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Multistep prediction of traffic state is a key technology for advanced transportation information system. The research results based on the principle of multistep prediction can provide more information about the traffic operating quality in advance. Considering that prediction error increases with the increasing numbers of multistep predictions, this research proposes the concept of dynamic predictability that is related to the characteristic of historical traffic flow data used. The traffic flow is characterized by randomness, regularity, and volatility according to the traffic flow theory. Therefore, three key indexes are firstly calculated to measure the characteristics of reliability series. Then a two-phase model is established based on wavelet neural network optimized by particle swarm optimization. The upper phase is a model to estimate the number of predictable steps, and the lower phase is the multistep prediction model of reliability. Compared with that of backpropagation neural network and support vector machine, results show that the convergence time of the wavelet neural network optimized by particle swarm optimization is the lowest, which only costs 256 and 291 seconds in both two-phase models under the same conditions. The average relative error of multistep prediction reached the lowest value, 8.91% and 12.01%, respectively, for weekday and weekend data used. Moreover, the prediction performance based on weekday is better than that of weekend. The research results lay a decision-making basis for managers in determining the key parts of road network to develop future improvement measures.

In recent decades, the increasing demand for road transportation has negatively affected the stability and reliability of traffic operation which caused a series of drawbacks, such as extensive waste of travel time, decreasing of environmental quality, and aggravated vehicle wear and tear. Meanwhile, the requirement for accurate prediction of traffic state is increasing. Prediction results serve not only as an important basis for traffic control and guidance but also as a decision support for travelers to adjust their travel plan.

Development of Advanced Traveler Information Systems usually divides traffic state into three states: smooth, congested, and blocked. It has oversimplified the problem since traffic state parameter is a continuous variable, e.g., flow, density, and speed. This study presents a new concept to describe traffic state. It is the traffic state reliability which is defined as the degree of actual traffic flow relative to the free flow, which assumes that the reliability of free flow is highest. In order to get the traffic flow trend in the future, the forecasting of traffic state reliability is needed to quantitatively measure the reliability of each alternative in future.

For example, there are five links A, B, C, D, and E for the travelers. The prediction result shows that links A, C, and D are in smooth state. However, the travelers do not know how to determine which is the best link among links A, C, and D. If the reliability prediction could provide the level of actual traffic flow relative to the most reliable free flow state of each link (that is, 0.9, 0.2, 0.92, 0.79, and 0.1). Obviously, the traveler will prioritize link C with the highest reliability value to travel.

Antoniou et al. [

In recent years, traffic data are currently collected through various sensors, including loop detectors, probe vehicles, cell phones, Bluetooth devices, video cameras, remote sensing applications, and public transport smart cards. Nantes et al. [

Traffic state estimation is a key problem with considerable implication for modern traffic management. Kong et al. [

Travelers should be able to select the best path to avoid traffic congestion. However, if travelers also avoid the link with high variability, then travelers will enjoy additional benefits, that is, the “reliability benefits.” To date, traffic reliability has introduced the idea of reliability into traffic research and is an important field of traffic problems analysis. Considerable research has been conducted on traffic reliability, covering from theory to practice and from model to algorithm. Frameworks for reliability analysis have also been developed.

Wang et al. [

Probabilistic forecasting of reliability can be used for risk-averse routing. Bezuglov and Comert [

Papathanasopoulou et al. [

In summary, most studies focus on traffic state estimation and travel time reliability, which mainly emphasize the traffic quality but not the reliability of traffic quality. Advanced Traveler Information System is one of the functional areas of Intelligent Transportation Systems and aims to provide forecasted traffic information for travelers to make better decisions. There are two methods to forecast the traffic data. One is one-step prediction, which predicts the reliability in the next one step. The other is fixed multistep prediction, in which the number of prediction steps is fixed. The results of fixed multistep prediction show that it easily leads to large prediction errors and the prediction errors gradually linearly increase with the number of multisteps [

The proposed concept of dynamic predictability needs to find the number

The definition of traffic state reliability (TSR) is the basis for setting a method to measure reliable level of traffic state. What is the relationship between past and present reliability and the future reliability? It is the discussion focus of this part.

For a link, intersection, or road network, TSR is defined as the degree of actual traffic flow state relative to the most reliable free flow state, which assumes that the reliability of free flow state is highest. It uses the TSR index to quantify evaluation, which is the ratio between actual speed and free flow speed in this research. TRS is classified to be equal to 1, when the actual speed is larger than or equal to the free flow speed and equal to

One-step prediction [

The above method can be used to offline predict the reliability of the next S steps by a kind of prediction method. The prediction results show that the average prediction errors gradually linearly increase with the number of steps. The graph shows the relationship between the average prediction errors and the number S of steps is seen in Figure

The relationship between the average prediction errors and the number S of steps.

Fluctuation index is used to measure the random fluctuations of TSR series data. It refers to the ratio of variance to average reliability series. The less

Tendency index is used to quantitatively measure the tendency characteristic of the continuous increasing or decreasing of TSR series data. It refers to the cumulative amplitude change between two consecutive TSR of data series. The less

This index is expressed as

Uncertainty characteristic index is used to quantitatively measure the disorder characteristic of the TSR series data. This index presents an inverse proportional relationship with the number of predictable steps. It can be quantitatively measured by a fuzzy entropy indicator based on fuzzy analysis theory. The less

Based on above description, it is required to find

Both the dynamic predictability and multistep prediction are nonlinear fitting problems. It is exactly the problem that wavelet neural network can solve. Wavelet neural network has the good capability of localization and nonlinear mapping [

A three-layer wavelet neural network is typically used to fit nonlinear correlations. This study uses a three-layer wavelet neural network to design the two-phase model for multistep prediction. First, in the upper-phase model, a three-layer wavelet neural network (Figure

Second, in the lower-phase model, a three-layer wavelet neural network (Figure

Structure of the wavelet neural network for upper phase.

The iterative optimization algorithm of wavelet neural network is algorithm with descent gradient. Notably, there is a local minimization problem. Particle swarm optimization can extract the transient characteristics of target as input, which has a better convergence and prediction accuracy. The principle of particle swarm optimization is to randomly generate a certain number of particle swarm optimizations, which has three attribute indexes: position, velocity, and fitness. The position represents a possible solution to the optimization problem. The velocity is a vector, which determines the direction and size of iteration. The fitness is used to measure the good or poor position. If evaluation result is poor, then the position should be updated through iteration. Supposing the position vector of the

The inertia factor

For a particular place, the current reliability series is

The three key indexes

The upper-phase model is constructed as shown in Figure

The training parameter is optimized by particle swarm algorithm.

The upper-phase model is trained by using the data sample from the training data set. The three key indexes

The upper-phase model is tested using the data sample from the testing data set until the average relative error satisfies the requirements that means the upper-phase model fits well. Then, the predictable step number

After obtaining the number

The lower-phase model is constructed as shown in Figure

Structure of the wavelet neural network for lower phase.

The training parameter is optimized using particle swarm algorithm and

The lower-phase model is trained by using the data sample from the training data set. The reliability series data

The lower-phase model is tested using the data sample from the testing data set. If the average relative error

In order to verify the effectiveness of the above model and algorithm, it is necessary to collect and analyze the actual traffic flow parameters, calculate proposed TSR, estimate the dynamic predictability, and predict the multistep reliability in future. Eventually, it is necessary to compare the research result with actual traffic state to confirm the validity of the proposed method.

The empirical data were obtained from a freeway of Shanghai. The structure of road network is shown in Figure

Diagram of the road experimental area.

Traffic data was collected by the loop detector from a freeway in Shanghai of China. The collected traffic parameters mainly include speed, traffic volume, and occupancy. The size of data set is 24 hours of every day from May to July. The time interval of collected data is 1 minute.

The reliability of each link was calculated using (

First, the step number of iterations is used to measure convergence effect of wavelet neural network optimized by particle swarm algorithm. Second, the mean relative error SAPE is calculated to measure performance of the upper-phase model. Third, the mean relative error MAPE is calculated to measure performance of the lower-phase model. As traffic state on weekday is usually very different from that on weekend, the number of neurons in the middle layer is separately determined, and the forecasting work is also verified separately.

Liu et al. [

The difference between wavelet neural network and classical artificial neural network (such as backpropagation neural network) is excitation function in middle hidden layer. The excitation function of the latter usually uses sigmoid function, which leads to nonconvergence or slow convergence. However, the wavelet neural network adopts wavelet function as excitation function. It introduces translation scale and scaling factor to extract local information. Thus, the convergence speed, approximation precision, and performance are improved compared with those of classical artificial neural network.

Support vector machine is one learning method for small sample. It ignores probability measure and law of large numbers unlike existing statistical methods. It is also based on the minimization principle of structural risk. The global optimal solution is obtained through the optimization problem of convex quadratic, which has high generalization capability and an advantage in terms of generalization and classification.

The convergence rate was trained and comparatively analyzed in upper-phase and lower-phase model using the same set of reliability data series and under the same conditions. The result is shown in Figures

Convergence of upper-phase model based on different methods.

Convergence of lower-phase model based on different methods.

Figures

In order to determine the number of neurons in hidden layers of upper-phase and lower-phase model, the number of neurons is set from 5 to 20 to train, respectively, based on weekday and weekend data. The result is shown in Figures

Relationship between MAPE/SAPE and the number of neurons on weekday.

Relationship between MAPE/SAPE and the number of neurons on weekend.

Based on weekday data, Figure

Based on weekday data, the number of predictable steps is estimated and analyzed comparatively based on wavelet neural network optimized by particle swarm algorithm, the backpropagation neural network, and the support vector regression. The result is shown in Table

SAPE of proposed method and referenced method for upper-phase model based on weekday data (%).

Reliability index | Calibration data set | support vector regression | back propagation neural network | Proposed method |
---|---|---|---|---|

reliability series | training sample | 15.41 | 19.87 | 7.92 |

testing sample | 20.75 | 20.74 | 8.63 |

SAPE of proposed method and referenced method for upper-phase model based on weekend data (%).

Reliability index | Calibration data set | support vector regression | back propagation neural network | Proposed method |
---|---|---|---|---|

reliability series | training sample | 16.51 | 22.91 | 10.07 |

testing sample | 24.73 | 20.62 | 12.73 |

Based on weekday data, the multistep prediction is analyzed comparatively based on wavelet neural network optimized by particle swarm algorithm, the backpropagation neural network, and the support vector regression. The result is shown in Table

MAPE of proposed method and referenced method for lower-phase model based on weekday data (%).

Reliability index | Data set | support vector regression | back propagation neural network | Proposed method |
---|---|---|---|---|

reliability series | training sample | 17.92 | 20.7 | 8.91 |

testing sample | 20.15 | 21.58 | 10.02 |

MAPE of proposed method and referenced method for lower-phase model based on weekend data (%).

Reliability index | Data set | support vector regression | back propagation neural network | Proposed method |
---|---|---|---|---|

TSR series | training sample | 22.32 | 23.89 | 12.01 |

testing sample | 26.13 | 23.91 | 13.54 |

Moreover, 30 steps in future are predicted, respectively, based on weekday and weekend data. The result is shown in Figures

Prediction results of 30 steps for weekday reliability based on different methods.

Prediction results of 30 steps for weekend reliability based on different methods.

Figures

Meanwhile, 24-step reliability is predicted, respectively, on Monday and Saturday for the direction link 080810 which is from south to north in the intersection. The forecasting period is from 08:00:00 to 08:23:00, and the time interval is 1 min. The result is shown in Figures

Multistep prediction of weekday reliability based on different methods.

Multistep prediction of weekend reliability based on different methods.

In view of the fact that the traffic state should be a continuous variable, this study proposes the concept of TSR to measure the stability of actual traffic flow state relative to the most reliable free flow state. In order to know TSR in future, a two-phase model of multistep prediction has been designed based on wavelet neural network optimized by particle swarm algorithm. In the upper-phase model, the number

The adjustment coefficient

The scaling factor

The volatility index of link or direction link

The scaling factor

The number of dimensions

The total number of data samples in training data set

The fuzzy entropy

The average predictive error of multistep prediction

The ambiguity

The number of continuous weeks

The tendency index of link or direction link p at period t

The number of predictable steps

The relative average predictive error of steps

The actual number of predictable steps of the

The predicted number of predictable steps of the

The actual number of predictable steps of data samples of historical reliability series

The total number of iterations

Reliability of link or direction link

the three key indexes, where

The velocity vector of the

The position vector of the

The actual value of reliability data sample in the future

The stretch factor of the

The stretch factor of the

The translation factor of the

The translation factor of the

The acceleration factor

The acceleration factor

The predictive function—

The best solution for the group obtained by all particles, that is, the global extreme value

The current times of iterations

The gradient calculated based on the slope of reliability series data—

The number of time intervals of reliability series

The best solution of the

A random value evenly distributed between 0 and 1

The current time interval

The predictive value of reliability in the next

The historical series of reliability

The actual reliability of the

The predicted reliability of the

The average value of reliability series of link or direction link

The reliability value of link or direction link

The reliability of link

The

The inertia factor

The traffic state values of link or direction link

The weight between the output layer and the

The weight between the

The weight between the

The weight between the

The Morlet wavelet function.

The authors (Jufen Yang, Zhigang Liu, and Guiyan Jiang) declare that there are no conflicts of interest regarding the publication of this paper. They confirm that the mentioned funding in the Acknowledgments did not lead to any conflicts of interest.

The authors greatly appreciate the support provided by the National Key Research and Development Plan of China (Grant no. 2017YFC0804900), the National Natural Science Foundation of China (Grant no. 71701124 and Grant no. 51278257), and the Special Fund of Training Scheme for Young Teachers of Universities in Shanghai (Grant no. ZZGCD15116).

The flow chart of the two-phase model of multistep prediction based on wavelet neural network optimized by particle swarm algorithm is shown in Supplementary Materials as an annex.