JOURNAL OF ADVANCED TRANSPORTATION J. Adv. Transp. (2011) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/atr.180
Practical approach for travel time estimation from point traffic detector data Luou Shen1* and Mohammed Hadi2 1
Department of Transportation Engineering, School of Civil and Transportation Engineering, South China University of Technology, 381 Wushan Road, Jiaotong Building, Tianhe, Guangzhou, Guangdong, 510641, China 2 Department of Civil and Environmental Engineering, College of Engineering and Computing, Florida International University, 10555 W. Flagler Street, EC 3605, Miami, FL 33174, U.S.A.
SUMMARY Accurate estimation of travel time is critical to the success of advanced traffic management systems and advanced traveler information systems. Travel time estimation also provides basic data support for travel time reliability research, which is being recognized as an important performance measure of the transportation system. This paper investigates a number of methods to address the three major issues associated with travel time estimation from point traffic detector data: data filling for missing or error data, speed transformation from time‐mean speed to space‐mean speed, and travel time estimation that converts the speeds recorded at detector locations to travel time along the highway segment. The case study results show that the spatial and temporal interpolation of missing data and the transformation to space‐mean speed improve the accuracy of the estimates of travel time. The results also indicate that the piecewise constant‐ acceleration‐based method developed in this study and the average speed method produce better results than the other three methods proposed in previous studies. Copyright © 2011 John Wiley & Sons, Ltd. KEY WORDS: travel time estimation; trajectory method; traffic detector; data filling; speed transformation
1. INTRODUCTION The estimation and short‐term prediction of travel time have attracted significant interests of researchers as well as practitioners for a variety of advanced traffic management systems and advanced traveler information systems. Estimated and/or predicted travel times can be disseminated to travelers, used as input to various traffic and incident management processes, and utilized in simulation/dynamic traffic assignment modeling. Travel time has also been widely used for real‐time and historical transportation system performance monitoring and evaluation. Moreover, the high variability in driver’s travel time experiences has become a concern for both travelers and management agencies. Because travel time variation affects the individual trip decision making, the strategic highway research plan has listed reliability as one of the major focal areas in 2009. The estimation of travel time provides [1] basic data support for reliability research. Therefore, there is a need for accurate and practical travel time estimation approaches. Travel time can be directly measured by probe vehicle methods such as license plate matching, electronic toll tag matching, and global positioning system tracking. Although these techniques are more direct measurements, they are more expensive and not as widely deployed as point detectors. Freeways in most North American metropolitan areas are or are in the process of being instrumented with point traffic detectors such as inductive loops and microwave detectors. Travel time can be indirectly estimated from the speed, flow, and/or occupancy data measured by these point traffic detectors. *Correspondence to: Luou Shen, Department of Transportation Engineering, School of Civil and Transportation Engineering, South China University of Technology, 381 Wushan Road, Jiaotong Building, Tianhe, Guangzhou, Guangdong, 510641, China. E‐mail:
[email protected]
Copyright © 2011 John Wiley & Sons, Ltd.
L. SHEN AND M. HADI
A number of issues are associated with travel time estimation based on point traffic detector measurements. First, missing and error data could be a problem. It is required to use data filling and data cleaning methods to correct the identified problems with the data. Second, traffic detectors can only measure time‐mean speed (TMS), which is the average speed at a point. However, travel time estimates should be derived from space‐mean speed (SMS), which represents the average speed along a highway section. Thus, proper speed transformation method needs to convert TMS to SMS. Third, a number of estimation methods have been proposed in the literature to estimate travel time for a segment based on detector measurements. There is a need to compare the quality of these methods and develop new methods in order to achieve accurate travel time estimation. The objective of this paper is to investigate the impacts of the methods used for data filling, speed transformation, and travel time estimation. First, the paper presents a review of the methods used in previous works followed by the methodology development process. After that, the paper discusses the procedure used to test the investigated methodology. The results of the test are then presented, and conclusions are drawn. 2. PREVIOUS STUDIES There are two types of missing/error data problems: random failures that occur because of temporary power‐off or data transfer problems and structure failures that occur mainly because of physical damage or maintenance problems [2]. The most common approach in previous studies is to replace the missing/error data by statistical values such as the mean, median, or other descriptive statistics [3,4]. Other studies have proposed advanced techniques including the use of traffic flow models, Kalman filters, and cross‐correlation algorithms [5–8]. These more advanced methods appear to be more appropriate than the simple statistics methods but require more modeling and calibration efforts. A spatial and temporal interpolation method was assessed by Van Lint et al. [2] so that data filling for off‐line travel time estimation can be performed. The spatial interpolation is carried out between the first available measurements from upstream and downstream detectors. The temporal interpolation fills the data gaps by interpolating between the first available measurements from past and future time intervals. For real‐time (online) applications, the temporal interpolation is not applicable because the future measurements are not yet available. Instead, an exponential moving average (EMA) method was proposed by Van Lint et al. [2]. In their simulation study, it showed that the interpolation method outperforms the statistics filling method and other complicated methods. As stated earlier, traffic detectors measure TMS. However, the travel time needs to be estimated from SMS. Thus, transformation from the measured TMS to SMS is needed. The SMS (μS) is always equal to or smaller than the TMS (μT) with the difference proportional to the speed variance (σ2S ; σ2T ). Wardrop [9] derived a relationship among the TMS, the variance, and the mean of the SMS. Rakha and Zhang [10] proposed a formulation for estimating SMS from the mean and the variance of TMS. However, the abovementioned relationships require the measurement of the variance of the speed, which currently is not a direct output from detectors. Garber and Hoel [11] derived a linear regression model between TMS and SMS. Later, Van Lint [12] presented a linear model in the following formula to estimate the variance from TMS for two traffic conditions on the basis of field data: uncongested conditions and congested conditions. ˆS ¼ σ
0:5μT −34 μT ≥74 0:02 μT þ 5 otherwise
Uncongested unit : km=h Congested
(1)
This regression method is very straightforward and will be further investigated in this study. In previous studies, two types of methods have been used to estimate the travel times on a highway segment on the basis of upstream and downstream detector measurements: the trajectory methods and the travel flow theory methods. Some widely used trajectory methods are listed as follows [13–16]: Copyright © 2011 John Wiley & Sons, Ltd.
J. Adv. Transp. (2011) DOI: 10.1002/atr
PRACTICAL APPROACH FOR TRAVEL TIME ESTIMATION
• The half‐distance strategy assumes that the speed measured by a detector is applicable to half the distances on both sides of the detector. • The average speed strategy uses the average speeds measured by the upstream and downstream detectors. • The minimum speed strategy uses the minimum of the speeds measured by the upstream and downstream detectors. Van Lint and Van der Zijpp [16] proposed an improved travel time estimation algorithm that they called the piecewise linear‐speed‐based (PLSB) method. The segment speed is assumed to be a linear spatial interpolation between the observed speeds at the upstream and downstream detector locations. On the basis of simulation results, Van Lint showed that the root mean squared error of the PLSB method was about half of the abovementioned half‐distance method. Meanwhile, models have been developed for the estimation of travel time from detector data on the basis of traffic flow theory. Examples of this approach include the work carried out by Nam and Drew [17–19], Petty et al. [20], Coifman [21], and Hoogendoorn [22]. The main advantage of these methods is that they can capture the dynamic characteristics of traffic and the growth and the dissipation of congestion. However, they require additional calibration efforts to ensure the model accuracy. 3. METHODOLOGY IN THIS STUDY In this study, two data filling methods are selected for further investigation: the average value replacement method and the spatial and temporal interpolation method including the EMA method. Given a route equipped with detectors n ∈ {1, …, N} located along the highway segment and the recorded traffic detector measurements M(n, t) at time periods t ∈ {1, …, T}, the location of each detector is denoted by xn. Suppose that no data is available for detector n during time period t, the average value replacement method will replace the missing value with the following formula: M fill ðn; t Þ ¼ M average
(2)
where Mfill(n, t) is the filled missing value and Maverage is the average value of the observed detector measurement. The spatial interpolation method will fill in this gap according to the following: 8 xn −xu > ðM ðnd ; t Þ−M ðnu ; t ÞÞ < M ðnu ; t Þ þ xd −xu spatial ðn; t Þ ¼ M M ðnd ; t Þ > : M ðnu ; t Þ
1 M ðn; tn Þ−M n; tp 1 : M n; t t¼T p
where Mtemporal(n, t) is the filled missing detector measurement, M(n, tp) and M(n, tn) are the first available past and next detector measurements, and tp and tn are the time intervals of the first available past and next detector measurements. The missing data were finally filled with the minimum value from the two interpolates because it was found in previous studies that the travel times were usually underestimated because the traffic interactions and weaving effects were difficult to model. Copyright © 2011 John Wiley & Sons, Ltd.
J. Adv. Transp. (2011) DOI: 10.1002/atr
L. SHEN AND M. HADI
M min ðn; t Þ ¼ min M spatial ðn; t Þ; M temporal ðn; t Þ
(5)
For online applications, the EMA method is applied to fill the missing detector measurements. This method recognizes the traffic measurements of a given location exhibiting strong autocorrelation over time. The missing or error traffic measurements M(n, t) from detector n at time instant t will be replaced by a forecast f(n, t) from a simple time series model (EMA) with the following equations: f ðn; t Þ ¼ f ðn; t−1Þ þ αðM ðn; t−1Þ−f ðn; t−1ÞÞ
0
