On the estimation of space-mean-speed from inductive loop detector data

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1 Transportation Planning and Technology ISSN: (Print) (Online) Journal homepage: On the estimation of space-mean-speed from inductive loop detector data Jiang Han, John W. Polak, Javier Barria & Rajesh Krishnan To cite this article: Jiang Han, John W. Polak, Javier Barria & Rajesh Krishnan (2010) On the estimation of space-mean-speed from inductive loop detector data, Transportation Planning and Technology, 33:1, , DOI: / To link to this article: Published online: 15 Dec Submit your article to this journal Article views: 687 View related articles Citing articles: 12 View citing articles Full Terms & Conditions of access and use can be found at Download by: [ ] Date: 21 November 2017, At: 11:18

2 Transportation Planning and Technology Vol. 33, No. 1, February 2010, On the estimation of space-mean-speed from inductive loop detector data Jiang Han a *, John W. Polak a, Javier Barria b and Rajesh Krishnan a a Centre for Transport Studies, Imperial College London, Exhibition Road, London SW7 2AZ, UK; b Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2AZ, UK (Received 9 March 2009; final version received 16 October 2009) Travel time is an important indicator of network performance used in traffic operations and management. Commonly deployed inductive loop detectors (ILDs) measure time-mean-speed (TMS), whereas space-mean-speed (SMS) is required to calculate the travel time. A well-known relationship between the TMS and the SMS was derived by Wardrop. However, this relationship cannot be used in practice to estimate travel times as it requires knowledge of the variance of the SMS. The variance of the SMS is not measured by the ILDs and is normally not available in practice. A novel formulation is presented in this paper to estimate the SMS using TMS obtained from ILDs. In addition, two additional models based on the formulation are developed to improve the estimation performance by taking traffic states into account. The initial results show that the proposed formulation can used to estimate the SMS, and hence the travel time, accurately using real-world data. Keywords: time-mean-speed; space-mean-speed; travel time estimation; ILD Introduction Travel time estimation (TTE) has been a high-interest topic in highway operation and management for years. Inductive loop detector (ILD) is the most widely deployed type of traffic sensor that provides data for TTE. ILDs provide a number of point-based measurements of traffic variables such as spot speed or time-mean-speed (TMS), flow and occupancy. These output measurements are used to evaluate traffic performance for traffic operation and management. Besides such point-based measurements, travel time is also an important indicator of traffic performance. However, ILDs cannot directly provide link travel times. Estimating travel time using data from widely deployed ILDs would be attractive to traffic management agencies, as models that use data from existing devices can minimise additional costs for obtaining more information. For example, every link in England s highway network is equipped with ILDs to monitor and record the traffic flow, occupancy and TMS (National Traffic Control Centre; NTCC 2009). Some roadway sections consisting of multiple links are installed with automatic number plate recognition (ANPR) cameras to measure travel time. Due to the high cost of procuring and installing ANPR cameras, relatively few road sections have ANPR cameras in the highway network. Hence, *Corresponding author. jiang.han06@imperial.ac.uk ISSN print/issn online # 2010 Taylor & Francis DOI: /

3 92 J. Han et al. there is no travel time information available for most of the links. Therefore, models that estimate travel time using data from ILDs are of particular interest for those links without ANPR cameras. Motivated by this practical problem, a simple and efficient method using ILD data to estimate the space-mean-speed (SMS), which can be used to calculate link travel times, is presented in this paper. Background Researchers and engineers have studied TTE since the late 1920s to evaluate transport performance and planning improvement. After decades of development, TTE is being used in the area of road network performance, traveller information systems and dynamic route guidance (Krishnan 2008). TTE models vary according to the input data used, such as ILD, probe vehicle technologies, license plate matching, GPS and aerial surveys. ILD is the most common technology and has been used since the 1960s (Klein et al. 2006). A number of TTE methods using ILD data have been developed by researchers. The traditional practice for estimating travel times using spot speed from single loop detectors is based on the assumption of a constant average effective vehicle length. Turner et al. (1998) investigated flow/ occupancy relationships (g-factor) to estimate speed with an assumption of constant vehicle length. Dailey (1999) presented an improved method that took the distribution of vehicle lengths into account for speed calculation to improve estimation accuracy. However, these techniques require access to the raw ILD output at a high temporal granularity, which is typically not available at a remote location. Moreover, such models provide an estimate of TMS and not SMS, which is required for TTE. Learning-based models such as linear regression (Sisiopiku and Rouphail 1994), artificial neural networks (ANN; Rouphail et al. 1993) and k-nearest neighbour (k-nn; Robinson and Polak 2004) can relate available ILD data to travel time. These models require reference travel time data and ILD output during a calibration period. The calibrated model can be subsequently used to estimate travel time using the ILD outputs for other time periods. However, the calibrated relationship between ILD output and travel time is specific to individual links. Kalman filter (Chen and Chien 2001) is able to apply the basic traffic flow formulations to adaptively estimate travel time. The performance largely depends on the quality of ILD data. Besides the above methods, investigation of the relationship between TMS and SMS also provides another way to estimate travel time. The earliest research in this area was Wardrop s (1952) well-known formulation. The author derived the relationship between TMS and SMS using the concepts of timespeed distribution and spacespeed distribution. The next section of this paper discusses this method in more detail. Garber and Hoel (2002) developed a linear relationship between TMS and SMS. Although their method is simple and direct, the model parameters are specific to a given link and traffic stream characteristics. In addition, the linearity of their proposed relationship may not be valid under all circumstances to provide a high level of accuracy. Rakha and Zhang (2005) presented a formulation similar to Wardrop s to approximate SMS from TMS directly. However, the variance of TMS required by their model is not available from typical ILD installations; e.g. the DATEX-II traffic data feed provided by the Highways Agency (HA) in England. Typically, only TMS, flow and occupancy are available from ILDs. Hence, their

4 Transportation Planning and Technology 93 method is of limited practical relevance. In order to overcome these drawbacks, a novel SMS estimation method is proposed in this paper. Methodology Time-mean-speed (TMS) versus space-mean-speed (SMS) As the proposed method aims to model the relationship between TMS and SMS, a number of concepts used in the formulation are outlined here. A comprehensive description can be found in Daganzo (1997). The mean speed measured by the ILD is TMS, defined as follows: X N j1 TMS (1) N where: u j : spot speed of the jth vehicle measured over the ILD; N: total number of vehicles measured during a given time period. From the Eq. (1), TMS is defined as the arithmetic mean of the speed of vehicles passing a point during a given time interval. Hence, TMS only reflects the traffic condition at one specific point. On the other hand, SMS is the arithmetic mean of the speed of all the vehicles occupying a given stretch of the road at a given instant, as given below: SMS M where: v i : instantaneous speed of the ith vehicle observed on the link; M: total number of vehicles on the link at the observed instance. Travel time can be calculated from SMS directly as: Average travel time L SMS X M i1 u j v i (2) (3) Modelling the relationship between time-mean-speed (TMS) and space-mean-speed (SMS) Both of TMS and SMS quantifies the vehicle speed for the same traffic flow in different ways. The well-known relationship between these two quantities was derived in Wardrop (1952) as follows: TMSSMS s2 sms SMS (4) A complete derivation of this equation can be found in Wardrop (1952). Eq. (4) assumes that the physical characteristics of the link are uniform along the length of the link. In Eq. (4), the s 2 sms is the variance in instantaneous vehicle speeds (v i) about

5 94 J. Han et al. the SMS, and is defined by Eq. (5) according to the derivation of Eq. (4) in Wardrop (1952): s 2 sms E[(v i SMS)2 ] (5) Consequently, Eq. (4) can be used to estimate TMS with the knowledge of SMS. However, in most of cases, s 2 sms cannot be obtained, which is why it is desirable to estimate SMS from the available TMS data. The main impediment to using Eq. (4) to estimate SMS from TMS is that the s 2 sms is unknown. In order to make use of the relationship given in Eq. (4), a further formulation needs to be derived. Based on Eq. (5), s 2 sms can be expanded as: s 2 sms E[(v i SMS)2 ] (6) E[v 2 i SMS2 2v i SMS] (7) E[v 2 i ]SMS2 2SMS E[v i ] (8) Assuming stationarity and uniformity of along the link for traffic conditions (Daganzo 1997), the instantaneous speed can be approximated to the spot speed measured by ILDs. The above assumption means that the physical characteristics of the link are uniform along its length, and that the traffic stream maintains same level of flow and speed along the given stretch of road during the period of observation. Based on this assumption, the vehicle instantaneous speed v i in Eq. (8) is approximately equivalent to the spot speed of ith vehicle measured by the ILD. Hence the instantaneous speed v i is approximated to the spot speed; in other words, E[v i ]:TMS. Hence, Eq. (8) can be re-written as follows: s 2 sms E[v2 i ]SMS2 2TMS SMS (9) Replacing s 2 sms in Eq. (4) using the expression in Eq. (9) and rewriting it: 2SMS 2 3TMS SMSE[v 2 i ]0 (10) If it is assumed that E[v 2 i ] has a known value, then Eq. (10) is a quadratic equation with SMS as the unknown variable. Using the quadratic formula, the solution to Eq. (10) is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SMS 3TMS9 9TMS 2 8E[v 2 i ] (11) 4 Eq. (11) formalises the relationship between SMS, TMS and /E[v 2 i ]: There are two solutions to Eq. (10), whereas SMS has only one value during any time interval. It was shown in May (1990) that the difference between TMS and SMS is of the order of 15% normally, using real traffic data as the supporting evidence. Slightly rewriting Eq. (11): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SMS0:75 TMS90:25 9TMS 2 8E[v 2 i ] (12) Since SMS should be above 75% of TMS, the sign for the second term must be positive. Therefore, the expression for SMS is as given in Eq. (13):

6 Transportation Planning and Technology 95 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SMS 3TMS 9TMS 2 8E[v 2 i ] 4 (13) Approximating E[v 2 i ] using time-mean-speed (TMS) In the above derivation, it is assumed that E[v 2 i ] is known; this is not necessarily true in practice. E[v 2 i ] can be considered as the expected value of the squared spot speeds measured by the ILD. Since the spot speed of each vehicle is not typically available from the ILD, E[v 2 i ] cannot be calculated using the ILD output. Hence, E[v2 i ] needs to be estimated using the known value of TMS. Using Eq. (1), TMS can be expressed as the expectation of v i during a given time interval: TMSE[v i ] (14) It can be reasonably assumed that the relationship between E[v 2 i ] and E[v i]is quadratic, which was supported by analysing a plot between the two quantities using ILD data from the English highway network. Hence, the following quadratic relationship was proposed: which is equivalent to: E[v 2 i ]ae2 [v i ]be[v i ]c (15) E[v 2 i ]atms2 btmsc (16) Assuming that the relationship between TMS and E[v i ] is transferrable across links for a given traffic stream, Eq. (15) or Eq. (16) can be solved to find the constant coefficients {a,b,c}. TMS from ILDs and SMS from ANPR cameras are available for certain links of the highway network in England. The relationship between SMS and TMS, specified through Eqs. (13) and (16), is calibrated using data from such links. The polynomial fitting algorithm by Weisstein (1999), which is a type of linear regression (Freedman 2005), is used to estimate the constant coefficients {a,b,c}. The empirical analysis presented below shows that the coefficients {a,b,c} estimated using data from the selected links are valid on other links on the network, and hence the approach is transferrable. Model evaluation Data description The proposed model is evaluated using ILD data from English highways obtained through the DATEX-II feed disseminated by the NTCC. The feed contains ILD data consisting of flow, TMS and occupancy, and travel time data from ANPR links. Both ILD data and ANPR data are provided at 5-minute intervals. This means that every 5 minutes, there is one TMS data point from ILDs and one average travel time data point from ANPR links. The ILDs used in this research are per-lane double-loops. The TMS is measured from double-loops using the time difference between the activations of the two loops by a given vehicle and the known distance between the two loops. In order to test the proposed model, only links with both ILD data and

7 96 J. Han et al. ANPR data were used; it must be noted that most of links have no ANPR data available. In order for the evaluation of the model to be as generic as possible, the links were chosen from a variety of road links with different locations and topologies, and shown in Figure 1. Data from the chosen links for a two week period between 15th July and 29th July 2008 was used in the evaluation. For the evaluation process, the TMS and SMS data samples are grouped pair-wise for each 5-minute time interval. Due to sensor or communication failures, either ILD or ANPR camera data were missing for some time period. Time periods when either of the data points were missing were removed Figure 1. Location and topology of chosen links: (a) Link-1; (b) Link-2; (c) Link-3; (d) Link- 4; (e) Link-5; (f) Link-6. Source: Google Earth.

8 Transportation Planning and Technology 97 from the data-set. The remaining data-set had a total of 9034 paired ILD (TMS) and ANPR (SMS) observations. Results on estimating E[v 2 i ] using time-mean-speed (TMS) The approximate relationship between E[v 2 i ] and TMS given in Eq. (16) is one of the basic building blocks of the proposed model. The E[v 2 i ] is calculated using Eq. (13), in which the value of SMS is obtained using the output of ANPR, calculated by Eq. (3) with the known link length and matched link travel time from ANPR. The resultant plot of E[v 2 i ] versus TMS approximated by Eq. (16) is shown in Figure 2. The approximately quadratic relationship between E[v 2 i ] and TMS is evident from the data points in Figure 2. This supports the earlier assumption that E[v 2 i ] is a quadratic function of TMS. This quadratic relationship is estimated using the polynomial fitting method. The fitted polynomial is shown as the red curve in Figure 2. Results on estimating space-mean-speed (SMS) at each link The constant coefficients {a, b, c} of the quadratic function were estimated to be {a1.22, b15.21, c207.95} with total 9034 samples and R The p-value for each of the coefficient {a,b,c} is 0 with t-statistic {13.04, 13.85, 11.10}, respectively. Using the relationships in Eqs. (16) and (13), SMS is estimated for each link. Results are presented only for Link-1 in this section; results for other links have similar patterns. The scatter plots between SMS and TMS (blue circles) and SMS and estimated SMS are shown in Figure 3(a). As expected, TMS is higher than SMS, while the estimated value of SMS is closer to the actual SMS. Figure 3(b) illustrates the comparison between the level of error before and after estimation (TMS and estimated SMS). The result shows that the proposed model reduced the error of Figure 2. Relationship between TMS and E[v 2 i ]:

9 98 J. Han et al. Figure 3. Estimation results from Link-1 data: (a) Scatter-plot between TMS and SMS and estimated SMS and SMS; (b) Comparison between errors from only using TMS and estimated SMS. speed measurement by approximately 10 km/h; quantitative error statistics is presented in Table 1. Model improvement The effects of varying traffic states The use of fixed coefficients {a,b,c} in Eq. (16) means that the estimated relationship between E[v 2 i ] and TMS holds good for all traffic conditions. Although the method based on this assumption shows better accuracy, the estimation performance can still be improved by refining the formulation. The calibrated relationship between E[v 2 i ] and TMS is valid for a given traffic state. However, when traffic states change, the coefficients in this relationship may vary. For example, consider two different traffic sates at one link: (1) high occupancy and high flow and (2) high occupancy but low flow. According to this description, the second state represents congested state, while the first state represented uncongested state. In the basic model, the same coefficients are used in Eq. (16) to estimate SMS regardless of the traffic state. However, it can be reasonably assumed that the variance of spot speed is different between the first and second states.

10 Table 1. Estimate performance comparison. Error type Estimation method Link-1 Link-2 Link-3 Link-4 Link-5 Link-6 MAPE TMS (%) Original estimation model (%) Segmentation based estimation model (%) Clustering based estimation model (%) RMSE TMS Original estimation model Segmentation based estimation model Clustering based estimation model Transportation Planning and Technology 99

11 100 J. Han et al. In order to investigate, how the difference between traffic states affect the accuracy of the proposed model, a parameter a is defined to represent the traffic state as follows: a1000 Occupancy Flow (17) Occupancy and flow are readily available from the ILD output, which are used to calculate the defined parameter a. Based on the value of the a, traffic states can be categorised into different groups. It is assumed that the parameters of the relationship between E[v 2 i ] and TMS in Eq. (16) will be different within each traffic state group. In order to test the assumption, the data-set is segmented into four groups to represent different traffic states based on the value of a: {a53, 3Ba55, 5Ba510, and a10}. Figure 4 shows the polynomial fitting results for these traffic states. The resulting plots are consistent with expectations. The fitted polynomial curves capture the relationship between E[v 2 i ] and TMS for a given traffic regime well. Using the above approach, flow and occupancy data from the ILDs can be used to determine the traffic states, and the relationship between E[v 2 i ] and TMS for the given traffic state can be used to estimate the SMS. Using mean absolute percentage error (MAPE) and root mean squared error (RMSE) as accuracy metrics, the performance of the original and segmentation-based estimation models is compared in Table 1. Applying k-means clustering technique The previous section explored how the traffic state affects the SMS estimation results. Although the state dependent method shows better estimation accuracy, the segmentation scheme used is arbitrary and may not be transferrable. The number and boundaries of the segments were determined based on prior analysis of data from the chosen links for which both TMS and SMS were available. For example, the occupancy value reported by an ILD will depend on its electromagnetic sensitivity. Hence the boundary values of a between traffic states could vary between ILDs. Using a fixed value of a for segmenting traffic states is hence not a transferable approach. In order to generalise the segmentation approach using flow and occupancy obtained from an arbitrary group of ILDs, traffic states are categorised into congested and uncongested regimes. The ideal relationship between flow and occupancy is given in Figure 5(a). The lower segment of the plot represents samples from the uncongested regime, while the upper segment represents the congested state. The scatter plot between real flow and occupancy data used in this study is given in Figure 5(b). It is clear that it is easy to determine if a given data point (flowoccupancy point) is congested or not from the scatter plot by visual inspection. However, this differentiation between the traffic states needs to be carried out in an automated fashion for the proposed approach. Clustering techniques such as the k-means clustering (Weisstein 2005) can be used to partition a data-set into a number of different clusters. However, direct application of the k-means clustering technique to partition the data points into two clusters representing congested and uncongested

12 Transportation Planning and Technology 101 Figure 4. /E[v 2 i ] vs. TMS under different traffic states: (a) when a53; (b) when 3Ba55; (c) when 5Ba510; (d) when a10. regime did not yield satisfactory results. Specifically, a number of data points in the congested regime were identified as uncongested. To solve this issue, a linear regression model is fitted to the points identified as uncongested. All data points that are identified as outliers are deemed as congested and are moved to the congested cluster. The result of this two-stage clustering is shown in Figure 6(a). For each cluster data, the polynomial fitting process was applied, and results are shown in Figure 6(b). The SMS estimation results based on the two-step clustering method is also shown in Table 1. Figure 5. Relationship between flow and occupancy: (a) theoretical relationship; (b) real-traffic data.

13 102 J. Han et al. Figure 6. Two states clustering results: (a) clustering result on scatter plot of flow vs. occupancy; (b) fitting result on each clustered state. Table 1 shows that SMS estimated using the models presented in this paper, and a clear improvement over using TMS to estimate travel times can be seen. When data points from Links 1, 4, 5 and 6 largely uncongested, the performance of the refined models (segmented and k-means clustering models) is only marginally better than the originally proposed model. In contrast, Links 2 and 3 have more data points from the congested state. Therefore, the segmentation and clustering-based models provide higher estimation accuracy than the originally proposed model. The MAPE values of Link 2 and Link 3 are and 17.51% based on the proposed clustering model during congestion period, while the values are and 34.84% for the traditional estimation method based on TMS. Although the overall performance of the segmentation-based model is better than that of the clustering-based model, the generality of the clustering-based model makes it transferable to other links. Moreover, the segmentation-based approach requires knowledge about SMS to accurately define the segments. Conclusions This paper presented a simple transferrable method to model the relationship between TMS and SMS. It was found that a quadratic function is able to model the relationship between TMS and the variance of SMS, which is normally an unknown quantity. Comparing with other works in this area, the proposed model has better accuracy than Garber and Hoel (2002) linear relationship between TMS and SMS. It is also easier and has fewer limitations to implement than ANN and k-nn-based methods from the aspect of traffic engineering. Rakha and Zhang (2005) s research is the most relevant in this area. Although it presents a new formulation to link TMS and SMS at second moments, it requires the knowledge of variance of TMS which is not available from typical ILD installations; e.g. the DATEX-II traffic data-set. The proposed model in this paper overcomes this limitation and only TMS value from ILDs is used to estimate SMS. It was also shown that differences in traffic state affect the parameters of this relationship. A location specific segmentation-based approach was proposed to partition the data into four different traffic states. In addition, a generic two-step clustering method based on the k-means clustering technique was used to partition the data into congested and uncongested states. The quadratic

14 Transportation Planning and Technology 103 relationship was calibrated for each partition separately. Based on the real-world data from English highways, it was shown that the SMS estimated using all the above models provide more accurate estimates of travel times compared to TTE using TMS directly. The segmentation-based models provided more accurate estimates of SMS compared to the original model. The original and clustering-based models can be used for real-world ITS applications easily. Further research on this topic could focus on two distinct issues. Firstly, more advanced methods can be developed to quantify the effect of traffic states on the relationship between TMS and E[v 2 i ]: Secondly, this paper presents results from limited-access highways. The proposed method can be extended to signalised urban roads with more complex traffic patterns. The main issue is that the proposed model is dependent on the type of ILDs. SCOOT loops are single loops and they do not provide accurate TMS. The g-factor approach (assuming an average vehicle length) is also not applicable since these loops are sampled at a relatively low frequency of 4 Hz. Moreover, a large number of SCOOT loops span two lanes and it is very difficult to obtain speed data from them. For this reason, the proposed method is not directly applicable to urban (SCOOT type) loops. References Chen, M. and Chien, S., Dynamic freeway travel time prediction using probe vehicle data: link based vs. path-based. Transportation Research Record: Journal of the Transportation Research Board, 1768, Daganzo, C.F., Fundamentals of transportation and traffic operations. Oxford: Pergamon. Dailey, D.J., A statistical algorithm for estimating speed from single loop volume and occupancy measurements. Transportation Research Part B: Methodological, 33 (5), Freedman, D., Statistical models: theory and practice. Cambridge: Cambridge University Press. Garber, N. and Hoel, L., Traffic and highway engineering. 3rd ed. California: Brooks- Cole. Klein, L.A., Mills, M.K., and Gibson, D.R., Traffic detector handbook. 3rd ed. Volume I. McLean, VA: Federal Highway Administration. Krishnan, R.K., Travel time estimation and forecasting on urban roads. Thesis (PhD). Centre for Transport Studies, Imperial College London, London. May, A.D., Traffic flow fundamentals. Englewood Cliffs, NJ: Prentice Hall. National Traffic Control Centre (NTCC) Road Network, Technical report. Publication code PR237/06. Birmingham, UK: Highways Agency Publications Group. Available from: pdf [Accessed 29 June 2009]. Rakha, H. and Zhang, W., Estimating traffic stream space-mean speed and reliability from dual and single loop detectors. Transportation Research Record: Journal of the Transportation Research Board, 1925, Robinson, S. and Polak, J., Some new perspectives on urban link travel time models: is the k-nearest neighbours approach the solution? In: 36th annual UTSG conference, 57 January Newcastle, UK: Universities Transport Study Group (UTSG), 4B2.1 4B2.12. Rouphail, N.M., et al., Travel time data fusion in ADVANCE: a preliminary design concept. ADVANCE Working Paper Series No. 21. Chicago, IL: University of Illinois at Chicago. Sisiopiku, V. and Rouphail, N., Toward the use of detector output for arterial link travel time estimation: a literature review. Transportation Research Record: Journal of the Transportation Research Board, 1457, Turner, S., et al., Travel time data collection handbook. Federal Highway Administration FHWA-PL Research Report F. Final Report.

15 104 J. Han et al. Wardrop, J.G., Some theoretical aspects of road traffic research. Proceedings of the Institute of Civil Engineers, 1 (2), Weisstein, E.W., Least squares fitting*polynomial. MathWorldA Wolfram Web Resource. Available from: html [Accessed 5 March 2009]. Weisstein, E.W., K-means clustering algorithm. MathWorldA Wolfram Web Resource. Available from: [Accessed 5 March 2009].