Chapter 5.Statistics and Application in Traffic Engineering Basic knowledge of statistics Three most-used distributions---Poisson,Binomial and Exponential M/M/1 model in queue theory Testing of hypothesis 2023/7/16
2023/7/16 Chapter 5. Statistics and Application in Traffic Engineering ◼ Basic knowledge of statistics ◼ Three most-used distributions --- Poisson, Binomial and Exponential ◼ M/M/1 model in queue theory ◼ Testing of hypothesis
1.Basic knowledge of statistics ■Population versus sample(全体与样本)- ■Discrete versus continuous(离散与连续)-- ■Statistical estimators(统计估计量)-- ■Data grouping(数据分组)- 2023/7/16
2023/7/16 1. Basic knowledge of statistics ◼ Population versus sample (全体与样本）--- ◼ Discrete versus continuous （离散与连续）--- ◼ Statistical estimators （统计估计量）--- ◼ Data grouping（数据分组）---
Frequency distribution of student's height Number of Height group(meter)Number of observation group 1 1.56-1.60 1 2 1.61-1.65 3 3 1.66-1.70 5 4 1.71-1.75 12 5 1.76-1.80 7 6 1.81-1.85 4 > 1.86-1.90 2 Total =34 2023/7/16
2023/7/16 Frequency distribution of student’s height Number of group Height group (meter) Number of observation 1 1.56 – 1.60 1 2 1.61 – 1.65 3 3 1.66 – 1.70 5 4 1.71 – 1.75 12 5 1.76 – 1.80 7 6 1.81 – 1.85 4 7 1.86 – 1.90 2 Total = 34
■mean(均值)，median(中位数)and mode(众数) are the measure of central tendency; ■variance(方差)and standard deviation are the measure of dispersion ■coefficient of variation(变异系数)indicates the spread of outcomes relative to the mean 2023/7/16
2023/7/16 ◼ mean （均值）, median （中位数）and mode （众数） are the measure of central tendency; ◼ variance （方差）and standard deviation are the measure of dispersion ◼ coefficient of variation （变异系数）indicates the spread of outcomes relative to the mean
skewness(非对称)is the measure of anomalies (不规则)in shape of distribution (mean-mod e) standard deviaton Ifa distribution is negatively skewed,it means that the data is concentrated to the left of the most frequent value;whereas if a distribution is positively skewed, then data is concentrated to the right. 2023/7/16
2023/7/16 ◼ skewness (非对称）is the measure of anomalies (不规则）in shape of distribution ◼ If a distribution is negatively skewed, it means that the data is concentrated to the left of the most frequent value; whereas if a distribution is positively skewed, then data is concentrated to the right. stan dard deviaton (mean − mod e)
Questions to address How many samples are needed to meet requirement for analysis? ■In what degree of confidence(置信区间)should we have in estimation of traffic characteristics? What statistical distribution can best picture the observed data? ■How to apply statistical model(统计模型)to work out traffic problems? 2023/7/16
2023/7/16 Questions to address ◼ How many samples are needed to meet requirement for analysis? ◼ In what degree of confidence （置信区间）should we have in estimation of traffic characteristics? ◼ What statistical distribution can best picture the observed data? ◼ How to apply statistical model （统计模型） to work out traffic problems?
2.Three most-used distributions Poisson distribution ---counting distribution Binomial distribution ---choice distribution Exponential distribution---headway distribution 2023/7/16
2023/7/16 2.Three most-used distributions ◼ Poisson distribution --- counting distribution ◼ Binomial distribution --- choice distribution ◼ Exponential distribution --- headway distribution
2.1 Poisson Distribution(泊松分布) P(X=x)---probability of x event occurring (vehicles arriving)at given T period P(X=x)=(AT)'e-ir x! T--given period in second ---the average rate of event occurring (vehicles arriving) x---variable of event occurring (number of vehicles) 2023/7/16
2023/7/16 2.1 Poisson Distribution (泊松分布） ◼ P(X=x) --- probability of x event occurring (vehicles arriving) at given T period ◼ T --- given period in second ◼ λ--- the average rate of event occurring (vehicles arriving) ◼ x --- variable of event occurring (number of vehicles) ( ) ( ) ! x T T e P X x x − = =
Features of Poisson Distribution Poisson distribution belongs to discrete function with only one variable In traffic engineering Poisson distribution equation is used to describe the arrivals of vehicles at intersections or toll booth(收费口)， as well as number of accident (crash) 2023/7/16
2023/7/16 Features of Poisson Distribution ◼ Poisson distribution belongs to discrete function with only one variable ◼ In traffic engineering Poisson distribution equation is used to describe the arrivals of vehicles at intersections or toll booth (收费口), as well as number of accident (crash)
■ Poisson distribution is appropriate to describe vehicle's arrival when traffic volume is not high When field data shows that the mean and variance (have significant difference,we can no longer apply Poisson distribution 2023/7/16
2023/7/16 ◼ Poisson distribution is appropriate to describe vehicle’s arrival when traffic volume is not high. ◼ When field data shows that the mean and variance (方差) have significant difference, we can no longer apply Poisson distribution