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Modeling phase diagrams as stochastic processes with application in vehicular traffic flow
Accepted Manuscript
Modeling phase diagrams as stochastic processes with application in vehicular traffic flow Daiheng Ni , Hui K Hsieh , Tao Jiang PII: DOI: Reference:
S0307-904X(17)30547-4 10.1016/j.apm.2017.08.029 APM 11941
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
28 October 2016 1 August 2017 22 August 2017
Please cite this article as: Daiheng Ni , Hui K Hsieh , Tao Jiang , Modeling phase diagrams as stochastic processes with application in vehicular traffic flow, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.08.029
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Highlights Proposed a methodology to model phase diagrams as stochastic processes. Formulated a generic procedure to determine homogeneous distributions with parameters derived from empirical data. Applied the methodology to address fundamental diagram of vehicular traffic flow. Conducted a verification on the resultant stochastic fundamental diagram using empirical data. Discussed the significance of the research and its applications beyond vehicular traffic flow.
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Modeling phase diagrams as stochastic processes with application in vehicular traffic flow Daiheng Ni a,*, Hui K Hsieh b, Tao Jiang,a a
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Department of Civil and Environmental Engineering, University of Massachusetts Amherst, MA 01003, USA b Department of Mathematics and Statistics, University of Massachusetts Amherst, MA 01003, USA * Corresponding author. Tel.: (413) 545-5408; fax: (413) 545-9569; E-mail address: [email protected] (D. Ni).
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Abstract:
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Motivated by the similarity between the fundamental diagram of vehicular traffic and the Maxwell-Boltzmann distribution of ideal gases, this paper proposed a methodology to model the fundamental diagram as a stochastic process which also applies to other real-world systems with similar nature. A concrete example is provided to illustrate the application of the methodology where the fundamental diagram of vehicular traffic is modeled as a stochastic process to capture the scattering effect in flow-density relationship. A verification study was conducted on the model using empirical data and the statistical analysis shows that the overall quality of the fitted stochastic process is acceptable. Related existing efforts are referenced to the proposed stochastic fundamental diagram where their similarities and differences are elaborated. Further discussion is carried out on the significance of the stochastic fundamental diagram as well as the proposed methodology with an additional real-world example to illustrate its applications.
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Keywords: vehicular traffic flow, fundamental diagram, Maxwell-Boltzmann distribution, stochastic process.
Introduction
Many real-world systems are governed by their underlying phase diagrams that are defined by a set of state variables representing the characteristics of such systems. A clear understanding of and practical solutions to these systems frequently depends on our ability to capture these phase diagrams which are typically stochastic processes. An example in this case is idealized gases under thermodynamic equilibrium in a container. The distribution of the speeds of gaseous molecules in this context was derived by Maxwell and further investigated later by Boltzmann, see an illustration of the Maxwell- Boltzmann
ACCEPTED MANUSCRIPT 3 distribution in Part 1 of Figure . When temperature is low (e.g., T1), the mean of molecule speeds is low and their variance is small. As temperature progressively increases (e.g., T2, T3, T4), a salient transition is that both the mean and the variance of molecular speeds increase accordingly. As a result, the corresponding distribution curves flatten out. Traffic speed v, m/s density
Idealized Gases
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Vehicle speeds v, m/s Part 3: Speed-density distribution Part 4: Stochastic fundamental diagram
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Figure 1. Examples of Engineering stochastic processes
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Another example of similar nature is the fundamental diagram of vehicular traffic, see Part 2 of Figure . The solid blue curve represents a typical equilibrium relationship between traffic speed (the mean of vehicular speeds in km/hr) and traffic density (measured as number of vehicles in a unit length of road, e.g., veh/km). Field observations reveal that, at each density level, drivers’ speed choices vary, but can generally be described as a speed distribution around the mean speed, see the three-dimensional representation in Part 4 of Figure . In addition, the distribution differs as traffic density changes. Also observed in the field is how the variance of vehicular speeds varies in the processes. For example, when density level is low, say veh/km, the variance of vehicular speeds is quite small. As density increases to the optimal level where capacity occurs, the variance peaks, but afterwards the variance reverts to small values as density further increases. Such a trend is shown as the read dash-dot curve in Part 2 of Figure . If one rotates Part 4 of Figure clockwise around its vertical (“probability”) axis, something similar
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The General Methodology
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to Part 3 of Figure is resulted. Strikingly, Parts 1 and 3 resemble each other except that, in the high-speed end, the curves in Part 1 keeps flattening out, while the curves in Part 3 revert to concentrated forms like those in the low-speed end. Similar to idealized gases and vehicular traffic, many real-world systems share the following features in common: (1) these systems evolve along a state variable (referred to as the “control variable” thereafter) such as temperature in idealized gases and density in vehicular traffic; (2) given a level of the control variable, another state variable such as molecule speed and vehicle speed (called the “target variable” thereafter) forms a distribution; (3) as the control variable evolves, the distribution of the target variable changes accordingly, but in general remains as the same type such as Gaussian or Beta whose underlying parameters vary with the control variable, e.g., the variance of vehicle speed is a function of density, ; (4) there exists an equilibrium relationship between the mean of the target variable and the control variable such as the traffic speed-density relationship . Therefore, the objective of this research is to develop a methodology that formulates system phase diagrams as stochastic processes, based on which the fundamental diagram of vehicular traffic flow is presented as a specific application of the methodology. The rest of the paper is arranged as follows. The next section proposes a methodology to address the problem in generic terms. The methodology is then applied to model stochastic fundamental diagram of vehicular traffic in Section 3, and a verification study using empirical data is carried out in Section 4. After that, this research is related to existing efforts in Section 5 and further discussion on the significance of this research is presented in Section 6 with an additional application example. Finally, conclusions are drawn in Section 7.
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Suppose the phase diagram of a real-world system is characterized by a set of state variables in time and space domain, among which the relationship between a control variable and a target variable is of interest. Also assume that empirical data of and can be collected from the real-world system under study. With sufficient amount of data, an empirical relationship between and can be graphically or numerically established, see an example in the left part of Figure where the data were collected from Georgia 400, a toll road to the north of Atlanta, Georgia. Such a relationship typically represents the phase diagram of the underlying system. The goal here is to model the phase diagram as a stochastic process using the following algorithm.
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Figure 2. An engineering phase process
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Step 1: Categorize empirical data and compute statistics
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The modeling approach starts with computing statistics of the empirical observations which are sorted according to the control variable . At each level of where takes a specific value if is discrete or represents an interval if is continuous, statistics of the target variable observations are computed: such as first moment (mean) , second moment (variance) , third moment (skewness) , fourth moment (kurtosis) , etc. Step 2: Map the statistics to the control variable and fit a function for the mapping
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As a result, a set of mapping is found for each of these statistics in the entire range of the control variable and the result is illustrated in the right part of Figure . Consequently, it is possible to fit each of the above statistics as a function of ̂ . the control variable: ̂ ̂ ,̂ ̂ , ̂ ̂ , ̂ Meanwhile, a certain type of statistical distribution is sought to represent the empirical distributions of the target variables at each level of the control variable. Assume that such a type of distribution is found with a set of underlying parameters such as: location , scale , shape , etc.
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Step 3: Relate the statistics to the parameters of chosen distribution Depending on the nature of the proposed distribution, there might be some relations between the statistics of the distribution and its parameters:
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Solving the above system of equations yields: ̂
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Step 5: Formulate the stochastic process
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Step 4: Solve for the parameters as functions of the statistics
Eventually, the statistical distribution can be represented as:
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Since each of the parameters is a function of the control variable, the above statistical distribution essentially represents a stochastic process as the control variable evolves:
Modeling Stochastic Fundamental Diagram
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An example to illustrate the application of the above methodology would be fundamental diagram that lies in the center of vehicular traffic flow theory. On highways, drivers adjust their speeds when following another vehicle, which constitutes the basis of car-following models. For example, Forbes [1] stipulated that the following driver observe a perception-reaction time behind the leading vehicle; Gipps [2] further added a buffer time to account for unexpected deceleration; Gazis et al [3] assumed a gravity effect between the leading and following vheicles; Ni [4] postulated that a driver perceives other vehicles as potential fields and proceeds along the valley of these fields. When aggregated over time and vehicles, each of these car-following models typically implies an equilibrium speed-density relationship which is part of the so-called fundamental diagram. From there, a flow-density relationship can be derived which plays an important role in dynamic traffic control and operations. Meanwhile, a speed-flow relationship can be derived which is essential to capacity analysis and determining level of service [5]. However, conventional approaches to the modeling of fundamental diagram is deterministic including two-parameter models [6] [7] [8], three-parameter models [9] [10], and four-parameter models [11], [12], and [4]. These models only capture the average behavior of traffic flow, e.g., the solid curve (“speed mean”) in the right part of Figure . Recognizing the scattering effect of
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the field observations as illustrated in the left part of Figure , a stochastic fundamental diagram might be more suited to represent the ground truth. According to the procedure presented in the Methodology section, the modeling of stochastic fundamental diagram takes the following steps: (1) categorizing empirical observations according to the control variable (i.e., density in this example) and computing statistics for each category, (2) modeling the mean of the target variable (e.g., speed) as a function of the control variable (e.g., equilibrium speed-density relationship), (3) modeling the variance of the target variable as a function of the control variable, and (4) modeling the fundamental diagram as a stochastic process. Categorization of empirical data
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First, the entire range of the control variable is divided into a series of categories if the control variable is discrete or intervals/bins if the control variable is continuous. Our case necessitates bins and the number of bins is chosen to ensure sufficient samples in each bin. In each bin (represented by the mid-value of the interval), there are a set of observations (e.g., speed samples ), based on which statistics are computed such as mean ̅ , variance , and number of observations . As a result, the data is aggregated into a new set ( ̅ ) . Note that the left part of Figure illustrates the original data set (i.e., the scatter plot), and the right part shows the processed statistics (i.e., mean and variance) of the data set.
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The mean of the target variable
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With the above result, speed mean is then fitted as a function of density, i.e., to establish an equilibrium speed-density relationship. Prior empirical studies [4] suggest that the macroscopic version of the Longitudinal Control Model (LCMx4) is a good option:
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where is free-flow speed, is aggressiveness, is perception-reaction time, and is effective vehicle length. Note that the nature of the problem requires that be expressed as a function of . Though LCMx4 has as an explicit function of , the other way around is difficult to obtain since the right hand side involves as can be seen in the second equation. Also note that this is a common issue to other state-of-the-art equilibrium models such as [11] and [12]. Fortunately, this problem can be easily resolved numerically. The variance of the target variable To fit the variance of speed, one has a variety of choices. A close analysis of field observation reveals that speed variance is apparently small when density is low. This is plausible since, at low density, drivers have a plenty of room to maintain their desired speeds which tend to be quite uniform around free-flow speed. Meanwhile, the speed variance also appears to be small at high density since there isn’t much room for drivers to improve their speeds even though they are so desired. In contrast, the speed variance peaks around optimal density, i.e., the density when
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The fundamental diagram as a stochastic process
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capacity is reached. This is so because on one hand drivers are likely to encounter slow drivers in the neighborhood of optimal density, and on the other hand there are still opportunities for drivers to change lane and find room to improve their speeds. Given the above nature of traffic flow and the particular shape of variance curve as illustrated in Figure , basically a function that features a peak with two tails would be suitable. Though fitting speed mean as a function of density is difficult, it is still possible to find equilibrium models with fine fitting quality. In contrast, there is no known solution to fitting speed variance. Among other choices, log-normal density function below with four parameters , , , and is found to be acceptable in terms of a combined consideration of function flexibility and proximity to empirical data used in this study. However, there is still much room for improvement as can be seen in the next section.
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With speed mean and variance as functions of density, it is time to search for a suitable type of distribution to represent speed distributions under varying density levels. Several considerations must be incorporated in the search. First, it is desirable to find a type of distribution that is supported on a bounded interval since vehicle speed ranges between 0 and some maximum speed . Otherwise, one might accidentally draw negative or impossible speed samples when applying the model. In addition, the type of distribution needs to be flexible enough to accommodate varying distribution curves, most of which feature a peak. Figure illustrates a spectrum of empirical distributions of vehicle speed (km/h) under different density levels (veh/km). These p.d.f. curves are generated from the Georgia 400 data set. It can be seen that the p.d.f. curves exhibit varying shapes ranging from a skinny pike to a regular bell shape to a flat hill. Meanwhile, the figure strikingly reminds us the schematic sketch in Part 3 of Figure .
Figure 4. A spectrum of empirical distributions of vehicle speed
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With these considerations in mind, we conducted a search for the desirable type of distribution. As a result, Beta distribution stands out to be promising and meets all the requirements. For example, its distribution function is bounded, continuous, and can be configured to varying shapes that fit the need of this study. In addition, it avoids issues of alternative distributions. For example, triangular distribution necessitates piece-wise formulation and pointy peak; truncated normal distribution has a complicated functional form that is difficult to manage; though attractive, logit-normal distribution does not have an analytical solution to the mean and the variance that is critical to implementing the proposed methodology. Therefore, an analysis of the properties has warranted the chosen type of distribution before running numerical tests. Beta distribution with parameters and has the following p.d.f:
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Beta distribution features a mean and variance in relatively simple terms:
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Note that the above formulation is for standard Beta distribution which supports a range of . However, the current application requires a speed range of , which means that the standard Beta distribution has to be transformed to suit the new range as follows. The standard Beta distribution has a total probability of 1 over range : ∫
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Similarly, for the new distribution with p.d.f. ∫
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Solving the above system of equation yields: ̂ ̂
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Since parameters and now becomes functions of the control variable , plugging them into the new Beta distribution p.d.f. results in:
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Empirical verification
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that models vehicular
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To verify the above methodology, a numerical study is conducted using empirical data collected from GA400, a toll road to the north of Atlanta, GA. The data consists of one year worth of observations at a mainline station equipped with a video image processing system. Each data entry represents a point observation in 20 seconds over 4 lanes. A variety of traffic information is collected including classified vehicle counts and average speed. Ideally, individual vehicle speeds are desirable in this application, but unfortunately such data was not logged. Alternatively, the average speed of vehicles observed in the 20-second interval is used as a surrogate to vehicle speed. Meanwhile, density is estimated from vehicle counts and average speed since direct measurement of density is not supported in point observations. Figure shows the raw data in the left and aggregated data in the right (the variance is halved to fit in).
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Figure 5. Fitting the fundamental diagram Figure shows the result of fitting LCMx4 model to the empirical data collected from station GA4000043. The dots are the data and the dashed curves represent the model. Model parameters are: free-flow speed = 27.8 m/s, effective vehicle length = 3.0 m, aggressiveness = 2 0.0435 s /m, and perception-reaction time = 1.47 s. The top left of the figure is the model fitted to the empirical mean of speed as a function density. The bottom right is the same plot but presented from a different angle by transforming density to its reciprocal, i.e., spacing defined as the average bumper-to-bumper distance between two consecutive vehicles. The top right and bottom left plots are speed-flow and flow-density relationships respectively derived from the speed-density relationship using the identity among flow , speed , and density : . This collection of plots is generally referred to as the fundamental diagram in vehicular traffic flow community. In this study, we employed from analysis of variance (ANOVA) to assess fitting quality where ranges between 0 and 1 with 1 being the best. The resultant for the speed mean fitting is 0.9986 which means that the model is able to explain 99.86% of the information contained in the data. Given the excellent performance of LCMx4 in modeling fundamental diagram, there are still a few small areas of under-fit. For example, in the beginning part of v-k relationship, the curve tends to be flat while the data slants down. As a result, in the top portion of the speed-flow relationship, the model fails to follow the slightly slanting trend of the data. Related to this, the capacity predicted by the model appears at an optimal speed that is a little higher than that suggested by the data. Also note that the tail of the flow-density relationship
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seems scattered to certain degree and the area right after the peak exhibit a little under-fit. Nevertheless, the overall fitting quality appears exceptional.
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Figure 6. Fitting the empirical speed variance Figure shows the result of using the four-parameter log-normal function to fit the empirical speed variance. The four parameters are configured as: = 37.80, = 0.43, = 8.44, and = 4580.42. The resultant = 0.8452 which is acceptable but not outstanding. Meanwhile, the empirical speed variance exhibits some visible features. First, speed variance under extremely low density ( ) is quite large, which is contrary to common sense. As density slightly increases, the variance decreases to a local minimal. As density continues to increase, the variance quickly increases to its peak at about 28 veh/km which is around the optimal density when capacity is reached. As density further increases, the variance drops and then asymptotically approaches zero at jam density. A few locations of under-fit are observable. First, the model fails to capture the decreasing effect of variance under very low density levels. Second, the model fails to hit the peak of the empirical variance, though they both peak at about the same density. Third, the model does not follow the asymptotically decreasing tail well, but rather flattens out toward the end. As it will become evident shortly, the above areas of lack of fit eventually find their way into the modeling of the stochastic process. Figure overlays the fitted distributions on top of the spectrum of empirical distributions of vehicle speed illustrated in Figure . Beta distribution is used to fit the data using the formulae of p.d.f. and parameters provided in the previous section. The figure reveals an acceptable fitting quality with rooms to improve.
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Figure 7. A spectrum of empirical distributions of vehicle speed fitted with Beta distribution Figure shows the overall fitting quality using as an indicator over the entire density range. At each density level, a Beta distribution is fitted to the data based on and at that density level. Then is computed. Normally a goodness-of-fit test such as test or Kolmogorov– Smirnov test is employed to compare two distributions. However, a close analysis of the nature of current problem suggests that is more relevant. This is so because, in a goodness-of-fit test, our major goal is to prove whether a set of samples come from a known distribution or two sets of samples come from the same distribution. This is not the case in our problem which aims to find a model to represent the data. As such, a model is acceptable as long as it is able to represent/replicate the data. Whether or not the model and the data are from the same distribution is not so critical.
Figure 8. Overall fitting quality of the stochastic process In Figure , the best fit appears in the mid-range of density between 50 and 150 with the two ends under fit. Note that, once the distribution type is determined, the fitting quality depends mainly on distribution parameters, i.e., and in our case. Since these two parameters are computed from speed mean ̂ and variance ̂ , the fitting quality of ̂ and ̂
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Relation between This Study and Existing Efforts
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directly affects that of the stochastic process. As clearly evident in Figure , the under fit at both ends is due primarily to the under fit of speed variance ̂ shown in Figure . Therefore, the key to improving the fitting of the stochastic process is to enhance the fitting of ̂ given that there is not much room for improvement in fitting ̂ . Nevertheless, the overall weighted on sample size is 0.8715 which is quite reasonable and acceptable. Note that factors such as vehicle composition, driver behavior, and weather condition can influence drivers’ speed choice and collectively speed distribution. One way to take these influencing factors into consideration is to adopt an explanatory approach that explicitly captures the mechanism how these factors affect speed. In contrast, this research takes a descriptive approach that attempts to fit a model to observed data. As such, the effect of the above influencing factors is captured indirectly through their resultant field observations.
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Historically, the conventional approach to fundamental diagram is deterministic modeling that features a functional relationship between any two of the three state variables: flow, speed, and density. These models may involve two-parameter models [6] [7] [8], three-parameter models [9] [10], and four-parameter models [11], [12], and [4]. More parameters may empower a model with better flexibility in fitting data. However, as indicated above, deterministic models only captures the average behavior of traffic flow and fail to capture the scattering effect of field observations. Obviously, a stochastic fundamental diagram such as the above proposed one is able to better capture the ground truth. Realizing the limitation of the deterministic approach, many researchers seek to represent fundamental diagram using a multiclass approach. For example, Saberi and Mahmassani [13] attributed the scattering effect as a result of loading and unloading processes that follow their own paths in fundamental diagram and may be represented using different models. Though capturing part of the scattering effect, such an approach is still deterministic in nature. Ngoduy [14] argued that the scattering effect might be caused by the random variations in driving behavior. To reproduce the hysteresis transitions and the wide scattering effect, Ngoduy utilized a multiclass first-order model with a stochastic setting in the model parameters. In an attempt to understand the reliability of LWR model [15] [16], Li et al [17] hypothesized a probabilistic fundamental diagram by postulating a flux function driven by a random free flow speed. Chen et al [18] proposed a stochastic fundamental diagram based on headway/spacing distributions. Though the above three models involve certain stochastic elements, they are not true stochastic models. In a closely related study, Wang et al [19] proposed a stochastic speed-density relationship. The model takes the form of traffic speed as the sum of a mean speed and a variance, each of which is treated as a function of density separately. As the deterministic term, the mean speed takes the form of a logistic function. Capturing the stochastic residuals, the variance term employs the Karhunen-Loeve expansion. In essence, this approach entails a deterministic mean and a stochastic variance, leading the underlying speed distribution undetermined in the entire range of density. In contrast, the current study systematically models fundamental diagram as a stochastic process captured completely in a Maxwell-Boltzmann-like distribution. Several other related studies are also identified. Helbing [20] concerned about the speed distribution of vehicles as well as its variance and skewness. Also of interest was the functional form of the speed-density relation and the variance-density relation at high density. Helbing used
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a Gaussian form to represent speed distribution and even presented in FIG. 1 of that paper a speed distribution diagram under varying density levels which is similar to Part 1 of Figure in this paper. Perhaps this seminal work was the first presentation of Maxwell- Boltzmann-liked distribution of vehicular speed, though it was not explicitly claimed. However, as evidenced in Figure and Figure of this study, FIG. 1 of [20] failed to show the full picture with the distribution reverting effect under low density levels being ignored. In a recent study, Jabari et al [21] used shifted gamma function to represent speed distribution and Fig. 12 of that paper does reflect the distribution reverting effect. Ben-Naim and Krapivsky [22] applied Boltzmann equation in studying vehicular traffic flow dynamics which is simplified as a one-dimensional problem. By making analogy to kinetic theory and introducing Maxwell model, they resulted in first-order differential equations which led to explicit expressions for time-dependent speed distributions in generic terms. In contrast, our search focuses on the steady-state properties of vehicular traffic flow and explicitly relate it to Maxwell-Boltzmann distribution with concrete forms. In a further effort, Galstyan and Lerman [23] formulated a joint speed-size distribution when applying the stochastic Master equation approach to study platoon formation. They pointed out that the distribution can’t be solved with speed dependent kernel, but the distribution in [22] is analytically tractable given the Maxwell model. The Significance of this Research
The contributions of this research can be interpreted at two levels with one regarding the stochastic fundamental diagram and the other pertaining to the general methodology.
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The implications of the stochastic fundamental diagram
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The significance of the stochastic fundamental diagram can be interpreted from multiple perspectives. Firstly, it offers a true stochastic means to traffic flow modeling and prediction. Our conventional approach to traffic flow simulation features deterministic models such as microscopic car following and macroscopic LWR. The so-called randomness in simulation, if any, is implemented by means of input data and model parameters such as a distribution of entry volume and a distribution of perception-reaction time. In contrast, the outcome of this research allows true stochastic traffic simulation because the underlying model itself is stochastic. While there is no guarantee what macroscopic behavior of traffic flow will look like when randomizing entry volumes and model parameters, the stochastic approach proposed in this study empowers traffic flow simulation with predictable macroscopic characteristics that converges to known behavior. In addition, such a stochastic fundamental diagram would allow us to make probabilistic prediction of traffic condition as opposed to the traditional deterministic prediction. Secondly, the wide scattering phenomenon in the fundamental diagram has inspired many school of thought on modeling. For example, some resorted to multi-class heterogeneous traffic flow [24] [14], some attributed to loading and unloading processes [13], and Kerner [25] described it as synchronized flow phase. Though these interpretations sound plausible, the ground truth has yet to reveal itself. Through the modeling of stochastic fundamental diagram, this research seems to offer yet another perspective: while it has long been believed that the central tendency of the speed-density distributions is related to the longitudinal motion of vehicles (e.g., the equilibrium speed-density relationship is related to car following), the variance appears have something to do with vehicle lateral motion. For example, vehicle lateral motion is essentially a sequential
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decision at two steps. The first step is the desire to make a lane change, e.g., after being blocked by a slow driver for some time. This is conventionally called a lane changing model and is typically treated as a discrete choice problem. Once the desire of lane change has been registered in the driver’s mind, the next step is to seek a good opportunity to switch to the target lane which is called gap acceptance in the community and modeled as another discrete choice problem. Considering that, at low density, drivers are free to travel at their desired speeds which tend to be around free-flow speed, very few drivers have the desire to change lane even though acceptable gaps are readily available. Consequently the probability of lane change is very low. On the other hand, at high density where the road is packed with vehicles, drivers’ desire for mobility is greatly suppressed and hence changing lanes to improve speed is strongly motivated. However, the probability of lane change is still very low since an acceptable gap in the target lane is rarely available. Only in certain range of density does lane change become frequent because there exists not only considerable desire to seek speed gain in other lanes but also ample gap opportunities to carry out the decision. In addition, such a frequency tends to peak around optimal density where capacity flow occurs. If one relates the frequency to speed variance and cross compare with empirical evidences in Figure and Figure , one would agree that the above theory and the field observations indeed bear some resemblance. Combined, the stochastic fundamental diagram seems to have captured information of vehicle motion in both directions in a single model. Thirdly, the proposed model can shed light on other engineering disciplines and help unite theories of vehicular traffic and other forms of traffic. For example, many engineering systems share similar features to that of vehicular traffic in their phase diagrams. The proposed methodology can serve as a unifying framework to formulate these stochastic processes which opens a door for cross-comparison and inspiring each other. As presented in Figure , the similarity between Maxwell-Boltzmann distribution and speed-density distribution seems to suggest a root of vehicular traffic in molecular dynamics and further physical science. Meanwhile, exploration of the difference brought about by their domain specifics especially the distribution reverting effect under low density levels might give birth to new discovery beyond Maxwell-Boltzmann distribution and fundamental diagram.
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The implications of the general methodology
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Beyond Maxwell- Boltzmann distribution and vehicular fundamental diagram, the proposed methodology can also be used to model other real-world systems where their phase diagrams are defined by a set of statistically related state variables. For example, Sociology concerns about life expectancy, a statistical measure of the average time an individual is expected to live. Among other influencing factors, gross domestic product (GDP) per capita plays a significant role in a country’s average life expectancy. The left part of Figure scatters the relationship between life expectancy and GDP per capita. Provided by the World Bank, the data include most countries in the world covering the period between 1960 and 2015. Each data point represents a country’s state (given by life expectancy and GDP per capita) in a specific year. The right part of Figure shows the mean and variance of the data point. It can be seen that at a certain level of GDP per capita, life expectancy is distributed with mean and variance as indicated. The striking resemblance between Figure and Figure suggests that the proposed methodology is readily applicable to the modeling of life expectancy. The resulting model can be very useful in
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Sociology to answer questions such as an estimate of the average number of additional years that people of a given age in a country with given level of GDP per capita can expect to live.
Figure 9 The relationship between life expectancy and GDP per capita (Source: World Bank) Conclusions
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Motivated by the similarity between the fundamental diagram of vehicular traffic and the Maxwell-Boltzmann distribution of ideal gases, this paper proposed a methodology to model the fundamental diagram as a stochastic process. Since the methodology is formulated in generic terms, it is also applicable to other systems whose phase diagrams exhibit similar nature. The methodology follows a data-to-model approach that starts from empirical data which consist of observations of the relationship between a control variable and a target variable. As such, the data are categorized according to the control variable and statistics of the target variable such as its mean and variance are computed based on the categorized data. With the above results, models are fitted to capture these statistics as functions of the control variable. With a good understanding of the empirical data, a distribution type is selected which is best suited to represent the distributions of the target variable under varying levels of the control variable. With that, the relationship between the parameters of the selected distribution and the statistics of the empirical data can be established. Therefore, a stochastic process can be formulated when the selected distribution type is instantiated with parameters expressed as functions of the control variable. A concrete example is provided to illustrate the application of the methodology that models the fundamental diagram of vehicular traffic. Following the procedure laid out in the methodology, a stochastic fundamental diagram is formulated that instantiates the generic model with options of specific forms. It turns out that mean of the target-control variables finds its root in vehicle longitudinal motion, while the variance might be closely related to vehicle lateral motion. Together the stochastic fundamental diagram may capture information of vehicle motion in both directions in a single model. An empirical verification was conducted using data collected from an observation station on GA400. Borrowed from the analysis of variance, the statistics is employed to evaluate the
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fitting quality. It appears that the overall quality of the stochastic process is determined by that of the fitted mean and variance of vehicle speed. While equilibrium traffic flow models with fine quality are available to fit the mean, good models are still being sought to fit the variance. Further discussion is carried out to relate this research to existing efforts as well as on the significance of the stochastic fundamental diagram. It appears that the resulting stochastic fundamental diagram may be used to capture information of vehicle motion in longitudinal and lateral directions in a single model that advances the state of the art of traffic flow modeling and simulation. Obviously, the proposed methodology is applicable to not only vehicular traffic but also other real-world systems with similar nature such as a population whose life expectancy varies with GDP level. Acknowledgements
This research is partly supported by the United States Department of Transportation (USDOT) University Transportation Center (UTC) grants.
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[1] T. Forbes, M. Simpson, Driver and Vehicle Response in Freeway Deceleration Waves, Transportation Science, 2 (1968) 1:77-104. [2] P. Gipps, A Behavioral Car Following Model for Computer Simulation, Transportation Research, Part B, 15 (1981) 105-111. [3] D. C. Gazis, R. Herman, R. W. Rothery, Non-Linear Follow the Leader Models of Traffic Flow, Operations Research, 9 (1961) 545-567. [4] Daiheng Ni, John D. Leonard, Chaoqun Jia, Jianqiang Wang, "Vehicle Longitudinal Control and Traffic Stream Modeling," Transportation Science, 50 (2015) 3:1016-1031. [5] TRB, Highway Capacity Manual.: Transportation Research Board, 2010. [6] B. D. Greenshields, A study in highway capacity, Highway Research Board, Proceedings 14 (1935) 448-477. [7] H. Greenberg, An Analysis of Traffic Flow, Operations Research, 7 (1959) 78-85. [8] R.T. Underwood, Speed, Volume and Density Relationships, Quality and Theory of Traffic Flow, Yale University Report, New Haven, Connecticut, 1961. [9] J. M. del Castillo, F. G. Benítez, On the Functional Form of the Speed-Density Relationship - I: General Theory, Transportation Research Part B: Methodological, 29 (1995) 5:373-389. [10] G. F. Newell, Nonlinear Effects in the Dynamics of Car Following, Operations Research, 9 (1961) 2:209-229. [11] Martin Treiber, Ansgar Hennecke, Dirk Helbing, Congested Traffic States in Empirical Observations and Microscopic Simulations, Phys. Rev. E, 62 (2000) 1805–1824. [12] M. Van Aerde, Single Regime Speed-Flow-Density Relationship for Congested and Uncongested Highways, the 74th Transportation Research Board Annual Meeting, Washington, D.C., 1995. [13]M. Saberi, H. Mahmassani, Hysteresis and Capacity Drop Phenomena in Freeway Networks: Empirical Characterization and Interpretation, Transportation Research Record, 2391 (2013) 4455. [14] D. Ngoduy, Multiclass first-order traffic model using stochastic fundamental diagrams, Transportmetrica, 7 (2011) 2:111-125.
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[15] Lighthill M. J., Whitham G. B., On Kinematic Waves: a Theory of Traffic Flow on Long Crowded Roads, Proc. of the Royal Society, Series A, 229 (1955) 317-345. [16] P. I. Richards, Shock Waves on the Highway, Operations Research, 4 (1956) 42-51. [17] Jia Li, Qian-Yong Chen, Haizhong Wang, Daiheng Ni, Analysis of LWR model with fundamental diagram subject to uncertainties, Transportmetrica, 8 (2012) 6: 387-405. [18] Xiqun (Michael) Chen, Li Li, Qixin Shi, Stochastic Fundamental Diagram Based on Headway/Spacing Distributions, Stochastic Evolutions of Dynamic Traffic Flow.: Springer Berlin Heidelberg, 2014, pp. 81-115. [19] Haizhong Wang, Daiheng Ni, Qian-Yong Chen, Jia Li, Stochastic Modeling of Equilibrium Speed-Density Relationship, Journal of Advanced Transportation, John Wiley & Sons, 47 (2013) 1:126-150. [20] Dirk Helbing, Fundamentals of traffic flow, Physical Review E, 55 (1997) 3:3735-3738. [21] Saif Eddin Jabari, Jianfeng Zheng, Henry X. Liu, A Probabilistic Stationary Speed-Density Relation Based on Newell's Simplified Car-Following Model, Transportation Research Part B: Methodological, 68 (2014) 205–223. [22] E. Ben-Naim, P. L. Krapivsky, Maxwell Model of Traffic Flows, Physical Review E, 59 (1999) 1:88-97. [23] A. Galstyan, K. Lerman, A Stochastic Model of Platoon Formation in Traffic Flow, Proceedings of the TASK workshop, Santa Fe, NM, 2001. [24] G.C.K Wong, S.C Wong, A multi-class traffic flow model – an extension of LWR model with heterogeneous drivers, Transportation Research Part A: Policy and Practice, 36 (2002) 9:827–841. [25] B. Kerner, The Physics of Traffic. Berlin, New York: Springer, 2004.
Modeling phase diagrams as stochastic processes with application in vehicular traffic flow Daiheng Ni , Hui K Hsieh , Tao Jiang PII: DOI: Reference:
S0307-904X(17)30547-4 10.1016/j.apm.2017.08.029 APM 11941
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
28 October 2016 1 August 2017 22 August 2017
Please cite this article as: Daiheng Ni , Hui K Hsieh , Tao Jiang , Modeling phase diagrams as stochastic processes with application in vehicular traffic flow, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.08.029
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Highlights Proposed a methodology to model phase diagrams as stochastic processes. Formulated a generic procedure to determine homogeneous distributions with parameters derived from empirical data. Applied the methodology to address fundamental diagram of vehicular traffic flow. Conducted a verification on the resultant stochastic fundamental diagram using empirical data. Discussed the significance of the research and its applications beyond vehicular traffic flow.
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Modeling phase diagrams as stochastic processes with application in vehicular traffic flow Daiheng Ni a,*, Hui K Hsieh b, Tao Jiang,a a
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Department of Civil and Environmental Engineering, University of Massachusetts Amherst, MA 01003, USA b Department of Mathematics and Statistics, University of Massachusetts Amherst, MA 01003, USA * Corresponding author. Tel.: (413) 545-5408; fax: (413) 545-9569; E-mail address: [email protected] (D. Ni).
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Abstract:
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Motivated by the similarity between the fundamental diagram of vehicular traffic and the Maxwell-Boltzmann distribution of ideal gases, this paper proposed a methodology to model the fundamental diagram as a stochastic process which also applies to other real-world systems with similar nature. A concrete example is provided to illustrate the application of the methodology where the fundamental diagram of vehicular traffic is modeled as a stochastic process to capture the scattering effect in flow-density relationship. A verification study was conducted on the model using empirical data and the statistical analysis shows that the overall quality of the fitted stochastic process is acceptable. Related existing efforts are referenced to the proposed stochastic fundamental diagram where their similarities and differences are elaborated. Further discussion is carried out on the significance of the stochastic fundamental diagram as well as the proposed methodology with an additional real-world example to illustrate its applications.
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Keywords: vehicular traffic flow, fundamental diagram, Maxwell-Boltzmann distribution, stochastic process.
Introduction
Many real-world systems are governed by their underlying phase diagrams that are defined by a set of state variables representing the characteristics of such systems. A clear understanding of and practical solutions to these systems frequently depends on our ability to capture these phase diagrams which are typically stochastic processes. An example in this case is idealized gases under thermodynamic equilibrium in a container. The distribution of the speeds of gaseous molecules in this context was derived by Maxwell and further investigated later by Boltzmann, see an illustration of the Maxwell- Boltzmann
ACCEPTED MANUSCRIPT 3 distribution in Part 1 of Figure . When temperature is low (e.g., T1), the mean of molecule speeds is low and their variance is small. As temperature progressively increases (e.g., T2, T3, T4), a salient transition is that both the mean and the variance of molecular speeds increase accordingly. As a result, the corresponding distribution curves flatten out. Traffic speed v, m/s density
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Vehicle speeds v, m/s Part 3: Speed-density distribution Part 4: Stochastic fundamental diagram
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Figure 1. Examples of Engineering stochastic processes
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Another example of similar nature is the fundamental diagram of vehicular traffic, see Part 2 of Figure . The solid blue curve represents a typical equilibrium relationship between traffic speed (the mean of vehicular speeds in km/hr) and traffic density (measured as number of vehicles in a unit length of road, e.g., veh/km). Field observations reveal that, at each density level, drivers’ speed choices vary, but can generally be described as a speed distribution around the mean speed, see the three-dimensional representation in Part 4 of Figure . In addition, the distribution differs as traffic density changes. Also observed in the field is how the variance of vehicular speeds varies in the processes. For example, when density level is low, say veh/km, the variance of vehicular speeds is quite small. As density increases to the optimal level where capacity occurs, the variance peaks, but afterwards the variance reverts to small values as density further increases. Such a trend is shown as the read dash-dot curve in Part 2 of Figure . If one rotates Part 4 of Figure clockwise around its vertical (“probability”) axis, something similar
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The General Methodology
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to Part 3 of Figure is resulted. Strikingly, Parts 1 and 3 resemble each other except that, in the high-speed end, the curves in Part 1 keeps flattening out, while the curves in Part 3 revert to concentrated forms like those in the low-speed end. Similar to idealized gases and vehicular traffic, many real-world systems share the following features in common: (1) these systems evolve along a state variable (referred to as the “control variable” thereafter) such as temperature in idealized gases and density in vehicular traffic; (2) given a level of the control variable, another state variable such as molecule speed and vehicle speed (called the “target variable” thereafter) forms a distribution; (3) as the control variable evolves, the distribution of the target variable changes accordingly, but in general remains as the same type such as Gaussian or Beta whose underlying parameters vary with the control variable, e.g., the variance of vehicle speed is a function of density, ; (4) there exists an equilibrium relationship between the mean of the target variable and the control variable such as the traffic speed-density relationship . Therefore, the objective of this research is to develop a methodology that formulates system phase diagrams as stochastic processes, based on which the fundamental diagram of vehicular traffic flow is presented as a specific application of the methodology. The rest of the paper is arranged as follows. The next section proposes a methodology to address the problem in generic terms. The methodology is then applied to model stochastic fundamental diagram of vehicular traffic in Section 3, and a verification study using empirical data is carried out in Section 4. After that, this research is related to existing efforts in Section 5 and further discussion on the significance of this research is presented in Section 6 with an additional application example. Finally, conclusions are drawn in Section 7.
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Suppose the phase diagram of a real-world system is characterized by a set of state variables in time and space domain, among which the relationship between a control variable and a target variable is of interest. Also assume that empirical data of and can be collected from the real-world system under study. With sufficient amount of data, an empirical relationship between and can be graphically or numerically established, see an example in the left part of Figure where the data were collected from Georgia 400, a toll road to the north of Atlanta, Georgia. Such a relationship typically represents the phase diagram of the underlying system. The goal here is to model the phase diagram as a stochastic process using the following algorithm.
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Figure 2. An engineering phase process
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Step 1: Categorize empirical data and compute statistics
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The modeling approach starts with computing statistics of the empirical observations which are sorted according to the control variable . At each level of where takes a specific value if is discrete or represents an interval if is continuous, statistics of the target variable observations are computed: such as first moment (mean) , second moment (variance) , third moment (skewness) , fourth moment (kurtosis) , etc. Step 2: Map the statistics to the control variable and fit a function for the mapping
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As a result, a set of mapping is found for each of these statistics in the entire range of the control variable and the result is illustrated in the right part of Figure . Consequently, it is possible to fit each of the above statistics as a function of ̂ . the control variable: ̂ ̂ ,̂ ̂ , ̂ ̂ , ̂ Meanwhile, a certain type of statistical distribution is sought to represent the empirical distributions of the target variables at each level of the control variable. Assume that such a type of distribution is found with a set of underlying parameters such as: location , scale , shape , etc.
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Step 3: Relate the statistics to the parameters of chosen distribution Depending on the nature of the proposed distribution, there might be some relations between the statistics of the distribution and its parameters:
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Step 4: Solve for the parameters as functions of the statistics
Eventually, the statistical distribution can be represented as:
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Since each of the parameters is a function of the control variable, the above statistical distribution essentially represents a stochastic process as the control variable evolves:
Modeling Stochastic Fundamental Diagram
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An example to illustrate the application of the above methodology would be fundamental diagram that lies in the center of vehicular traffic flow theory. On highways, drivers adjust their speeds when following another vehicle, which constitutes the basis of car-following models. For example, Forbes [1] stipulated that the following driver observe a perception-reaction time behind the leading vehicle; Gipps [2] further added a buffer time to account for unexpected deceleration; Gazis et al [3] assumed a gravity effect between the leading and following vheicles; Ni [4] postulated that a driver perceives other vehicles as potential fields and proceeds along the valley of these fields. When aggregated over time and vehicles, each of these car-following models typically implies an equilibrium speed-density relationship which is part of the so-called fundamental diagram. From there, a flow-density relationship can be derived which plays an important role in dynamic traffic control and operations. Meanwhile, a speed-flow relationship can be derived which is essential to capacity analysis and determining level of service [5]. However, conventional approaches to the modeling of fundamental diagram is deterministic including two-parameter models [6] [7] [8], three-parameter models [9] [10], and four-parameter models [11], [12], and [4]. These models only capture the average behavior of traffic flow, e.g., the solid curve (“speed mean”) in the right part of Figure . Recognizing the scattering effect of
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the field observations as illustrated in the left part of Figure , a stochastic fundamental diagram might be more suited to represent the ground truth. According to the procedure presented in the Methodology section, the modeling of stochastic fundamental diagram takes the following steps: (1) categorizing empirical observations according to the control variable (i.e., density in this example) and computing statistics for each category, (2) modeling the mean of the target variable (e.g., speed) as a function of the control variable (e.g., equilibrium speed-density relationship), (3) modeling the variance of the target variable as a function of the control variable, and (4) modeling the fundamental diagram as a stochastic process. Categorization of empirical data
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First, the entire range of the control variable is divided into a series of categories if the control variable is discrete or intervals/bins if the control variable is continuous. Our case necessitates bins and the number of bins is chosen to ensure sufficient samples in each bin. In each bin (represented by the mid-value of the interval), there are a set of observations (e.g., speed samples ), based on which statistics are computed such as mean ̅ , variance , and number of observations . As a result, the data is aggregated into a new set ( ̅ ) . Note that the left part of Figure illustrates the original data set (i.e., the scatter plot), and the right part shows the processed statistics (i.e., mean and variance) of the data set.
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With the above result, speed mean is then fitted as a function of density, i.e., to establish an equilibrium speed-density relationship. Prior empirical studies [4] suggest that the macroscopic version of the Longitudinal Control Model (LCMx4) is a good option:
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where is free-flow speed, is aggressiveness, is perception-reaction time, and is effective vehicle length. Note that the nature of the problem requires that be expressed as a function of . Though LCMx4 has as an explicit function of , the other way around is difficult to obtain since the right hand side involves as can be seen in the second equation. Also note that this is a common issue to other state-of-the-art equilibrium models such as [11] and [12]. Fortunately, this problem can be easily resolved numerically. The variance of the target variable To fit the variance of speed, one has a variety of choices. A close analysis of field observation reveals that speed variance is apparently small when density is low. This is plausible since, at low density, drivers have a plenty of room to maintain their desired speeds which tend to be quite uniform around free-flow speed. Meanwhile, the speed variance also appears to be small at high density since there isn’t much room for drivers to improve their speeds even though they are so desired. In contrast, the speed variance peaks around optimal density, i.e., the density when
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The fundamental diagram as a stochastic process
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capacity is reached. This is so because on one hand drivers are likely to encounter slow drivers in the neighborhood of optimal density, and on the other hand there are still opportunities for drivers to change lane and find room to improve their speeds. Given the above nature of traffic flow and the particular shape of variance curve as illustrated in Figure , basically a function that features a peak with two tails would be suitable. Though fitting speed mean as a function of density is difficult, it is still possible to find equilibrium models with fine fitting quality. In contrast, there is no known solution to fitting speed variance. Among other choices, log-normal density function below with four parameters , , , and is found to be acceptable in terms of a combined consideration of function flexibility and proximity to empirical data used in this study. However, there is still much room for improvement as can be seen in the next section.
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With speed mean and variance as functions of density, it is time to search for a suitable type of distribution to represent speed distributions under varying density levels. Several considerations must be incorporated in the search. First, it is desirable to find a type of distribution that is supported on a bounded interval since vehicle speed ranges between 0 and some maximum speed . Otherwise, one might accidentally draw negative or impossible speed samples when applying the model. In addition, the type of distribution needs to be flexible enough to accommodate varying distribution curves, most of which feature a peak. Figure illustrates a spectrum of empirical distributions of vehicle speed (km/h) under different density levels (veh/km). These p.d.f. curves are generated from the Georgia 400 data set. It can be seen that the p.d.f. curves exhibit varying shapes ranging from a skinny pike to a regular bell shape to a flat hill. Meanwhile, the figure strikingly reminds us the schematic sketch in Part 3 of Figure .
Figure 4. A spectrum of empirical distributions of vehicle speed
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With these considerations in mind, we conducted a search for the desirable type of distribution. As a result, Beta distribution stands out to be promising and meets all the requirements. For example, its distribution function is bounded, continuous, and can be configured to varying shapes that fit the need of this study. In addition, it avoids issues of alternative distributions. For example, triangular distribution necessitates piece-wise formulation and pointy peak; truncated normal distribution has a complicated functional form that is difficult to manage; though attractive, logit-normal distribution does not have an analytical solution to the mean and the variance that is critical to implementing the proposed methodology. Therefore, an analysis of the properties has warranted the chosen type of distribution before running numerical tests. Beta distribution with parameters and has the following p.d.f:
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Note that the above formulation is for standard Beta distribution which supports a range of . However, the current application requires a speed range of , which means that the standard Beta distribution has to be transformed to suit the new range as follows. The standard Beta distribution has a total probability of 1 over range : ∫
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Similarly, for the new distribution with p.d.f. ∫
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Solving the above system of equation yields: ̂ ̂
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Since parameters and now becomes functions of the control variable , plugging them into the new Beta distribution p.d.f. results in:
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To verify the above methodology, a numerical study is conducted using empirical data collected from GA400, a toll road to the north of Atlanta, GA. The data consists of one year worth of observations at a mainline station equipped with a video image processing system. Each data entry represents a point observation in 20 seconds over 4 lanes. A variety of traffic information is collected including classified vehicle counts and average speed. Ideally, individual vehicle speeds are desirable in this application, but unfortunately such data was not logged. Alternatively, the average speed of vehicles observed in the 20-second interval is used as a surrogate to vehicle speed. Meanwhile, density is estimated from vehicle counts and average speed since direct measurement of density is not supported in point observations. Figure shows the raw data in the left and aggregated data in the right (the variance is halved to fit in).
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Figure 5. Fitting the fundamental diagram Figure shows the result of fitting LCMx4 model to the empirical data collected from station GA4000043. The dots are the data and the dashed curves represent the model. Model parameters are: free-flow speed = 27.8 m/s, effective vehicle length = 3.0 m, aggressiveness = 2 0.0435 s /m, and perception-reaction time = 1.47 s. The top left of the figure is the model fitted to the empirical mean of speed as a function density. The bottom right is the same plot but presented from a different angle by transforming density to its reciprocal, i.e., spacing defined as the average bumper-to-bumper distance between two consecutive vehicles. The top right and bottom left plots are speed-flow and flow-density relationships respectively derived from the speed-density relationship using the identity among flow , speed , and density : . This collection of plots is generally referred to as the fundamental diagram in vehicular traffic flow community. In this study, we employed from analysis of variance (ANOVA) to assess fitting quality where ranges between 0 and 1 with 1 being the best. The resultant for the speed mean fitting is 0.9986 which means that the model is able to explain 99.86% of the information contained in the data. Given the excellent performance of LCMx4 in modeling fundamental diagram, there are still a few small areas of under-fit. For example, in the beginning part of v-k relationship, the curve tends to be flat while the data slants down. As a result, in the top portion of the speed-flow relationship, the model fails to follow the slightly slanting trend of the data. Related to this, the capacity predicted by the model appears at an optimal speed that is a little higher than that suggested by the data. Also note that the tail of the flow-density relationship
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seems scattered to certain degree and the area right after the peak exhibit a little under-fit. Nevertheless, the overall fitting quality appears exceptional.
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Figure 6. Fitting the empirical speed variance Figure shows the result of using the four-parameter log-normal function to fit the empirical speed variance. The four parameters are configured as: = 37.80, = 0.43, = 8.44, and = 4580.42. The resultant = 0.8452 which is acceptable but not outstanding. Meanwhile, the empirical speed variance exhibits some visible features. First, speed variance under extremely low density ( ) is quite large, which is contrary to common sense. As density slightly increases, the variance decreases to a local minimal. As density continues to increase, the variance quickly increases to its peak at about 28 veh/km which is around the optimal density when capacity is reached. As density further increases, the variance drops and then asymptotically approaches zero at jam density. A few locations of under-fit are observable. First, the model fails to capture the decreasing effect of variance under very low density levels. Second, the model fails to hit the peak of the empirical variance, though they both peak at about the same density. Third, the model does not follow the asymptotically decreasing tail well, but rather flattens out toward the end. As it will become evident shortly, the above areas of lack of fit eventually find their way into the modeling of the stochastic process. Figure overlays the fitted distributions on top of the spectrum of empirical distributions of vehicle speed illustrated in Figure . Beta distribution is used to fit the data using the formulae of p.d.f. and parameters provided in the previous section. The figure reveals an acceptable fitting quality with rooms to improve.
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Figure 7. A spectrum of empirical distributions of vehicle speed fitted with Beta distribution Figure shows the overall fitting quality using as an indicator over the entire density range. At each density level, a Beta distribution is fitted to the data based on and at that density level. Then is computed. Normally a goodness-of-fit test such as test or Kolmogorov– Smirnov test is employed to compare two distributions. However, a close analysis of the nature of current problem suggests that is more relevant. This is so because, in a goodness-of-fit test, our major goal is to prove whether a set of samples come from a known distribution or two sets of samples come from the same distribution. This is not the case in our problem which aims to find a model to represent the data. As such, a model is acceptable as long as it is able to represent/replicate the data. Whether or not the model and the data are from the same distribution is not so critical.
Figure 8. Overall fitting quality of the stochastic process In Figure , the best fit appears in the mid-range of density between 50 and 150 with the two ends under fit. Note that, once the distribution type is determined, the fitting quality depends mainly on distribution parameters, i.e., and in our case. Since these two parameters are computed from speed mean ̂ and variance ̂ , the fitting quality of ̂ and ̂
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Relation between This Study and Existing Efforts
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directly affects that of the stochastic process. As clearly evident in Figure , the under fit at both ends is due primarily to the under fit of speed variance ̂ shown in Figure . Therefore, the key to improving the fitting of the stochastic process is to enhance the fitting of ̂ given that there is not much room for improvement in fitting ̂ . Nevertheless, the overall weighted on sample size is 0.8715 which is quite reasonable and acceptable. Note that factors such as vehicle composition, driver behavior, and weather condition can influence drivers’ speed choice and collectively speed distribution. One way to take these influencing factors into consideration is to adopt an explanatory approach that explicitly captures the mechanism how these factors affect speed. In contrast, this research takes a descriptive approach that attempts to fit a model to observed data. As such, the effect of the above influencing factors is captured indirectly through their resultant field observations.
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Historically, the conventional approach to fundamental diagram is deterministic modeling that features a functional relationship between any two of the three state variables: flow, speed, and density. These models may involve two-parameter models [6] [7] [8], three-parameter models [9] [10], and four-parameter models [11], [12], and [4]. More parameters may empower a model with better flexibility in fitting data. However, as indicated above, deterministic models only captures the average behavior of traffic flow and fail to capture the scattering effect of field observations. Obviously, a stochastic fundamental diagram such as the above proposed one is able to better capture the ground truth. Realizing the limitation of the deterministic approach, many researchers seek to represent fundamental diagram using a multiclass approach. For example, Saberi and Mahmassani [13] attributed the scattering effect as a result of loading and unloading processes that follow their own paths in fundamental diagram and may be represented using different models. Though capturing part of the scattering effect, such an approach is still deterministic in nature. Ngoduy [14] argued that the scattering effect might be caused by the random variations in driving behavior. To reproduce the hysteresis transitions and the wide scattering effect, Ngoduy utilized a multiclass first-order model with a stochastic setting in the model parameters. In an attempt to understand the reliability of LWR model [15] [16], Li et al [17] hypothesized a probabilistic fundamental diagram by postulating a flux function driven by a random free flow speed. Chen et al [18] proposed a stochastic fundamental diagram based on headway/spacing distributions. Though the above three models involve certain stochastic elements, they are not true stochastic models. In a closely related study, Wang et al [19] proposed a stochastic speed-density relationship. The model takes the form of traffic speed as the sum of a mean speed and a variance, each of which is treated as a function of density separately. As the deterministic term, the mean speed takes the form of a logistic function. Capturing the stochastic residuals, the variance term employs the Karhunen-Loeve expansion. In essence, this approach entails a deterministic mean and a stochastic variance, leading the underlying speed distribution undetermined in the entire range of density. In contrast, the current study systematically models fundamental diagram as a stochastic process captured completely in a Maxwell-Boltzmann-like distribution. Several other related studies are also identified. Helbing [20] concerned about the speed distribution of vehicles as well as its variance and skewness. Also of interest was the functional form of the speed-density relation and the variance-density relation at high density. Helbing used
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a Gaussian form to represent speed distribution and even presented in FIG. 1 of that paper a speed distribution diagram under varying density levels which is similar to Part 1 of Figure in this paper. Perhaps this seminal work was the first presentation of Maxwell- Boltzmann-liked distribution of vehicular speed, though it was not explicitly claimed. However, as evidenced in Figure and Figure of this study, FIG. 1 of [20] failed to show the full picture with the distribution reverting effect under low density levels being ignored. In a recent study, Jabari et al [21] used shifted gamma function to represent speed distribution and Fig. 12 of that paper does reflect the distribution reverting effect. Ben-Naim and Krapivsky [22] applied Boltzmann equation in studying vehicular traffic flow dynamics which is simplified as a one-dimensional problem. By making analogy to kinetic theory and introducing Maxwell model, they resulted in first-order differential equations which led to explicit expressions for time-dependent speed distributions in generic terms. In contrast, our search focuses on the steady-state properties of vehicular traffic flow and explicitly relate it to Maxwell-Boltzmann distribution with concrete forms. In a further effort, Galstyan and Lerman [23] formulated a joint speed-size distribution when applying the stochastic Master equation approach to study platoon formation. They pointed out that the distribution can’t be solved with speed dependent kernel, but the distribution in [22] is analytically tractable given the Maxwell model. The Significance of this Research
The contributions of this research can be interpreted at two levels with one regarding the stochastic fundamental diagram and the other pertaining to the general methodology.
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The implications of the stochastic fundamental diagram
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The significance of the stochastic fundamental diagram can be interpreted from multiple perspectives. Firstly, it offers a true stochastic means to traffic flow modeling and prediction. Our conventional approach to traffic flow simulation features deterministic models such as microscopic car following and macroscopic LWR. The so-called randomness in simulation, if any, is implemented by means of input data and model parameters such as a distribution of entry volume and a distribution of perception-reaction time. In contrast, the outcome of this research allows true stochastic traffic simulation because the underlying model itself is stochastic. While there is no guarantee what macroscopic behavior of traffic flow will look like when randomizing entry volumes and model parameters, the stochastic approach proposed in this study empowers traffic flow simulation with predictable macroscopic characteristics that converges to known behavior. In addition, such a stochastic fundamental diagram would allow us to make probabilistic prediction of traffic condition as opposed to the traditional deterministic prediction. Secondly, the wide scattering phenomenon in the fundamental diagram has inspired many school of thought on modeling. For example, some resorted to multi-class heterogeneous traffic flow [24] [14], some attributed to loading and unloading processes [13], and Kerner [25] described it as synchronized flow phase. Though these interpretations sound plausible, the ground truth has yet to reveal itself. Through the modeling of stochastic fundamental diagram, this research seems to offer yet another perspective: while it has long been believed that the central tendency of the speed-density distributions is related to the longitudinal motion of vehicles (e.g., the equilibrium speed-density relationship is related to car following), the variance appears have something to do with vehicle lateral motion. For example, vehicle lateral motion is essentially a sequential
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decision at two steps. The first step is the desire to make a lane change, e.g., after being blocked by a slow driver for some time. This is conventionally called a lane changing model and is typically treated as a discrete choice problem. Once the desire of lane change has been registered in the driver’s mind, the next step is to seek a good opportunity to switch to the target lane which is called gap acceptance in the community and modeled as another discrete choice problem. Considering that, at low density, drivers are free to travel at their desired speeds which tend to be around free-flow speed, very few drivers have the desire to change lane even though acceptable gaps are readily available. Consequently the probability of lane change is very low. On the other hand, at high density where the road is packed with vehicles, drivers’ desire for mobility is greatly suppressed and hence changing lanes to improve speed is strongly motivated. However, the probability of lane change is still very low since an acceptable gap in the target lane is rarely available. Only in certain range of density does lane change become frequent because there exists not only considerable desire to seek speed gain in other lanes but also ample gap opportunities to carry out the decision. In addition, such a frequency tends to peak around optimal density where capacity flow occurs. If one relates the frequency to speed variance and cross compare with empirical evidences in Figure and Figure , one would agree that the above theory and the field observations indeed bear some resemblance. Combined, the stochastic fundamental diagram seems to have captured information of vehicle motion in both directions in a single model. Thirdly, the proposed model can shed light on other engineering disciplines and help unite theories of vehicular traffic and other forms of traffic. For example, many engineering systems share similar features to that of vehicular traffic in their phase diagrams. The proposed methodology can serve as a unifying framework to formulate these stochastic processes which opens a door for cross-comparison and inspiring each other. As presented in Figure , the similarity between Maxwell-Boltzmann distribution and speed-density distribution seems to suggest a root of vehicular traffic in molecular dynamics and further physical science. Meanwhile, exploration of the difference brought about by their domain specifics especially the distribution reverting effect under low density levels might give birth to new discovery beyond Maxwell-Boltzmann distribution and fundamental diagram.
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The implications of the general methodology
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Beyond Maxwell- Boltzmann distribution and vehicular fundamental diagram, the proposed methodology can also be used to model other real-world systems where their phase diagrams are defined by a set of statistically related state variables. For example, Sociology concerns about life expectancy, a statistical measure of the average time an individual is expected to live. Among other influencing factors, gross domestic product (GDP) per capita plays a significant role in a country’s average life expectancy. The left part of Figure scatters the relationship between life expectancy and GDP per capita. Provided by the World Bank, the data include most countries in the world covering the period between 1960 and 2015. Each data point represents a country’s state (given by life expectancy and GDP per capita) in a specific year. The right part of Figure shows the mean and variance of the data point. It can be seen that at a certain level of GDP per capita, life expectancy is distributed with mean and variance as indicated. The striking resemblance between Figure and Figure suggests that the proposed methodology is readily applicable to the modeling of life expectancy. The resulting model can be very useful in
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Sociology to answer questions such as an estimate of the average number of additional years that people of a given age in a country with given level of GDP per capita can expect to live.
Figure 9 The relationship between life expectancy and GDP per capita (Source: World Bank) Conclusions
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Motivated by the similarity between the fundamental diagram of vehicular traffic and the Maxwell-Boltzmann distribution of ideal gases, this paper proposed a methodology to model the fundamental diagram as a stochastic process. Since the methodology is formulated in generic terms, it is also applicable to other systems whose phase diagrams exhibit similar nature. The methodology follows a data-to-model approach that starts from empirical data which consist of observations of the relationship between a control variable and a target variable. As such, the data are categorized according to the control variable and statistics of the target variable such as its mean and variance are computed based on the categorized data. With the above results, models are fitted to capture these statistics as functions of the control variable. With a good understanding of the empirical data, a distribution type is selected which is best suited to represent the distributions of the target variable under varying levels of the control variable. With that, the relationship between the parameters of the selected distribution and the statistics of the empirical data can be established. Therefore, a stochastic process can be formulated when the selected distribution type is instantiated with parameters expressed as functions of the control variable. A concrete example is provided to illustrate the application of the methodology that models the fundamental diagram of vehicular traffic. Following the procedure laid out in the methodology, a stochastic fundamental diagram is formulated that instantiates the generic model with options of specific forms. It turns out that mean of the target-control variables finds its root in vehicle longitudinal motion, while the variance might be closely related to vehicle lateral motion. Together the stochastic fundamental diagram may capture information of vehicle motion in both directions in a single model. An empirical verification was conducted using data collected from an observation station on GA400. Borrowed from the analysis of variance, the statistics is employed to evaluate the
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fitting quality. It appears that the overall quality of the stochastic process is determined by that of the fitted mean and variance of vehicle speed. While equilibrium traffic flow models with fine quality are available to fit the mean, good models are still being sought to fit the variance. Further discussion is carried out to relate this research to existing efforts as well as on the significance of the stochastic fundamental diagram. It appears that the resulting stochastic fundamental diagram may be used to capture information of vehicle motion in longitudinal and lateral directions in a single model that advances the state of the art of traffic flow modeling and simulation. Obviously, the proposed methodology is applicable to not only vehicular traffic but also other real-world systems with similar nature such as a population whose life expectancy varies with GDP level. Acknowledgements
This research is partly supported by the United States Department of Transportation (USDOT) University Transportation Center (UTC) grants.
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References
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