To test the performance of the spatiotemporal gradient–boosted regression tree (STGBRT) method, we compared the predictive performance of STGBRT with that of the Autoregressive Integrated Moving Average [12
], Random Forest [54
] and Gradient Boosting [27
] methods in terms of their absolute percentage errors (MAPE). The Gradient Boosting Method (GBM) considers the time correlation of a target link without regard for the influence of spatial correlation or big data describing historic traffic conditions in estimating link travel time. The Autoregressive Integrated Moving Average Model (ARIMA) model is a generalization of the autoregressive moving average (ARMA) model and is one of the most widely recognized methods for traffic parameter forecasting. The model is fitted to time series data to understand the data better or to predict future points in the series. ARIMA is applied in cases where data show evidence of non-stationarity. It converts non-stationary time series to stationary time series. The model is constructed using the dependent variable, its lag value, and the present value of the random error; predictions from ARIMA are based on regression of current and past data. Non-seasonal ARIMA models are generally denoted as ARIMA (p, d, q) where parameters p, d, and q are non-negative integers, p is the order of the autoregressive model, d is the degree of differencing, and q is the order of the moving average model. Optimization of the ARIMA model involves order selection and parameter estimation. Detailed information on the theoretical background underlying ARIMA, and the steps involved in fitting an ARIMA model, can be found in the literature [55
]. The Random Forest (RF) method is another widely used ensemble method whose extension was developed by Leo Breiman [54
] and is different from the gradient–boosted regression tree method.
To compare these four methods for predicting link travel time, we obtained statistical data collected by probe vehicles traversing the regional road network in Wuhan on weekdays, Monday to Friday, except holidays, from January to May 2014. We extracted the spatiotemporal features of links within the network. The data from 21 July 2014 to 22 July 2014 were used as test data to compare the prediction performance among the four models (STGBRT, GBM, RF, and ARIMA). The prediction accuracy of these four models was compared based on their predictions one and two time steps (that is, 30 and 60 min) after the present time. The experiment discussed in Section 4.2
showed that the MAPE of the STGBRT model achieved a minimum value when the learning rate was set to 0.01 and the number of basic regression trees was set to 500. Therefore, in the comparative experiment, we set the corresponding experimental parameters to 0.01 and 500, respectively. For GBM and ARIMA, we tested different combinations of variables during the training process and selected the parameters that achieved the minimum MAPE values.
We used traffic big data representing historic traffic conditions from January to May in 2014 and real data obtained from the 11 weeks between 5 May 2014 and 20 July 2014 as training data. We used two days of data (21 and 22 July 2014) as test data to compare the prediction performance among STGBRT, GBM, and ARIMA. The line charts in Figure 12
and Figure 13
illustrate the variation among predictions made 30 min and one hour ahead from the four models on 21 July 2014 and 22 July 2014, respectively. The blue line in the two figures represents the true link travel time, while the red line represents prediction results from the STGBRT model, the green line represents the prediction results from GBM, the orange line represents the prediction from RM and the purple line represents the prediction results from the ARIMA model. As shown, the STGBRT model and GBM model fit the true link travel time most closely. ARIMA provided the least favorable match to the true link travel time among the four models. Under the same conditions, the predictions of STGBRT outperform those from the random forest method in our experiments, as depicted in Figure 12
, Figure 13
and Figure 14
. Figure 14
shows a comparison of the MAPE values for the performance of these four models for predictions made 30 min and one hour ahead. As illustrated in Figure 14
, the prediction results of STGBRT outperformed those of the other three models. The MAPE for STGBRT (7.43%) was superior to the MAPE values corresponding to half-an-hour predictions for GBM, RF, and ARIMA, which were 9.37%, 15.83%, and 33.79%, respectively. At the same time, the STGBRT half-an-hour prediction performance had a significantly better MAPE value (7.43%) than the one-hour prediction (9.49%). Figure 15
illustrates the standard deviations of predictions made 30 min and one hour ahead by the four models for 21 July 2014 and 22 July 2014. As illustrated in Figure 15
, the prediction results of STGBRT had a small MAPE value and outperformed the other three models except in terms of the one-hour predictions made for 21 July 2014. Figure 16
gives the computational performance of different models under the same conditions, that is, using the same training and prediction data. The figure shows that STGBRT, GBM, and RF require similar amounts of computational time: 5.09 s, 5.73 s, and 5.24 s, respectively. The ARIMA model requires the smallest amount of computation time; however, it had poor prediction performance compared to the other three models, as depicted in Figure 14
. A Wilcoxon test showed that the differences between true link travel time and the results from the STGBRT, GBM, and RF models are all symmetrically distributed about zero except for predictions made one hour ahead by the RF model for 21 July 2014. However, the differences between true link travel time and predicted values from ARIMA are not symmetrically distributed about zero except for predictions made one hour ahead for 22 July 2014. Therefore, the STGBRT, GBM, and RF models yield better predictions than the ARIMA model. Figure 17
shows five days (Monday, 21 July 2014–Friday, 25 July 2014) of predicted link travel times from the STGBRT model. The blue line represents the true link travel time, and the red line represents the predicted link travel time. Table 9
shows the MAPE values for the travel time prediction obtained from the STGBRT model from Monday to Friday; the STGBRT model had high MAPE values. Figure 17
reflects overall trends, as well as how well the models captured sudden changes in travel time. For example, on 21 July 2014 (upper panel of Figure 17
), the STGBRT model captured changes especially well during the morning rush hour when congestion is likely to occur. Theoretically, the STGBRT model can handle complex interactions among input variables and can fit the complex nonlinear relationships found in dynamic traffic systems for superior prediction performance.