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Short-term traffic flow prediction is one of the most important issues in the field of adaptive traffic control system and dynamic traffic guidance system. In order to improve the accuracy of short-term traffic flow prediction, a short-term traffic flow local prediction method based on combined kernel function relevance vector machine (CKF-RVM) model is put forward. The C-C method is used to calculate delay time and embedding dimension. The number of neighboring points is determined by use of Hannan-Quinn criteria, and the CKF-RVM model is built based on genetic algorithm. Finally, case validation is carried out using inductive loop data measured from the north–south viaduct in Shanghai. The experimental results demonstrate that the CKF-RVM model is 31.1% and 52.7% higher than GKF-RVM model and GKF-SVM model in the aspect of MAPE. Moreover, it is also superior to the other two models in the aspect of EC.

Short-term traffic flow prediction is an important basis for intelligent transportation systems (ITS). Real-time and accurate prediction information can be directly applied to the advanced traffic management system (ATMS) and advanced traffic information service system (ATIS). Because of its importance, short-term traffic flow predication has generated great interest among the scientific community and a large number of relevant methods exist in the literature. These include the spectral analysis model [

In order to get the accurate prediction results, we need to find the nonlinear prediction function. However, it is hard to get the accurate function due to the interference of inside and external excitations. But determining the linear function is not hard since detecting linear relations has been focus of much research in statistics and machine learning fields for decades and the resulting algorithms are well understood, well developed, and efficient. So if we could combine both, it will solve the problem. Instead of trying to fit a nonlinear model, we can map the problem from the input space to the feature space by doing a nonlinear transformation using suitably chosen basis functions and then use a linear model in the feature space. The basis function is called kernel function. The linear model in the feature space corresponds to a nonlinear model in the input space. This is the main idea of relevance vector machine (RVM) model. Due to RVM theoretical advantages, it has gained special attention in recent years, such as [

For these reasons, and with the goal of improving the accuracy of short-term traffic flow prediction, we put forward a short-term traffic flow local prediction method based on combined kernel function relevance vector machine model. The remainder of this paper is structured as follows: Section

Phase space reconstruction theory proposed by Packard et al. [

Phase space can be reconstructed using delay coordinate method. The basic idea of delay coordinate method is that the evolution of any single variable of a system is determined by the other variables with which it interacts. Information about the relevant variables is thus implicitly contained in the history of any single variable. For a time series

Embedding dimension and delay time are the key parameters for phase space reconstruction. At present, there are two kinds of views about the selection of these two parameters. One view is that the two parameters are independent and could be determined separately. The methods of calculating delay time include Average Displacement method [

The time series

The test statistics is

As

For fixed embedding dimension

Consider

According to the BDS statistic result, we select

The relevance vector machine (RVM) model proposed by Tipping [

Consider a data set

Because there are many parameters in the model, the maximum likelihood estimates of

Because we have defined the prior probability distribution and the likelihood distribution, the posterior probability distribution is as follows according to the Bayesian theory:

Posterior covariance matrix and mean value are as follows, respectively:

According to the maximum expected hyperparameter estimation, the value of

The noise variance

Given a new sample

The traditional relevance vector machine model mostly adopts single kernel function to complete the process of feature space mapping, which has achieved good performance in many practical applications. But the single kernel function has great limitations when the sample data contains heterogeneous information. Therefore, this paper integrates the Gaussian kernel function and polynomial kernel function to construct a new combination kernel function. The form of combination kernel function is as follows:

Different kernel functions have different advantages; if the weight coefficient of combination kernel function is inappropriate, the performance of combination kernel function may be lower than single kernel function. Therefore, proper weight coefficient is of great importance for the combined kernel function.

There are three parameters that need to be optimized in the combined kernel function. The commonly used parameter optimization methods mainly include cross validation method [

Therefore, genetic algorithm is used to obtain the optimal parameters of combination kernel function. The specific steps are as follows.

The population size and maximal generation count: the population size is 20, and the maximal generation count is 100.

The parameters to be optimized

The cross validation method is used to prevent overfitting and underfitting. The training data set is randomly divided into

Selection, crossover, and mutation are carried out to generate population. The chromosomes with better fitness function values are selected using the roulette wheel method. The crossover probability of creating new chromosomes is set to 0.8. Mutation probability is set to 0.05.

If the generation count reaches its maximum value, the iteration is stopped. Otherwise, the process is repeated from Step

The experimental traffic flow data come from loop detectors located on the north–south viaduct expressway in Shanghai, China. This segment includes 24 mainline detecting sections and 30 ramp detecting sections, equipped with 88 mainline loop detectors and 60 ramp loop detectors, respectively. The experimental data are collected on five consecutive Mondays from September 1, 2008, to September 29, 2008. The original time interval of collected data is 5 min. Figure

The traffic flow time series data from five consecutive Mondays.

Phase space reconstruction is the basis of chaotic time series analysis which affects the prediction performance directly. This paper selects C-C method to complete phase space reconstruction. Figure

The curve graph between

The curve graph between

From Figure

Figure

The 2D attractor of the reconstructed phase space for traffic flow time series.

From Figure

Among the wide variety of methods available for chaos identification, the most popular one is the largest Lyapunov exponent method. The main methods of calculating largest Lyapunov exponent include Wolf method [

The result of small data sets method.

The number of neighboring points is one of the most important parameters which affects the prediction accuracy and the amount of calculation. If the number of neighboring points is too little, the nonlinear fitting advantage of relevance vector machine model will not be reflected. However, if the number of neighboring points is too much, the amount of calculation will increase greatly and the overfitting phenomenon will appear. Therefore, the Hannan-Quinn criteria [

The number of neighboring points based on Hannan-Quinn criteria.

According to Hannan-Quinn criteria, when

Genetic algorithm is used to optimize

The fitness curve of GA.

From Figure

In order to evaluate the performance of the proposed method, two different types of measurements are introduced: the mean absolute percentage error denoted by MAPE and equal coefficient denoted by EC. The equations for the MAPE and EC are as follows:

Data collected from September 1 to September 22 are used as training samples, and data collected on September 29 are used as test samples to evaluate the performance of prediction model. In order to illustrate the predictive performance of the proposed method intuitively, Figure

The prediction performance based on the proposed method.

As shown in Figure

To describe the superiority of the proposed method detailedly, comparative analysis is carried out. This paper selects Gaussian kernel function relevance vector machine (GKF-RVM) model and Gaussian kernel function support vector machine (GKF-SVM) model as comparative approaches. For the sake of comparison and analysis in terms of macroscopic and microscopic aspects, Figure

Prediction performance comparison of different methods.

Model | East mainline | West mainline | ||
---|---|---|---|---|

MAPE | EC | MAPE | EC | |

CKF-RVM | 5.0% | 0.987 | 5.4% | 0.982 |

GKF-RVM | 7.2% | 0.958 | 7.9% | 0.965 |

GKF-SVM | 10.6% | 0.940 | 11.4% | 0.935 |

The microscopic comparative results of different methods.

As shown in Figure

From Table

This paper proposes a new short-term traffic flow local prediction method based on combined kernel function relevance vector machine model. The proposed method is more in line with the short-term traffic flow characteristic, which are nonlinear, chaotic, and nonstationary. The main contribution of this paper is not the specific techniques but rather the demonstration that the forecasting model should take the dynamic characteristics of short-term traffic flow into consideration. The most important contribution is that this paper provides the new idea and methodology to the relevance vector machine model on how to construct the combined kernel function for the short-term traffic flow forecasting model and how to optimize and identify the model structure parameters efficiently and effectively.

Traffic flow data collected from expressway are employed to evaluate the prediction performance of the proposed method, and the results are encouraging. The theoretical advantage and better performance from our studies indicate that the CKF-RVM model has good potential to be developed and is feasible in applying for short-term traffic flow prediction. In order to have more general and robust conclusions, traffic data from different roadways require further exploration. And future studies need to apply the model to other traffic variable data sets (such as traffic speed, travel time, and average occupancy; this study chooses the traffic flow as the demonstration). Moreover, it will be interesting to test traffic data set in different time intervals in the model.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors express their sincere appreciation to the Chinese National High Technology Research and Development Program Committee for the financial support provided under Grant no. 2014BAG03B03, National Natural Science Foundation of China (no. 51408257 and no. 51308248), and Jilin Province Science and Technology Development Plan of Youth Research Fund Project no. 20140520134JH.