﻿ 高速公路交通流量的齐次泊松过程模型检验 Test for the Homogeneous Poisson Process Model of Highway Traffic Flows

Statistics and Application
Vol.06 No.05(2017), Article ID:23244,7 pages
10.12677/SA.2017.65065

Test for the Homogeneous Poisson Process Model of Highway Traffic Flows

Ya’nan Li, Shuhe Lei*, Fenglin Sun

School of Mathematical Sciences, Ocean University of China, Qingdao Shandong

*通讯作者。

Email: *lyn_stat@163.com

Received: Dec. 7th, 2017; accepted: Dec. 22nd, 2017; published: Dec. 29th, 2017

ABSTRACT

In order to build a traffic model of vehicles arrival times of highway, using the observed traffic data, a stochastic point pattern is composed by recording vehicles arrival times. Based on the theory of spatial point process, the Ripley’s K function is applied to one-dimensional stochastic point process. According to the characteristics of Homogeneous Poisson Process, a graphical approach is proposed to test Homogeneous Poisson Process combined with envelope test. And the effectiveness of the test is verified by simulation. Then the graphical approach is applied to the patterns, and the result indicates that arrival times of vehicles composed a Homogeneous Poisson Process. Finally, the traffic model of Chang-Tai freeway is obtained by estimating the parameter of the Homogeneous Poisson Process.

Keywords:Homogeneous Poisson Process, Traffic Flows, K Function, Envelope Test

Email: *lyn_stat@163.com

1. 引言

2. 理论与方法

2.1. 空间点过程与Ripley的K函数

$K\left(r\right)=\frac{1}{\lambda }E\left[{N}_{r}\left(x\right)\right],\text{\hspace{0.17em}}r>0$ (1)

$K\left(r\right)=\pi \cdot {r}^{2},\text{\hspace{0.17em}}r>0$ (2)

 (3)

$\stackrel{^}{K}\left(r\right)=\frac{1}{\stackrel{^}{\lambda }}\frac{\underset{i=1}{\overset{N\left(\phi \right)}{\sum }}\underset{j=1}{\overset{N\left(\phi \right)}{\sum }}I\left\{|{x}_{i}-{x}_{j}| (4)

$K\left(r\right)=2r$ (5)

2.2. 图形化检验方法

$\mathrm{log}\left\{{\beta }^{n}\mathrm{exp}\left[-\beta \underset{i=1}{\overset{n}{\sum }}{T}_{i}\right]\mathrm{exp}\left[-\beta \left(T-{t}_{n}\right)\right]\right\}=n\mathrm{log}\beta -\beta T$ (6)

$\stackrel{^}{\beta }=n/T$ (7)

${F}_{i}\left(r\right)={\stackrel{^}{K}}_{i}\left(r\right)-{K}_{0}\left(r\right)$ (8)

${F}_{0}\left(r\right)=0$ ，并定义上下包络为：

$U\left(r\right)=\mathrm{max}\left\{{F}_{i}\left(r\right)\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=2,3,\cdots ,s+1$ (9)

$L\left(r\right)=\mathrm{min}\left\{{F}_{i}\left(r\right)\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=2,3,\cdots ,s+1$ (10)

$L\left(r\right)<{F}_{i}\left(r\right) , ${r}_{\mathrm{min}} (11)

2.3. 随机模拟

3. 实证分析

$t=m-\frac{p}{v}$ (12)

Figure 1. The envelope test of Homogeneous Poisson Process

Figure 2. The envelope test of the arrival times in Chang-tai Highway

4. 结论与讨论

Test for the Homogeneous Poisson Process Model of Highway Traffic Flows[J]. 统计学与应用, 2017, 06(05): 576-582. http://dx.doi.org/10.12677/SA.2017.65065

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