Abstract—Time series of travel speed on multilane freeways are considered complex and irregular particularly when addressing the variability across lanes. Literature shows evidence of interactions between speed variability and traffic mix and inclement weather, without extending these results to addressing speed predictability across lanes. We propose the development of a Bayesian system of equations in order to concurrently treat time series collected from each lane in an autoregressive methodological framework. Exogenous variables such as volume, percentage of trucks per lane, as well as precipitation levels are integrated into the model. The proposed approach improves on the predictability of travel speeds across lanes over the commonly used ARIMA models.

I. INTRODUCTION

reeway speed modeling has been a subject of significant research over the years. Previous research has revealed a

complex interaction between traffic mix (the percentage of cars, trucks, buses and so on) and the freeway travel speed –volume relationship [1]-[4]. The special case of trucks and their effect to speed distributions has also been systematically studied [5]-[7]. Dey et al. [7] have recently discussed that, as traffic becomes heterogeneous and congested, speed data may not follow a unimodal distribution; they also stated that the speed distribution could be altered from unimodal to bimodal (or even multimodal) distribution when speed variation became large (such as under congested traffic conditions).

A number of other papers have addressed the impact of weather conditions on freeway speed variability. It is a general observation that freeway speeds decrease in inclement weather [8], [11]. Ibrahim and Hall [9] provided

Manuscript received March 15, 2010. This work was funded by the

NTUA Basic Research Program 2009. Eleni I. Vlahogianni is a Lecturer at the Department of Transportation

Planning and Engineering, School of Civil Engineering, National Technical University of Athens, 5 Iroon Polytechniou Str, 15773, Zografou, Greece (e-mail: elenivl@central.ntua.gr).

Matthew G. Karlaftis is an Associate Professor at the Department of Transportation Planning and Engineering, School of Civil Engineering, National Technical University of Athens, 5 Iroon Polytechniou Str, 15773, Zografou, Greece (e-mail: mgk@central.ntua.gr).

Pantelis Kopelias is the Head of Traffic Management at the Attica Tollway Operations Authority, Attikes Diadromes S.A. 41, 9 km Attiki Odos Motorway, 19002, Peania, Greece (e-mail: pkopelia@attikesdiadromes.gr).

quantitative information on the impact of light and heavy rain on operating speeds and maximum observed flow rates. Moreover, the decrease in freeway capacity has been found to be between 4 and 10% under light rain conditions (intensity of 0.01 - 0.25 inches/hour) and 25-30% under heavy rain (intensity of 0.25 inches/hour or greater) [10]. Ranka [10] also stated that under rainy weather, regardless of intensity, operating speeds decrease by approximately 5.0-6.5%. No impacts were found on traffic stream jam density, but both rain and snow did impact traffic free-flow speed, speed at capacity [10].

Although both traffic mix and weather conditions have been found to be critical in terms of the speed variability, limited interest has been given to the distributional features of such interrelations across lanes. Prior research has revealed significant volume dependencies across different lanes and result in different underlying speed-flow relationships [11]. Moreover, speed variability per lane has been studied in relation to the probability of crash occurrence [12]. No systematic effort to account for the speed variability in relation to the traffic mix and weather conditions in prediction models has been found in literature.

From a modeling perspective, most approaches up to date implement classical autoregressive moving average models ([14]-[22]) or conditional heteroscedastic models [23]-[24]. The issue with such approaches is the univariate nature of the dependent variable as well as the lack of accounting for the effect of exogenous variables synchronously in a multivariate setting. Until now the multivariate modeling in traffic flow has been treated almost exclusively using neural network techniques [1], [25]-[30].

In the present paper, we aim at developing a system of autoregressive equations in order to concurrently address the modeling of travel speed in each lane in a multilane freeway. The specific approach will also integrate information on traffic mix and weather conditions in the above model with the scope of evaluating their effect in the prediction accuracy.

II. VECTOR AUTOREGRESSIVE PROCESSES WITH

EXOGENOUS VARIABLES

A. Vector Autoregression

Vector autoregression (VAR) is commonly used for

Modeling Freeway Travel Speed Across Lanes: A Vector Autoregressive Approach

Eleni I. Vlahogianni, Ph.D. and Matthew G. Karlaftis, Ph.D. and Pantelis Kopelias, Ph.D.

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2010 13th International IEEEAnnual Conference on Intelligent Transportation SystemsMadeira Island, Portugal, September 19-22, 2010

MC2.4

978-1-4244-7659-6/10/$26.00 ©2010 IEEE 569

forecasting systems of interrelated time series and for analyzing the dynamic impact of random disturbances on the system of variables. The VAR approach sidesteps the need for structural modeling by treating every endogenous variable in the system as a function of the lagged values of all of the endogenous variables in the system. Let ty be a k-

dimensional time series vector of random variables of interest. The pth-order VAR process is written as [31]:

 

1

p

t i t i ti

y δ y ε

(1)

where t t 2t kt, ,..., ε is a vector white noise process

such that ( ) 0t ε , ( )t t ε ε and ( ) 0t t ε ε for

t s , 1 2 k, ,..., δ is a constant vector and i is a

k k matrix. Analyzing and modeling the series jointly enables to understand the dynamic relationships over time among the series and to improve the accuracy of forecasts for individual series by using the additional information available from the related series and their forecasts.

A VAR process can be affected by other observable variables that are determined outside the system of interest. Such variables are called exogenous (independent) variables. Exogenous variables can be stochastic or non-stochastic. The process can also be affected by the lags of exogenous variables. A model used to describe this process is known as VARX(p,s) model and can be written as:

 

1 1

p p

t i t i i t i ii i

y δ y x ε

(2)

where ty is an r-dimensional time series vector and i is a

k r matrix.

B. Bayesian VARX

The Bayesian vector autoregressive model with exogenous variables is called the BVARX(p,s) model can be written as the form of Eq (2) and the parameter estimates can be obtained by representing the general form of the multivariate linear model:

  kX I y β e

(3)

when the parameter vector β has a prior multivariate normal

distribution with known mean β and covariance matrix

V . In practice, β and V need to be specified. If all the

parameters are considered to shrink toward zero, the null prior mean should be specified. According to Litterman [32], the prior variance can be given by:

  2

2

/ if ( )

( / ) if ij

ij ij

l i jv l

i j

(4)

where ( )ijv l is the prior variance of the ( , )i j th element of

l , is the prior standard deviation of the diagonal

elements of i , is a constant in the interval (0,1), and 2ij

is the ith diagonal element of posterior covariance matrix.

III. APPLICATION AND FINDINGS

A. The Data

Attica Tollway is a 65 km urban motorway, connecting 2 major interurban motorways, the Athens international airport and the city center (Fig. 1).

Traffic operations are managed by an advanced traffic management centre. Traffic monitoring and management is conducted via a broad ITS equipment network encompassing 220 CCTV cameras, inductive loops placed every 500m in open road and 50m in tunnels, Emergency Roadside Telephones (ERT), overheight vehicle detection system at entrance points and more than 70 Variable Message Signs (VMS).

Fig. 1. Attica Tollway, in Athens Greece.

The available data comes from detectors located on the

busiest section of Attica Tollway and consist of time series of volume and travel speed collected on 10 minute intervals per lane (Fig. 2). Along with these time series, a traffic mix dataset is also available for processing. In this paper, information on the percentage of trucks is being integrated to the time series in order to investigate the effect of truck circulation in the distribution of speeds per lane.

Regarding the weather data, meteorological data is available from the METEONET network (http://meteonet.chi.civil.ntua.gr/en/divs.html) that is developed by members of the Laboratory of Hydrology and Water Resources Management of the National Technical University of Athens. The available data is collected and stored every 10 minutes and consist of time series of precipitation (mm). Fig.2 demonstrates time series of travel speed for each of the three lanes in a typical rainy day, where 5 precipitation episodes - with varying intensity - were observed; apart from the early morning hours, where an increased variability on the speeds of the left lane is observed, speeds seem follow similar patterns - with

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differences in their magnitude- during a typical day in all lanes. This is also observed in the speed-volume relationships.

Regarding the statistical characteristics of the speed time series (Fig. 3), it is evident that speeds are not normally distributed. Moreover, the autocorrelation graph that does not decay exponentially in all lanes [33], revealing that all speed time series are expected to be non-stationary. These two remarks on normality and stationarity of the time series are critical for the application of autoregressive models and need to be treated before modeling.    

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Fig. 2. Time series of travel speed along with the speed-volume

relationship for each travel lane for a typical day that rainfall is observed.

 

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Travel Speed (Left lane)N(s=24.3)

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Fig. 3. Distribution of freeway speed per lane and the corresponding

autocorrelation graphs.

B. Testing for Cointegration

The question that arises from a modeling perspective is whether the series can be integrated and linearly combined in order to result to a system that is stationary. If a series yt with no deterministic components can be represented by a stationary and invertible ARMA process after differencing d times, the series is integrated of order d, that is, I(d) and if

all elements of the vector yt are I(d) and there exists a cointegrating vector 0 such that ( )ty I d b for

any 0b , the vector process is said to be cointegrated ( , )CI d b [34]. Cointegration signifies a correlation among

variables. Multicointegration extends the cointegration technique beyond two variables and occasionally to variables integrated at different orders.

In the present paper, we apply a cointegration test using the methodology developed in Johansen [35]. The results of using the time series of speed for each one of the available lanes the test are depicted in Table I. The Johansen Trace test indicates 3 cointegrating equations at the 0.05 level.

TABLE I UNRESTRICTED COINTEGRATION RANK TEST (TRACE)

Hypothesized Trace 0.05 No. of CE(s) Eigenvalue Statistic Critical ValueProb.**

None * 0.214320 91.64252 29.79707 0.0000At most 1 * 0.061262 23.38146 15.49471 0.0026At most 2 * 0.019214 5.490592 3.841466 0.0191

* denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis [36] p-values

C. Modeling

In this section the Bayesian VAR model is compared to a VAR approach as well as to a typical ARIMA model for speed time series approximation. The first step of the analysis is to develop VARX (p,s) models in order to test the impact of different exogenous variables to both the efficiency and accuracy of the model. In the modeling, we consider only first order VARX processes (p=1) with s=1 lag for the exogenous variables vector.

Results are depicted in Table II in relation to the r-squared value, the Schwarz criterion (SC) and the Mean Absolute Percent Error (MAPE).

TABLE II VARX(1,1) ESTIMATION RESULTS FOR DIFFERENT EXOGENOUS VARIABLES

COMBINATIONS Endogenous

Variables Exogenous Variables

R2 SC MAPE

Speed (LL)

Traffic

0.83 7.6 6.9

Speed (ML) 0.90 6.5 5.8

Speed (RL) 0.86 6.5 5.9

Speed (LL) Traffic

%Trucks

0.84 7.5 6.4

Speed (ML) 0.92 6.4 5.7

Speed (RL) 0.86 6.5 5.9

Speed (LL) Traffic

%Trucks Precipitation

0.84 7.5 6.1

Speed (ML) 0.92 6.4 5.4

Speed (RL) 0.88 6.4 5.8

LL: Left lane, ML: middle lane, RL: right lane

As can be observed, both traffic, mix and precipitation

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information in the exogenous variable vector provide a slight improvement in the estimation capabilities of the VARX(1,1) model for speeds of all lanes and especially the left and right lane where the largest degree of variability is observed.

Next, a BVARX(1,1) model is constructed with λ=0.9 and θ=0.2 using as exogenous variables the volume, the percent of trucks, as well as information on precipitation. Estimation results are seen in Table III with respect to the mean absolute percent error (MAPE). In the same table (Table III), the BVAR model is compared to the ARIMA for univariate the prediction of the available speed series. As can be observed for Table III, BVARX model provide more accurate speed predictions than the univariate ARIMA time series approach for all lanes.

Moreover, the graphical representation of the actual and the predicted speed time series per lane, as well as the scatter plots of the actual versus the predicted values of speed per lane are seen in Fig. 4.The fit as described by the time series and the scatter plots of Fig. 4 reveal an efficient representation of the data. However, the improvement in terms of accuracy over the VARX models cannot be considered as significant. As in the case of VARX, speeds the left lane are more difficult to predict.

Moreover, assessing on the contribution of each exogenous variable to the prediction of speed per lane, speeds in the left lane seem to be mostly affected by volume in the entire section, as well as the percentage of trucks in the left and the middle lane.

TABLE III

COMPARATIVE RESULTS OF BVARX(1,1) AND ARIMA MODELS WITH

RESPECT TO THE MAPE

Endogenous Variables

Exogenous Variables

BVARX ARIMA

Speed (LL) Traffic

%Trucks Precipitation

5.9 7.03

Speed (ML) 5.4 6.6

Speed (RL) 5.6 6.4

Regarding speeds in the right lane, precipitation along

with volume in the middle and right lane, as well as composition in the right lane were found to be critical. Finally, in the middle lane, speed is strongly cross-correlated with volume of the neighboring lanes, as well as strongly related to the composition in the right lane. A weaker relation exists for precipitation and speed in the middle lane.

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R² = 0,8126

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R² = 0,8653

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Fig. 4 Time series of actual and predicted freeway speeds per lane and the scatter plots of actual versus predicted freeway speeds per lane using the

BVARX(1,1) model.

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A thorough look at the properties of the BVARX model reveals that all the eigenvalues lie inside the unit circle. This statement signifies that the BVAR satisfies stability condition. However, the LM ARCH test [37] with the null hypothesis of no ARCH effect in the BVARX residuals is rejected leading to assume that the series of speed exhibit volatility that is not treated using the BVAR approach. This was - in part- expected as the proposed modeling captures variability in the conditional mean and not the conditional variance. This remark suggest the plausibility of dual memory models previously applied to short-term volume forecasting [24], but in a multivariate framework.

IV. CONCLUSIONS

The present paper addresses the speed variability across lanes in freeways and attempts to model the relation between the temporal evolution of freeway speeds across lanes and other exogenous variables, such as the volume per lane, the percentage of trucks, as well as the effect of adverse weather conditions defined by precipitation. The Bayesian vector autoregressive model with exogenous variables was developed and showed better accuracy than a classical univariate ARIMA model for the prediction of lane speed values. Moreover, some critical interactions of the speeds per lane with volume, traffic mix and weather are reported.

From a methodological standpoint, although the improvement on the model’s accuracy is not considerable, the joint analysis and modeling of the speed series across lanes provided a comprehensive framework that may enable the better understand the dynamic relationships over time among the dependent variables, as well as the interactions of the dependent and the exogenous variables. However, several extensions of the proposed approach should be reported. First, it would be interesting to evaluate the prediction improvement from jointly treating the conditional mean and variance of a system of autoregressive equations; the joint consideration of mean and variance in the modeling may enhance the predictability in cases of sudden shifts to extreme speed values and speed volatile behavior. Second, as traffic flow is a spatio-temporal phenomenon, the modeling should be extended in order to incorporate both speed and exogenous variables’ data at least from upstream locations from the location of interest.

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