The results in the previous section are from a path (trip) with three links. The sensitivity of the re- sults to trip distances and road types is also of interest. Statistical tests, unimodality and spatiotem- poral correlation, were conducted to examine the underlying characteristics of link TTDs. The uni- modality of a distribution was tested using the Hartigan dip test (Hartigan and Hartigan, 1985). A high dip significance value provides support that the distribution is unimodal.
Trip distance
Figure 7-13 shows the KL performance contour maps of distribution estimation as a function of dis- tance from the first stop (stop order) in different time 30 minutes time intervals (e.g. 7:00 = 7:00- 7:30 am). Four models are presented: convolution, GMC_GMMS_NC (Hofleitner,Herring and Bayen, 2012), GMC_GMMS_CLN and MGF_CLN (Srinivasan et al., 2014). The MC_GMMS model is not shown as it was not feasible to estimate the distribution for trips with more than 9 links due to computational reasons which underlies one of the major limitations.
In general, the GMC models provide more accurate estimations than the convolution and MGF models for trips in peak hours, especially for trips between stops 1 to 3 on Motorway links. The reason for the lower performance of the convolution and MGF models are mainly due to the highly significant link correlations and multimodal distributions in peak hours on motorways. How- ever, the MGF model performs well and relatively better than the GMC models for trips in off-peak hours (9:00-14:00), when the link and trip running times have unimodal distributions. The comple- mentary performance between the GMC and MGF models points to the potential of developing a hybrid approach that can make full use of their advantages by balancing accuracy and computation burden. The GMC_GMMS_CLN model is more accurate and robust than the GMC_GMMS_NC model by further taking into consideration the spatial correlations along a Markov path. The superi- or performance of the former occurs for trips from stop 1 to stops 11 and 12 in peak hours. Stop 11 is a major spot near the CBD area where bus bunching happens frequently in peak hours. The corre- lation between its adjacent links was found to be significant. This could be explained by the fact that a bus having to stop would spend more time on both the adjacent upstream and downstream links than a bus driving through.
Table 7-5 summarizes the performance of the various models. The tested cases include trips on routes 555 and 60 with different distances from 7:00 to18:00 in 30 minutes intervals on week- days. The results verify that the proposed GMC model provides more accurate and robust TTD es- timation than the alternatives and it can fit the empirical distributions accurately for 88% of the test- ed cases.
Figure 7-13: KL performance metric as a function of trip distance and time of day (R555, inbound, weekdays). (a) Convolution, (b) GMC_GMMS_NC, (c) GMC_GMMS_CLN, and (d) MGF_CLN Table 7-5: Performance summary
Method Average KL Max KL Percentage cases passed KS test Convolution 0.083 0.190 61%
GMC_GMMS_NC 0.027 0.126 82% GMC_GMMS_CLN 0.014 0.089 88%
MGF_CLN 0.044 0.156 69%
Road types
Table 7-6 compares the distribution estimations for trips on routes 555 and 60, on different types of roads on weekdays between 8:00-8:30 am. The facilities include Motorway, Busway, CBD, Arterial, and Residential roads. For Motorway, CBD, and Arterial trips, the link travel times have multimod- al distributions (unimodal sig < 0.05) and are highly correlated (ST-ACF).
The convolution and MGF models fail to approximate the ground-truth empirical distribu- tions. The proposed GMC model M3 incorporating link correlations along a Markov path performs relatively better than M2 assuming no correlation. For Busway trips, the travel times have an insig- nificant correlation and a unimodal distribution. In this case, the four models all approximate well the empirical distribution, with the MGF model M4 performing the best. For residential trips, the travel times have a significant correlation and a unimodal distribution. All models passed the KS test except the convolution model. If balancing simplicity and accuracy important, it is reasonable to use the convolution method for Busway trips. The MGF model can also be applied when the trip travel times have a unimodal distribution.
Table 7-6: Comparison of estimation performance for trips on different types of roads
Road types Links* Distance (km) Unimodal sig1 ST-ACF (conf.)2 KL_M1 (pass)3 KL_M2 (pass) 3 KL_M3 (pass) 3 KL_M4 (pass) 3 Motorway 1_3 14.4 0.00 0.25(.08) 0.140(0) 0.016(1) 0.011(1) 0.081(0) Busway 3_10 14.4 0.82 0.03(.04) 0.018(1) 0.021(1) 0.014(1) 0.003(1) CBD 6_9 2.57 0.01 0.21(.04) 0.273(0) 0.033(1) 0.023(1) 0.238(0) Arterial 9_11 1.25 0.00 0.11(.04) 0.192(0) 0.041(1) 0.032(1) 0.139(0) Residential 1_5 2.54 0.38 0.13(.06) 0.096(0) 0.014(1) 0.005(1) 0.011(1)
Note: * 1_3 indicates trips from stop 1 to stop 3.
1. sig = significance value. 2. conf. = 95% confidence boundary; ST-ACF = spatiotemporal autocorrelation function. 3. M1-M4 stands for Convolution, GMC_GMMS_NC, GMC_GMMS_CLN and MGF_CLN methods.
7.6
Discussions and applications
Figure 7-14 shows the implementation of the proposed GMC model for transit trip TTDs estimation. The data input is solely from the AVL system. For a trip between OD pair (i, j) at 7:00-7:30, the source data is partitioned according to trip attributes and then cleaned to exclude abnormal observa- tions. The GMMS algorithm clusters the link running times and outputs the distributions condition- al on current link states. The MNL TPM model estimates the transition probabilities with inputs of RCI on the current link, CI on the previous link at current time, and CI on the current link at the preceding time interval.
The Markov chain process constructs the Markov paths and calculates the probability of each path using the estimated transition probabilities. The Markov path distribution is approximated using the MGF algorithm with consideration of link correlations along the Markov paths. The co- variance matrix conditional on the states between any two links can be derived from empirical data or estimated from a model. The trip running time distributions are estimated as the sum of Markov path distributions weighted by Markov path probabilities. The trip dwell time distribution is esti- mated as the convolution of stop dwell time distributions fitted using a non-parametric kernel model. Finally, the trip TTD is derived as the convolution of the trip running time and trip dwell time dis- tributions.
Figure 7-14: The implementation of the proposed GMC structure for transit application As the proposed GMC structure is modular, it can be easily adapted for different applica- tions. The upper part of the implementation structure can be readily used to estimate TTDs for car trips as well since all the inputs required are link based. The GMC structure can also be extended to provide real-time predictions of the bus arrival time distribution at downstream stops using a similar idea as in Noroozi and Hellinga (2014). For example, the state probabilities on the next link can be estimated using the MNL model given the current link state. The corresponding TTDs conditional on link states can be derived from historical data. Then the TTD on the next link can be estimated as the sum of these distributions weighted by the estimated probabilities.
To examine the effectiveness of the above mentioned approach in predicting the down- stream link TTD, an alternative model with the fixed transition probabilities and a naïve historical data based model are developed for comparison. The fixed transition probabilities are calculated using the count-based approach for two successive links in 30 minutes under different cases (e.g. a case is weekday inbound from 7:00-7:30 am). The naïve model predicts the link TTD as the empiri- cal distributions in 30 minutes under different cases. The lower and upper bounds of an interval prediction with theoretical coverage100(1−α) whereα∈
( )
0,1 are calculated as the100( / 2)α and 100(1−α/ 2)percentiles of the distribution, respectively (Woodard et al., 2015).To measure the accuracy of the deterministic (mean) predictions, the mean absolute error (MAE) and the mean absolute percentage error (MAPE) are used. To measure the accuracy of in- terval predictions, the empirical coverage percentage and the average width of the interval are re- ported. The empirical coverage measures the percentage of test trips for which the observed travel time is inside the predicted interval with a specific theoretical coverage. If two methods can both predict the variability correctly, the average interval width is used to indicate which one is better (Woodard et al., 2015).
1 1 1 MAE 1 MAPE= 100% n act pred i i i act pred n i i act i i T T n T T n T = = = − − ×
∑
∑
(7.21) where, act iT = actual travel time for observation i;
pred i
T = predicted travel time for observation i; n= number of observations.
Table 7-7 provides the summary of deterministic and interval predictions. The tested data are link running times for weekday inbound trips from 7:00-18:00 over a 6 months period. General- ly, the models with the predicted and fixed TPMs perform better than the naïve historical data based model since they can both reflect the influence from the upstream link. The model with the predict- ed TPMs provides a relatively more accurate prediction performance than the one with the fixed TPMs in terms of both deterministic and interval predictions, since it further incorporates the real- time information from the preceding time intervals than can better adjust the predicted state proba- bilities on the successive link.
Table 7-7: Summary of deterministic and interval predictions performance
Accuracy measures Model with predicted TPMs
Model with fixed TPMs
Naïve historical mod- el
Mean absolute error* 4.9 5.4 8.5
Mean percentage error 6.9% 7.8% 13.2% Empirical coverage# 92.9% 93.2% 95.0%
Average interval width# 25.5 28.7 40.9 Note: The bold values indicate the best prediction performance;
* the unit is seconds per kilometres;
# the theoretical coverage is 95% (the interval is between 2.5 percentile and 97.5 percentile of the distribution).
Figure 7-15 shows the mean and 95% confidence interval prediction results from the models with the predicted and fixed TPMs. The model with fixed TPMs tends to give relatively more con- stant mean predictions for all trips in the same time of a day which is counterintuitive in reality. The model with the predicted TPMs can reflect the various trip running times by taking advantage of the real-time information from the preceding time intervals, but still rather limited. The model with the predicted TPMs can provide relatively narrower interval predictions than the one with the fixed TPMs, especially in terms of the lower confidence boundary.
Figure 7-15: Predictions of means and intervals on a sample Motorway link for weekdays inbound trips over different times of day across six months period. [Model 1 uses the predicted probability; Model 2 uses the fixed probability; 95% conf. indicates 95% confidence intervals]
7.7
Summary
Travel time distribution along paths in transportation networks can provide comprehensive infor- mation for planning, operations monitoring and control, as well as travel planning. Previous studies focus on directly fitting the link or route travel times. However, the applicability of these methods may be limited due to the small number of direct observations at the Origin-Destination level. Fur- thermore, many of the methods assume link independence or correlated unimodal link distributions. The research proposes a GMC approach to estimate trip TTDs by aggregating link TTDs that takes into account the spatial-temporal correlations among link travel times. The proposed GMC ap- proach captures the correlations among link travel times conditional on the underlying traffic states. The method is applicable under general conditions, as the link distributions are derived conditional on the states of the current link and the transition probabilities are estimated as a function of ex- planatory covariates using logit models.
The proposed approach has been demonstrated and validated in a transit case-study using AVL data. The results confirm that the GMC approach provides an effective and efficient way to estimate TTDs compared to other methods. The performance of the GMC method is promising spe- cially when link correlations conditional on states and multimodal distributions exist. The method is also computationally more efficient than other methods proposed in the literature.
The use of transition probabilities as a function of explanatory variables makes the model general (and requires less data for calibration purposes than other methods). However, there is a small cost in estimation accuracy, as the estimated transition probabilities and the fitting of para- metric link distribution models (required by the MGF algorithm), inevitably introduce errors that influence its estimation performance. The sensitivity analysis on trip distances and road types high- light the complementary performance of the GMC and MGF methods. Since dwell times are diffi- cult to model at stop-level, being mainly determined by demand, the research has undertaken pre- liminary analysis on demand modelling and proposed an interactive multiple models (IMM) based pattern hybrid approach to predict short-term passenger demand at a route-level (Ma, Xing, et al., 2014).