Modeling and Simulation
Vol.07 No.02(2018), Article ID:25214,7 pages
10.12677/MOS.2018.72012

Calibration of Car-Following Model Based on Cross-Entropy Method

Kaiyan Fu1, Jiandong Qiu1, Jiajie Pan2

1Shenzhen Urban Transport Planning Center Co., Ltd., Shenzhen Guangdong

2Traffic Information Engineering & Technology Research Center of Guangdong Province, Shenzhen Guangdong

Received: May 5th, 2018; accepted: May 23rd, 2018; published: May 30th, 2018

ABSTRACT

Calibration of car following models seeks for a more realistic representation of car following behavior in complex driving situations to improve traffic safety and to better understand several puzzling traffic flow phenomena, such as stop-and-go oscillations. However, calibrating these models is never a trivial task. This is caused by the fact that some parameters are generally not directly observable from traffic data. Moreover, conventional deterministic calibration methods always result in a large number of local optima. This contribution puts forward a framework of calibration based on Cross-Entropy Method (CEM), which approaches the optimal probability density function with monte carlo and important sampling strategy. Empirical cases calibrate the intelligent driving model with synthetic data and NGSIM data. The results not only verify the ability of CEM to search global optima, but also confirm the great potential of CEM to adopt into actual traffic measurements.

Keywords:Car-Following Models, Model Calibration, Cross-Entropy Method, Global Optima

1深圳市城市交通规划设计研究中心有限公司，广东 深圳

2广东省交通信息工程技术研究中心，广东 深圳

1. 引言

2. 智能驾驶模型

${\stackrel{˙}{v}}_{IDM}\left(s,v,\Delta v\right)=a\left[1-{\left(\frac{v}{{v}_{0}}\right)}^{4}-{\left(\frac{{s}^{*}\left(v,\Delta v\right)}{s}\right)}^{2}\right]$ (1)

${s}^{*}\left({v}_{f},\Delta v\right)={s}_{0}+{v}_{f}T-\frac{{v}_{f}\Delta v}{2\sqrt{ab}}$ (2)

3. 交叉熵算法

${X}_{*}=\mathrm{arg}\mathrm{min}PI\left(X\right),X\in \Omega$ (3)

$\stackrel{^}{l}\left(z\right)=\frac{1}{N}\sum _{i=1}^{N}I\left(PI\left({X}_{i}\right)\le z\right)\frac{f\left({X}_{i}\right)}{g\left({X}_{i}\right)}$ (4)

${X}_{i}$ 表示服从概率分布函数 $f\left(X\right)$ 的样本， $i=1,2,\cdots ,N$$I\left(PI\left({X}_{i}\right)\le z\right)$ 表示指示变量，当事件 $PI\left({X}_{i}\right)\le z$ 成立时取值1，其他情况取值0。当z越来越接近最优值 ${z}_{*}$ 时，满足事件 $PI\left({X}_{i}\right)\le z$ 的样本非常少，概率 $l\left(z\right)$ 的取值非常小，因此优化问题的求解转换成小概率事件的概率估计问题。

1) 迭代计数器k初始化为0，并令概率分布函数的参数 $\gamma ={\gamma }_{0}$

2) 当 $k=0,1,2,\cdots ,$ 生成一系列服从概率分布函数 $f\left(X|{\gamma }_{k}\right)$ 的随机样本 $\left\{{X}_{i,k}\right\}$ ，( $i=1,2,\cdots ,N$ )，并且计算各个样本的目标函数值 $PI\left({X}_{i,k}\right)$ 。将得到的PI值从小到大排序，并且挑选取得最小目标值的 $100\rho %$ (一般 $\rho$ 取0.05)作为重要样本，用 ${s}_{best}$ 表示，同时令 ${z}_{k}$ 等于排序好的 $100\rho %$ 分为的目标函数值。

3) 利用重要样本 $\left\{{X}_{j,k}\right\}\left(j\in {s}_{best}\right)$ 的相关信息更新概率分布函数的参数 $\gamma$ ，使得 ${\sum }_{i}\mathrm{ln}\left(f\left({X}_{j,k}|{\gamma }_{k}\right)\right)$ 取最小值，用 ${\gamma }_{k}^{new}$ 表示。更新参数的原则如下所述：

${\gamma }_{k+1}=\beta {\gamma }_{k}^{new}+\left(1-\beta \right){\gamma }_{k}$ (5)

4) 计数器k加1，并返回第二步，直到满足预设的迭代停止条件。

4. 跟驰模型标定的目标函数

$PI\left(X\right)=\sum _{t=1}^{{T}_{t}}{\left(\mathrm{ln}{s}_{t}^{cali}\left(X\right)-\mathrm{ln}{s}_{t}^{obse}\right)}^{2}$ (6)

5. 案例分析

5.1. 标定合成路径数据

5.2. 标定NGSIM路径数据

NGSIM路径数据收集的是加利福利亚州内名为Interstate 80高速公路的一段路径，时间是从下午4点到4点15分，日期是2005年4月13号。该路段由6条车道和一条汇流闸道组成。在这个例子中，我们随机挑选了该路径上的一对跟车对作为标定数据源，验证交叉熵算法在标定实测路径数据的有效性。得到的结果如表2图2所示。

Table 1. The calibration result with synthetic data

Table 2. The calibration result with NGSIM data

Figure 1. The comparison between simulated data and synthetic data

Figure 2. The comparison between simulated data and NGSIM data

6. 结束语

Calibration of Car-Following Model Based on Cross-Entropy Method[J]. 建模与仿真, 2018, 07(02): 96-102. https://doi.org/10.12677/MOS.2018.72012

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