# Traffic Flow Forecasting Using a Spatio-Temporal Bayesian Network Predictor

## Abstract

A novel predictor for traffic flow forecasting, namely spatio-temporal Bayesian network predictor, is proposed. Unlike existing methods, our approach incorporates all the spatial and temporal information available in a transportation network to carry our traffic flow forecasting of the current site. The Pearson correlation coefficient is adopted to rank the input variables (traffic flows) for prediction, and the best-first strategy is employed to select a subset as the cause nodes of a Bayesian network. Given the derived cause nodes and the corresponding effect node in the spatio-temporal Bayesian network, a Gaussian Mixture Model is applied to describe the statistical relationship between the input and output. Finally, traffic flow forecasting is performed under the criterion of Minimum Mean Square Error (M.M.S.E.). Experimental results with the urban vehicular flow data of Beijing demonstrate the effectiveness of our presented spatio-temporal Bayesian network predictor.

## 1 Introduction

Short-term traffic flow forecasting, which is to determine the traffic volume in the next time interval usually in the range of five minutes to half an hour, is an important issue for the application of Intelligent Transportation Systems (ITS) [1]. Up to the present, some approaches ranging from simple to elaborate on this theme were proposed including those based on neural network approaches, time series models, Kalman filter theory, simulation models, non-parametric regression, fuzzy-neural approach, layered models, and Markov Chain models [1][8]. Although these methods have alleviated difficulties in traffic flow modelling and forecasting to some extent, from a careful review we can still find a problem, that is, most of them have not made good use of spatial information from the viewpoint of networks to analyze the trends of the object site. Though Chang et al utilized the data from other roadways to make judgmental adjustments, the information was still not used to its full potential [9]. Yin et al developed a fuzzy-neural model to predict the traffic flows in an urban street network whereas it only utilized the upstream flows in the current time interval to forecast the selected downstream flow in the next interval [7].

The main contribution of this paper is that we proposed an original spatio-temporal Bayesian network predictor, which combines the available spatial information with temporal information in a transportation network to implement traffic flow modelling and forecasting. The motivation of our approach is very intuitive. Although many sites may be located at different even distant parts of a transportation network, there exist some common sources influencing their own traffic flows. Some of the distributed sources include shopping centers, home communities, car parks, etc. People’s activities around these sources usually obey some consistent laws in a long time period, such as the usually common working hours. To our opinion, these hidden sources imply some information useful for traffic flow forecasting in different sites. Therefore, construct a causal model (Bayesian network) among different sites for traffic flow forecasting is reasonable. This paper covers how to use the information from a whole transportation network to design feasible spatio-temporal Bayesian networks and carry our traffic flow forecasting of the object sites. Encouraging experimental results with real-world data show that our approach is rather effective for traffic flow forecasting.

## 2 Methodology

In a transportation network, there are usually a lot of sites (road links) related or informative to the traffic flow of the current site from the standpoint of causal Bayesian networks. However, using all the related links as input variables (cause nodes) would involve much irrelevance, redundancy and would be prohibitive for computation. Consequently, a variable selection procedure is of great demand. Up to date many variable selection algorithms include variable ranking as a principal or auxiliary selection mechanism because of its simplicity, scalability, and good empirical success [10]. In this article, we also adopt the variable ranking mechanism, and the Pearson correlation coefficient is used as the specific ranking criterion defined for individual variables.

### 2.1 Variable Ranking and Cause Node Selection

Variable ranking can be regarded as a filter method: it is a preprocessing step, independent of the choice of the predictor [11]. Still, under certain independence or orthogonality assumptions, it may be optimal with respect to a given predictor [10]. Even when variable ranking is not optimal, it may be preferable to other variable subset selection methods because of its computational and statistical scalability [12]. This is also the motivation of our using the best-first search strategy to select the most relevant traffic flows from the ranking result as final cause nodes of a Bayesian network.

Consider a set of samples consisting of input variable and one output variable . Variable ranking makes use of a scoring function computed from the value and . By convention, we assume that a high score is indicative of a valuable variable and that we sort variables in decreasing order of . Furthermore, let denote the random variable corresponding to the component of input vector , and denote the random variable of which the outcome is a realization. The Pearson correlation coefficient between and can be estimated by:

(1) |

where the bar notation stands for an average over the index [10].

In this article, we use the norm as a variable ranking criterion. After the variable ranking stage, a variable selection (cause node selection) process is adopted to determine the final cause nodes (input variables) for predict the effect node (output). Here we use the best-first search strategy to find the cause nodes as the first several variables in the ranking list because of its fastness, simplicity and empirical effectiveness.

### 2.2 Flow Chart for Traffic Flow Forecasting

Given the derived cause nodes and the effect node in a Bayesian network, we utilize the Gaussian Mixture Model (GMM) and the Competitive EM (CEM) algorithm to approximate their joint probability distribution. Then we can obtain the optimum prediction formulation as an analytic solution under the M.M.S.E. criterion. For details about the GMM, CEM algorithm and the prediction formulation, please refer to articles [8][13][14].

Now we describe the flow chart of our approach for traffic flow forecasting. First the data set is divided into two parts, one serving as training set for input variable (cause node) selection and parameter learning, and the other test set. The flow chart can be given as follows: 1) Choose an object road site whose traffic flow should be forecasted (effect node) and collect all the available traffic flows in a traffic network as the original input variables; 2) Compute the Pearson correlation coefficients between the object traffic flow (effect node) and the input variables on the training set with different time lags respectively, and then select several most related variables in the ranking list as the final cause nodes of the spatio-temporal Bayesian network; 3) Derive the optimum prediction formulation using GMM and CEM algorithm detailed in articles [8][14]; 4) Implement forecasting on the test set using the derived formulation.

Conveniently, the flow chart can be largely reduced and for real-time utility when forecasting a new traffic flow, because the cause node selection and the prediction formulation need only be computed one time based on the historical traffic flows (learning stage), and thus can be derived in advance.

## 3 Experiments

The field data analyzed in this paper is the vehicle flow rates of discrete time series recorded every 15 minutes along many road links by the UTC/SCOOT system in Traffic Management Bureau of Beijing, whose unit is vehicles per hour (veh/hr). Fig. 1 depicts a real patch used to verify our proposed predictor. The raw data for utility are of 25 days and totally 2400 sample points taken from March, 2002. To validate our approach, the frist 2112 points (training set) are employed to carry out input cause node selection and to learn parameters of the spatio-temporal Bayesian network, and the rest (test set) are employed to test the forecasting performance.

In addition, we utilize the the Random Walk method and Markov Chain method as base lines to evaluate our presented approach [8]. Random Walk is a classical method for traffic flow forecasting. Its core idea is to forecast the current value using its last value, and can be formulated as:

(2) |

Markov Chain method models traffic flow as a high order Markov chain. It has shown great merits over several other approaches for traffic flow forecasting [8]. In this paper the joint probability distribution for the Markov Chain method is also approximated by the GMM whose parameters are estimated through CEM algorithm. The number of input variables is also taken as 4 (same as in [8]) for each object site in our approach. This entire configuration is to make an equitable comparison as much as possible. Now the only difference between our Bayesian network predictor and the Markov Chain method is that we utilize the whole spatial and temporal information in a transportation network to forecast while the latter only uses the temporal information of the object site.

We take road link as an example to show our approach. represents the vehicle flow from upstream link to downstream link . All the available traffic flows which may be informative to forecast in the transportation network includes . Considering the time factor, to forecasting the traffic flow (effect node), we need judge the above sites with different time indices, such as , etc. In this paper, is taken as 100 empirically. Four most correlated traffic flows which are selected with the correlation variable ranking criterion and the best-first strategy for five different object traffic flows and the corresponding correlation coefficient values are listed in Table 1.

Object traffic flows | Strongly correlated traffic flows (cause nodes) | |||
---|---|---|---|---|

0.971 | 0.968 | 0.967 | 0.966 | |

0.963 | 0.961 | 0.959 | 0.959 | |

0.983 | 0.978 | 0.964 | 0.961 | |

0.967 | 0.962 | 0.962 | 0.957 | |

0.967 | 0.967 | 0.967 | 0.966 |

Methods | Ch | Dd | Fe | Gd | Ka |
---|---|---|---|---|---|

Random Walk | 79.85 | 70.99 | 157.60 | 177.57 | 99.20 |

Markov Chain | 68.51 | 66.15 | 122.65 | 151.31 | 80.46 |

Spatio-Temporal Bayesian Network | 65.95 | 57.46 | 115.07 | 141.37 | 73.02 |

With the selected cause nodes (input traffic flows), we can approximate the joint probability distribution between the input and output with GMM, then derive the optimum prediction formulation for road link . In addition, we also conducted experiments on four other traffic flows. Table 2 gives the forecasting errors denoted by Root Mean Square Error (RMSE) of all the five road links through Random Walk method, Markov Chain method and our predictor. In the same column of Table 2, the smaller RMSE corresponds to the better forecasting accuracy. From the experimental results, we can find the significant improvements of forecasting capability brought by the spatio-temporal Bayesian network predictor which integrates both spatial and temporal information for forecasting.

## 4 Conclusions and Future Work

In this paper, we successfully combine the whole spatial with temporal information available in a transportation network to carry out short-term traffic flow forecasting. Experiments show that distant road links in a transportation network can have high correlation coefficients, and this relevance can be employed for traffic flow forecasting. This knowledge would greatly broaden people’s traditional knowledge about transportation networks and the transportation forecasting research. Many existing methods can be illuminated and further developed on the scale of a transportation network. In the future, how to extend the presented spatio-temporal Bayesian network predictor to forecast traffic flows in case of incomplete data would be a valuable direction.

## Acknowledgements

The authors are grateful to the anonymous reviewers for giving valuable remarks. This work was supported by Project 60475001 of the National Natural Science Foundation of China.

### Footnotes

- email: sunsl02@mails.tsinghua.edu.cn, {zcs,zhyi}@mail.tsinghua.edu.cn
- email: sunsl02@mails.tsinghua.edu.cn, {zcs,zhyi}@mail.tsinghua.edu.cn
- email: sunsl02@mails.tsinghua.edu.cn, {zcs,zhyi}@mail.tsinghua.edu.cn

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