# Modeling Driver Behavior near Intersections in Hidden Markov Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Hidden Markov Model

- N is the number of possible hidden states in the model. Each individual state can be denoted as ${S}_{i}$, i.e., 1 ≤ I ≤ N. And the state symbol at time t is defined as q
_{t}. - M is the number of observable symbols per state ${v}_{k}$, i.e., 1 ≤ k ≤ M. And the observation symbol at time t is denoted as O
_{t}. - The state transition probability distribution $A=\left\{{a}_{ij}\right\}$ is denoted as:$${a}_{ij}=P\{{q}_{t+1}={S}_{j}\uff5c{q}_{t}={S}_{i}\},\text{\hspace{1em}}\begin{array}{c}1\le i,j\le N\end{array}$$
_{ij}representing the transition probability from state S_{i}to state S_{j}, have the following two constraints:$${a}_{ij}\ge 0,\text{\hspace{1em}}\begin{array}{c}1\le i,j\le N\end{array}$$$$\sum _{j=1}^{N}{a}_{ij}}=1,\text{\hspace{1em}}\begin{array}{c}1\le i\le N\end{array$$The constraint ${a}_{ij}\ge 0$ indicates that the state S_{i}can reach any other state S_{j}in one step. - The observation probability distribution $B=\left\{{b}_{j}\left(k\right)\right\}$ can be indicated as:$${b}_{j}(k)=P\{{v}_{k}\text{}\mathrm{at}\text{}t\uff5c{q}_{t}={S}_{j}\},\text{\hspace{1em}}\begin{array}{c}1\le j\le N,1\le k\le M\end{array}$$
_{j}(k) represents the probability of the state value j at time t with the observation symbol v_{k}. - The initial state probability distribution $\pi =({\pi}_{i})$, where$${\pi}_{i}=p\{{q}_{1}={S}_{i}\},\text{\hspace{1em}}\begin{array}{c}1\le i\le N\end{array}$$
_{i}are the probabilities of S_{i}being the initial state in a state sequence.

_{i}and S

_{j}at time t and t + 1, respectively, given the observation sequence O and the model parameters λ. ${\alpha}_{t}(i)$ is the partial observation sequence ${O}_{1},{O}_{2},{O}_{3},\cdots ,{O}_{t}$ given state S

_{i}at time t. ${\beta}_{t+1}(j)$ represents the remainder of the observation sequence ${O}_{t+1},{O}_{t+2},{O}_{t+3},\cdots ,{O}_{T}$ given state S

_{j}at time t + 1.

_{i}at time t given the observation sequence O and the model parameters λ. And then, parameters can be updated as follows [34,40]:

_{i}at time t (t = 1).

_{i}to S

_{j}. ${\sum}_{t=1}^{T-1}{\gamma}_{t}(i)$ is the expected number of transitions from state S

_{i}.

_{k}. The denominator is the expected number of times in state j.

#### 2.2. Hidden Markov Driving Model

_{i}at time t.

_{jk}represents the corresponding probability of dangerous situations.

_{t}are chosen when they are individually most likely to occur. However, this could still result in an invalid state sequence [40].

_{i}. And define ${\phi}_{t}(i)$ to record the state sequence.

## 3. Data Description

#### 3.1. Data Collection

#### 3.2. Vehicle State Sequence

_{ab}) and b to c (t

_{bc}) can be calculated:

_{ab}and n

_{bc}are respectively the number of frames when the vehicle passes through detection line a to b and the number of frames when vehicle passes through detection line b to c. N is the number of frames defined to represent 1 s, which is set to 30 frames per second in this study. Then the vehicle speed at detection line b can be calculated:

#### 3.3. Model Development

_{1}, p

_{2}, p

_{3}and p

_{4}, respectively. They are the initial probability distribution of π, namely, π

_{1}, π

_{2}, π

_{3}, and π

_{4}in the HMM model. In this study, the Baum-Welch algorithm is adopted to estimate the value of π

_{i}, a

_{ij}and b

_{j}(k). Then the supervised learning algorithm based MLE method is utilized to test the results of the B-W algorithm; that is, the estimation of the parameters of the HMM $\lambda =(A,B,\pi )$.

_{ij}is the number of observations with the state value at time t and t + 1 are S

_{i}and S

_{j}respectively. B

_{jk}is the frequency of observations with the state value j and the observation symbol v

_{k}at time t.

## 4. Results and Discussion

#### 4.1. Estimation of Driver Behavior

#### 4.1.1. Parameter Calibration Results

#### 4.1.2. Stability of Driver Behavior

#### 4.1.3. Risk of Driver Behavior

_{jk}represents the corresponding probability of the dangerous combinations. The larger values of the risk index indicate more dangerous conditions. Table 6 shows the risk index of the whole road and the three predefined zones.

#### 4.2. Predicting Driver Behavior

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- National Highway Traffic Safety Administration. Traffic Safety Facts 2010; National Highway Traffic Safety Administration: Washington, DC, USA, 2012.
- Schorr, J.P.; Hamdar, S.H. Safety propensity index for signalized and unsignalized intersections: Exploration and assessment. Accid. Anal. Prev.
**2014**, 71, 93–105. [Google Scholar] [CrossRef] [PubMed] - Guo, F.; Wang, X.; Abdelaty, M.A. Modeling signalized intersection safety with corridor-level spatial correlations. Accid. Anal. Prev.
**2010**, 42, 84–92. [Google Scholar] [CrossRef] [PubMed] - Chin, H.C.; Quddus, M.A. Applying the random effect negative binomial model to examine traffic accident occurrence at signalized intersections. Accid. Anal. Prev.
**2003**, 35, 253–259. [Google Scholar] [CrossRef] - Wang, X.; Abdel-Aty, M.; Brady, P. Crash estimation at signalized intersections: Significant factors and temporal effect. Transp. Res. Rec.
**2006**, 10–20. [Google Scholar] [CrossRef] - Porter, B.E.; England, K.J. Predicting red-light running behavior—A traffic safety study in three urban settings. J. Saf. Res.
**2000**, 31, 1–8. [Google Scholar] [CrossRef] - Poch, M.; Mannering, F. Negative binomial analysis of intersection-accident frequencies. J. Transp. Eng.
**1996**, 122, 105–113. [Google Scholar] [CrossRef] - Abdelaty, M.A.; Radwan, A.E. Modeling traffic accident occurrence and involvement. Accid. Anal. Prev.
**2000**, 32, 633–642. [Google Scholar] [CrossRef] - Jovanis, P.P.; Chang, H.L. Modeling the Relationship of Accidents to Miles Traveled. In Transportation Research Record; No. 1068; Transportation Research Board: Washington, DC, USA, 1986; pp. 42–51. [Google Scholar]
- Jones, B.; Janssen, L.; Mannering, F. Analysis of the frequency and duration of freeway accidents in Seattle. Accid. Anal. Prev.
**1991**, 23, 239–255. [Google Scholar] [CrossRef] - Joshua, S.C.; Garber, N.J. Estimating truck accident rate and involvements using Linear and Poisson Regression models. Transp. Plan. Technol.
**1990**, 15, 41–58. [Google Scholar] [CrossRef] - Xie, K.; Wang, X.; Huang, H.; Chen, X. Corridor-level signalized intersection safety analysis in Shanghai, China using Bayesian hierarchical models. Accid. Anal. Prev.
**2013**, 50, 25–33. [Google Scholar] [CrossRef] [PubMed] - Miaou, S.P. The relationship between truck accidents and geometric design of road sections: Poisson versus negative binomial regressions. Accid. Anal. Prev.
**1994**, 26, 471–482. [Google Scholar] [CrossRef] - Miaou, S.P.; Lum, H. Modeling vehicle accidents and highway geometric design relationships. Accid. Anal. Prev.
**1993**, 25, 689–709. [Google Scholar] [CrossRef] - Shankar, V.; Mannering, F.; Barfield, W. Effect of roadway geometrics and environmental factors on rural freeway accident frequencies. Accid. Anal. Prev.
**1995**, 27, 371–389. [Google Scholar] [CrossRef] - Abdel-Aty, M.; Wang, X. Crash estimation at signalized intersections along corridors: Analyzing spatial effect and identifying significant factors. Transp. Res. Rec.
**2006**, 1953, 98–111. [Google Scholar] [CrossRef] - Lord, D.; Persaud, B. Accident prediction models with and without trend: Application of the generalized estimating equations procedure. Biochem. Int.
**2000**, 1717, 102–108. [Google Scholar] [CrossRef] - Ahmed, M.; Huang, H.; Abdel-Aty, M.; Guevara, B. Exploring a Bayesian hierarchical approach for developing safety performance functions for a mountainous freeway. Accid. Anal. Prev.
**2011**, 43, 1581–1589. [Google Scholar] [CrossRef] [PubMed] - Anastasopoulos, P.C.; Mannering, F.L. A note on modeling vehicle accident frequencies with random-parameters count models. Accid. Anal. Prev.
**2009**, 41, 153–159. [Google Scholar] [CrossRef] [PubMed] - Dinu, R.R.; Veeraragavan, A. Random parameter models for accident prediction on two-lane undivided highways in India. J. Saf. Res.
**2011**, 42, 39–42. [Google Scholar] [CrossRef] [PubMed] - Huang, H.; Abdelaty, M. Multilevel data and bayesian analysis in traffic safety. Accid. Anal. Prev.
**2010**, 42, 1556–1565. [Google Scholar] [CrossRef] [PubMed] - Sabey, B.E.; Taylor, H. The Known Risks We Run: The Highway. In Societal Risk Assessment; Session I; Springer: New York, NY, USA, 1980; pp. 43–70. [Google Scholar]
- Chen, H.; Cao, L.; Logan, D.B. Analysis of risk factors affecting the severity of intersection crashes by logistic regression. Traffic Inj. Prev.
**2012**, 13, 300–307. [Google Scholar] [CrossRef] [PubMed] - Cooper, P.J. Differences in accident characteristics among elderly drivers and between elderly and middle-aged drivers. Accid. Anal. Prev.
**1990**, 22, 499–508. [Google Scholar] [CrossRef] - Holubowycz, O.T.; Kloeden, C.N.; Mclean, A.J. Age, sex, and blood alcohol concentration of killed and injured drivers, riders, and passengers. Accid. Anal. Prev.
**1994**, 26, 483–492. [Google Scholar] [CrossRef] - Kim, K.; Brunner, I.M.; Yamashita, E. Modeling fault among accident—Involved pedestrians and motorists in Hawaii. Accid. Anal. Prev.
**2008**, 40, 2043–2049. [Google Scholar] [CrossRef] [PubMed] - Shinar, D.; Schechtman, E.; Compton, R. Self-reports of safe driving behaviors in relationship to sex, age, education and income in the US adult driving population. Accid. Anal. Prev.
**2001**, 33, 111–116. [Google Scholar] [CrossRef] - Zhang, G.; Yau, K.K.W.; Chen, G. Risk factors associated with traffic violations and accident severity in China. Accid. Anal. Prev.
**2013**, 59, 18–25. [Google Scholar] [CrossRef] [PubMed] - Mogens, F. Speed and income. J. Transp. Econ. Policy
**2005**, 39, 225–240. [Google Scholar] - Factor, R.; Mahalel, D.; Yair, G. Inter-group differences in road-traffic crash involvement. Accid. Anal. Prev.
**2008**, 40, 2000–2007. [Google Scholar] [CrossRef] [PubMed] - Aoude, G.S.; Desaraju, V.R.; Stephens, L.H.; How, J.P. Driver behavior classification at intersections and validation on large naturalistic data set. IEEE Trans. Intell. Transp. Syst.
**2012**, 13, 724–736. [Google Scholar] [CrossRef] - Bougler, B.; Cody, D.; Nowakowski, C. California Intersection Decision Support: A Driver-Centered Approach to Left-Turn Collision Avoidance System Design. Available online: http://www.path.berkeley.edu/sites/default/files/publications/PRR-2008-01.pdf (accessed on 31 August 2016).
- Zou, X.; Levinson, D. Modeling pipeline driving behaviors: A Hidden Markov Model approach. Transp. Res. Rec.
**2006**, 1980, 16–23. [Google Scholar] [CrossRef] - Gadepally, V.; Krishnamurthy, A.; Ozguner, U. A framework for estimating driver decisions near intersections. IEEE Trans. Intell. Transp. Syst.
**2014**, 15, 637–646. [Google Scholar] [CrossRef] - Oliver, N.; Pentland, A.P. Graphical Models for Driver Behavior Recognition in a Smartcar. In Proceedings of the IEEE Intelligent Vehicles Symposium 2000 (Cat. No.00TH8511), Dearborn, MI, USA, 5 October 2000; pp. 7–12.
- Mitrovic, D. Reliable method for driving events recognition. IEEE Trans. Intell. Transp. Syst.
**2005**, 6, 198–205. [Google Scholar] [CrossRef] - Crawford, A. Driver judgment and error during the amber period at traffic light. Ergonomics
**1962**, 5, 513–532. [Google Scholar] [CrossRef] - Gazis, D.; Herman, R.; Maradudin, A. The problem of the amber signal light in traffic flow. Oper. Res.
**1960**, 8, 112–132. [Google Scholar] [CrossRef] - Baum, L.E.; Petrie, T. Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Stat.
**1966**, 37, 1554–1563. [Google Scholar] [CrossRef] - Rabiner, L.R. A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition. Available online: http://www.ece.ucsb.edu/Faculty/Rabiner/ece259/Reprints/tutorial%20on%20hmm%20and%20applications.pdf (accessed on 31 August 2016).
- Cheshomi, S.; Rahati-Q, S.; Akbarzadeh-T, M.R. Hybrid of chaos optimization and Baum-Welch algorithms for HMM training in continuous speech recognition. In Proceedings of the 2010 International Conference on Intelligent Control and Information Processing, Dalian, China, 13–15 August 2010; pp. 83–87.
- Meng, X.; Lee, K.K.; Xu, Y. Human driving behavior recognition based on Hidden Markov Models. In Proceedings of the 2006 IEEE International Conference on Robotics and Biomimetics, Kunming, China, 17–20 December 2006; pp. 274–279.
- Zegeer, C.V. Effectiveness of Green-Extension Systems at High-Speed Intersections. Available online: http://uknowledge.uky.edu/cgi/viewcontent.cgi?article=2066&context=ktc_researchreports (accessed on 31 August 2016).

Lanes | Traffic Volume (vph) | Speed Limit (km/h) | Cycle Length (s) | Green Time (s) |
---|---|---|---|---|

straight direction | 1900 | 60 | 190 | 35 |

Speed | Headway | Queue Length | Signal Light | ||||
---|---|---|---|---|---|---|---|

Before (m/s) | After | Before (s) | After | Before | After | Before | After |

≤8 | 1 | Head Car | 1 | Head Car | 1 | Green | 1 |

(8,16) | 2 | (0,6) | 2 | No preceding car stopped | 2 | Red | 2 |

≥16 | 3 | ≥6 | 3 | Others | 3 | Yellow | 3 |

Sequence Number | Speed | Headway | Queue Length | Signal Light |
---|---|---|---|---|

1 | 1 | 1 | 1 | 1 |

2 | 1 | 1 | 1 | 2 |

3 | 1 | 1 | 1 | 3 |

4 | 1 | 1 | 2 | 1 |

5 | 1 | 1 | 2 | 2 |

6 | 1 | 1 | 2 | 3 |

7 | 1 | 1 | 3 | 1 |

8 | 1 | 1 | 3 | 2 |

9 | 1 | 1 | 3 | 3 |

Classification | Included Variables |
---|---|

Hidden State Variables | Acceleration |

Deceleration | |

Maintain Speed | |

Stop | |

Observed Variables | Speed |

Headway | |

Queue Length | |

Signal Light |

Road Range | 1st Zone | 2nd Zone (Dilemma Zone) | 3rd Zone |
---|---|---|---|

2-norm of Matrix B | 0.595 | 0.448 | 0.518 |

Road Range | 1st Zone | 2nd Zone (Dilemma Zone) | 3rd Zone |
---|---|---|---|

Risk Index α | −5.437 | −3.343 | −8.881 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, J.; He, Q.; Zhou, H.; Guan, Y.; Dai, W. Modeling Driver Behavior near Intersections in Hidden Markov Model. *Int. J. Environ. Res. Public Health* **2016**, *13*, 1265.
https://doi.org/10.3390/ijerph13121265

**AMA Style**

Li J, He Q, Zhou H, Guan Y, Dai W. Modeling Driver Behavior near Intersections in Hidden Markov Model. *International Journal of Environmental Research and Public Health*. 2016; 13(12):1265.
https://doi.org/10.3390/ijerph13121265

**Chicago/Turabian Style**

Li, Juan, Qinglian He, Hang Zhou, Yunlin Guan, and Wei Dai. 2016. "Modeling Driver Behavior near Intersections in Hidden Markov Model" *International Journal of Environmental Research and Public Health* 13, no. 12: 1265.
https://doi.org/10.3390/ijerph13121265