Dynamic Prediction Method with Schedule Recovery Impact for Bus Arrival Time

Transcription

1 Dynamic Prediction Method with Schedule Recovery Impact for Bus Arrival Time Mei Chen, Xiaobo Liu, and Jingxin Xia This study develops a dynamic bus arrival time prediction model using the data collected by the automatic vehicle location and automatic passenger counter systems. It is based on the Kalman filter algorithm with a two-dimensional state variable in which the prediction error in the most recent observation is used to optimize the arrival time estimate for each downstream stop. The impact of schedule recovery is considered as a control factor in the model to reflect the driver s schedule recovery behavior. The algorithm performs well when tested with a set of automatic vehicle location automatic passenger counter data collected from a real-world bus route. The algorithm does not require intensive computation or an excessive data preprocessing effort. It is a promising approach for real-time bus arrival time prediction in practice. The objective of this study is to develop a dynamic prediction methodology that is capable of providing bus arrival time at downstream major stops listed on the timetable, called time points (TPs), for real-time implementation. This methodology will be able to interface with the data collected by the automatic vehicle location (AVL) and automatic passenger counter (APC) and provide updates on bus arrival time for each downstream stop when the newest information on the bus location and time becomes available. Furthermore, the schedule recovery effort by the bus drivers will be incorporated into the model. Bus transit plays an important role in transportation systems, especially in urban areas. In today s tough competition with automobiles, transit service needs to improve its quality. Traditionally, passengers rely on published timetables to make decisions about departure time to the bus station and transfer activities. However, due to shared right-of-way, bus services often experience delays that result from unexpected congestion along the routes. Thus, passengers often find themselves waiting a long time at bus stops. This delay may also foil their plans to transfer by incurring additional interruption in their schedules. A more reliable source for bus arrival information is needed to increase the confidence of passengers on bus service, especially in the event of unexpected congestion along the route. More transit agencies are offering bus arrival time information to help transit users make smarter decisions about their departure time from home or work to shorten their waiting time. On the other hand, such information can also help transit operators conducting performance evaluation, operational control, and service planning. M. Chen and J. Xia, Department of Civil Engineering, University of Kentucky, 267 Raymond Building, Lexington, KY X. Liu, LS Engineering Associates Corporation, 23 U.S. Highway 26, Flanders, NJ Transportation Research Record: Journal of the Transportation Research Board, No. 1923, Transportation Research Board of the National Academies, Washington, D.C., 25, pp Among those transit agencies with such information are AC Transit in Almeda County, California (1); the City-University-Energysaver bus system in Fairfax, Virginia (2); the Vail Bus Service in Vail, Colorado (3); the municipal railway system in San Francisco, California (4); the Tri-met Transit Tracker system in Portland, Oregon (5); and King County Metro Transit in the state of Washington (6). Transit agencies have started to use advanced sensing and communication technologies to improve transit service quality. Various technologies such as APC and AVL have been implemented (7) nationwide, among which the APC technology is drawing increasing attention. The AVL-APC system is considered an efficient tool in collecting bus operating information (e.g., numbers of passengers boarding and alighting at each stop as well as the corresponding time and location) that is critical to transit operation analysis and service planning. The cost of obtaining such information as provided by the AVL-APC system using traditional data collection methods in the same quantity and quality is prohibitive. Using the rich data made available by such devices installed on buses, a methodology was developed to predict bus arrival time. It is dynamic in that it can incorporate the most recent bus travel information (e.g., current location and time), whenever it becomes available, into the model for arrival time prediction at downstream stops. The algorithm is simple enough to not require intensive computation, which would be desirable in real-time applications. Furthermore, the model takes into consideration the impact of the driver s schedule recovery effort as a control factor. PREDICTION METHODOLOGY Various models have been developed to predict bus arrival time. Generally, the techniques used can be categorized as follows: regression (8), artificial neural network (9 11), Kalman filter (9, 11 13), and a combination of the preceding techniques (9). Lin and Zeng (8) developed a set of algorithms to predict bus arrival time based on Global Positioning System (GPS) data. Regression models were built for different combinations of independent variables. The prediction accuracy was limited because of some inherent features of GPS data, such as an inconsistent sample period. Prediction models based on an artificial neural network algorithm were also developed through various studies such as those of Chen et al. (1, 11) and Chien et al. (9). The advantage of such an algorithm is that it does not require an explicit function form or independence among the input variables. However, large amounts of data are required to train the network to achieve a reliable prediction. With its dynamic feature to update the estimation of state variables, a Kalman filter has been used widely in various fields including forecasting traffic parameters. Dailey et al. (12) and Cathey and Dailey (14) developed a Kalman filter-based algorithm for transit 28

2 Chen, Liu, and Xia 29 arrival time prediction using data collected by onboard AVL units. The prediction was performed at a fixed time interval using observations of vehicle location that were obtained asynchronously. The algorithm has been implemented by King County Metro, a transit agency in the metropolitan Seattle area. A Kalman filter algorithm was also used in studies by Chien et al. (9) and Shalaby and Farhan (13). However, these models were developed based on the assumption that all buses traveling on the route were equipped with APC or AVL devices. In reality, the high initial investment associated with these devices limits their application in real-world practice. In this study, the Kalman filter technique is also used in developing a model to predict bus arrival time. This model will be based on APC data collected from the studied bus route. The data contain not only the information on passenger activities but also information about when and where these activities occurred. A Kalman filter is a powerful mathematical tool that can estimate the future states of variables even without knowing the precise nature of the system modeled. It is a recursive procedure that corrects its estimates whenever new observations become available, with the objective of minimizing the estimated error covariance. The filter procedure developed in this study is designed to predict arrival time at downstream TPs for each bus trip. The starting TP is defined as the origin, and each downstream TP is treated as a destination. Assume there are N TPs numbered sequentially along the route with the origin labeled as 1. The general concept of the prediction algorithm is presented in Figure 1. Let J denote the set of TPs (excluding the origin) along the route; thus, j (2, N), j J. Each of these TPs may be a destination to which the travel time will be estimated. Let t k,j denote the travel time from TP k (i.e., the current TP from which arrival time prediction is performed) to destination j (i.e., the downstream TP for which arrival time prediction is performed, j J and k < j ); τ k,k+1 denote the estimated travel time from TP k to TP k + 1; and s k denote the travel time from the origin (i.e., TP1) to TP k. Then the travel time from TP k + 1 to destination j ( j J and k + 1 < j ), t k+1,j, can be calculated by Equation 1 as follows: t k+ 1, j = tk, j τk, k+ 1 () 1 And the travel time from the origin to TP k + 1 can be calculated by Equation 2 as follows: s 1 = s + τ, 1 ( 2) k+ k k k+ One of the operational goals of a transit agency is to keep buses on schedule. In reality, transit operators tend to constantly adjust vehicle speeds to maintain good schedule adherence. Through examining a set of AVL-APC data collected in 22 from a reputable transit agency in the northeastern United States, it was found that a schedule recovery effort can be observed on at least half the segments on a transit route. For instance, if a bus is delayed at TP k, around 5% of trips can be observed to have a shorter travel time than scheduled between TPs k and k + 1. Lin and Bertini (15) used the Markov chain concept to model bus operators behavior in the schedule recovery process. This study attempted to use the Kalman filter to account for the impact of such an effort. The term τ k,k+1 is defined as the estimated travel time between TPs k and k + 1. It consists of two elements; one is the schedule travel time from k to k + 1, T k,k+1, and the other is the driver adjustment D k based on the schedule adherence status of the bus. Then τ k,k+1 can be estimated as follows: τ kk, + 1 = T kk, D k () 3 s 1 = t TP 1 t 1,2 TP 2 1,k t 1,14 TP k TP 13 TP 14 TP 1 TP s 2 2 t 2,k TP k TP 13 t 2,14 TP 14.. TP TP 2 1 s k TP k t k,13 TP 13 t k,14 TP 14 FIGURE 1 Arrival time prediction procedure.

3 21 Transportation Research Record 1923 The delay experienced by the bus at TP k can be expressed as T 1,k s k. Assume the driver adjustment between k and k + 1, D k, is proportional to the delay at TP k. Then, if β k represents the adjustment factor for the segment between TPs k and k + 1, one can estimate its value with historical trip information. Therefore, the driver adjustment can be estimated as follows: Dk = βk( T1, k sk) ( 4) If z k denotes the observed travel time from the origin to TP k, then z k = s k. For each destination j J, let x k,j = (t k,j s k ) T denote a twodimensional state variable, and a Kalman filter can be formulated as follows: xk+ 1, j = Φk+ 1xk, j + uk + wk, j ( 5) zk = Hkxk, j + vk, j ( 6) where 1 Φ k+1 =, 1 H k = ( 1), 1 u k = τ kk. + 1, and w k, j and v k, j = white noise associated with the transition process and measurement, respectively, and are assumed to have zero mean and variances of Q k,j and R k,j, respectively. Given Equations 3 and 4, the control input term u k can be converted as follows: u k = 1 τ k, k+ 1 = 1 [ Tkk, βk( T1, k sk) ] = 1 ( T, β T1, ) β ( 1) x, k = 1 β ( Tkk, βkt1, k) + x [ kk k k k kj] Let βk A k+1 = βk and Equation 5 becomes k, j β k xk+ 1, j = Φ k+ 1xk, j + uk + wk, j ( 7) where Φ k+1 =Φ k+1 + A k+1, and 1 u k = ( Tkk, βkt1, k). Equations 7 and 5 form a Kalman filter that accounts for the impact of drivers schedule recovery effort. It projects the bus progression process using a form of feedback control. Equation 7 describes the time update process in which a bus travels from one TP to its downstream TP. It predicts the state of the variable (e.g., travel time to the destina- tion) for the next segment based on its current state and error covariance estimates. Equation 5 provides a measurement update function that realizes the feedback process that is, the newly measured travel time (from origin) at the current TP is used to adjust the predicted travel time (to the destination). The overall filtering procedure is a recursive prediction correction process outlined as follows: Step 1. Initialize. Set k = 1 and j = 2 Step 2. Initialize state variables. Let xˆk,j = (tˆk,j ŝ k ) T, where tˆk,j is the estimated total travel time from the origin to destination j, and ŝ k is the travel time from the origin to TP k. Step 3. Initialize covariance P k,j. Step 4. Extrapolate state variable. xˆ = Φ xˆ + u k+ 1, j k+ 1 k, j k 1 βk t = 1 β s where T k,k+1 is the scheduled travel time between TP k and k + 1. Step 5. Extrapolate covariance. T Pk+ 1, j = Φ k+ 1Pk, jφ k+ 1 + Qk, j ( 9) Step 6. Compute Kalman gain (K). T T Kk+ 1, j = Pk+ 1, jhk+ 1 Hk+ 1Pk+ 1, jhk+ 1 + Rk+ 1, j ( 1) Step 7. Update state variable. xˆ = xˆ + K z H xˆ ( ) k+ 1, j k+ 1, j k+ 1, j k+ 1 k+ 1 k+ 1, j 11 Stop if k + 1 = j (i.e., when TP k + 1 is the destination) and j = N. Otherwise, if j < N, go to Step 8; else, if j = N, go to Step 9. Step 8. Update destination. j = j + 1 Go to Step 2 Step 9. Update covariance. Pk+ 1, j = Pk+ 1, j Kk+ 1, jhk+ 1Pk+ 1, j ( 12) k = k + 1, j = k + 1 Go to Step 2. This Kalman filter algorithm starts with a baseline estimate of travel time from the origin to each downstream destination j ( j J). It uses the most recent observation of travel time between the origin and the last stop k (k < j ) to adjust the predicted travel time from k to each destination j. The predictions are updated whenever the bus reaches the next downstream TP that is, when a new measurement z k becomes available. CASE STUDY Data Collection k k, j k Tkk, + kt, k + 1 ( 1 + β 1 ) ( ) ( ) The AVL-APC data collected in 22 were obtained from a reputable transit agency in the northeastern United States. The pattern 1 () 8

4 Chen, Liu, and Xia 211 selected for this study has 14 TPs. The main attributes in each APC record used for the model development are summarized in Table 1. The ideal data structure for observing bus operations and developing prediction models is a set of successive TP records. Thus, the actual bus travel times between TPs can be used to update the estimates from the prediction model. However, the APC units record activities on all stops (whether a TP or not). In other words, there will be records on stops made between TPs to pick up or drop off passengers for the stop-on-demand type of operation. On the other hand, there might not be a record for a particular TP if the bus did not stop because no passenger demanded existed there. Therefore, travel time interpolation was performed for those skipped TPs to make the APC data set consistent for prediction model development. Schedule Recovery Phenomenon It is a common understanding that transit operators tend to actively pursue schedule recovery if the bus is delayed. On the basis of AVL- APC data and the timetable, the delay at each TP was calculated and then correlated with the travel time deviation (from the schedule) on the next segment (from the current TP to the next downstream TP). Schedule recovery was observed on all segments and was particularly frequent on the upstream portions of the route. The passenger boarding alighting activities along the route, as recorded by the APC devices, can offer some explanation of the pattern of schedule recovery. Generally, fewer stop activities and less passenger demand could make the driver s schedule recovery effort easier. Further exploration showed that besides traffic impact, the schedule recovery phenomenon usually occurred on segments with fewer stops made between two adjacent TPs and a smaller number of passengers boarding alighting. For example, because there is no TABLE 1 Variable Sched arrival time Transit day Time of day Open time Close time Stop sequence Time point ID Trip status Lat Lon On Off Stop distance Dwell time Leg time Leave psgr load Arrive psgr load Pattern ID Trip index Main Index in APC Data Set Description Scheduled arrival time at each time point Date of the service Time period of bus operation Recorded bus door opening time Recorded bus door closing time A unique number to all intended stops along the route Time point indicator number Trip status (start or end) Latitude Longitude Number of boarding passengers at a stop Number of alighting passengers at a stop Travel distance between two consecutive stops The bus door open time at any stop Inter-stop travel time Number of onboard passengers when the bus leaves a stop Number of onboard passengers when the bus arrives at a stop Unique index associated with a pattern in each pick data Unique index associated with a trip intermediate stop between TP4 and 5 (they are two airport terminals), about 85% of the trips recovered a certain loss in travel time on this segment. On the other hand, 73% of the trips experienced additional schedule deviation on the segment between times 1 and 11. The APC data showed that on this segment an average of two stops per mile were made to allow, on average, three passengers to board and four passengers to alight at each stop. Using the historical trip data, one can get the distribution of the adjustment factor (β k ) along the route. On most segments, the average adjustment factors are mostly between.5 and.5. In this study, the average of all historical adjustment factors was used on segment k as the β k. Next, an example is presented to demonstrate implementation of this prediction algorithm with the schedule recovery process on an individual bus trip. This is followed by a discussion of the overall performance of the algorithm and comparative analysis with other bus travel time estimation models. Individual Trip Analysis Following the steps of the Kalman filter algorithm, travel times from the current stop to every downstream TP are predicted. The predictions are then updated when the bus reaches the next TP. For illustration purposes, an arbitrarily selected trip is used as an example to analyze the algorithm performance. This trip was made on Wednesday, October 2, 22. It was scheduled to depart TP1 on 1:32 p.m. and arrive at TP14 at 3:19 p.m. Table 2 presents the prediction results using the Kalman filter algorithm on this trip. There are 14 TPs in total along the route, with the first one defined as the origin. When the bus reaches the next TP, 1 is added to the k value until the bus reaches the final destination (i.e., TP14). Each cell in the table represents the estimated state variable xˆk,j = (tˆk,j ŝ k ) T. For example, when k = 3, the bus has already reached TP3. For destination TP1 (i.e., j = 1), the estimated travel time from TP3 to TP1, tˆ3,1, is 3,851 seconds, and the bus has spent 953 seconds traveling from the origin to TP3 (i.e., ŝ 3 = 953 seconds). When k = 1, the filter initializes itself using baseline estimates of travel time between the origin and each of the downstream TPs (destination j J ). The timetable is a good source for such estimates. For example, when j = 2, the estimated travel time from TP1 to TP2, tˆ1,2, is 48 seconds according to the timetable, and the estimated travel time from the origin (i.e., TP1) to the current TP (i.e., TP1) is. When the bus reaches the next TP, k + 1, the travel time predictions to the downstream TPs, k + 2 through N (N = 14), are adjusted using the actual arrival information at the current TP, k + 1. This process is repeated until the bus arrives at the final destination, TP14. One should note that because s k is defined as the travel time from the origin to the current TP k, it always reflects the travel time that has already been recorded and is not related to the downstream destination. Therefore, its estimates for all downstream destinations are the same for a given k. In addition, if one lets L k,j = t k,j + s k for any j J and k < j, L k,j represents the travel time estimate, made when the bus reaches TP k, from the origin to destination j. The prediction error is defined as the difference between the predicted and the actual travel time for each pair of TPs. Figure 2 shows the distribution of the prediction error of the studied trip. Markers on each line represent the prediction errors for corresponding downstream destinations. One can observe that, for a given current location, prediction error tends to be larger at destinations that are farther away from the current bus location. Typically, with the bus proceeding

5 212 Transportation Research Record 1923 TABLE 2 Travel Time Prediction Output for Individual Trip (seconds) k\j * ** *: ˆt k, j **: Ŝk 1 8 Prediction Error (seconds) FIGURE 2 Error distributions for prediction performed at each TP.

6 Chen, Liu, and Xia 213 along the route (i.e., when k approaches the destination j ), the prediction error generally decreases for a given downstream destination. Using TP14 as an example, Figure 3 shows that the prediction error decreases as the bus proceeds and new arrival information becomes available and is applied in correcting the estimates. For the same trip, the predicted arrival times on each TP and actual arrival times are illustrated on a time space diagram in Figure 4a. Figure 4b is an enlarged local view (from TP9 to TP14) of Figure 4a. The figure shows that the predicted trajectories consistently approach the actual trajectory when the bus proceeds along the route. This results from the continuous adjustment of the predicted travel time. The power of the Kalman filter model is clearly demonstrated in this application. Performance Analysis The performance of the algorithm is tested on all bus trips for which AVL-APC data are available. Figure 5 shows the distribution of the prediction error, where TP1 is taken as the origin and each of the other TPs along the route is a destination. A comparative analysis is conducted to demonstrate the superior performance of the Kalman filter. Considering the ultimate goal of achieving a real-time application, two additional models, historical average and smoothing, that are relatively easy to implement are built and tested with the same set of data. The historical average approach relies on the analog between past and future traffic conditions. It categorizes the traffic condition by time of day for each segment and then estimates the future travel time on a segment by averaging historical travel times recorded for the same segment during the same time of day. The smoothing method also depends on historical data. However, it is designed to predict future travel time on a segment by calculating a weighted average between the estimated travel time, based on the latest trip information, and the historical travel time. The weight parameter used in the model is determined by minimizing the mean square error between the projected and actual travel times based on the historical data set. The root-mean-square error (RMSE) is computed as a performance measure to evaluate the average variation between the actual and predicted travel times by each model. The RMSE is defined as follows. RMSE = 1 N N i = 1 2 ( y yˆ ) ( 13) i i where N = number of test samples, y i = actual travel time of sample i, and ŷ i = predicted travel time of sample i. For each travel time prediction method, the RMSEs are computed for travel times from origin to each downstream TP. Their distributions are shown in Figure 6, from which one can conclude that the Kalman filter algorithm provides better predictions of travel times than the timetable as well as the historical average and the smoothing algorithms. The overall RMSEs aggregated from all segments are presented in Figure 7. Compared with the Kalman filter algorithm, the travel times indicated on the bus timetable show a larger deviation from the actual travel time measured, especially for destinations farther away from Prediction Error (seconds) FIGURE 3 Prediction error variation for travel time (from TP1 to TP14).

7 214 Transportation Research Record Travel Time (sec) Actual Predicted at TP1 Predicted at TP2 Predicted at TP3 Predicted at TP4 Predicted at TP5 Predicted at TP6 Predicted at TP7 Predicted at TP8 Predicted at TP9 Predicted at TP1 Predicted at TP11 Predicted at TP12 Predicted at TP13 (a) Travel Time (sec) Actual Predicted at TP1 Predicted at TP2 Predicted at TP3 Predicted at TP4 Predicted at TP5 Predicted at TP6 Predicted at TP7 Predicted at TP8 Predicted at TP9 Predicted at TP1 Predicted at TP11 Predicted at TP12 Predicted at TP13 (b) FIGURE 4 Time space diagram of one sample trip.

8 Chen, Liu, and Xia Prediction Error (sec) FIGURE 5 Prediction error distribution RMSE (sec) Historical Average Kalman Filter Smoothing Schedule FIGURE 6 Model performance comparison RMSE (sec) Historical Average Kalman Filter Smoothing Schedule FIGURE 7 Overall performance comparison.

9 216 Transportation Research Record 1923 the origin. Because the Kalman filter algorithm is initialized based on the timetable, the prediction shows a larger variation (from the actual travel time) for the first couple of segments, compared with the historical average and smoothing methods. However, with the bus proceeding along the route, the Kalman filter algorithm consistently generates better prediction of travel time than the others. The Kalman filter algorithm also requires minimum effort for data preprocessing when compared with other algorithms. It does not require historical travel time at each step of estimation if it is initialized based on the timetable. However, both historical average and smoothing methods would demand continuous reference to the historical travel time. Other methods for bus travel time prediction, such as that of Chen et al. (11), may involve significant modeling effort (e.g., neural network model development) as part of the algorithm. This certainly increases the difficulty for implementation of such models. Even with the drivers schedule recovery effort, the absolute prediction errors still tend to propagate when the bus runs over a longer distance because there are other stochastic factors (such as traffic condition) affecting the bus on time performance. Therefore, the relative prediction error is calculated to evaluate the performance of the prediction model, as indicated in Figure 8. It is observed that the relative prediction error does show a decreasing trend with the bus approaching the final destination. Implementation Issue The Kalman filter algorithm developed in the study does not require intensive calculation. It uses the time and location information at each stop, once they become available, to adjust the prediction dynamically. The computation time is minimal and can be ignored in real-time applications. With a simple initialization based on the bus timetable, the Kalman filter algorithm outperforms the historical average and smoothing models. Compared with these alternatives as well as other existing methods involving the Kalman filter such as that of Chen et al. (11), this algorithm is more straightforward and requires less preprocessing of data. Currently, the APC data are collected along the route and stored at the onboard computer before they are downloaded to the central database after the buses return to the garage. The real-time implementation of the algorithm will become feasible once the live communication mechanism is established to transfer data to the control center where the computation is expected to be carried out. Even though this algorithm does not require a particular trip length, it is shown that, for longer trips, the benefit of the algorithm would be more obvious that is, the relative prediction errors have a strong tendency of decreasing along the trip. CONCLUSIONS AVL-APC data contain rich information about various bus operating and service characteristics. In this study, a Kalman filter algorithm is developed to predict bus arrival times based on the time space information extracted from such data. Using the feature of dynamic prediction adjustment, the algorithm updates estimated arrival times at downstream TPs whenever new observation of travel time becomes available. When it is initialized based on the timetable, this algorithm is not computationally intensive. This is especially beneficial to its potential application in real-world transit systems. In addition, the impact of schedule recovery efforts by bus operators is also taken into consideration in the model. The AVL-APC data greatly facilitate the understanding of the schedule recovery effect as well as its relationship to the number of stops made and passenger on and off counts. The driver adjustment factor estimated from historical trip data is then used as a control input in the prediction model. The overall performance of the Kalman filter model is quite satisfactory; the decreasing trend of relative prediction error along the route demonstrates its dynamic optimization capability. It especially can bring more benefit for longer trips, which cannot be realized by other models such as historical average and smoothing algorithms. 6 4 Relative Error (%) FIGURE 8 Distribution of relative prediction error.

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