Network-Level Travel Time Prediction Considering The Effects of Weather and Seasonality

N E T W O R K - L E V E L T R A V E L T I M E P R E D I C T I O N C O N S I D E R I N G T H E E F F E C T S O F W E A T H E R A N D S E A S O N A L I T Y Y ufei Ai Department of Civil & En vironmental Engineering Univ ersity of Houston yufeiai91@gmail.com Y ao Y u, Ph.D. Meta Inc. yuyao08@gmail.com W enjing Pu, Ph.D. Boston Consulting Group Pu.Wenjing@bcg.com Lu Gao, Ph.D. Department of Civil & En vironmental Engineering Univ ersity of Houston lgao5@central.uh.edu Y ihao Ren Department of T ransportation, Logistics and Finance North Dakota State Uni versity yihao.ren@ndsu.edu A B S T R AC T Accurately predicting trav el time information can be helpful for trav elers. This study proposes a framew ork for predicting network-le vel trav el time index (TTI) using machine learning models. A case study was performed on more than 50,000 TTI data collected from the W ashington DC area ov er 6 years. The proposed approach is also able to identify the ef fects of weather and seasonality . The performances of the machine learning models were assessed and compared with each other . It was shown that the ridge regression model outperformed the other models in both short-term and long-term predictions. K eywords T rav el T ime Index, Machine Learning, Seasonality , T ravel T ime Reliability , Network Performance 1 Introduction Researches on predicting travel time can be di vided into three categories by the time scale of prediction: long-term, short- term, and real-time tra vel time prediction. The generally adopted methods can be categorized into parametric (linear regression models and time series analysis) and nonparametric approaches (neural networks and pattern searching) [ 1 ]. 1.1 Real-Time Prediction Real-time trav el prediction is aimed at pro viding the most up-to-date traf fic information to the driv ers with the input of GPS information which can be retriev ed periodically to the server . W isitpongphan et al. [2] conducted a study based on the GPS data of 297 probing v ehicles driving on a 22km-long road section. The study used artificial neural network (ANN) model with multi-layer feed forw ard algorithm. Gi ven the time and v ehicles’ coordinate, speed, and heading, their model is able to predict the trav el time on any gi ven road in real-time. The predictions during non-rush hour periods are very close to the actual results whereas during rush hours, it still remains unstable due to the traf fic congestion. Moreover , significant effects of different weekday and time of the day were observed in the data sets. Gurmu and Fan [3] dev eloped a multilayer perceptron (MLP) artificial neural network (ANN) model to predict bus trav el time at a given do wnstream bus stop. They used arri val and departure times at each bus stop of the selected buses as input and obtain their best accuracy with the error of less than 10%. The authors pointed out that ANN models show advantages on its good robustness and capacity of predicting complicated models while it contains some unknown, highly nonlinear relationship, or noisy information in the data sets. Howe ver , the data size has to be lar ge enough to ensure the accuracy which could be an obstacle for data acquisition in practical. 1.2 Short-T erm Prediction A considerable number of researches has been done by many researchers on the subject of short-term travel time forecasting. In these researches trav el time at a gi ven period of time in the future is predicted based on the historical and current traffic data. The most common used methodologies include: (1) linear regression models [ 4 , 5 ]; (2) auto-regressi ve integrated moving av erage (ARIMA) models [ 6 , 7 ]; (3) artificial neuron networks (ANN) models [ 8 , 9 ]. Rice and V an Zwet [5] utilized a simple linear regression model with time-v arying coef ficients. The model was proposed in [ 10 ] to perform trav el time predictions on a given section of free w ay . T rav el time was predicted for a certain time lag ahead departure; gi ven the historical a verage tra vel time data in all the periods in pre vious 34 weekdays, and the instantaneous trav el time data obtained from multiple types of measurement de vices, such as single-loop detector or probe vehicles, the model w as tested to be v alid for the time lag ranging from 0 to 60 minutes. W u et al. [11] applied support vector progression model for predicting travel time for high ways near T aipai and analyzing the daily and weekly patterns. The accuracy measures in terms of relative mean errors (RME) and root-mean-squared errors (RMSE) were sho wn to be within 5 to 8%. The satisfying results were believ ed to be credited with SVR’ s great generalization ability and guaranteed global minima of the results whereas only local minima can be located with neural netw orks. Oda [7] predicted the trav el time with ARIMA model, in which the tra vel time observ ations were arranged in stationary time series and decomposed into three termsan auto-regressi ve term (AR), a moving-a verage term (MA), and an integrating term that accounts for the data noises. The prediction was completed through minimizing the noise term. The best accuracy w as achiev ed for the morning peak data where the error w as controlled within 6%. Saito and W atanabe [6] predicted trav el time 60 minutes ahead with last 30 minutes’ observations. The average error of the prediction results is less than 3 minutes apart from actual v alues. As a non-parametric approach, neural networks also play an important role in the research on short term trav el time prediction. Huisken and v an Berkum [12] proposed a multilayer feed forward based ANN model for predicting tra vel time at a gi ven highway and compared its performance with the results from tw o nai ve methods: dynamic tra vel time estimation (DTTE) and static tra vel time estimation (STTE). The model w as trained by the flo w and speed data on a one-minute basis. The information was collected from inducti ve loop detectors at 21 locations on a gi ven freew ay section, at a corresponding time of the day . It was found that ANN models outperform the two nai ve models significantly . In order to extend the tra vel time prediction model to road traf fic networks rather than a single road section, W ang et al. [13] integrated space-time neuron into the neural networks by adding a hidden layer . The road traffic netw ork was represented numerically in the form of a spatial weight matrix. Performance of the space-time delay neural networks was compared against and prov ed to be better than the other three models, Naive, ARIMA, and space-time ARIMA, at the 5, 15 and 30 minutes forecasting horizons. Moreira et al. [14] did a series of works to predict trav el time in a three-day prediction horizon based on a 244-day data set. They utilized and compared three non-parametric regression methods in R package: Projection Pursuit Regression (PPR), Support V ector Machine (SVM) and Random Forest (RF). Departure time, weekday , day of the year, and day type (holiday , bridge day , etc.) were found to be the most v aluable independent variables. In their following work [ 15 ], study on 128 combinations which composed of the three models and respectiv e parameter sets was done for each of the six studied routes by using a heterogeneous ensemble approach with dynamic selection. The results showed that the ensemble approach is able to increase accuracy by 8.2% against the use of a single algorithm and parameter set. 1.3 Long-T erm Prediction Klunder et al. [16] conducted a long-term travel time prediction by using K-nearest neighbor (K-NN) approach. This is a typical pattern searching pattern approach. As a non-parametric method, it mak es predictions by searching out the traffic patterns on similarities among data instead of defining parameters and/or distrib utional assumptions on input and output variables. Case study was done on a single route of three 25-km successiv e motorways with date and trav el time data for e very 15 minutes. School holidays, major e vents, special days and accurate precipitation data were also considered. The prediction in non-rush hour (8:00am and 10:00pm), with the RME of the median 4.6% and 3.3%, perform better than that in rush hour (12:00pm and 5:30pm) 7.6% and 19.4%. 1.4 Existing Research on Seasonality and W eather Numerous studies ha ve in vestigated the role of seasonality in transportation systems [ 17 – 36 ]. For example, Nookala [37] used linear regression model with time-v arying coefficients to conduct a short-time tra vel time prediction with an emphasis on inv estigating the weather impacts. The linearity between a set of weather indexes, traffic volume, and dynamics was in vestigated by measurement of correlation coefficients. Conclusion was drawn that the daily total traffic volume decreased under non-ideal weather inde xes while the congest ion increased as the free way capacity dropped. The results of trav el time prediction showed that the prediction errors become more significant under changing weather index es. El Faouzi et al. [38] integrated weather ef fect in vestigation into their tra vel time prediction by using database search for similar profiles extraction based on toll collection data. W eather data analyzed in this work include rain, 2 temperature, wind direction and speed. The data sets in this study consist of the whole hourly measurements in 43 days. It was sho wn that it provides more accurate predictions on trav el time in junction with weather index es. Klunder et al. [16] indicated that the accuracies of predicted tra vel time were improv ed by including the accurate precipitation data into input variable set. RME of median decreased from 15% to 7.6% for the morning rush hour period and from 34.4% to 19.4% for the ev ening peak hours. 2 Case Study In this research, hourly TTI data collected between Jan 1, 2010 and Jun 26, 2016 in the DC area and corresponding daily weather index data were used. 2.1 Descriptive Statistics In order to in vestigate data v ariation caused by effect of dif ferent time, ef fect of different weekday , effect of dif ferent season, and ef f ect of dif ferent year , the mean historical TTI has been calculated under dif ferent time scale for descriptive statistical analysis. Figure 1 shows the daily–a veraged TTI during the gi ven time period. Further in vestigation has been done with regards to se veral significant peak v alues of TTI. As labeled in the figure, there exists a significant relationship between sno w ev ent and high value of daily av eraged TTI. It can be seen that weather indexes, especially sno w , play an important role to the significant high TTI values on a daily basis. Figure 1: Daily Tra vel T ime Index from 1/1/2010 to 1/31/2016 Ho wev er , the relationship between daily TTI and sno w e vent does not lead to the conclusion that TTI v alues in winter is generally higher than in the other seasons. As shown in Figure 2, the maximum monthly-a veraged historical TTI occurs in June. May and October are also associated with relativ ely high TTI values. Significance of monthly effect is not observed from this study as the v ariation TTI over dif ferent month is small. Figure 3 shows the historical av eraged TTI as a function of time. The red line is resulted from the data in the days with no precipitation, and the blue line is resulted in the days that has precipitation. It can be seen that TTI values reach the peak values at 08:00 and 17:00, which are within the morning and e vening rush hours. The variation of av erage TTI in dif ferent time of a day is very significant. Therefore, it can be concluded that the hourly pattern is an important f actor to be considered in data preparation. It can also be observed that the historical a veraged TTI based on the data from wet days hav e slightly greater values than that from the days with no precipitation. Daily pattern is another significant factor in the study of historical data of TTI. The mean historical TTI is plotted against each weekday in Figure 4, where 0 through 6 on X–axis represent Sunday to Saturday . As observed from the figure, TTI is generally highest on W ednesday and lo west on Saturday . TTI on weekdays are higher than weekends. It can also be observed that the historical av eraged TTI on days with high precipitation are higher than TTI on days with low precipitation. This may be attributed to the traffic congestion caused by the wet weather . The effect of different year is also studied by calculating the yearly-average TTI from 2010 to 2015. It is plotted in Figure 5. No regular pattern has been witnessed from the figure. The relatively small v ariations indicate that the TTI lev els are relatively stable during the years of interest. 3 Figure 2: A veraged T rav el T ime Index as a Function of Month Figure 3: A veraged T rav el T ime Index o ver T ime of a Day 2.2 Model evaluation criterion In this study , the prediction of TTI is considered as a machine learning regression problem. For continuous-valued objects, there are v arieties of regression algorithms designed to minimize the dif ference between the output from the predictor and the actual v alues. Therefore, the predictor can be used for future predictions with a promising accuracy . Coefficient of determination ( R 2 ), named as score, is used as a measurement of the model accurac y . It is defined by Equations (1)–(3). R 2 = 1 − S S res S S tot (1) S S res = X i ( y i − f i ) 2 (2) S S tot = X i ( y i − ¯ y ) 2 (3) where y i is the actual v alue at testing/validation data point i ; f i is the predicted v alue at testing/validation data point i ; ¯ y is the mean of actual values of all testing/validation data points; S S res is the residual sum of squares, as defined by Equation (2); and S S tot is the total sum of squares, as defined by Equation (3). 4 Figure 4: A veraged T rav el T ime Index per W eekday Figure 5: A veraged T rav el T ime Index per Y ear As an indicator of how good the model fits the real data, the R 2 score ranges from 0 to 1 . The best possible score is 1 , and it can be negati ve if the model performs arbitrarily worse than a baseline. A constant model that always predicts the expected v alue of y , regardless of the input features, w ould obtain an R 2 score of 0 . The ev aluation of the model’ s accuracy is completed by generating scores based on a separate data set from the one that was used for learning the parameters of prediction function for avoiding o ver–fitting. Instead of performing tests on the trained regression model with the held-out part of data called testing data set, the prediction scores in this study are obtained by using k-fold cross validation. The av ailable data set is split into k smaller sets. The following procedure is followed for each of the k folds: (1) A model is trained using k-1 of the folds as training data; (2) The resulting model is v alidated on the remaining part of the data to gi ve a performance measure such as score in this case; (3) The score reported by k-fold cross-validation is then the a verage of the values computed in the loop. Although this approach need more computational resource than the conv entional method, it enhances the accuracy of the model by avoiding the potential biases caused by an arbitrary test set while minimizing reduction of the size of training data set. In this case, k is taken as 5. It means the av ailable data set was split into 5 smaller sets, each of which takes turns to be used as training data set while the rest of the data used as validation set. In order to save as much computation time as possible, while ensuring enough prediction accurac y , we use random permutation function to pick up 1000 data as a sample for one e xperiment (modeling training and v alidation) and repeat the process 10 times to conduct 10 experiments with dif ferent samples. In this way , the sample size reduces significantly compared to the original data set containing 56791 data points. The prediction score is computed by averaging the scores resulted from the 10 experiments with random picked samples to k eep the test impartial. 5 T able 1: Best prediction results and corresponding parameters and number of variables used in the prediction based on historical data up to 1 hour before. Model Best parameters # V ariables Best score ( R 2 ) Ridge Regression α = 1 . 0 24 0.9025 Linear Regression – 21 0.8839 SVR C = 2 . 8 , ϵ = 0 . 1 10 0.8322 Decision T ree Regressor max _ depth = 2 . 2 21 0.6824 Lasso Regression α = 0 . 1 19 0.4265 2.3 Featur e selection In this research, there are 93 possible independent v ariables. Other than 5 time and date parameters (hour , day , weekday , month and year), 34 parameters re garding weather information and 11 parameters of historical TTI, the rest 43 v ariables are generated by transforming the date and time information from numerical into indicator v ariables. On one hand, there exist man y redundant and/or insignificant information within the v ariable sets. On the other hand, it is tremendously time and space-consuming to deal with so many x-variables, especially when polynomial features are also in volved. It is obvious that b uilding regression model based on the whole set of v ariables is neither necessary nor feasible. In order to identify and remov e as much irrelev ant independent variables as possible from the training data set, an algorithm called Recursi ve Feature Elimination (RFE) is e xecuted in the data preparation of this study . As a feature selection algorithm, RFE follows a procedure that have three components. First, instead of exhaustiv ely proposing subsets of features and attempting to find an optimal subset from them, RFE uses a backward stepwise selection algorithm which starts with all attributes in the set and gradually removes them one at a time. Second, an external estimator is used to assign coefficients to the proposed features and to ev aluate the accuracy of the model. Third, a stopping criterion, which is that the addition or deletion of an y feature will not produce a better subset (Karagiannopoulos et al., 2007), has to be met to stop iterations and establish the optimal solution. The procedure is recursi vely repeated on the pruned set until the desired number of features to select is ev entually reached. In this study , linear regression model is defined as external estimator . The desired number of features is iterated from 1 to 24. In other word, 24 optimal subsets at different sizes (1 to 24 elements) are tested conjunctively with different other factors (model, parameter, polynomial degree, etc.). The final model is selected out of those combinations by comparing their cross-validation scores. 2.4 Model Comparison In order to find out the best model to predict TTI in DC area with high accuracy , 5 regression models provided by Scikit-learn [ 39 ] are tested for performance comparison, including: linear regression, Lasso re gression, Ridge regression, support vector machines–SVR and decision tree re gressor . For the re gression models whose accurac y might be sensiti ve to one or more parameters, we iterate each of the paramet ers within the pre-determined range based on the restrictions of algorithm and nature of the problem. For each regression model with different parameter v alues, it runs with optimized combination of independent variables selected by the feature selection algorithm. Also, polynomial features are taken into consideration. Feature matrices consisting of all polynomial combinations of the features with degree less than or equal to the specified maximum degree are tested. For example, if the specified maximum degree of the input sample is 2 and of the form [ a, b ] , the degree-2 polynomial features are [1 , a, b, a 2 , ab, b 2 ] . The maximum degree takes its v alue in the range from 1 to 5 . In ev ery single case, as a combination of different re gression model, parameter value, independent v ariable set, and polynomial degree, the predictions are repeated for 10 times based on 1000 data points randomly selected from the historical TTI data. The final score in each case is calculated by averaging the resulted scores from the 10 calculations. T wo cases are ev aluated separately based on different time scales of historical data: (1) Case 1: historical TTI data used are up to 1 hour before the data point under prediction which is called short-term prediction case herein-belo w; (2) Case 2: historical TTI data up to 1 day before is called long-term prediction case. The best cases of short-term prediction case for each model are output and summarized in T able 1. The results from long-term predictions are summarized in T able 2. As shown in T able 1 and T able 2, ridge regression is the best model out of the fi ve models we hav e tested to predict TTI in both short- and long-term prediction cases. In short-term prediction case, a best score of 0.9025 is achieved 6 T able 2: Best prediction results and corresponding parameters and number of variables used in the prediction based on historical data up to 1 day before. Model Best parameters # V ariables Best score ( R 2 ) Ridge Regression α = 1 . 9 13 0.6086 Linear Regression – 15 0.6041 SVR C = 1 . 6 , ϵ = 0 . 1 15 0.5802 Decision T ree Regressor max _ depth = 2 . 2 11 0.5367 Lasso Regression α = 0 . 19 15 0.6062 when the parameter alpha has a value of 1.0 and the best 21 independent variables are selected to train the second-order polynomial regression model. In long-term prediction case, howe ver , the best score achiev ed is 0.6086. It is significantly lower than the score in short-term case. This difference is attributed to the candidacy of the historical TTI values 1 to 3 hours before as independent variables which can therefore be extrapolated to be an important factor influencing the TTI prediction. The best value of parameter alpha is 1.9. T otally 15 variables together with their second-order terms and cross-product terms are in volved in the best polynomial regression model. 3 CONCLUSION In this research, instead of using TTI data for only tens of days, TTI data collected for 6 years are being used to provide more comprehensi ve information. Other than that, more weather index es are also being used than the pre vious works to offer the research team a better insight into the relationship between weather and TTI. The increased amount of data in volved will significantly impro ve the reliability and accurac y of the result reached by the research. During the research, it is surprising to find that instead of minimum visibility , the mean visibility is the weather index that has a more prominent effect on TTI. It is also surprising to find that the Ridge regression model outperform the linear regression model which is widely used in the previous works. This improved model, together with the findings in weather index es, will provide industries a better chance to predict their travel time and therefore the monetary resource that is required to in vest into new projects. In conclusion, this paper presents trav el time index (TTI) prediction using historical data. Fi ve regression models provided by Scikit-learn package are tested for performance comparison, including: linear regression, Lasso regression, Ridge regression, support vector machines–SVR and decision tree regressor . Ridge regression is the best model out of the fiv e models to predict TTI in both short and long-term prediction cases. In short-term prediction case, a best score of 0.9025 is achiev ed when the parameter alpha has a value of 1.0 and the best 21 independent variables are selected to train the second-order polynomial re gression model. In long-term prediction case, the best score achiev ed is 0.6086. 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