (2015), speed differentials are a subtle effect which is not easy to detect in any macroscopic analysis. In fact, to detect this subtle effect, Ponnu and Coifman (2015) focus on an HOV lane against its immediately adjacent GP lane where the speed differential effect is the greatest and is much higher than the speed differential between the GP lanes. In addition, they make use of a very large dataset and also carefully control for many irrelevant traffic factors via the single vehicle passage (svp) method. More specifically, the data used in Ponnu and Coifman (2015) consists of traffic measurements across several eastbound dual loop detection stations over 69 weekdays for a two mile stretch with five lanes near Oakland. With the svp method developed in Coifman (2014), they group vehicles according to their traffic characteristics to control for potential confounding factors and to reduce random noise, so that a clean and meaningful fundamental relationship can be revealed. In the literature, various other vehicle grouping methods are also used based on driving behaviour (e.g. Daganzo 2002) or vehicle class (e.g. Punzo and Tripodi 2007; Yang et al. 2013). A serious consequence of using the svp method, however, is that it may lead to a very small sample size after vehicle grouping and data aggregation. For example, the final sample size for the spacing-speed regression analysis in Ponnu and Coifman (2015) reduces extraordinarily to 25, 29, 22, 17, 10 and 4 respectively for the evening peak hours, and to 20, 25, and 4 respectively for a non-peak period. Apparently, the reduced sample sizes make formal statistical testing difficult, if not possible, to detect subtle relations.

Ponnu and Coifman (2015) focus on an HOV lane and its immediately adjacent GP lane when investigating the speed differential effect. Extending the analysis to interior GP lanes is more challenging because a GP lane usually cannot sustain a high relative speed to both of its immediately adjacent lanes. Recently, Coifman and Ponnu (2020) have investigated speed differential effect for GP lanes. They present empirical findings that show drivers in a study lane become more conservative in response to slower speeds in either adjacent lane and conversely, they become less conservative in response to higher speeds in either adjacent lane. Coifman and Ponnu (2020) have used the same analytical approach applied in Ponnu and Coifman (2015), i.e. the svp method, to detect the speed differential effect for GP lanes. As it is the case in Ponnu and Coifman (2015), however, the small sample sizes resulted from the svp method in Coifman and Ponnu (2020) make it difficult to statistically test the speed differential effect and to accurately quantify the impact of the speed differential effect on the spacing-speed relation.

In this paper, rather than to use the svp method that may lead to the small-sample problem, we will develop a new statistical approach, i.e. the mixture spacing-speed model, that allows us to statistically control for confounding factors and to test the speed differential effect. This mixture spacing-speed model is then further related to the relative speeds to the adjacent lanes to quantitatively measure the speed differential effect. The developed approach to quantifying the speed differential effect may help scientists/engineers develop algorithms for automated/autonomous driving in future research, like the studies in Zhao, Oya, and El Kamel (2009), Campolo et al. (2017) and Xu et al. (2019).

Our empirical analysis provides strong evidence that the speed differential effect is a phenomenon that may exist across GP lanes, even when the speed difference between two GP lanes is relatively small. We also investigate the scenarios where the two immediately adjacent lanes of a study lane exhibit some complicated traffic conditions including: (a) the speed of the inner GP lane is much higher than the study lane, whereas the speed of the outer GP lane is much lower than the study lane; and (b) the speed differential between the inner GP lane and the study lane is of a similar magnitude to the speed differential between the outer GP lane and the study lane.

This paper is structured as follows. In the next section, we first investigate probabilistic modelling for the spacing-speed relation of traffic flow, and then, to quantify the speed differential effect, we further relate this spacing-speed relation to the relative speeds to the adjacent lanes. With the developed model, we statistically test the speed differential effect in Section 3. This paper concludes in Section 4 with discussion about future research. The proof of the theorem is given in Appendix A.

2. Probabilistic models for the spacing-speed relation with speed differentials

In this section, we first develop a new probabilistic approach to the modelling of the spacing-speed relation for traffic flow, and then we extend it to account for the speed differential effect.

2.1. Probabilistic model for the spacing-speed relation

Consider a number of consecutive vehicles, indexed by $n-1,n,n+1,\dots$, travelling on a roadway. For each vehicle $n$, let vehicle $n-1$ denote the vehicle ahead. In addition, let $v_n$ denote the travelling speed of vehicle $n$ and $s_n$ the spacing between vehicles $n$ and $n-1$.

The mixture spacing-speed model to be developed in this paper is inspired by Newell’s car-following model (2002) below: (1) $s_n=d_n+{\tau }_n{v}_{n-1}$(1) where $v_{n-1}$ is the speed of the lead vehicle. ${\tau }_n$ and $d_n$ are time and space displacements of vehicle $n$. Equation (1)(1) $s_n=d_n+{\tau }_n{v}_{n-1}$(1) is a linear, deterministic model of the spacing-speed relation for vehicles travelling in the car-following mode, although Newell (2002) also considers sampling issues and measurement noise.

To statistically test the speed differential effect, the Newell’s model needs to be transformed from its deterministic form to a stochastic one to account for the stochastic nature of traffic flow. For this end, we follow Newell (2002) and consider a sample of $N$ vehicles, each with space and time displacements $d_n$ and ${\tau }_n$ ($n=1,\dots ,N$) respectively. Newell (2002) defines average space displacement and average time displacement as their sample means, i.e. $\overline{d}=\frac{1}{N}\sum _{n=1}^N{d}_n$ and $\overline{\tau }=\frac{1}{N}\sum _{n=1}^N{\tau }_n$, respectively. For probabilistic modelling purposes, we also consider the corresponding population values of the average space displacement and average time displacement as follows: $d:=E\overline{d}=E\left[\frac{1}{N}\sum _{n=1}^N{d}_n\right]\text{and}\tau :=E\overline{\tau }=E\left[\frac{1}{N}\sum _{n=1}^N{\tau }_n\right],$ where $E\left(x\right)$ represents the mathematical expectation of a random variable $x$.

Now, we decompose the spacing $s_n$ in Equation (1)(1) $s_n=d_n+{\tau }_n{v}_{n-1}$(1) into the systematic component, ${\overline{s}}_n:=d+\tau v_{n-1}$, plus a zero-mean stochastic component $e_{1n}$, i.e. (2) $s_n={\overline{s}}_n+e_{1n}=d+\tau v_{n-1}+e_{1n},$(2) where, following Newell (2002), we assume that the error term $e_{1n}$ is normally distributed with zero mean and variance ${\sigma }_1^2$. Under the car-following condition, the stochastic version of Newell’s car-following model can be written as $s_n|v_{n-1}\sim N\left({\overline{s}}_n,{\sigma }_1^2\right)$, with probability density function $g_1\left(s_n|v_{n-1}\right)=\phi \left(s_n;{\overline{s}}_n,{\sigma }_1^2\right)$, where $\phi \left(s;\mu ,{\sigma }^2\right)=\frac{1}{\sigma \sqrt{2\pi }}\text{exp}\left\{-\frac{{\left(s-\mu \right)}^2}{{\sigma }^2}\right\}$ denotes the probability density function of the normal distribution with mean $\mu$ and variance ${\sigma }^2$. We use $\mathrm{\Phi }\left(s;\mu ,{\sigma }^2\right)$ to denote the corresponding cumulative distribution function. The error term $e_{1n}$ in Equation (2)(2) $s_n={\overline{s}}_n+e_{1n}=d+\tau v_{n-1}+e_{1n},$(2) stands for random noise that consists of the measurement error, un-modelled determinants of the spacing, etc.

Real traffic is usually composed of vehicles that follow their lead vehicle and of vehicles that travel at the free speed. To account for vehicles travelling under the free-flowing condition, we consider a vehicle $n$ travelling at the constant, free speed $v_F$. Let $h_n$ denote vehicle time headway between vehicles $n$ and $n-1$. We suppose that the spacing $s_n$ follows a normal distribution for the given time headway $h_n$, i.e. $s_n|h_n\sim N\left(v_F{h}_n,{\sigma }_F^2\right)$ with mean $v_F{h}_n$ and variance ${\sigma }_F^2$. In the literature, a widely used vehicle time headway distribution is Cowan’s M3 model (Cowan 1975). Under the free-flowing condition, the Cowan’s M3 model stipulates a shifted exponential distribution $\text{Exp}\left(\lambda ,\kappa \right)$ for time headway $h_n$: $p\left(h_n\right)=\left(1/\lambda \right)\text{exp}\left(-\left(h_n-\kappa \right)/\lambda \right)$ for $h_n\ge \kappa$, where $\kappa$ is the minimum time headway and $1/\lambda$ is the arrival rate of vehicles. The following theorem gives the distribution of vehicle spacings under the free-flowing condition.

Theorem 1.

For traffic under the free-flowing condition, if: (a) vehicle time headway $h_n$ follows a shifted exponential distribution $\text{Exp}\left(\lambda ,\kappa \right)$; and (b) spacing $s_n$ follows a normal distribution $N\left(v_F{h}_n,{\sigma }_F^2\right)$ for a given time headway $h_n$, then the unconditional distribution of spacing is given by (3) $g_2\left(s_n\right)=(\lambda v_F{)}^{-1}\exp \left(-\frac{s_n-\kappa v_F-{\sigma }_F^2/\left(2\lambda v_F\right)}{\lambda v_F}\right)\mathrm{\Phi }\left\{\frac{s_n-\kappa v_F-{\sigma }_F^2/\left(\lambda v_F\right)}{{\sigma }_F}\right\}$(3)

See Appendix A for proof. We point out that the distribution in Equation (3)(3) $g_2\left(s_n\right)=(\lambda v_F{)}^{-1}\exp \left(-\frac{s_n-\kappa v_F-{\sigma }_F^2/\left(2\lambda v_F\right)}{\lambda v_F}\right)\mathrm{\Phi }\left\{\frac{s_n-\kappa v_F-{\sigma }_F^2/\left(\lambda v_F\right)}{{\sigma }_F}\right\}$(3) is an instance of the exponentially modified Gaussian (EMG) distribution (Grushka 1972); see Appendix B for further discussion about the EMG distribution $g_{\mathrm{EMG}}\left(s;\gamma ,\rho ,\sigma \right)$. Following Equation (A1) in Appendix B, we obtain the mean and variance of vehicle spacings under the free-flowing condition: ${\mu }_2:=E{s}_n=\left(\lambda +\kappa \right)v_F\text{and}{\sigma }_2^2:=\text{var}\left(s_n\right)={\sigma }_F^2+(\lambda v_F{)}^2$ Now, combining the two spacing models under the car-following and free-flowing conditions together, we may obtain a mixture vehicle spacing model for traffic flow that consists of both car-following and free-flowing vehicles. Specifically, let $\theta$ be the probability that a vehicle is under the car-following condition. Then, by the law of total probability, for a given $v_{n-1}$, the conditional distribution of vehicle spacing $s_n$ for the traffic stream is (4) $\begin{array}{rl}g\left(s_n|v_{n-1}\right)& =\theta g_1\left(s_n|v_{n-1}\right)+\left(1-\theta \right)g_2\left(s_n\right),\\ & =\mathrm{\theta \phi }\left(s_n;d+\tau v_{n-1},{\sigma }_1^2\right)+\left(1-\theta \right)(\lambda v_F{)}^{-1}\\ & \times \exp \left(-\frac{s_n-\kappa v_F-{\sigma }_F^2/\left(2\lambda v_F\right)}{\lambda v_F}\right)\mathrm{\Phi }\left\{\frac{s_n-\kappa v_F-{\sigma }_F^2/\left(\lambda v_F\right)}{{\sigma }_F}\right\}\end{array}$(4) We may rewrite Equation (4)(4) $\begin{array}{rl}g\left(s_n|v_{n-1}\right)& =\theta g_1\left(s_n|v_{n-1}\right)+\left(1-\theta \right)g_2\left(s_n\right),\\ & =\mathrm{\theta \phi }\left(s_n;d+\tau v_{n-1},{\sigma }_1^2\right)+\left(1-\theta \right)(\lambda v_F{)}^{-1}\\ & \times \exp \left(-\frac{s_n-\kappa v_F-{\sigma }_F^2/\left(2\lambda v_F\right)}{\lambda v_F}\right)\mathrm{\Phi }\left\{\frac{s_n-\kappa v_F-{\sigma }_F^2/\left(\lambda v_F\right)}{{\sigma }_F}\right\}\end{array}$(4) as the following more familiar format, with two separately regression equations, one for each traffic mode: (5) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+e_{1n}\text{ with probability }\theta \\ {\text{β}}_{20}+e_{2n}\text{ with probability }1-\theta \end{array},$(5) where ${\text{β}}_{10}=d$ and ${\text{β}}_{11}=\tau$ are regression coefficients of the first component of the mixture model. $e_{1n}$ is a zero-mean error term with variance ${\sigma }_1^2$ that follows a normal distribution i.e. $e_{1n}\sim N\left(0,{\sigma }_1^2\right)$. In addition, ${\text{β}}_{20}={\mu }_2=\left(\lambda +\kappa \right)v_F$ is the regression coefficient of the second component. $e_{2n}$ is a zero-mean error term with variance ${\sigma }_2^2$ that follows a shifted EMG distribution $g_{\text{shiftedEMG}}\left(s;\gamma ,\sqrt{{\sigma }_2^2-{\gamma }^2}\right)$, with a shape parameter $\gamma =\lambda v_F$. See Equation (A2) in Appendix B for the mathematical expression of the probability density function $g_{\text{shiftedEMG}}\left(s;\gamma ,\sigma \right)$.

In this paper, Equation (5)(5) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+e_{1n}\text{ with probability }\theta \\ {\text{β}}_{20}+e_{2n}\text{ with probability }1-\theta \end{array},$(5) is termed a mixture spacing-speed model. Unlike the spacing-speed relations in the existing studies, this model probabilistically accounts for both the car-following and free-flowing driving modes of a traffic stream. In some applications, the mixture model (5) with two traffic modes may collapse to a single-mode model under some certain traffic conditions. In particular, it includes the following two important special cases:

1. $\theta =0$ for very light traffic, e.g. during early morning periods and/or on rural roads;

2. $\theta =1$ for very heavy traffic, e.g. during peak hours in some metropolitan areas. Note that this is the case that many existing studies consider, e.g. Duret, Buisson, and Chiabaut (2008) and Ma and Ahn (2008).

We point out that the mixture spacing-speed model (5) follows an existing research line in the traffic literature where the car-following and free-flowing travelling behaviours are modelled as a unified, mixture model; see, e.g. Buckley (1968) and Cowan (1975). In contrast, some existing studies in the literature do not consider the spacing-speed relation in separate modes; rather, they use a single equation, e.g. Equation (2)(2) $s_n={\overline{s}}_n+e_{1n}=d+\tau v_{n-1}+e_{1n},$(2) , to fit all data on vehicle spacings and speeds (see, e.g. Duret, Buisson, and Chiabaut 2008; Ma and Ahn 2008). As we will see in the empirical analysis in the next section, the developed mixture spacing-speed model is able to better control for the variation coming from different types of driving behaviour for statistical testing purposes, where drivers in different travelling modes respond to speed differentials differently.

When traffic becomes more and more congested, vehicles are forced to reduce their speed to a relatively low level, and hence it becomes impossible for vehicles to drive at the free speed; rather, the speed of any vehicle in this case is influenced by the vehicle ahead to some extent, depending on its spacing. In the literature, some researchers have identified three different traffic states: free, synchronised, and congested (see, e.g. Kerner 2021, and the references therein), where in the synchronised flow mode, there is a tendency towards synchronisation of vehicle speeds in each of the road lanes. In this case, model (5) needs to be modified to reflect such synchronisation of vehicle speeds. Here we consider two broad categories of vehicles, where vehicles in the first driving mode closely follow its lead vehicle, whereas vehicles in the second driving mode follow the vehicle ahead more loosely. Consequently, we extend Equation (5)(5) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+e_{1n}\text{ with probability }\theta \\ {\text{β}}_{20}+e_{2n}\text{ with probability }1-\theta \end{array},$(5) as follows: (6) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+e_{1n}\mathrm{with}\mathrm{probability}\theta \\ {\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}+e_{2n}\mathrm{with}\mathrm{probability}1-\theta \end{array}$(6)

Clearly, Equation (6)(6) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+e_{1n}\mathrm{with}\mathrm{probability}\theta \\ {\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}+e_{2n}\mathrm{with}\mathrm{probability}1-\theta \end{array}$(6) reduces to model (5) when ${\text{β}}_{21}=0$, and hence model (6) can accommodate a more general traffic scenario than Equation (5)(5) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+e_{1n}\text{ with probability }\theta \\ {\text{β}}_{20}+e_{2n}\text{ with probability }1-\theta \end{array},$(5) . In practice, we may start our modelling with the more general model (6), where model selection between model (5) and model (6) may be undertaken by testing the hypothesis $H_0:{\text{β}}_{21}=0$. It should be noted that, by definition, vehicles in the second traffic mode of model (6) no longer travel at the free speed; rather, they loosely follow the vehicle ahead. The shift from model (5) to model (6) reflects the traffic scenario where vehicles begin the tendency towards synchronisation of their speeds.

It is of interest to note that the error terms $e_{1n}$ and $e_{2n}$ in Equation (6)(6) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+e_{1n}\mathrm{with}\mathrm{probability}\theta \\ {\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}+e_{2n}\mathrm{with}\mathrm{probability}1-\theta \end{array}$(6) are distributed very differently. For vehicles in the first driving mode that closely follow its lead vehicle, the error term $e_{1n}$ is assumed to follow a normal distribution for which its tail probabilities will vanish very rapidly. This means that almost all vehicles travelling in this driving mode will closely follow the spacing-speed relation ${\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}$. More precisely, according to the properties of normal distributions, almost all vehicles in this driving mode are within two standard deviations ($2{\sigma }_1$) of either side of the spacing-speed relation. On the other hand, for vehicles travelling in the second driving mode that follow the vehicle ahead more loosely, the error term $e_{2n}$ is assumed to have a shifted EMG distribution in Equation (A2). This distribution has a very long right tail (see Figure A1 in Appendix B). Hence, vehicles in this driving mode usually have a spacing varying in a much wider range. This means that a substantial number of vehicles retain a long spacing. Overall, when traffic is relatively heavy (which is usually the case of interest), each driver, no matter which mode she/he is in, has a vehicle ahead, where drivers in the first mode track the lead vehicle very closely, and drivers in the second mode have a substantially larger spacing than those in the first mode. In such a situation, drivers drive in the second mode are usually more cautious than that of the first mode, because they choose not to closely track the vehicle ahead and keep a substantial separation on purposes.

In the statistics literature, the mixture model (6) belongs to an important class of models, i.e. finite mixture of regression; see, e.g. Skrondal and Rabe-Hesketh (2004), for an overview of finite mixture of regression. We note, however, that in statistics, usually both of the two component distributions in mixture regression are chosen to be normal distributions (Skrondal and Rabe-Hesketh 2004). To the best of our knowledge, mixture regression of a normal and an EMG component distribution has never been investigated in both of the statistics and traffic literature. Since the EMG is highly skewed, as shown in Figure A1 in Appendix B, replacing the EMG distribution in the mixture model (5) or (6) with a normal distribution will lead to biased parameter estimation.

Models (5) and (6) involve several parameters and hence they need to be estimated in practical applications. We briefly discuss the parameter estimation in Appendix C.

2.2. The spacing-speed relation with the speed differential effect

In this subsection, we consider the spacing-speed modelling in relation to the speed differential effect. To quantitatively measure the speed differential effect, we relate the mixture spacing-speed relation in the previous subsection to the relative speeds to the adjacent lanes, through which we further consider the statistical testing of the speed differential effect.

To start with, we define the speeds of a study lane relative to its two adjacent lanes as follows: $u_{n-1}^O=v_{n-1}-v_{n-1}^O\text{and}{u}_{n-1}^I=v_{n-1}-v_{n-1}^I,$ where for each vehicle $n$ in the study lane, $v_{n-1}$ is the speed of vehicle $n-1$ (i.e. the vehicle ahead) travelling in the study lane. $v_{n-1}^O$ (or $v_{n-1}^I$) is the vehicle speed of the immediately preceding vehicle in the outer (or inner) adjacent lane. $u_{n-1}^O$ (or $u_{n-1}^I$) therefore is the speed of the study lane relative to the outer (or inner) lane. Note that when the study lane is the innermost or outermost lane, there is only one adjacent lane.

Now consider the spacing-speed Equation (6)(6) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+e_{1n}\mathrm{with}\mathrm{probability}\theta \\ {\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}+e_{2n}\mathrm{with}\mathrm{probability}1-\theta \end{array}$(6) . To quantitatively measure the speed differential effect, we assume that drivers in the study lane respond to the speed differentials $u_{n-1}^O$ and $u_{n-1}^I$ by changing the shape of the spacing-speed curve. Ponnu and Coifman (2015) and Coifman and Ponnu (2020) show that both of the intercept and slope parameters of the spacing-speed curve in the study lane may differ due to the speed differential effect. This motivates us to develop a model to test the speed differential effect as follows. Mathematically, we consider the coefficients of the spacing-speed relationship in Equation (6)(6) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+e_{1n}\mathrm{with}\mathrm{probability}\theta \\ {\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}+e_{2n}\mathrm{with}\mathrm{probability}1-\theta \end{array}$(6) being affected by the speed differentials relative to its immediately adjacent lanes in response to the traffic conditions in the adjacent lanes as follows: (7) ${\text{β}}_{\mathrm{ij}}={\alpha }_{\mathrm{ij}}+{\delta }_{\mathrm{ij}}^I{u}_{n-1}^I+{\delta }_{\mathrm{ij}}^O{u}_{n-1}^O\text{for}i=1,2;j=0,1$(7) Substituting Equation (7)(7) ${\text{β}}_{\mathrm{ij}}={\alpha }_{\mathrm{ij}}+{\delta }_{\mathrm{ij}}^I{u}_{n-1}^I+{\delta }_{\mathrm{ij}}^O{u}_{n-1}^O\text{for}i=1,2;j=0,1$(7) into (6), we obtain (8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) Note that, when the study lane is the outermost lane, we set ${\delta }_{\mathrm{ij}}^O\equiv 0$ ($i=1,2;j=0,1$) in Equations (7)(7) ${\text{β}}_{\mathrm{ij}}={\alpha }_{\mathrm{ij}}+{\delta }_{\mathrm{ij}}^I{u}_{n-1}^I+{\delta }_{\mathrm{ij}}^O{u}_{n-1}^O\text{for}i=1,2;j=0,1$(7) and (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) . On the other hand, we set ${\delta }_{\mathrm{ij}}^I\equiv 0$ ($i=1,2;j=0,1$) in Equations (7)(7) ${\text{β}}_{\mathrm{ij}}={\alpha }_{\mathrm{ij}}+{\delta }_{\mathrm{ij}}^I{u}_{n-1}^I+{\delta }_{\mathrm{ij}}^O{u}_{n-1}^O\text{for}i=1,2;j=0,1$(7) and (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) when the study lane is the innermost lane.

We now examine the relationships of the variables in Equation (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) . From Equation (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) , the mathematical expectation of spacing $s_n$ conditional on traffic mode $i$ is given by: (9) $E\left[s_n|i\right]={\alpha }_{i0}+{\alpha }_{i1}{v}_{n-1}+({\delta }_{i0}^I+{\delta }_{i1}^I{v}_{n-1})u_{n-1}^I+({\delta }_{i0}^O+{\delta }_{i1}^O{v}_{n-1})u_{n-1}^O$(9) Equation (9)(9) $E\left[s_n|i\right]={\alpha }_{i0}+{\alpha }_{i1}{v}_{n-1}+({\delta }_{i0}^I+{\delta }_{i1}^I{v}_{n-1})u_{n-1}^I+({\delta }_{i0}^O+{\delta }_{i1}^O{v}_{n-1})u_{n-1}^O$(9) shows a clear mathematical relationship for each of the three variable pairs, i.e. spacing $s_n$versus each of $v_{n-1}$, $u_{n-1}^I$ and $u_{n-1}^O$. More specifically, we have

1. for fixed $v_{n-1}$, the speed differential effect of $u_{n-1}^I$ on the spacing $s_n$ is characterised by $\mathrm{\partial E}\left[s_n|i\right]/\mathrm{\partial }{u}_{n-1}^I={\delta }_{i0}^I+{\delta }_{i1}^I{v}_{n-1};$

2. for fixed $v_{n-1}$, the speed differential effect of $u_{n-1}^O$ on the spacing $s_n$ is characterised by $\mathrm{\partial E}\left[s_n|i\right]/\mathrm{\partial }{u}_{n-1}^O={\delta }_{i0}^O+{\delta }_{i1}^O{v}_{n-1};$

3. the speed effect of the vehicle ahead on the spacing $s_n$ is characterised by $\mathrm{\partial E}\left[s_n|i\right]/\mathrm{\partial }{v}_{n-1}={\alpha }_{i1}+{\delta }_{i1}^I{u}_{n-1}^I+{\delta }_{i1}^O{u}_{n-1}^O.$

Note that for different traffic modes, these effects usually differ. For example, $\mathrm{\partial E}\left[s_n|i\right]/\mathrm{\partial }{u}_{n-1}^I$ differs for different traffic mode $i=1$ or 2, i.e. ${\delta }_{10}^I+{\delta }_{11}^I{v}_{n-1}$ for the first mode and ${\delta }_{20}^I+{\delta }_{21}^I{v}_{n-1}$ for the second mode. In addition, $\mathrm{\partial E}\left[s_n|i\right]/\mathrm{\partial }{v}_{n-1}$ may be approximated to be ${\alpha }_{i1}$ for small speed differentials.

From a traffic perspective, with positive speed differential effects, i.e. $\frac{\mathrm{\partial }E\left[s_n|i\right]}{\mathrm{\partial }{u}_{n-1}^I}>0$ and $\frac{\mathrm{\partial }E\left[s_n|i\right]}{\mathrm{\partial }{u}_{n-1}^O}>0$, Equation (9)(9) $E\left[s_n|i\right]={\alpha }_{i0}+{\alpha }_{i1}{v}_{n-1}+({\delta }_{i0}^I+{\delta }_{i1}^I{v}_{n-1})u_{n-1}^I+({\delta }_{i0}^O+{\delta }_{i1}^O{v}_{n-1})u_{n-1}^O$(9) implies that when the speed differential between the study lane and the inner (or outer) lane becomes larger, drivers in the study lane will become more conservative by increasing their spacing. Conversely, Equation (9)(9) $E\left[s_n|i\right]={\alpha }_{i0}+{\alpha }_{i1}{v}_{n-1}+({\delta }_{i0}^I+{\delta }_{i1}^I{v}_{n-1})u_{n-1}^I+({\delta }_{i0}^O+{\delta }_{i1}^O{v}_{n-1})u_{n-1}^O$(9) also shows that when the speed differential between the study lane and the inner (or outer) lane becomes smaller, drivers in the study lane will become less conservative by decreasing their spacing.

Equation (9)(9) $E\left[s_n|i\right]={\alpha }_{i0}+{\alpha }_{i1}{v}_{n-1}+({\delta }_{i0}^I+{\delta }_{i1}^I{v}_{n-1})u_{n-1}^I+({\delta }_{i0}^O+{\delta }_{i1}^O{v}_{n-1})u_{n-1}^O$(9) also suggests that the speed differential effects from the adjacent lanes are additive in the sense that drivers are the most conservative when both adjacent lanes are slower than the study lane, and the least conservative when both adjacent lanes are faster than the study lane. This is in line with the observations made in Coifman and Ponnu (2020).

Apart from the above two cases, Equation (9)(9) $E\left[s_n|i\right]={\alpha }_{i0}+{\alpha }_{i1}{v}_{n-1}+({\delta }_{i0}^I+{\delta }_{i1}^I{v}_{n-1})u_{n-1}^I+({\delta }_{i0}^O+{\delta }_{i1}^O{v}_{n-1})u_{n-1}^O$(9) also indicates that the combined impact of the speed differential effects from the inner lane and the outer lane may cancel out from each other when the speed differentials relative to the two adjacent lanes have opposite signs. More specifically, consider the case that the speed of the inner lane is higher than the speed of the study lane (i.e. $u_{n-1}^I<0$), and the speed of the study lane is higher than that of the outer lane (i.e. $u_{n-1}^O>0$). If $\frac{\mathrm{\partial }E\left[s_n|i\right]}{\mathrm{\partial }{u}_{n-1}^I}>0$ and $\frac{\mathrm{\partial }E\left[s_n|i\right]}{\mathrm{\partial }{u}_{n-1}^O}>0$, then from Equation (9)(9) $E\left[s_n|i\right]={\alpha }_{i0}+{\alpha }_{i1}{v}_{n-1}+({\delta }_{i0}^I+{\delta }_{i1}^I{v}_{n-1})u_{n-1}^I+({\delta }_{i0}^O+{\delta }_{i1}^O{v}_{n-1})u_{n-1}^O$(9) we can see that the combined impact of the speed differential effects, i.e. $({\delta }_{i0}^I+{\delta }_{i1}^I{v}_{n-1})u_{n-1}^I+({\delta }_{i0}^O+{\delta }_{i1}^O{v}_{n-1})u_{n-1}^O$, will mostly cancel out. In practice, this is a very typical scenario and it may partly explain why it is not easy to discover the speed differential effect.

We also note that, with ${\delta }_{i0}^I>0$ and ${\delta }_{i1}^I<0$, the speed differential effect $\frac{\mathrm{\partial }E\left[s_n|i\right]}{\mathrm{\partial }{u}_{n-1}^I}={\delta }_{i0}^I+{\delta }_{i1}^I{v}_{n-1}$ from the inner lane will vanish when $v_{n-1}$ increases gradually. The same is also true for the outer lane. This makes sense since the vehicle ahead has a greater impact on the driving behaviour of the following vehicle, in comparison with vehicles in the adjacent lanes. When the speed of the vehicle ahead gradually becomes higher, the following vehicle will usually try to retain the speed synchronisation, regardless of the speed differences relative to the adjacent lanes.

The mixture spacing-speed model with the speed differential effect in Equation (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) provides an approach to quantifying the speed differential effect. This quantitative relation may help us develop algorithms for automated/autonomous driving in future research.

Statistically speaking, based on Equation (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) , we may examine the relation between the spacing $s_n$ and the relative speeds to the adjacent lanes, i.e. $u_{n-1}^I$ and $u_{n-1}^O$. If the spacing $s_n$ is statistically related to the speed differentials $u_{n-1}^I$ and $u_{n-1}^O$, it provides evidence to support the speed differential effect. Hence, Equation (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) also provides an approach to statistically testing the speed differential effect. More specifically, we consider the following null hypothesis: (10) $H_0:{\delta }_{\mathrm{ij}}^I={\delta }_{\mathrm{ij}}^O=0\text{for}i=1,2;j=0,1$(10) In empirical analysis, if the above null hypothesis (10) is rejected for a study lane, the evidence is in favour of the speed differential effect.

We also point out that with Equation (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) , we may control for the variation coming from different types of driving behaviour exhibited in different driving modes, and control for different types of response to the speed differentials relative to the inner lane and outer lane. In doing so, this will enhance the sensitivity when testing hypothesis (10).

3. Statistical test for the speed differential effect

In the literature, the speed differential effect concerns the relationship between vehicle spacings in a study lane and the speeds relative to its immediately adjacent lanes. Ponnu and Coifman (2015) focus on an HOV lane versus its immediately adjacent GP lane when investigating the speed differential effect. Recently, Coifman and Ponnu (2020) have examined the speed differential effect for GP lanes via a graphical method. In this section, we will use a different approach, the probabilistic modelling approach detailed in Section 2, to statistically test the speed differential effect. We consider the scenario where the study lane is any GP lane. In the following analysis, we are concerned with the following research questions:

1. is there any statistical evidence of the speed differential effect for GP lanes?

2. what if the speed of one adjacent lane (usually the inner lane) is much higher than the speed of the study lane, whereas the speed of the other adjacent lane (usually the outer lane) is much lower than the speed of the study lane?

3. what if the speed difference between the study lane and one adjacent lane is of a similar magnitude to that of the study lane and the other adjacent lane?

3.1. Data

The analysis in this section is based on the dataset investigated in Ponnu and Coifman (2015) and Coifman and Ponnu (2020). The original data in Ponnu and Coifman (2015) consists of traffic measurements across several eastbound dual loop detection stations over 69 weekdays during September to December for a two mile stretch of I-80 with five lanes near Oakland; see, e.g. Li (2010) for further information about dual loop detection stations.

The developed statistical approach in the previous section is efficient for the detection of the speed differential effect, and hence it does not require data as big as the one used in Ponnu and Coifman (2015) and Coifman and Ponnu (2020). Hence, we selected only a small portion of the data used in Ponnu and Coifman (2015) in our analysis. Specifically, we extracted the data measured by the first detection station during the first 10 weekdays in November in the dataset in Ponnu and Coifman (2015); see Figure 1 for a diagram of the road layout. The data used in the following analysis was pre-screened using the method in Ponnu and Coifman (2015). Based on the data measured from the dual loop detection station, we followed Ponnu and Coifman (2015) and calculated the required vehicle speed and spacing variables; see Ponnu and Coifman (2015) and Coifman and Ponnu (2020) for a detailed description about the calculation of these traffic variables. In the following analysis, to avoid any potential serial correlation problems, only every kth vehicle in each study lane was selected as our study vehicles, where k was taken as 5 in the analysis.

Figure 1. Illustration of the road layout for the road segment under investigation in I-80 with five lanes.

We consider two different traffic scenarios: (a) the peak-hour traffic from 15:15 pm to 18:45 pm, and (b) the traffic after the evening peak hours, i.e. 19:15 pm–21:00 pm.

For the data measured for the evening peak hours (15:15 pm–18:45 pm), Table 1 displays summary statistics for traffic volume, vehicle spacing and vehicle speed. Note that during this time period, lanes 2–5 were GP lanes, whereas lane 1 was an HOV lane.

Table 1. Dataset for the evening peak-hour traffic (15:15 pm–18:45 pm): Summary statistics for traffic volume, vehicle spacing and vehicle speed of all lanes.

		Spacing (m)		Speed (km/h)
	Traffic volume (veh/h)	Mean	S.D.	Mean	S.D.
Lane 1 (HOV)	1255	65.5441	66.8114	81.8673	12.7364
Lane 2 (GP)	1836	32.0554	18.4883	60.3202	17.4039
Lane 3 (GP)	1620	32.4776	21.9973	54.6041	16.8255
Lane 4 (GP)	1782	27.3504	17.8219	50.8647	16.2374
Lane 5 (GP)	1690	29.1252	20.8798	50.1835	14.6063

Table 1 shows that the mean vehicle speed is 81.9, 60.3, 54.6, 50.9 and 50.2 km/h for lanes 1, 2, 3, 4, and 5 respectively during this study period. Clearly, except for the speed differential between lanes 1 and 2 (81.9 versus 60.3 km/h), the speed differentials for all the other lanes are small. This is particularly the case between lanes 4 and 5 where the difference in average speed is only 50.9-50.2 = 0.7 km/h. We also note that the traffic volume is high (1620 veh/h or above) for all the GP lanes, whereas it is much lower for the HOV lane (1255 veh/h) due to the HOV effect. Overall, the average spacing for the GP lanes is relatively small (around 30 metres) with a small standard deviation (between 17 and 22 metres). This indicates that the traffic was congested in these GP lanes at the time and vehicles followed the vehicle ahead relatively closely.

Next, we turn our attention to the traffic after the peak hours, i.e. 19:15 pm–21:00 pm. For the traffic data measured during this period, Table 2 displays summary statistics for traffic volume, vehicle spacing and vehicle speed. Note that after 19:00 pm, lane 1 reverted to a GP lane.

Table 2. Dataset for the traffic after peak hours (19:15 pm–21:00 pm): Summary statistics for traffic volume, vehicle spacing and vehicle speed of all lanes.

		Spacing (m)		Speed (km/h)
	Traffic volume (veh/h)	Mean	S.D.	Mean	S.D.
Lane 1 (GP)	1043	95.6664	108.4300	109.0829	13.0353
Lane 2 (GP)	1383	73.3667	58.9653	103.5396	11.6710
Lane 3 (GP)	1302	73.7525	56.6283	96.9987	11.6655
Lane 4 (GP)	1405	63.8753	52.5654	92.4875	12.2312
Lane 5 (GP)	1199	73.3550	72.5354	87.7417	13.4462

Table 2 shows that the mean vehicle speed is 109.1, 103.5, 97.0, 92.5 and 87.7 km/h for lanes 1, 2, 3, 4, and 5 respectively during this study period. Clearly, for all the GP lanes, the average speed is higher than the level of the peak hours. In addition, the traffic volume for all the lanes is lower than the peak hours. Moreover, we can see that the average spacing for each GP lane is more than doubled the level of the peak hours, with a much larger standard deviation. This indicates that the traffic was much less congested at the time and there was a group of vehicles that did not follow the vehicle ahead very closely.

It is also worth noting that the speed limit at the time of data collection was 65 MPH (104.6074 km/h). Hence, the average speed for lane 1 (109.0829 km/h) is higher than the speed limit. In addition, the average spacing for lane 1 is 95.7 metres with a very large standard deviation of 108.4 metres. This suggests that a substantial portion of vehicles in lane 1 travelled in the free speed during this time period.

3.2. Empirical results for the evening peak-hour traffic

In this subsection, we consider the evening peak hours, 15:15 pm–18:45 pm. As discussed in the previous subsection, the traffic was tightly-spaced with a high traffic volume during this time period. We now focus on the four GP lanes, i.e. lanes 2-5. We applied Equation (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) to analyse the traffic. The obtained results are displayed in Table 3, where figures displayed in boldface represent statistical significance at the 5% significance level. From Table 3, we may observe the following properties for the regression coefficients that are significant at the 5% significance level: ${\alpha }_{i1}>0,{\delta }_{i0}^I>0,{\delta }_{i0}^O>0,{\delta }_{i1}^I<0,{\delta }_{i1}^O<0,\text{for},i=1,2.$ First, we discuss the obtained results in relation to the ones in Coifman and Ponnu (2020). With the svp method, Coifman and Ponnu (2020) use lane 2 as the study lane to examine the speed differential effect relative to the inner lane (lane 1) and outer lane (lane 3) respectively. Specifically, they consider the spacing $s_n$ of each vehicle $n$ in lane 2. For several fixed speed levels of the lead vehicle $v_{n-1}$, they investigate the relationship between spacing $s_n$ and speed differential $u_{n-1}^I$ in graphs (a)–(e) of their Figure 6, and the relationship between spacing $s_n$ and speed differential $u_{n-1}^O$ in graphs (f)–(k) of their Figure 6, respectively. They find that:

Table 3. Dataset for the evening peak-hour traffic (15:15 pm–18:45 pm): regression coefficients of Equation (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) with the corresponding t-values in the parentheses.

	Lane 2	Lane 3	Lane 4	Lane 5
	Effect of the vehicle ahead
${\alpha }_{11}$	0.1276	0.1080	0.0847	0.0878
	(10.6525)	(11.6009)	(7.0296)	(13.3948)
${\alpha }_{21}$	0.3681	0.3875	0.3418	0.4025
	(20.0293)	(38.5823)	(18.9917)	(38.1603)
	Effect othe inner outer lane
${\delta }_{10}^I$	0.0245	0.0946	0.0649	0.0655
	(1.1245)	(2.7654)	(2.0641)	(2.3997)
${\delta }_{11}^I$	−0.0003	−0.0019	−0.0009	−0.0006
	(−0.7644)	(−2.9369)	(−1.4361)	(−1.2377)
${\delta }_{20}^I$	0.1724	0.1356	0.1795	0.3124
	(4.5060)	(2.7595)	(3.4572)	(4.2343)
${\delta }_{21}^I$	−0.0016	−0.0020	−0.0037	−0.0041
	(−2.4206)	(−2.2270)	(−3.7031)	(−3.0320)
	Effect of the adjacent outer lane
${\delta }_{10}^O$	0.0202	−0.0358	0.0668	–
	(0.5975)	(−1.0340)	(2.4938)
${\delta }_{11}^O$	0.0001	−0.0000	−0.0013	–
	(0.1351)	(−0.0618)	(−2.7342)
${\delta }_{20}^O$	−0.0271	0.0697	0.1074	–
	(−0.4128)	(1.2180)	(2.0070)
${\delta }_{21}^O$	0.0005	−0.0015	−0.0017	–
	(0.4656)	(−1.4254)	(−1.7483)

(a) with a fixed speed level of $v_{n-1}$, when $u_{n-1}^I$ increases, $s_n$ will increase;

(b) with a fixed speed level of $v_{n-1}$, when $u_{n-1}^O$ increases, $s_n$ will increase;

(c) with a higher level of $v_{n-1}$, the spacing $s_n$ is also larger.

Overall, these graphs in Coifman and Ponnu (2020) suggest a positive relation between $s_n$ and each of $u_{n-1}^I$, $u_{n-1}^O$ and $v_{n-1}$. However, because this is essentially a multivariate relation among these four variables, it is challenging to gain a full picture for the relationships of these variables via the graphical method used in Coifman and Ponnu (2020).

We now examine the relationships of these variable pairs via Equation (9)(9) $E\left[s_n|i\right]={\alpha }_{i0}+{\alpha }_{i1}{v}_{n-1}+({\delta }_{i0}^I+{\delta }_{i1}^I{v}_{n-1})u_{n-1}^I+({\delta }_{i0}^O+{\delta }_{i1}^O{v}_{n-1})u_{n-1}^O$(9) . We focus on the first mode; the analysis for the second mode is similar. Let us take lane 4 as an example. Consider the average speed in lane 4, i.e. $v_{n-1}=$50.8647 km/h, and assume an average speed differentials $u_{n-1}^I=50.8647-54.6041=-3.7394$ and $u_{n-1}^O=50.8647-50.1835=0.6812$ (see Table 1). Then from Table 3, we obtain that:

the speed differential effect of $u_{n-1}^I$ on the spacing $s_n$ is positive and given by $\frac{\mathrm{\partial }E[s_n|\text{mode}=1]}{\mathrm{\partial }{u}_{n-1}^I}={\delta }_{10}^I+{\delta }_{11}^I{v}_{n-1}=0.0649+\left(-0.0009\right)\times 50.8647=0.0191;$ the speed differential effect of $u_{n-1}^O$ on the spacing $s_n$ is positive and given by $\frac{\mathrm{\partial }E[s_n|\text{mode}=1]}{\mathrm{\partial }{u}_{n-1}^O}={\delta }_{10}^O+{\delta }_{11}^O{v}_{n-1}=0.0668+\left(-0.0013\right)\times 50.8647=0.0007;$ the effect of the lead vehicle speed $v_{n-1}$ on the spacing $s_n$ is positive and given by $\begin{array}{rl}\frac{\mathrm{\partial }E[s_n|\text{mode}=1]}{\mathrm{\partial }{v}_{n-1}}& ={\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O\\ & =0.0847+\left(-0.0009\right)\times \left(-3.7394\right)+\left(-0.0013\right)\times 0.6812=\mathrm{0.0872.}\end{array}$ Further, we can see that the speed differential effect for the inner lane (0.0191) is much larger than that of the outer lane (0.0007). Given the fact that the speed differential between lanes 3 and 4 is higher than that for lanes 4 and 5 in magnitude (3.7394 versus 0.6812), this is not surprising. Similar analysis can be undertaken for the other lanes and for the second mode. These results are in line with the results obtained in Coifman and Ponnu (2020) via their graphical method.

Next, we examine the results in Table 3 in more detail and compare and contrast the differences in the two driving modes, as well as the impacts from the inner and outer lanes respectively.

We first consider each GP lane against its inner lane. From Table 3, we can see that most of the regression coefficients ${\delta }_{\mathrm{ij}}^I(i=1,2;j=0,1)$ are significant at the 5% significance level. Furthermore, we also notice from Table 3 that there are some important differences in driving behaviour between the two modes: the estimated regression coefficients of ${\delta }_{20}^I$ and ${\delta }_{21}^I$ are all significant at the 5% significance level for vehicles travelling in the second traffic mode. Mathematically, we may observe the following interesting empirical phenomenon: ${\alpha }_{21}>{\alpha }_{11}>0,{\delta }_{20}^I>{\delta }_{10}^I>0\text{and}|{\delta }_{21}^I|>|{\delta }_{11}^I|.$ The above inequalities show that: (a) drivers in the second mode tend to have a larger sensitivity to the speed of the vehicle ahead, i.e. for the same increment in the speed of the vehicle ahead, they respond more conservatively by retaining larger spacing; (b) drivers travelling in the second mode are more sensitive to the relative speed to the inner lane.

From the perspective of statistical testing, the above differences between the two driving modes also show the importance of the mixture model developed in this paper, as a tool to control for the variation in responding to the speed differentials and to the speed of the vehicle ahead. If we did not differentiate the two driving modes in the modelling, the relations among these variables could become less clear and/or un-detectable.

In terms of the influence from the outer lane, we see the observations made for the inner lane still hold to some extent. However, except for lane 4, the regression coefficients ${\delta }_{\mathrm{ij}}^O(i=1,2;j=0,1)$ are insignificant at the 5% significance level for all the other GP lanes. Hence, for these lanes, we do not have evidence that the spacing-speed relation in the study lane is affected by the speed relative to their adjacent outer lane. This is not surprising: according to the analysis in the previous subsection, on average the speed differential between the inner lane and the study lane is much higher than that of the study lane and the outer lane. Hence, Table 3 indicates that, when the speed of one adjacent lane (usually the inner lane) is substantially higher than the speed of the study lane, and the speed of the other adjacent lane (usually the outer lane) is lower than the speed of the study lane, it is the higher speed differential that will mainly affect the driving behaviour in the study lane. This in general shows the differences in the speed differential effect between the inner lane and outer lane.

To better understand the interplay of vehicles in a study lane against vehicles in the two adjacent lanes, we take a close look at two interesting scenarios, i.e. lane 2 and lane 4, and highlight the differences between these two scenarios.

First, we consider lane 2 as a study lane. From Table 1, we notice that the difference in the average speed between lanes 1 and 2 is much higher (i.e. 81.9-60.3 = 21.6 km/h) than that of lanes 2 and 3 (i.e. 60.3-54.6 = 5.7 km/h). Table 3 shows that the estimates of ${\delta }_{\mathrm{ij}}^I(i=1,2;j=0,1)$ for lane 2 are significant at the 5% significant level, whereas the estimates of ${\delta }_{\mathrm{ij}}^O(i=1,2;j=0,1)$ for lane 2 are insignificant. This indicates that statistically the outer lane has no significant impact on the spacings of the study lane (lane 2). Hence, in this case, it is the speed differential between lane 1 and lane 2 that affects the driving behaviour in lane 2.

Turning our attention to lane 4 as a study lane, we see that the speed differences for lane 4 against its two adjacent lanes are much close to each other, i.e. 54.6-50.9 = 3.7 km/h and 50.9-50.2 = 0.7 km/h respectively. Table 3 shows that in this case the speed differentials from both of its inner and outer lanes have an impact on the spacings of the study lane. This is in stark contrast to the case of lane 2 as the study lane.

In summary, the analysis in this subsection suggests that at the 5% significant level, we have strong evidence against the null hypothesis in (10), hence supporting the speed differential effect for GP lanes. This shows that for peak-hour traffic, the speed differential effect exists for GP lanes, even when the speed differential is relatively small for some lanes.

3.3. Empirical results for the traffic after the peak hours

In this subsection, we investigate the scenario where traffic is less congested. We focus on the time period after the evening’s peak hours, i.e. 19:15 pm–21:00 pm.

We applied Equation (8(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) ) to analyse the traffic during this time period. The obtained results are displayed in Table 4. Overall, we can see that the results in Table 4 exhibit similar patterns to that we observed in the previous subsection. We summarise the main conclusions as follows.

Table 4. Dataset for the traffic after the peak hours (19:15 pm–21:00 pm): regression coefficients of Equation (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) with the corresponding t-values in the parentheses.

	Lane 1 (GP)	Lane 2	Lane 3	Lane 4	Lane 5
	Effect of the vehicle ahead
${\alpha }_{11}$	0.2214	0.1650	0.1454	0.1014	0.1136
	(7.3841)	(6.1538)	(6.5245)	(5.4036)	(4.1489)
${\alpha }_{21}$	0.1133	0.3744	0.3120	0.3182	0.3155
	(3.6584)	(10.8050)	(11.0906)	(17.4391)	(5.9247)
	Effect of the inner outer lane
${\delta }_{10}^I$	–	0.0245	0.0946	0.0649	0.0655
		(1.1245)	(2.7654)	(2.4600)	(2.3997)
${\delta }_{11}^I$	–	−0.0003	−0.0019	−0.0009	−0.0006
		(−0.7644)	(−2.9369)	(−1.6679)	(−1.2377)
${\delta }_{20}^I$	–	0.1724	0.1356	0.1795	0.3124
		(4.5060)	(2.7595)	(2.3947)	(4.2343)
${\delta }_{21}^I$	–	−0.0016	−0.0020	−0.0037	−0.0041
		(−2.4206)	(−2.2270)	(−3.1884)	(−3.0320)
	Effect of the adjacent outer lane
${\delta }_{10}^O$	0.0263	0.0202	−0.0358	0.0668	–
	(0.0089)	(0.5975)	(−1.0340)	(2.6611)
${\delta }_{11}^O$	0.0115	0.0001	−0.0000	−0.0013	–
	(0.3485)	(0.1351)	(−0.0618)	(−2.8439)
${\delta }_{20}^O$	0.0797	−0.0271	0.0697	0.1074	–
	(0.4194)	(−0.4128)	(1.2180)	(1.2663)
${\delta }_{21}^O$	−0.0007	0.0005	−0.0015	−0.0017	–
	(−0.3062)	(0.4656)	(−1.4254)	(−1.3298)

First, we can see from Table 4 that overall, there is strong evidence supporting the speed differential effect for lanes 2-5. In particular, the regression coefficients ${\delta }_{20}^I$ and ${\delta }_{21}^I$ associated with the inner lane are all significant at the 5% significance level for vehicles travelling in the second traffic mode. In addition, the following interesting empirical results hold for lanes 2-5: ${\alpha }_{21}>{\alpha }_{11}>0,{\delta }_{20}^I>{\delta }_{10}^I>0\text{and}|{\delta }_{21}^I|>|{\delta }_{11}^I|.$ Again, this shows that the speed differential effect has a stronger impact on the vehicles travelling in the second mode.

Interestingly, we also see that the regression coefficients for lane 1 are insignificant at the 5% significance level. Lane 1 was an HOV lane during the peak-hour period (15:00 pm–19:00 pm) and reverted to a GP lane after 19:00 pm. As observed in Section 3.1, the average speed for lane 1 during this time period was higher than the speed limit. In addition, most vehicles had a very large spacing. This indicates that a large portion of vehicles in this lane travelled at the free speed, and hence it may explain why the drivers’ travelling behaviour was not impacted by the speed differential relative to the adjacent lane.

Next, Table 4 also shows that, for the scenario where the average speed in the inner lane is much higher than the study lane, and the average speed in the outer lane is much lower than the study lane, there is strong evidence of the speed differential effect between the inner lane and the study lane, but no statistical evidence of the speed differential effect between the outer lane and the study lane. This can be seen clearly for lane 2 and lane 3.

Third, when the average speed is of a similar magnitude across the inner lane, study lane, and outer lane, there is evidence of the speed differential effect for either of the adjacent lanes, as exhibited for lane 4 as the study lane.

In summary, the analysis in this subsection provides statistical evidence that the speed differential effect exists for GP lanes for traffic immediately after the evening peak hours.

4. Conclusions and discussion

In this section, we briefly summarise this paper and then outline some further potential extensions of the research.

This paper has investigated mixture spacing-speed modelling to capture the spacing and speed relationship for traffic flow, in relation to the speed differential effect. Applying the developed method to analyse real traffic data, we have statistically tested the speed differential effect discovered in Ponnu and Coifman (2015), and Coifman and Ponnu (2020). Overall, the obtained results have provided strong statistical evidence to support the speed differential effect for GP lanes.

In the literature, the speed differential effect was not discovered until very recently when Ponnu and Coifman (2015) have carefully examined a huge amount of traffic data. They focus on an HOV lane versus its adjacent GP lane, where the speed differential effect is the greatest. As pointed out by Ponnu and Coifman (2015), it is not easy to detect such a subtle effect in any macroscopic analysis. In fact, the discovery of the speed differential effect is possible only if the other factors are well controlled, for example, as done by the grouping method, svp, in Ponnu and Coifman (2015). In this paper, we provide an alternative approach to detecting the speed differential effect via an advanced regression-based method.

Due to the very small sample sizes for the spacing-speed regression in their study, Ponnu and Coifman (2015) and Coifman and Ponnu (2020) have not performed any statistical test regarding their discovered speed differential effect. In contrast, the analysis undertaken in this paper provides strong statistical evidence to support the speed differential effect. Furthermore, in addition to a qualitative examination for the existence of the speed differential effect, here in this paper we have also quantitatively measured how the speed differential of each adjacent lane impacts on the spacing-speed relation.

This paper has made theoretical contributions to the literature. The speed differential effect investigated in this paper will have practical impacts in the future. For example, the method of quantifying the speed differential effect on the shape of the spacing-speed curve may help engineers/scientists improve on the existing automated/autonomous driving algorithms. Another example is the calibration of practical parameters when using a car-following-model based simulation studies. An improved car-following model by taking into consideration the speed differential effect would make it better describe the way that drivers interact with other vehicles, and hence make these parameters more accurately calibrated.

Before we conclude this paper, we discuss a couple of extensions and the related future research.

First, we would like to differentiate two different variations of the hypothesis for the speed differential effect, i.e. the weak form and generic form, which represent two different assumed levels of drivers’ response to speed differentials. The weak form of the speed differential effect assumes that the speeds relative to the adjacent lanes affect the intercept and slope parameters of the spacing-speed curve of the study lane. This weak form may be tested straightforwardly via Equation (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) . On the other hand, the generic form of the speed differential effect assumes that the speeds relative to the adjacent lanes affect the response curve of the study lane in a way that may or may not be through the impact on the intercept and slope parameters. Clearly, this generic form of the speed differential effect may not be directly testable via Equation (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) . In this paper, we have focused on the weak form. The generic form could be much more difficult to detect in practice. This will be a research topic for future research.

Secondly, we point out that the mixture spacing-speed model in Equation (6)(6) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+e_{1n}\mathrm{with}\mathrm{probability}\theta \\ {\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}+e_{2n}\mathrm{with}\mathrm{probability}1-\theta \end{array}$(6) may be further extended to accommodate more complicated traffic scenarios. For example, taking the advantages of multiple regression, we may further expand the linear mixture model (6) to account for other traffic/roadway factors denoted by $x_j$ ($j=1,\dots ,J$): (11) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+\sum _{j=1}^J{\omega }_{1j}{x}_{\mathrm{nj}}+e_{1n}\text{ with probability }\theta \\ {\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}+\sum _{j=1}^J{\omega }_{2j}{x}_{\mathrm{nj}}+e_{2n}\text{ with probability }1-\theta \end{array},$(11) where ${\omega }_{1j}$ and ${\omega }_{2j}$ ($j=1,\dots ,J$) are coefficients, and $x_{\mathrm{nj}}$ ($j=1,\dots ,J$) are the values measured on factors $x_j$ for vehicle $n$. The above extended mixture spacing-speed model provides us with an opportunity to accommodate multiple factors when investigating the spacing-speed relation, without the need to split data into multiple subgroups of vehicles. Specifically, we may use some indicator variables $x_j(j=1,\dots ,J)$ to examine the impact of various traffic/roadway factors on vehicle spacings. For example, if we set $x_1=0/1$ as an indicator of dry/rainy road surface, we may measure and test its effect on vehicle spacings. We also point out that Equation (8)(8) $s_n=\left\{\begin{array}{l}({\alpha }_{10}+{\delta }_{10}^I{u}_{n-1}^I+{\delta }_{10}^O{u}_{n-1}^O)\\ +({\alpha }_{11}+{\delta }_{11}^I{u}_{n-1}^I+{\delta }_{11}^O{u}_{n-1}^O{)}_{v_{n-1}}+e_{1n}\text{ with probability }\theta \\ ({\alpha }_{20}+{\delta }_{20}^I{u}_{n-1}^I+{\delta }_{20}^O{u}_{n-1}^O)\\ +({\alpha }_{21}+{\delta }_{21}^I{u}_{n-1}^I+{\delta }_{21}^O{u}_{n-1}^O{)}_{{}_{v_{n-1}}}+e_{2n}\text{ with probability }1-\theta \end{array}\right\}.$(8) may be considered a special case of model (11), where each factor $x_j$ in Equation (11)(11) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+\sum _{j=1}^J{\omega }_{1j}{x}_{\mathrm{nj}}+e_{1n}\text{ with probability }\theta \\ {\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}+\sum _{j=1}^J{\omega }_{2j}{x}_{\mathrm{nj}}+e_{2n}\text{ with probability }1-\theta \end{array},$(11) is either a speed differential variable ($u_{n-1}^I$ or $u_{n-1}^O$) or its interaction with $v_{n-1}$.

The linear relation between spacings and speeds in Equation (11)(11) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+\sum _{j=1}^J{\omega }_{1j}{x}_{\mathrm{nj}}+e_{1n}\text{ with probability }\theta \\ {\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}+\sum _{j=1}^J{\omega }_{2j}{x}_{\mathrm{nj}}+e_{2n}\text{ with probability }1-\theta \end{array},$(11) may be further relaxed to take into consideration potential nonlinearity between spacing and speed as follows: (12) $s_n=\{\begin{array}{l}{f}_1\left(v_{n-1}\right)+\sum _{j=1}^J{\omega }_{1j}{x}_{\mathrm{nj}}+e_{1n}\text{ with probability }\theta \\ f_2\left(v_{n-1}\right)+\sum _{j=1}^J{\omega }_{2j}{x}_{\mathrm{nj}}+e_{2n}\text{ with probability }1-\theta \end{array},$(12) where $f_1(.)$ and $f_2(.)$ in Equation (12)(12) $s_n=\{\begin{array}{l}{f}_1\left(v_{n-1}\right)+\sum _{j=1}^J{\omega }_{1j}{x}_{\mathrm{nj}}+e_{1n}\text{ with probability }\theta \\ f_2\left(v_{n-1}\right)+\sum _{j=1}^J{\omega }_{2j}{x}_{\mathrm{nj}}+e_{2n}\text{ with probability }1-\theta \end{array},$(12) are two potentially nonlinear functions. The extension to a nonlinear spacing-speed relation makes sense not only from a purely statistical modelling perspective, but is also related to the traffic literature. In the literature, Newell’s model (1) is typically considered a simplified model for the spacing-speed relation; there are some important, more advanced car-following models in the literature, including Gazis-Herman-Rothery (GHR) model (Chandler, Herman, and Montroll 1958; Gazis, Herman, and Rothery 1961), Gipps model (Gipps 1981), full velocity difference model (Jiang, Wu, and Zhu 2001), the L-L model (Laval and Leclercq 2010) and the related behavioural car-following model (Chen et al. 2012), and the intelligent driver model (Kesting, Treiber, and Helbing 2010; Treiber and Kesting 2013). These car-following models capture the relationship between vehicles’ acceleration and speed, as well as spacing, implying a complicated, nonlinear relation for spacing and speed under the car-following condition. Hence, replacing the linear spacing-speed relation with potentially nonlinear functions $f_1(.)$ and $f_2(.)$ in Equation (12)(12) $s_n=\{\begin{array}{l}{f}_1\left(v_{n-1}\right)+\sum _{j=1}^J{\omega }_{1j}{x}_{\mathrm{nj}}+e_{1n}\text{ with probability }\theta \\ f_2\left(v_{n-1}\right)+\sum _{j=1}^J{\omega }_{2j}{x}_{\mathrm{nj}}+e_{2n}\text{ with probability }1-\theta \end{array},$(12) may reduce the approximation error in the modelling when real traffic follows any of these advanced car-following models.

In practice, the nonlinear functions $f_1(.)$ and $f_2(.)$ in Equation (12)(12) $s_n=\{\begin{array}{l}{f}_1\left(v_{n-1}\right)+\sum _{j=1}^J{\omega }_{1j}{x}_{\mathrm{nj}}+e_{1n}\text{ with probability }\theta \\ f_2\left(v_{n-1}\right)+\sum _{j=1}^J{\omega }_{2j}{x}_{\mathrm{nj}}+e_{2n}\text{ with probability }1-\theta \end{array},$(12) may be approximated via a polynomial approximation. For example, the simple linear mixture spacing-speed model (5) is a special case of Taylor’s expansions of $f_1(.)$ and $f_2(.)$ to the first- and zero-orders respectively. In addition, Ponnu and Coifman (2015) consider a quadratic polynomial model for the spacing-speed relation. In statistical modelling, the nonlinear functions $f_1(.)$ and $f_2(.)$ may also be approximated using a neural network or splines approach.

We also point out that in this paper, we have focused on mixture models based on the assumption that there are only up to two traffic modes. This assumption may not be true in practice. As discussed previously, some researchers have identified three different traffic states in the literature, i.e. free-flowing, synchronised, and the congested states; see, e.g. Kerner (2021). Hence, when the time period under investigation is sufficiently long such that the traffic involves all three states, one may extend the developed model in this paper into a mixture of three components to provide a more accurate representation of real traffic: $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+e_{1n}\text{ with probability }{\theta }_1\\ {\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}+e_{2n}\text{ with probability }{\theta }_2\\ {\text{β}}_{30}+e_{3n}\text{ with probability }1-{\theta }_1-{\theta }_2\end{array}$ where ${\theta }_1$ and ${\theta }_2$ are probabilities of a vehicle travelling in the congested and synchronised modes respectively. Mode 3 is the free-flowing mode for which the spacing of a vehicle does not depend on the speed of the vehicle ahead. $e_{1n}\sim N\left(0,{\sigma }_1^2\right)$. $e_{\mathrm{in}}\sim g_{\text{shiftedEMG}}\left(s;{\gamma }_i,\sqrt{{\sigma }_i^2-{\gamma }_i^2}\right)$ for $i=2,3$. Clearly, the above three-component mixture model includes both models (5) and (6) as special cases: it reduces to model (5) when ${\theta }_2=0$, and it collapses to model (6) when ${\theta }_1+{\theta }_2=1$.

Finally, we note that Coifman and Ponnu (2020) show in their Figure 7 that speed differentials also impact on the flow variable. Future research could extend the spacing-speed model in this paper to investigate the speed differential effect in relation to the corresponding flow variable.

Acknowledgements

The author thanks Professor Benjamin Coifman at The Ohio State University for making the data analysed in this paper publicly available. The author also thanks the anonymous reviewers for their insightful comments that have improved on the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix A

Proof of Theorem 1.

First, we note that, given the conditional distribution $g_2\left(s_n|h_n\right)=\phi \left(s_n;v_F{h}_n,{\sigma }_F^2\right)$ and the marginal distribution $p\left(h_n\right)=\text{exp}\left(-\left(h_n-\kappa \right)/\lambda \right)/\lambda$, we can obtain the joint distribution of $s_n$ and $h_n$: $p(s_n,h_n)=g_2\left(s_n|h_n\right)p\left(h_n\right)$. Next, we integrate out $h_n$ from the joint distribution $p(s_n,h_n)$ and obtain the following marginal distribution of $s_n$ under the free-flowing condition: $\begin{array}{rl}{g}_2\left(s_n\right)& ={\int }_{\kappa }^{+\mathrm{\infty }}{g}_2\left(s_n|h_n\right)p\left(h_n\right)d{h}_n={\lambda }^{-1}{\int }_{\kappa }^{+\mathrm{\infty }}\phi \left(s_n;v_F{h}_n,{\sigma }_F^2\right)\text{exp }\left(-\left(h_n-\kappa \right)/\lambda \right)d{h}_n\\ & =\frac{e^{\kappa /\lambda }}{{\sigma }_F\lambda \sqrt{2\pi }}{\int }_{\kappa }^{+\mathrm{\infty }}\text{exp }\left[-\frac{\lambda {(s_n-v_F{h}_n)}^2+2{\sigma }_F^{2}h}{2{\sigma }_F^2\lambda }\right]d{h}_n.\end{array}$

The exponent of the integrand in the above equation can be rewritten as $-\frac{\lambda {(s_n-v_F{h}_n)}^2+2{\sigma }_F^{2}h}{2{\sigma }_F^2\lambda }=-\frac{v_F^2}{2{\sigma }_F^2}[h_n+\frac{{\sigma }_F^2-v_F\lambda s_n}{\lambda v_F^2}{]}^2-\frac{s_n}{\lambda v_F}+\frac{{\sigma }_F^2}{2{\lambda }^2{v}_F^2}.$ We note $\frac{v_F}{{\sigma }_F\sqrt{2\pi }}{\int }_{\kappa }^{+\mathrm{\infty }}\text{exp}\{-\frac{v_F^2}{2{\sigma }_F^2}[h_n+\frac{{\sigma }_F^2-v_F\lambda s_n}{\lambda v_F^2}{]}^2\}d{h}_n=\mathrm{\Phi }\left(\frac{s_n-\kappa v_F-{\sigma }_F^2/\left(\lambda v_F\right)}{{\sigma }_F}\right)$. Hence, we obtain $g_2\left(s_n\right)=(\lambda v_F{)}^{-1}\exp \left(-\frac{s_n-\kappa v_F-{\sigma }_F^2/\left(2\lambda v_F\right)}{\lambda v_F}\right)\mathrm{\Phi }\left\{\frac{s_n-\kappa v_F-{\sigma }_F^2/\left(\lambda v_F\right)}{{\sigma }_F}\right\}.$ This completes the proof.

Appendix B. Exponentially modified Gaussian (EMG) distributions

In the statistical literature, the probability density function of the EMG is given by: $g_{\text{EMG}}\left(s;\gamma ,\rho ,\sigma \right)={\gamma }^{-1}\exp \left(-\frac{s-\rho -{\sigma }^2/\left(2\gamma \right)}{\gamma }\right)\mathrm{\Phi }\left\{\frac{s-\rho -{\sigma }^2/\gamma }{\sigma }\right\}$ with the corresponding cumulative distribution function: $G_{\text{EMG}}\left(s;\gamma ,\rho ,\sigma \right)=\mathrm{\Phi }\left\{\frac{s-\rho }{\sigma }\right\}-\exp \left(-\frac{s-\rho -{\sigma }^2/\left(2\gamma \right)}{\gamma }\right)\mathrm{\Phi }\left\{\frac{s-\rho -{\sigma }^2/\gamma }{\sigma }\right\}.$ Figure A1. displays $g_{\text{EMG}}\left(s;\gamma ,\rho ,\sigma \right)$ for several different parameter values. We can see that in general, EMG distributions are highly skewed with a long, heavy right tail.

Grushka (1972) shows that an EMG random variable $s$ with distribution $g_{\text{EMG}}(s;\gamma ,\rho ,\sigma )$ can be expressed as the sum of two independent random variables $s=s_1+s_2$, such that $s_1\sim \text{Exp}\left(\gamma ,0\right)$ and $s_2\sim N\left(\rho ,{\sigma }^2\right)$. Hence, following this nice property, we obtain $\mathrm{Es}=\gamma +\rho$ and $\mathrm{var}\left(s\right)={\gamma }^2+{\sigma }^2$.

Let $\rho =\kappa v_F$, $\gamma =\lambda v_F$ and $\sigma ={\sigma }_F$. Applying the above important properties, we may calculate the mean and variance of vehicle spacings under the free-flowing condition: (A1) ${\mu }_2:=E{s}_n=\left(\lambda +\kappa \right)v_F\text{and}{\sigma }_2^2:=\text{var}\left(s_n\right)=(\lambda v_F{)}^2+{\sigma }_F^2.$(A1) Finally, for a random variable $s$ that follows an EMG with probability density function $g_{\text{EMG}}\left(s;\gamma ,\rho ,\sigma \right)$, we define a demeaned (centred) random variable: $\tilde{s}=s-(\gamma +\rho )$. This new random variable $\tilde{s}$ has a zero mean and has the following shifted EMG distribution: (A2) $g_{\text{shiftedEMG}}\left(s;\gamma ,\sigma \right)={\gamma }^{-1}\exp \left(-\frac{s+\gamma -{\sigma }^2/\left(2\gamma \right)}{\gamma }\right)\mathrm{\Phi }\left\{\frac{s+\gamma -{\sigma }^2/\gamma }{\sigma }\right\}.$(A2)

Figure A1. Probability density function $g_{\mathrm{EMG}}(s;\gamma ,\rho ,\sigma )$ of the EMG distribution for: (a) $\gamma =42$, $\rho =15$ and $\sigma =3$ (——); (b) $\gamma =45$, $\rho =6$ and $\sigma =3$ (− − −); and (c) $\gamma =30$, $\rho =24$ and $\sigma =9$ (− · − ·).

Appendix C. Model estimation

We briefly outline a model estimation method in this appendix. For simplicity, we focus on Equation (6)(6) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+e_{1n}\mathrm{with}\mathrm{probability}\theta \\ {\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}+e_{2n}\mathrm{with}\mathrm{probability}1-\theta \end{array}$(6) . Suppose that a random sample consisting of $N$ vehicles is selected. For each vehicle $n=1,\dots ,N$, let $s_n$ denote the measurement on the vehicle spacing, and $v_{n-1}$ denote the measured speed of the vehicle ahead. From Equation (6)(6) $s_n=\{\begin{array}{l}{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1}+e_{1n}\mathrm{with}\mathrm{probability}\theta \\ {\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}+e_{2n}\mathrm{with}\mathrm{probability}1-\theta \end{array}$(6) , we can write out the likelihood function as follows: $\begin{array}{rl}L& =\prod _{n=1}^{N}g\left(s_n|v_{n-1}\right)=\prod _{n=1}^N\{\phantom{\frac{s_n+\gamma -\left({\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}\right)-{\eta }^2/\gamma }{\eta }}\mathrm{\theta \phi }\left(s_n;{\text{β}}_{10}+{\text{β}}_{11}{v}_{n-1},{\sigma }_1^2\right)+\left(1-\theta \right){\gamma }^{-1}\\ & \times \exp \left(-\frac{s_n+\gamma -\left({\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}\right)-{\eta }^2/\left(2\gamma \right)}{\gamma }\right)\mathrm{\Phi }\left\{\frac{s_n+\gamma -\left({\text{β}}_{20}+{\text{β}}_{21}{v}_{n-1}\right)-{\eta }^2/\gamma }{\eta }\right\}\},\end{array}$ with $\eta =\sqrt{{\sigma }_2^2-{\gamma }^2}>0$. Maximising the above likelihood function yields the estimated coefficients of the linear mixture spacing-speed model (6), including ${\text{β}}_{10}$, ${\text{β}}_{11}$, ${\text{β}}_{20}$, ${\text{β}}_{21}$, $\theta$, $\gamma$, ${\sigma }_1$ and ${\sigma }_2$.