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Directing against the shortcoming of low accuracy in short-term traffic flow prediction caused by strong traffic flow fluctuation, a novel method for short-term traffic forecasting based on the combination of improved grey Verhulst prediction algorithm and first-order difference exponential smoothing is proposed. Firstly, we constructed an improved grey Verhulst prediction model by introducing the Markov chain to its traditional version. Then, based on an introduced dynamic weighting factor, the improved grey Verhulst prediction method, and the first-order difference exponential smoothing technique, the new method for short-term traffic forecasting is completed in an efficient way. Finally, experiment and analysis are carried out in the light of actual data gathered from strong fluctuation environment to verify the effectiveness and rationality of our proposed scheme.

In recent years, the popularization of seamless links among heterogeneous traffic equipment brought about higher requirements on the real-time and reliability of short-term traffic flow prediction. With continuous improvement of traffic information processing, how to predict the short-term traffic flow accurately and effectively has aroused wide attention of scholars domestically and abroad [

In order to solve the aforementioned problems, domestic and foreign scholars have contrived an improved model to realize fusion prediction with the advantages of different models integrated of short-term traffic flow. For example, Xie et al. [

However, thanks to the diversified developments of traffic information processing and data transmission techniques within heterogeneous traffic network, as well as the impact caused by dynamic changes of road topology, traffic accidents, severe weather, driving styles, and so forth, short-term traffic data are instantaneous and the irregular volatility is always changing [

The layout of this paper is arranged as follows. Firstly, based on the introduction of the traditional grey Verhulst model, an improved grey Markov forecasting model is devised by introducing Markov chain. Secondly, combining the advantages of forecasting by utilizing first-order difference exponential smoothing algorithm and introducing a dynamic weighting factor, a new method for short-term traffic forecasting is concretely constructed according to the afore contrived models. Finally, comparative analysis of the examples illustrated that our proposed model and algorithm are more effective.

The grey system theory was first put forward by Professor Deng Julong, a Chinese scholar, in 1980s [

Classical grey system theory mainly includes GM

The nonnegative data sequence is defined as

Then, assuming that

The grey Verhulst model and its whitening equation can be defined as follows.

The grey Verhulst model is

The whitening equation of the grey Verhulst model is

As deduced in [

If the grey Verhulst model is defined as formula (

The least squares estimator of the parameter column satisfies

If the grey Verhulst model is defined as definition (

According to Theorems

The time response sequence of grey Verhulst model can be defined as

Let

The reduction formula can then be defined as

Pointing at the traffic flow with the trend of increasing saturation, numerous traffic prediction algorithms have been proposed taking advantage of the grey Verhulst model to achieve preferable prediction effects. However, as a complicated nonlinear system involving multitudinous uncertainties, the probability of fortuitous events on urban roads is highly fluctuant, which leads to the deviation of forecasting results. Therefore, it is necessary to take the nonlinearity and time-varying characteristics of overall interactions into account in line with various influential factors. So, we improve the algorithm aiming at the accuracy of prediction model next.

During the process of traffic data aggregation, which is coordinated by complex human-vehicle-environment interaction, the current traffic flow is often affected by previous moments. Therefore, in order to improve the prediction accuracy of grey Verhulst model, mathematical description of traffic flow aggregation is given in advance by constructing the Markov state transition probability matrix [

It is obvious that the state transition probability matrix should be updated over time when Markov is used to optimize the grey model. That is to say, at time point

The grey Markov forecasting model is shown in Figure

Grey forecasting model based on Markov chain.

The adjustment factor

Exponential smoothing [

For data sequence (

Herein,

We use dynamic weighting factors to weight the models of grey Markov model and first-order difference exponential smoothing model. The predictive value of the grey exponential smoothing model is thus achieved:

In formula (

In formula (

According to the synthetic forecasting algorithm, the prediction process of grey difference exponential smoothing prediction model is shown in Figure

Grey difference exponential smoothing forecasting model.

In order to verify the prediction effect of our model, the benchmarks of mean absolute error (MAE), mean square error (MSE), mean absolute percentage error (MAPE), and mean square percentage error (MSPE) are taken as evaluation indexes. Those evaluation indexes are defined as formulas (

Mean absolute error is

Mean square error is

Mean absolute percentage error is

Mean square percentage error is

The urban expressway in Nan’an District of Chongqing Municipality in China is selected in observation phase to verify the actual prediction effect of the grey difference exponential smoothing model, which is composed of 6 bidirectional lanes. The schematic diagram of the section observation is shown in Figure

Road observation diagram.

The road traffic flow is counted every 5 minutes, and the 226 observation data points of No. 1 Lane are used as the basic data for verification. The fluctuation trend of the data is shown in Figure

Historical traffic flow data.

As illustrated in Figure

Evaluation index of first-order difference exponential smoothing algorithm.

| Index | |||
---|---|---|---|---|

MAE | MSE | MAPE | MSPE | |

| | | | |

| 27.191 | 2.601 | 0.392 | 0.046 |

| 27.451 | 2.627 | 0.394 | 0.046 |

| 28.828 | 2.760 | 0.408 | 0.047 |

| 33.704 | 3.220 | 0.470 | 0.052 |

According to the results in Table

Predictive results of first-order difference exponential smoothing model.

The initial state probability transfer matrix is calculated from the historical traffic sequence when the grey Markov model is used to predict the short-term traffic flow. The state is divided according to the same interval that every 30 vehicles are used as a state. In the following prediction process, we added real-time traffic data to update Markov state probability transfer matrix in real time. The prediction result of grey Markov model is determined by formula (

Predictive results of grey Markov model.

Comparing Figure

Therefore, combining the advantages of grey Markov model and first-order difference exponential smoothing, a new predictive value is obtained by using dynamic weighting factors according to formula (

Predictive results of grey difference exponential smoothing model.

Based on the short-term traffic data of the observed road sections, the test data are predicted, respectively, by formulas (

Fitting curves between model predictive values and actual test values (the first 120 data points).

Fitting curves between model predictive values and actual test values (the latter 106 data points).

The evaluation indexes of the above prediction models are shown in Table

Evaluation index of forecasting model.

Model | Index | |||
---|---|---|---|---|

MAE | MSE | MAPE | MSPE | |

Grey Markov model | 25.520 | 2.220 | 0.445 | 0.049 |

First-order difference exponential smoothing model | 26.820 | 2.563 | 0.389 | 0.046 |

Grey difference exponential smoothing model | | | | |

Compared with the above models, the first-order difference exponential smoothing algorithm can closely reflect the trend of the original data sequence in the prediction trend of the data, but its predicted value depends on the value of

This paper mainly contributes to the prediction effect of short-term traffic flow. To overcome the shortcoming of low accuracy in short-term traffic flow prediction caused by strong traffic flow fluctuation, a novel method for short-term traffic forecasting based on the combination of improved grey Verhulst prediction algorithm and first-order difference exponential smoothing is proposed. The main conclusions are as follows.

(1) The grey differential exponential smoothing model combines the advantages of the grey Markov model and the first-order differential exponential smoothing model. The experimental results illustrated that this method is suitable for forecasting short-term traffic flow with large fluctuation. Therefore, this method is practical and feasible.

The analysis shows that this method has smaller prediction error compared with the grey Markov model and the first-order difference exponential smoothing model, and the prediction result is closer to the actual value.

(2) In a word, the method proposed in this paper has obvious advantages. However, due to high uncertainty and nonlinearity of the short-term traffic flow, how to improve the prediction accuracy with fluctuation remains an open problem. So this will be our continuous research orientation.

The authors declare that they have no conflicts of interest.

This work is supported by the National Science Foundation of China (NSFC) under Grants 61573076 and 61703063; the Scientific Research Foundation for the Returned Overseas Chinese Scholars under Grant 2015-49; the Program for Excellent Talents of Chongqing Higher School under Grant 2014-18; Science and Technology Research Project of Chongqing Municipal Education Commission of China under Grants KJ1705121 and KJ1705139; the Chongqing Natural Science Foundation of China under Grant CSTC2017jcyjA1665.