A Stochastic Model for Traffic Flow Prediction and Its Validation

-00 A Stochastic Model for Traffic Flow Prediction and Its Validation Li Yang Romesh Saigal Department of Industrial and Operations Engineering The University of Michigan Ann Arbor, MI Chih-peng Chu Department of Business Administration National Dong Hwa University Hualien, Taiwan Yat-wah Wan Graduate Institute of Logistics Management National Dong Hwa University Hualien, Taiwan No. of words: No. of Tables: 1 No. of Figures: September, 0 1

Abstract This research presents and explores a stochastic model to predict traffic flow along a highway and provides a framework to investigate some potential applications, including dynamic congestion pricing. Traffic congestion in urban areas is posing many challenges, and a traffic flow model that accurately predicts traffic conditions can be useful in responding to them. Traditionally, deterministic partial differential equations, like the LWR model and its extensions, have been used to capture traffic flow conditions. However, a traffic system is subject to many stochastic factors, like erratic driver behavior, weather conditions etc. A deterministic model may fail to capture the many effects of randomness, and thus may not adequately predict the traffic conditions. The model proposed in this research uses a stochastic partial differential equation to describe the evolution of traffic flow on the highway. The model as calibrated and validated by real data shows that it has better predictive power than the deterministic model. Such a model has many uses, including dynamic toll pricing based on congestion, estimating the loss of time due to congestion, etc. 1

1 Introduction 1 1 1 1 1 1 1 1 0 1 Just as the normal blood circulation necessitates a healthy body, the smooth traffic flow is necessary for the heathy business and community development, in a city as well as a region. Yet problems such as traffic congestion that inflicts uncertainties, drains resources, reduces productivity, stresses commuters, and harms environment are haunting communities. A study estimated that % of the daily travel in major US urban areas occurred under congested traffic conditions [1]. Also, a study by the Texas Transportation Institute [] found that in 00, congestion caused urban Americans to travel an extra. billion hours and to purchase an additional. billion gallons of fuel thus incurring an additional cost due to congestion of. billion dollars - an increase of more than 0% over the costs incurred a decade ago. Thus it is not surprising that major efforts at reducing traffic congestion have been undertaken. Various forms of Intelligent Transportation Systems (ITS) that take real-time traffic data for decision support have been developed for this purpose, and congestion pricing has been implemented on highways for New Jersey, New York, London, Paris, Singapore, and San Diego []. The success of these efforts at controlling congestion requires an accurate prediction of the evolution of traffic flow. For this purpose, reliable and robust traffic flow models are indispensable. There have been two main mathematical approaches to model the traffic flow. One approach takes the microscopic view to study the movements of individual vehicles. The approach considers the interaction between individual vehicles and the stochastic driving behavior of the drivers []. One example of the microscopic approach is the car-following

1 1 1 1 1 1 1 model []. This approach usually uses simulation models, such as the cellular model to understand the traffic flow. For example, Boel [] proposes a compositional stochastic model for real time freeway traffic simulation. Chrobok [] predicts the traffic flow at a fixed location in the highway using an online cellular-automation simulator. Park and Schneeberger [] creates a simulation based model and also describes the methods for model calibration and validation. The macroscopic approach studies properties induced by the interaction of a collection of vehicles. The approach ignores the detailed identities of individual vehicles. The kinetic models adopt ideas from statistical mechanics. They treats a vehicle as a particle, and the traffic flow is the result of the interaction of such particles. The distributions of the positions and velocities of vehicles are deduced from those of particles in gas. The fluid-dynamical models treat traffic flow as a compressible fluid. In such models, the basic variables are flow rate q, traffic density ρ and speed v, which are considered as a functions of time and space. The basic idea is to build a partial differential equation for the variables and then solve the equation for the evolution of the variables. A classical macroscopic model is the LWR model [], []. Other methods of forecasting traffic flow have been explored by many researchers. Ahmed and Cook [] apply time series methods to provide a short-term forecast of 1 traffic occupancies for incident detection. Okutani and Stephanedes [1] employ the 0 1 Kalman filtering theory in dynamic prediction of traffic flow. Davis et al. [1] use pattern recognition algorithms to forecast freeway traffic congestion. Lu [1] develops a model of adaptive prediction of traffic flow based on the least-mean-square algorithm. Siegel and Belomestnyi [1] give an overview of the stochastic properties of the single-car road

traffic flow. They suggest that the traffic dynamics is a stochastic process on two fixed points, which reflect the jam and free-flow regime. Minazuki et al. [1] consider the dynamics of traffic flow as geometric Brownian motion. In recent years, abundant traffic data has become available with the extensive use of the detection and surveillance devices on the road. In many states of the United States, the administrators of the transportation departments have constructed databases that can provide historical and real-time traffic data to the public. Thus, these data sets provide researchers an opportunity to build data-driven models for predicting the traffic flow. Gripping such an opportunity, we propose a stochastic partial differential equation (SPDE) model for the traffic flow. Building on the classic LWR model for modeling 1 1 1 1 1 1 1 1 0 1 traffic flow, the model takes real-life data and predicts the traffic flow across time and space. The results from the numerical tests suggest that the SPDE model is capable of making accurate predictions of the traffic flows. A preliminary investigation of its uses as a primary building block within an ITS system to provide intelligent decision support gives promising results, and we report some of these in the future development section of this paper. These include a dynamic congestion toll pricing system and a model to predict the total system congestion cost. The rest of the paper is organized as follows. The formulation is presented in Sections and, the calibration and the validation of the stochastic traffic model are presented in Section. Some other possible applications of the model and future work are discussed in Section.

Modeling of Traffic Flow as a SPDE Our macroscopic model is built on four ideas, the LWR model, the fundamental flow relationship, the speed-density function, and the generalized Ornstein-Uhlenbeck process. The classical LWR model is a partial differential equation on traffic flow which we generalize to a SPDE. The fundamental flow relationship and the speed-density function reduce the degree of freedom of the variable space of the traffic flow problem to one; finally, the stochasticity is introduced as a forcing term, incorporating a Brownian Sheet, Walsh [1] and a term incorporating an Ornstein-Uhlenbeck (OU) mean-reverting property (we will say more about this property in Section.). We adapt the following notation: (x, t) is the space and time pair where location x [0, L] and time t [0, T ]; 1 1 Q(x, t), the flow, also called volume, is the rate (i.e, number per unit time) of vehicles passing through location x at time t; 1 ρ(x, t), the density is the number of vehicle per unit distance at location x, time t; 1 v(x, t) is the average velocity of all vehicles at location x, time t. 1 1 W (x, t) is the Brownian Sheet, a Gaussian process indexed by two parameters x and t. The definition and properties of the sheet are developed in Walsh [1]. 1 1 0 For simplicity, sometimes we use Q, ρ, v in subsequent presentation with the understand- ing that these quantities are dependent on x and t. When multiple locations x i, x j are considered, x i > x j for i > j.

.1 The LWR Model Since proposed in s, the classical LWR model, [], [] has been the building block of many macroscopic traffic flow models. It is a first-order fluid dynamical model on a single-lane highway, and is based on the following: for x 1 < x, by the conservation of flow, x ρ(x, t) dx = Q(x 1, t) Q(x, t). t x 1 Further assume that the flow rate Q is determined by the density ρ alone, Q(x, t) Q(ρ(x, t)). (1) With x 1 approaching x and with ρ differentiable, the conservation of flow equation can be stated in the form ρ(x, t) + Q(x, t) = 0. () t x Given the initial condition of ρ(x, 0) for all x at t = 0, () can generally be solved, often in closed-form, for some appropriate choices of the form of the function Q(ρ(x, t)).. The Fundamental Flow Relationship 1 1 1 There is a fundamental relationship between Q, ρ, and v: Q = ρ v. In our case Q is determined by ρ, as in (1). The fundamental flow equation can be expressed explicitly in ρ as

Table 1: Potential Speed-Density Functions Functions v(ρ) Q(ρ) = ρ v(ρ) Greenshields v f (1 ρ/ρ j ) ρv f (1 ρ/ρ j ) Greenberg v 0 ln(ρ j /ρ) ρv 0 ln(ρ j /ρ) Underwood v f exp( ρ/ρ j ) ρv f exp( ρ/ρ j ) Log Piecewise Linear min{v f, αρ m } ρ min{v f, αρ m } Q(ρ) = ρv(ρ), () 1 or expressing explicitly the dependence on the space-time variables as Q(ρ(x, t)) = ρ(x, t)v(ρ(x, t)).. The Speed-Density Function 1 As shown in (), given any forms or values for any two of Q, ρ, and v the form or value of the remaining third can be determined. If the speed-density function v(ρ) is known, the PDE is determined only by the functional ρ. Several functional forms of the speed and density relation have been proposed in the literature. Table 1 summarizes the relationships that have been frequently used in the literature. The fourth function, v = min{v f, αρ m }, which we call the log piecewise linear model, is adopted as the speed-density function for our SPDE model. Please refer to Section.1, the calibration of speed density functions on the data, for the reason of our choice.

. The Stochastic Modeling of the Traffic Flow 1 1 1 1 1 1 1 1 0 The LWR model is a first-order partial differential equation that assumes that the traffic flow reaches an equilibrium immediately. It predicts traffic flow well in relatively heavy traffic when the effects of individual driver behavior are minimal. See [1] and [1] for the critics of the LWR model. There are second-order adjustments based on fluid dynamics, proposed to enhance LWR; however, as shown in [1], any model that smoothens the discontinuity of density may predict backward flow and negative velocity. Traffic data collected on a highway reflects the cumulative effects of random microscopic phenomenon occurring along the highway and it suggests that the traffic flow may be stochastic in nature, Saigal and Chu [0]. In this paper, we propose a stochastic enhancement of the LWR model that accounts for the stochasticity of the traffic flow along the highway. These microscopic and other phenomenon that leads to the stochastic behavior include the effects of sudden acceleration/deceleration, lane shifts, lane surface conditions, change in weather etc. In addition, the traffic that leaves or enters at various exits along the highway is stochastic in nature, and depends on the time of the day and the day of the week. This thus contributes to the drift in the mean of the density and its volatility. We capture these effects by adding a stochastic forcing function g to the right side of () with the hope that the forcing function g(ρ, x, t) incorporates many of the above described factors that affect the flow conservation of the traffic flow at (x, t). Thus the SPDE model we consider is:

ρ(x, t) + Q(ρ(x, t)) = g(ρ(x, t), x, t), t x Q(ρ(x, t)) = ρ(x, t) v(ρ(x, t)), () g(ρ(x, t), x, t) = a(x, t) + b(x, t) ρ(x, t) + σ(x, t) W (dx, dt). 1 Here a, b and σ are deterministic functions and W is a Brownian Sheet, Walsh [1]. A note on the form of the forcing function g is in order here. The deterministic constant function a(x, t) is the drift term designed to capture the effects of the means of inflow\ outflow and other factors that may affect flow conservation. The function σ(x, t) is 1 1 1 1 1 1 designed to capture the magnitude of the resulting volatility of the disturbance to the flow conservation. The Brownian sheet W is a Gaussian process indexed by two parameters x and t with mean 0 and covariance E(W (x, s)w (y, t)) = min(x, y) min(s, t), and the term U(dx, dt) = b(x, t) ρ(x, t) + σ(x, t) W (dx, dt), borrowing from the nomenclature of the stochastic differential equation literature, converts the Brownian Sheet into an Ornstien-Uhlembek (OU) Sheet, U. This choice makes the process mean-reverting, i.e., after a disruption moves the process away from the its mean behavior, the effect of this disruption dissipates in time and space and the process returns to its mean behavior. Related and other properties of the OU Sheet can be found in Walsh [1]. An important property of note is that if we restrict U to the line segment x = u + vt of the sheet, the process indexed by the parameter t, V (t) = U(u + vt, t), is an OU Process. Though W (u+vt, t) is not a Brownian motion in parameter t, it is a Matringale with independent but non-stationary increments. Thus there is some justification in calling V (t) an OU

Process. As is known for the LWR model, which is a quasi-linear transport partial differential equation (PDE), Evans [1], global solutions do not exist, and one can prove the existence of a weak solution to an integral form of the equation. Such solutions exhibit the phenomenon of shock waves generated by these PDEs, which are a solution to the Reimann problem; see for example [1]. The Solution and Applications of the SPDE model 1 1 1 1 1 1 1 1 0 1 It is shown in Gunnarsson [] that the SPDE of Section has a unique weak solution under some very general conditions on the functions a, b and σ. In this section we show how these functions can be estimated using data collected along a highway, and how the model can be used to predict the traffic flow in a sheet [0, L] [0, T ], with (0, 0) the reference point, given the traffic flow along the length of the highway at time 0, i.e, in [0, L] {0}. It is unlikely that an analytical solution to () exists, however the SPDE can be readily solved numerically to obtain a good approximate solution using a finite difference scheme. This scheme first discretizes the domain [0, L] [0, T ], generates a sample of the Brownian sheet, and uses the Godunov scheme to numerically solve the resulting PDE. The prediction is then made using the mean and variance of the numerical solution found for some large number of samples of the Brownian sheet. For an application, the values of a(x, t), b(x, t), and σ(x, t) for specific x and t and the function form of the speed-density function v(ρ) must be determined. The methodology for determining the values of the parameter functions a(x, t), b(x, t) and σ(x, t) is

developed in Section.1. The Godunov scheme to numerically evaluate the stochastic differential equation is discussed in Section.. The application of the Godunov scheme on the SPDE model is in Section.. The calibration of the best fitted speed-density function v(ρ) is postponed to Section.1 in the section for results..1 The Estimation of the Parameter Values of a, b and σ To derive the numerical or analytic solution of (), the values of a(x, t), b(x, t) and σ(x, t) at any location x in the required segment of highway (0, L) and any time t in the required time horizon (0, T ) must be determined. When a finite-difference method is used, we only need these values at the specific finite number of grid points. Refer to Figure 1 for an example of the grid points. Figure 1: Finite difference method 1 1 1 1 1 We assume that for the specific grid point (x, t) we record the value of ρ and Q for a number of days. Given such data for an adequate number of days, we treat the left hand side of the SPDE (see equation () below) as the dependent variable and the right hand side as a linear function of the independent variable ρ(x, t); we treat the Brownian sheet term as the noise in a linear regression model. The values of the dependent variable are approximated using the finite difference formulas on the grid of Figure 1, resulting in

the equations () and () that follow: ρ(x, t) + Q(ρ(x, t)) t x = a(x, t) + b(x, t)ρ(x, t) + σ(x, t)w (dx, dt), () ρ(x, t + t) ρ(x, t) ρ(x, t) =, t t () Q(ρ(x + x, t)) Q(ρ(x x, t)) Q(ρ(x, t)) =. x x () Using the data for ρ(x, t) and v(x, t), the values of a, b and σ are estimated by solving the linear regression problem at each grid point (x, t). The validation and the results are discussed in Section... The Godunov Scheme The Godunov scheme is proposed by S. K. Godunov [], in 1 for numerically solving partial equations representing flow conservation, like the transport equation we are dealing with. This method is applied to a discretization of the space and on each cell it assumes that the flow and the density are constant. In the application of the 1 1 1 1 1 Godunov s scheme, in cell [x, x + x] [t, t + t] at (x, t) the estimated values of the parameters a(x, t), b(x, t) and σ(x, t) are assumed to be constant. At the boundaries of the cells, this method solves a local Reimann problem, and the discontinuities at the cell interfaces are smoothened by averaging the effect of the wave formations. A good reference for this method is Hirsch []. The solution to the SPDE is a random process, and thus we can numerically determine its various statistics like the mean and the variance, as well as its distribution function. 1

This is done by randomly choosing a certain number of sample sheets, and solving the PDE for each such choice. The mean and variance of the random density process can thus be generated by taking the mean and variance of the results obtained by solving the PDEs for various samples of the Brownian sheets.. The Prediction from the Stochastic Model Section. proposes that a finite difference procedure of Godunov be used to solve the SPDE, and thus obtaine the future evolution of the traffic density. Assume that at t = T 0, the density is known at each location along the highway. This serves as the boundary condition for the SPDE. We assume that this initial condition is perfectly observable. After the initial condition is observed, the model parameters have been 1 1 1 1 1 1 1 calibrated, and a random sample of the Brownian sheet has been generated, Godunov s scheme is applied to generate a numerical solution of the resulting PDE at t = T 1. By simulating the Brownian sheet multiple times and solving the resulting PDEs under these different samples, the distribution of the flow density at t = T 1 can be obtained. In order to investigate the advantage of the stochastic model with respect to the deterministic model, we compare the prediction accuracy between the two models. In the stochastic case, the mean of the simulated density in the future would be taken as the prediction. The result is in Section.. 1 Results 0 1 The data used to fit and test the model are obtained from the Virginia Department of Transportation. The data consist of the traffic flow and average speed at sensor 1

stations along a stretch of highway Interstate I heading Northbound to Washington DC. A sample point is taken by a sensor for every one minute aggregated time interval during each day from April to June in 00. The distance of adjacent sensor stations is about half a mile, and the total length of the highway section is about 1 miles (between mileposts 1-1). The map of the highway section and the location of sensors and exit locations are shown in Figure. The diagram at the bottom of this figure shows the 1 sensor locations (identified by black dots) and exits (identified by red arrows). Of the sensors on the highway, only 1 shown were functioning and collecting data. Figure : The map of the highway and locations of sensors and exits The following (sub-)sections contain three types of results. Section.1 shows how the speed-density function is chosen. Section. discusses the calibration of the SPDE model. Finally, Section. compares the results of the SPDE over the conventional 1 deterministic model (the LWR model). 1

.1 Speed Density Relationship (a) Greenshields (b) Greenberg (c) Underwood (d) Log piecewise Linear Figure : Speed-Density function calibration The collected data is used to choose one of the four speed-density functions discussed in Section. for our SPDE. For this testing, the data at each of the 1 working sensor locations is randomly partitioned into two sets, one for fitting and the other for testing the functions. After fitting the functions, i.e., determining the parameters, the function is used to predict the velocity for each velocity value in the data set. The prediction error is then the sum of the squares of the differences in the observed and predicted values of the density, and the percentage errors are compared among the potential functions. The function with the smallest error is chosen as the speed-density function for the SPDE. 1

When fitting the piecewise linear function, we take the natural logarithm transformation on both sides of equation, as presented in equation (). Then ln(v) is a piecewise linear function of ln(ρ) and piecewise linear regression is applied to fit the data. ln(v) = min{ln(v f ), ln α + m ln(ρ)}. () The average percentage error for this location for the four functions we tested are as follows: Greenshields 0.%; Greenberg 1.01%; Underwood.% and log piecewise linear function.1%. At all the 1 locations tested, the log piecewise linear function gave the smallest percentage error. Thus the speed-density function is used in the SPDE model calibrations.. The Calibration of the SPDE model 1 1 1 1 1 1 1 1 As described in Section.1, a(x, t), b(x, t) and σ(x, t) are calibrated using the linear regression between ρ t + Q x and ρ. To calibrate the parameters at sensor location x and time slot [t, t ], the data during this time slot collected each day over the two months is used. For example, if the parameters in the time slot, :00 am to :0 am at sensor location 1, are to be calibrated, all the flow and average speed data at sensor location 1 in the time slot between :00 am and :0 am are extracted from the database. Standard least squares technique is then used to estimate the parameters for the time slot between :00 am and :0 am at sensor location 1. Figure a shows an example of the calibration of the SPDE at location from :1 am to :1 am. The R of 0. in the regression shows that the model explains more 1

(a) SPDE calibration (b) Constant variance (c) QQplot of the residual (d) histogram of the residual Figure : SPDE calibration 1 than % of the variability in the data, and is a good fit. The linear regression finds a = 1., b = 1., σ = 1. at sensor location from :1am to :1am. The % confidence interval is [1, 1] for a and [., 0.] for b. Because 0 is not included in the confidence interval for both a and b, this means that ρ is significant for ρ t + Q x. The scatter diagram Figure b shows the residuals to be consistent with a normal distribution. This is confirmed by the QQ-plot in Figure c, where the hypothesis of the residual as normally distributed is accepted. Figure d plots the histogram of the residuals, again confirming the normality assumption on the residuals. The parameters of a(x, t), b(x, t) and σ(x, t) are obtained using the above proce- 1

dure for every location and every non-overlaping time slot. For each of such tests, the normality assumption on the residuals is accepted, and the parameters a, b and σ are significant.. Comparison of Stochastic Model with Deterministic Model (a) Percentage error of deterministic model (b) Percentage error of stochastic model Figure : Comparison of the stochastic model and the deterministic model Figure shows the prediction results of stochastic model and deterministic model. The percentage error of the stochastic model is 0% less than the deterministic model, i.e., the prediction accuracy of our SPDE model is better. The predictions are made at 1 of the 1 sensor locations, because data at the the first and the last sensor locations are used for the finite difference estimate of ρ t + Q x.. Future Traffic Conditions 1 1 1 Based on the current traffic conditions along the 1 mile stretch of the highway, the SPDE can be used to predict the traffic conditions for the near future along the entire length of the highway. To demonstrate the predictive power of the SPDE we arbitrarily pick two time periods: a time when the highway is congested (:0am - :0am); and 1

a time when the highway is not congested (:0am - :0am). Starting with the initial conditions observed at the begining of the time period, we apply the Godunov Scheme with 00 sample Brownian sheets. The results for the average and the standard deviation of the density are shown in the color coded Figure, where the time axis is the minutes after midnight.: (a) Mean of Density evolution congested (b) SD of Density evolution congested (c) Mean Density evolution un-congested (d) SD of Density evolution un-congested Figure : The density evolution at :0am (congested) and :0am (un-congested) The shock waves, as predicted by the Riemann s initial conditions, are clearly seen in Figure a, with sudden increase in density in several locations, most prominently around milepost (where SR 1 meets I ). The high standard division of the prediction at 1

the top left hand corner of the Figures b and d are due to the missing initial conditions required for an accurate prediction of density at these zones. Conclusion and Future Work 1 1 1 1 1 1 1 1 0 1 In this paper we presented a stochastic generalization of the LWR model, using data collected on a highway, we calibrated the speed-density function and the estimated parameters required for the application of this model. We observe that the stochastic model provides a more accurate prediction than the deterministic model. The speeddensity function and SPDE provide a framework for many intelligent transportation applications as the SPDE captures the evolution of the traffic flow. As an illustration we show how the model can be used for traffic control. Consider a highway with two parallel lanes, with one a toll lane. Towards generating a policy that sets toll based on congestion, we show how the SPDE performs in predicting the evolution of density when a certain amount of flow switches from the general lane to the toll lane. For this, we assume that the parameters of the SPDE that determines the density evolution on the general lane are the same as the ones estimated for the time :0am to :0 am in Section.; and for the time :0am to :0am for the traffic on the toll lane. We assume that 0% of the flow on the general lane switches to the toll lane at the mile post 1. (. miles from the start of the 1 mile highway), and that this switch occurs instantaneously. The mean and standard deviation of the resulting traffic density, as predicted by the SPDEs with 00 random samples of Brownian sheet, are presented in the Figure. 0

(a) Mean of Density evolution general (b) SD of Density evolution general (c) Mean Density evolution toll (d) SD of Density evolution toll Figure : Density evolution with 0% switch of flow from general to toll lane 1 As can be seen from the figures a and c, around the switch location, the predicted mean density decreases in the general lane and correspondingly, increases in the toll lane. A future topic of research, based on the SPDE, is to develop an optimal control model for determining the toll amount based on the predicted congestion. For this model to be effective in controlling, a switch response function to the toll price is required. Obtaining this function is under consideration. Given the velocity function v(ρ(x, t)) and the free flow velocity v f the time lost due to congestion at location (x, t), S(x, t), can be readily seen as 1

S(x, t) = max{0, v(ρ(x, t)) 1 }. v f 1 This is an option on the process ρ(x, t), and can be priced. Pricing this option gives an alternate method for setting toll based on congestion. methods to see which is more effective. We will compare the two thus The function Q(x, t) S(x, t) is the time lost by all vehicles at the location (x, t), and t l t 1 l 1 Q(x, t) S(x, t)dxdt is the time lost due to congestion by all vehicles in the sheet [l 1, l ] [t 1, t ]. The mean and variance of this function can be obtained using the SPDE. We will investigate the use of this function by policy makers of transportation systems. References [1] Status of the Nation s Highways, Bridges, and Transit: 00 Conditions and Per- formance. 00. 1 1 [] D. Schrank and T. Lomax. 00 Urban Mobility Report. Texas Transportation Institute, 00. 1 1 [] T. Litman. Road Pricing, Congestion Pricing, Value Pricing, Toll Roads, and HOT Lanes. TDM Encyclopedia. Victoria Policy Institute, 00.

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