Development of Travel Time Prediction Models Using Statistical and GEP Techniques for Public Transportation Bus Routes in Amman-Jordan

2.1 Factors affecting bus travel time

The are many important factors increasing bus travel time, thus leading to a reduced number of passengers using buses. Countries around the world seek to reduce private vehicles in road networks by enhancing public transportation systems to encourage people to use it instead of their private vehicles. Due to the importance of bus travel time, many types of research have studied the factors affecting bus travel time.

2.1.1 Types of bus lanes

Buses can operate in different lane types (Exclusive or mixed with traffic). Tu et al. [1] assessed the effectiveness for three types of bus lane including: (the roadside exclusive bus lane, bus priority lane, and mixed traffic bus lane) at an urban street under a high volume of traffic and a large number of buses in Nagaoka-Japan. The data was collected on bus route segments no.351 and no.36 leading to Nagaoka Station by using ten cameras; four cameras were installed at four intersections to observe traffic flow and turning movements. The other cameras were placed at bus stops to record video footage at each stop. All cameras started at the same time to ensure the accuracy of data used to estimate bus travel time during morning peak hours from 7:30 AM to 9:30 AM. All passengers in the segment were counted as 19.4 passengers per bus and 1.25 passengers per non-bus (observed vehicles were classified as bus and non-bus vehicles) it was found that the travel time for all passengers (bus and non-bus passenger) in priority lane type minimized the passenger travel time by 1.2 sec along the 500 meters segment versus ordinary type lane (mixed with traffic). But the exclusive lane raised all passenger travel time along the 500 meters segment by 1.3 sec because of the elasticity to choose lane in the priority bus lane.

Wei and Chong [2], studied the performance comparison between an old bus that was operated with mixed traffic lane and the first exclusive bus lane that was implemented in Kunming-China in 1999. The result showed a 58% increase in average bus speed from 9.6 to 15.2 km/hr, after two years of implementation of bus lanes.

2.1.2 The delay at intersections with traffic signs and signals

Generally, the intersection with a traffic sign forms time of a delay for all vehicles more than an intersection with stop signs because traffic signs are used on high traffic volume when stop signs fail.

Figliozzi et al. [3] used 104010 data points for route no.15 in Portland to anticipate the two regression models for bus travel time with R-square 0.47. The results summarized the average bus delay due to stop signs is 13s/stop. Also, the signalized intersection adds an average of 4.5,20,38s to total bus travel time for passing through, right turn, and left-turn movements in series. Mazloumi et al. [4] explored the public transportation travel time variability for buses operating on a 27 km circumferential route in the eastern suburbs of Melbourne- Australia by using GPS records for each bus in the route. Their model with 0.70 adjusted R square and 0.95 confidence level showed that 22% is the increase of total travel time per each signalized intersection. Evans and Skiles [5] observed the average delay formed by traffic signs is 10-15 percent of the total bus trim time bus. On the other hand, their results in LOS ANGELES showed bus priority signals decreased bus passenger delay by 70-76% for a segment that has two signalized intersections.

2.1.3 Bus stops and dwelling time

Busses stops were defined as an area located along the bus route to the boarding and alighting passengers. It contains one or more loading areas; bus stops have two main types: On-Street Bus stops and Off-line bus stops that supply greater vehicle capacity (TCRP 1996) [6]. There are three types of On-Street Bus Stop Locations, as shown in Figure 1:

• Near-side stop: bus stop located directly before an intersection.
• Far-side stop: bus stop located immediately after an intersection.
• Mid-block stop: bus stop located within the block.

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Figure 1. On street bus stop locations

*Source: slideplayer.com/slide/12569638/

Period of time that starts from bus closes doors to depart a stop (clearance time) effect on total travel time on off-line stops more than On-Street Bus stops, because the requirement for a suitable gap in traffic to allow the bus to re-enter the traffic stream and accelerate (re-entry delay) is increased in high traffic volume (TCRP 1996). According to TCRP Report 19 (Fitzpatrick et al. 1996), bus dwell time and time lost serving stop will be affected by the layout of the bus stop. Fitzpatrick and Nowlin [7] discussed the impacts of bus stop location on the suburban arterial road and the relationship between bus travel time, speed, and traffic volume for different designs of bus stops and locations. There are different dwell time and time lost serving stop for each type because the number of stops varied by each type. For instance, at far-side stop bus maybe stop twice, at a red signal and on its stop. But at the near side bus perhaps to make one stop: bus can load/unload passengers while waiting for the signal to change to green light. Wu and Murray [8] in Columbus, Ohio applied a proposed model for routes 6 and 7; which compose main services for this region to eliminate the redundant service stops, and to enhance the transit service quality by proposing a multiple-route, maximal covering/shortest-path model (MRMCSP) considering access coverage in the selection of stops. Their model evaluated three components of bus stops delay times by considering closing-opening door, bus speed, and acceleration-deceleration rate. The result showed 50% of stops area redundant because only 21 out of 49 stops on route 6 and 33 out of 74 stops on route 7 area desired for coverage current demand.

2.2 Bus travel time and delay estimation and prediction models

Many types of the research proposed models in varied ways models to estimate travel time and time losses that occur due to dwelling time, intersections, traffic volume … etc. Chen et al. [9] calculated the delay at bus stops and used the microscopic analysis from three bus routes 105, 651, and 919 in Beijing-China, relying on queuing, impeding, and waiting. Their results concluded that there are significant impacts of bus load factors on bus dwelling time that are expressed as binomial and exponential equations with R square 0.83 and 0.86 in series for curbside stops.

Kumar et al. [10] established a prediction model to estimate the bus's travel time. They examined the special and temporal variation to predict the travel time. They developed a prediction model by using the flow macroscopic traffic equation (speed, density, flow.) This model is created depending on the field data. The presented paper relies on the time-space discretizing the partial differential equation (PDE) using the numerical scheme and then the predicted output updated by an Ensemble Kalman Filter (EnKF). This approach is considered both special and temporal evaluations to predict accurate travel time information which aims to enhance a public transport system. In this study, they used GPS (Global Positioning System) technology to collect real-time data. A Metropolitan Transport Corporation (MTC) buses in the city of Chennai (India) were applied to collect the GPS data. They inspected two (MTC) bus routes (19B,5C) with different geometric characteristics, volume level, and land use characteristics. The 19-bus route is 30 km length with 20 bus stops and 13 intersections. The average headway and the average travel time were 30 min and 90 min, respectively. On the other hand, the 5C bus route has a 15 km length with 10 bus stops and 14 signalized intersections. The average headway and average travel time were 45 min and 70 min respectively. The surveying period was from 4 AM to 10 PM, the GPS data which includes ID and GPS unit were collected every (5s). These data were arranged and saved in the Structured Query Languages (SQL) database. Then the distance between two entries was computed using the Eq. (1):

Distance $(\mathrm{d})=2 \mathrm{r}$ arcsine $\left(\sqrt{\text { haversine }\left(\emptyset_2-\emptyset_1\right)+\cos \emptyset_2 \cos \emptyset_1 \text { haversine }\left(\lambda_2-\lambda_1\right)}\right.$          (1)

where, r is the radius of the earth (6378.1 km), $\lambda_1, \lambda_2$ indicate the longitude of point 1 and point 2 in series, $\emptyset_1, \emptyset_2$ indicate the latitude of point 1 and point 2 respectively.

A linear interpolation technique (MATLAB) was performed to calculate the travel time after dividing the obtained routes into a subsection of (100m) length. Considering the delay due to signalized intersection, bus stop, and traffic congestion. To evaluate the performance of the time-space discretization model they compared with the space discretization and time discretization method. They found that the result obtained from the modified discretization approach is more realistic and closely matching to the real data than the predicted values obtained from time discretization and space discretization methods. In addition to the lower value of errors resulting from (Spatio-temporal) evaluation in comparing with temporal evaluation alone and spatial evaluation alone for each subsection. The data obtained from the EnKF numerical scheme outperform and achieves more accurate results than the ANN, historical average, and regression analysis approaches.

Chien et al. [11] developed a prediction model for estimating bus arrival time in an urban transit network. They employed the Artificial Neural Network (ANN) to predict the arrival time for buses. This study was conducted in New Jersey route 39 with 4.4-mile length and include (30) intersections, (26) of them are signalized intersections while the remaining are stops per directions.

Many data are utilized training the ANN (traffic volumes, delay, and speed). In this study, two kinds of ANN were applied to establish the prediction model. The first network is Link-Based artificial neural network. This network is deliberated to estimate the arrival time for buses at the downstream stations by collecting the travel time of buses across two-stop stations. The second network is Stop-Based, which is dissimilar to a Link-Based. This network is initiated by training the means and standard deviation for speed, delay, and volumes between two stops.

The real-time data were collected using the advanced public transportation system (APTS) and advanced traveler information system (ATIS). These data were introduced to the ANN to develop an arrival time prediction model.

For evaluation purposes, they applied for the CORSIM microscopic simulation program that deals with overtaking, merging, maneuvering, and commuter arrival allocation to accomplish reasonable results of arrival time by examining the real-time data in the transit system.

The ANN prediction model provides a precise dynamically estimation model for the bus arrival time, and the ANN model is perfectly applicable for single and multiple stop predictions. For multiple intersections, the Stop-Based is desired, while in stops with few intersections the Link-Based is preferred.

Chien and Kuchipudi [12] studied the role of real-time and historical data to estimate bus travel time. This study has been situated on the interchange between New York State Thruway (NYST) and New Jersey Garden State Parkway (GSP). The route contains (8) on-ramps and (5) off-ramps with a 10.57-mile length. They employed two types of data in the travel time predictive model. The first data is path-based, and the second data is time-based.

Since the travel time data is acquired using traffic control devices such as (loop and microwave detectors). And these techniques are not practical to put on different parts on the roadway due to these problems; wireless techniques have been developed to obtained travel time.

In this study the data collected for the selected route were processed to enhance the quality using professional computer programmers. These data contain (travel time, speed, standard deviation, and traffic volume) the details of these data were performed to promote the travel time prediction model. The result of this process gave: all vehicle data, time-based data, and vehicle travel time.

To continuously update the data, they used the Kalman filtering algorithm when the next inspection would be obtainable.

This technique includes two groups: estimation and execution of estimator. The result of this study shows that the historic path-based data are more applicable for peak hours on travel time prediction than the link-based data.

Moridpour et al. [13] developed a prediction model of the travel time for buses by employing the Least Squares-Support Vector Machine (LS-SVM) method. It depends on the Linear regression technique instead of quadratic programming. For precision purposes, they used the Genetic Algorithm (GA) to obtain the optimal parameters for the LS-SVM approach. SVM technique is based on estimating the travel time, either large or small database. The result of this approach highly depends on the quality of the input parameter because they focused on the right sitting of the parameter using the GA. In this study, they observed the data for the bus route in Melbourne, Australia. The length of the route is 8 km it was divided into four sections (same length). The period for the study was six months and the travel time data set was taken for 1800 weekdays. They employed 80% of the dataset for training which includes 6761 observations and the remaining data used for testing purposes. They divided the period into five main intervals: morning peak, mid-day off-peak, afternoon peak, late night-off peak, and day off-peak). This study has seven independent variables were occupied (month of observing the travel time; day of the week when travel time was observed; time of day in different 15-minute time intervals; delay at the upstream timing point; the degree of saturation in the previous 15-minutes intervals for the intersections within the section; traffic counts in the previous 15-minutes intervals for the intersections within the section; and scheduled travel time). The result of this study shows the efficiency of the LS-SVM approach in predicting travel time and the advantages of GA to find the optimal setting with lower errors to estimate the travel time. Eventually, these results were compared with the ANN model results. However, the proposed model shows more accurate results with lower error and less computational complexity for the data.

Farhan et al. [14] developed a travel time prediction model that gives real-time information for the passenger about the bus arrival time. They employed the AVL (Automatic Vehicle Location) and APC (Automatic Passenger Counter) techniques to observe the empirical data used in the travel time model and to apply the proactive control strategies (holding and expressing) in the transit time and compare the performance of the various model. Each model contains simple independent variables to establish the model. This study was performed on bus route number 5 in downtown Toronto in May 2001. Bus routes have a length of (6.5km) and 27 bus stops in each direction. It also has a 6-time point stop. The bus route involves 21 signalized intersections, the headway for the bus at the peak period ranged from 12 min to 30 min, and the total duration of the study was five weekdays. Historical average, regression analysis, and Artificial Neural Network (ANN) were used to develop the travel time prediction models. The Historical average model depends on the historical average travel time between time-point along the bus route. This model is appropriate for the travel time propagation to passengers, and it is not incorporate independent variables, thus this model cannot be used to evaluate the control strategies. While the regression model includes various independent variables (distance, average bus speed, dwell time, and intersection delay) which are also used in the neural network model. In the ANN model, the data consists of three different groups: for training, cross-validation, and testing. 75% of data for training, 10% for validation, and the remaining data were used to evaluate the performance of the model. The result of this study shows that the ANN model outperforms in the accuracy and lowest error which indicates a high performance compared with other models.

Qi et al. [15] predicted a model bus inter-stop travel time considering the signalized intersection influence by using data collected from buses at rout 63 Harbin-China. It contained 21 bus stops, 10.1-km operating distance and 29 signalized intersections with a maximum number of intersections between two neighboring stops is 3. The variables inputs were stopping distance, historical inter-stop travel times, intersections number and traffic volumes, and signal timing at the intersection. All these factors were used to develop a novel prediction model for inter-stop bus travel time. The developed model was tested by the Lagrange Multiplier (LM) test to examine the autocorrelation in the model residuals. The model resulted that the signalized intersection between two neighboring stops has a noteworthy effect on inter-stop bus travel time, bus travel time also has exceptionally pertinent to the various four factors referenced previously.

Yu et al. [16] developed a prediction model for bus travel time using Random Forests Based on Near Neighbors (RFFN) by using (14182 and 7623 data points for bus routes 232 and 249, respectively) that has located in Shenyang-China. Bus route 232 has an operating distance of about 10.7 Kilometers with 19 bus stops, a running speed of 15.6 km/hr, and a total travel time is 60 minutes with a bus frequency of 2.5 minutes. Bus route 249 includes 27 bus stops with a total length of about 15 kilometers, running speed of 14.9 km/hr, and 7 minutes of bus frequency. Each bus route reaches out from the suburban to the Center of Shenyang without timetables. The factors used in the developed model were only the bus running data without focusing on other factors such as (waiting passengers’ number and weather conditions). It was compared with four models: linear regression (LR), k-nearest neighbors (KNN), support vector machine (SVM), and classic random forest (RF). The result of the comparison indicated that the RFFN has the highest accuracy with mean absolute error (MAE) 13.65,13.77 and route mean squared error (RMSE) 26.37,29.01 for routes 232 and 249, respectively. Furthermore, it can handily reach out to evaluate bus arrival time at each bus stop dependent on current traffic conditions. The RFFN has a superior presentation inexactness, however, not computation time.

4.1 Statistical models

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Figure 2. Plot of predicted operating bus speed against measured for model 1

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Figure 3. Plot of predicted operating bus speed against measured for model 2

Multiple regression analysis was improved using SPSS and Microsoft excel programs using 43 data points from all routes, to improve two models. The first model has developed a relationship between operating bus speed (V_o) and the other independent variables affecting bus speed with R square 0.89. The second model has predicted the relation between bus operating speed, private vehicle speed, and the number of stops with R square 0.96. Twenty-one data points have been used in validation (testing) for two models. Figure 2 and Figure 3 show plots for predicted versus measured values for operating bus speed for models 1 and 2, respectively.

The summary for the best result for multiple regression models are:

Model 1: correlation between bus operating speed time and other independent variables as shown in Eq. (7).

$\mathrm{V}_{\mathrm{o}}=44+0.56 * \mathrm{~L}-0.62 * \mathrm{~S}-1.2 * \mathrm{I}-0.02 * \mathrm{~V}$          (7)

where,

V_o: Bus operating speed (km/hr).

L: Route length (km).

S: Number of stops.

I: Number of intersections

V: Average traffic volume (vehicle/hr/lane).

Table 4 shows the regression statistics for model 1.

The parameter estimates for operating bus speed; the p-value showed the factors are statistically significant, the negative sign of coefficients explains that the bus operating speed will decrease when the stops, intersections, and traffic volume increased, on the other side the positive sign shows that the operating speed increase as the length of the route increase, as illustrate in Table 5.

Although p-values are provided to assess statistical significance, we recognize the value of including confidence intervals in these estimates. Confidence intervals provide a more accurate understanding of the precision and potential range of the regression coefficients. In future analyses, we aim to include confidence intervals to provide readers with a more complete assessment of the strength and practical relevance of our results.

Model 2: predicting bus operating speed using private vehicle speed at the same route as shown in Eq. (8).

$\mathrm{V_o}=3.76+0.78 \mathrm{V_p}-0.211 \text{S}$              (8)

where,

V_o: Bus operating speed (km/hr).

V_p: Privet vehicle speed (km/hr).

S: Number of bus stops.

Table 6 shows regression statistics for model 2.

Table 7 illustrates the parameter estimates for operating bus speed, P-value showed the factors are statistically significant. Including confidence intervals alongside p-values for the regression coefficients would offer a more precise understanding of the estimates' reliability.

The negative coefficients related with stops, intersections, route length, bus operating speed, and traffic volume provide identified insights with an essential implication for transportation planning:

Stops and Intersections: The unexpected negative relationship between stops, intersections, and shorter bus travel times prompts a reassessment of conventional assumptions.

Route Length and Bus Operating Speed: Negative coefficients for route length and bus operating speed emphasizes the potential gains in travel time efficiency with shorter, more direct routes and ideal bus speeds.

Traffic Volume: The negative coefficient associated with traffic volume challenges assumptions and suggested a counterintuitive relationship between higher traffic density and shorter bus travel times.

In conclusion, these findings advocate for a reevaluation of traditional paradigms in transportation planning. The study underscores the importance of adaptive strategies, grounded in these nuanced insights, to optimize public transportation systems for heightened operational efficiency.

ANOVA Test: Analysis Of Variance (ANOVA), it has been performed at a 95% confidence level to decide if there is a significant relationship between the dependent variable and independent variables. Calculations were made in case there is a committed relationship between the dependent variable and independent variables. Table 8 shows a significant relationship for models 1 and 2.

Our ANOVA results reveal a significant relationship between the variables being studied, but it is crucial to consider the assumptions that underlie this analysis. One key assumption is the normality of residuals. After a closer examination, we observed a slight deviation from normality in the distribution of residuals. Nonetheless, it is worth noting that ANOVA can withstand deviations from normality, especially with larger sample sizes.

Table 4. Regression statistics model 1

Table 5. Parameter estimates for model 1

Table 6. Regression statistics model 2

Table 7. Parameter estimates for model 2

A validation approach must be implemented to check the model reality. In this study, two methods of validation have been applied on different 21 trips selected randomly from four routes.

Figures 4 and 5 show a small difference between actual and predicted bus operation speed for models 1 and 2, respectively.

The R2 for validation data has been determined for each model. The result showed a high value of R2 with 0.915 and 0.97 for models 1 and 2 in series, as shown in Figures 6 and 7 respectively.

Also, Thiel's inequality coefficient (U) was employed to validate the model. This parameter provides a measure of how well-estimated values compare to corresponding observed values. The U value varies from 0 to1. The closer the value of U is to zero, the better the forecast method and best fit between observed and predicted values. A value of 1 means the worst forecast and bad fit between observed and predicted values Toledo and Koutsopoulos [20].

$U=\frac{\sqrt{\frac{1}{N} * \sum_{n=1}^N\left(V_{\text {o obs }}-V_{\text {o est }}\right)^2}}{\sqrt{\frac{1}{N} \sum_{n=1}^N\left(V_{\text {o obs }}\right)^2}+\sqrt{\frac{1}{N} \sum_{n=1}^N\left(V_{\text {o est }}\right)^2}}$            (9)

The results of U-coefficient values and average percentage error for two models shown in Table 9 describe a good, predicted model.

As mentioned previously, the U-coefficient value has varied from 0 to 1. The large value of U reflects the model's poor ability to predict accurately by Naghawi et al. [21]. For a good predicting model, the U-coefficient should be less than 0.3. In this study, the U value for both training and testing data records was very small, so the developed model is accurate and efficient.

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Figure 4. Actual bus operating speed vs. predicted for model 1

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Figure 5. Actual bus operating speed vs. predicted for model 2

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Figure 6. Plot of predicted operating bus speed against measured for validation data model 1

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Figure 7. Plot of predicted operating bus speed against measured for validation data model 2

Table 9. Thiel's inequality coefficient (U) and average percentage error % result summary

We selected Gene Expression Programming (GEP) due to its adaptability, ability to construct features automatically, scalability, interpretability, and resilience. In contrast to alternative approaches, GEP can manage intricate connections, adjust to data traits, scale effectively, generate models that are easy to understand, and mitigate overfitting. These benefits align well with our research objectives.

Two models have been predicted using (GeneXpro5.0) software that is based on the Darwinian principle to solve problems. Sixty-four data points were used to predict correlations between operating bus speed and other independent variables for model 1, also for model 2 to predict and operate bus speed depending on private vehicle speed and the number of stops. There are two major stages in the GEP software process: training and validation (testing), (43) 70% of data records were randomly selected from input data, and employed to train the model and learn the relationship between input variables and output variables. In the testing phase, the remaining 21 independent data records which formed 30% of input data were used to test the developed GEP model.

The model development is based on a number of genes that have non-linear behaviors but their combination in a linar form shapes the final structure of the goal model. This GEP model is composed of 30 chromosomes, 3 genes, and 2 genes for model 1 and model 2, respectively. as mentioned previously. The head and intermediate nodes represent mathematical function while the tail nodes represent the independent variables or constant values, in this model the head and tail sizes were 8 and 9, respectively. Table 10 shows Gene Expression parameter estimates for model l and model 2, respectively.

The models have been exhibited in Figures 8 and 9 that show the regression plot for predicted bus operating speed versus measured for training and testing data for models 1 and 2, respectively.

Our research provides valuable insights into the dynamics of bus travel time in our specific urban area. It is important to consider how well our models can be applied to different urban settings and bus systems. While our models provide a basic for understanding the factors influencing travel times, the applicability of our models may vary depending on factors such as infrastructure, population density, traffic patterns, and public transportation policies in different cities or regions. Future research could examine the transferability of the model to different urban environments, perhaps through cross-validation with data from other locations or external validation.

Table 10. Gene Expression Parameter estimates for models 1 and 2

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Figure 8. The regression plot of predicted bus operating speed against measured for testing and training data for gene expression model 1

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