# 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

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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 |

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**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