SENCER Summer Institute 2008 Traffic!. Woodbury University »Small »Professional focus »Architecture »Professional Design »Business »Liberal Arts »Burbank

SENCER

Summer Institute

2008

Traffic!Traffic!

Woodbury UniversityWoodbury University

» SmallSmall» Professional focusProfessional focus

» ArchitectureArchitecture» Professional DesignProfessional Design» BusinessBusiness» Liberal ArtsLiberal Arts

» Burbank and San DiegoBurbank and San Diego

Course DevelopmentCourse Development

» Why?Why?» Limitations of a single Limitations of a single disciplinediscipline» Integrate scientific knowledgeIntegrate scientific knowledge» To solve complex real world issuesTo solve complex real world issues

» Improved critical thinking Improved critical thinking skillsskills» Transdisciplinary thinkingTransdisciplinary thinking» Team teachingTeam teaching


» Who developed the course?Who developed the course?» Nageswar Rao Chekuri - physicsNageswar Rao Chekuri - physics» Nick Roberts - architectureNick Roberts - architecture» Marty Tippins - mathematicsMarty Tippins - mathematics» Zelda Gilbert - psychologyZelda Gilbert - psychology» Ken Johnson - Ken Johnson -

» traffic engineer, City of Burbanktraffic engineer, City of Burbank

» Anil Kantak - Anil Kantak - » communications engineer, JPLcommunications engineer, JPL


» WhatWhatSC 370.3 - TRAFFICSC 370.3 - TRAFFICTopics courseTopics courseTeam taughtTeam taughtProject orientedProject orientedTransdiciplinaryTransdiciplinaryMeets upper division G.E. Meets upper division G.E. requirementrequirement

Course DescriptionCourse DescriptionA team taught class covering both overall

implications and consequences of traveling by personal vehicle as well as more specific issues. Topics include the history of traffic in cities in the American West, the role of communications in alleviating traffic problems, the mathematics and the physics of traffic, and psychological issues such as aggressive driving and road rage. The course will also allow students to explore the challenges facing the existing system in the next few years, including population growth, congestion, the end of oil and the economic effects of carbon emissions.

Course PrerequisitesCourse Prerequisites

» WritingWriting» SpeechSpeech» MathematicsMathematics» ScienceScience» PsychologyPsychology

EnrollmentEnrollment

» Who enrolled?Who enrolled?» Six architecture majorsSix architecture majors

TrigonometryTrigonometryTwo semesters of physicsTwo semesters of physics

» One interior architecture majorOne interior architecture major» College AlgebraCollege Algebra» BiologyBiology

» One fashion design majorOne fashion design major» College AlgebraCollege Algebra» Human BiologyHuman Biology

Course ElementsCourse Elements

» Examples of presentationsExamples of presentations» MathematicsMathematics

» The Mathematics of TrafficThe Mathematics of Traffic

» PsychologyPsychology» Road RageRoad Rage

» Field TripField Trip» Burbank Traffic Command CenterBurbank Traffic Command Center

The Mathematics of Traffic

Marty Tippens

Introduction

» Mathematics as communication

» No one all-encompassing way to

model traffic.

Topics of Discussion

» Deriving the flow equations» Probability and Statistics» Queue Theory» Wave analysis and traffic» Chaos

I. Deriving the Flow Equation

» Traffic Flow as Fluid» Derivation of flow equation

(1.1) d rtDistance = (rate)(time)

Let c = number of cars

(1.2) cd crtcrt

cd

(1.3)

If d = r t, then r = d/t Substituting for r in equation (1.3) , we get

c d

c td t

c c d

t d t

(1.4)

(1.5)

(Number of cars per time) = (Number of cars per distance)(Distance per time)

• Number of cars per time is called flow. • Number of cars per distance is called density. • Distance per time is speed.

Let q = flow, k = density and

= speed.

Then equation (1.5) becomes

q k(1.6)

Reassign speed as v = average speed

(1.7) q = kv

Flow and average speed are functions of density

(1.8) q(k) = kv(k)

Traffic flow goes to zero in two instances

1. No traffic on the road2. Traffic is jam-packed

These two cases give us “boundary conditions”

Figure 1 – Flow as a function of density

Probability and Statistics are involved as various distributions are used to compute q, k and v

Common distributions used in traffic analysis

• Normal• Binomial• Poisson

II. Wave Propagation

• Freeway traffic appears to move in waves

• Road quality and/or the human element can cause a shift in traffic flow rate q and corresponding density k.

• Waves are backward moving as vehicles exert an influence only on the vehicles behind them.


Animation Video

QuickTime™ and aGIF decompressor

are needed to see this picture.





http://amasci.com/amateur/traffic/tanim.html


QuickTime™ and aH.264 decompressor



QuickTime™ and aH.264 decompressor


Velocity equals the difference in flow over the difference in density.

1 2

1 2

q qv

k k

(1.17)

Three characteristics of wave propagation:

1. The range of zero flow at zero density to maximum flow corresponds to relatively uncongested traffic flow. A small increase in domain moves forward along the road.

2. The range from maximum flow to zero flow at “jam” density corresponds to congested stop and go traffic.

3. Any transition from one steady state flow to another is associated with wave propagation given by

the slope of the segment CD in Figure 1.

III. The Normal Distribution

• History of the normal distribution.

• Properties of the normal distribution

• Applications to traffic on the 405

Brief History

The Normal Probability Distribution

• The normal distribution is the most commonly observed probability distribution.

• First published by Abraham de Moivre in 1733.

• Used by Carl Friedrich Gauss in the early 19th century in astronomical applications

• AKA Gaussian Distribution and Bell Curve

Car Crashes: In a study of 11,000 car crashes, it was found that 5720 of them occurred within 5 miles of home (based on data from Progressive Insurance). Use a 0.01 significance level to test the claim that more than 50% of car crashes occur within 5 miles of home. Are the results questionable because they are based on a survey sponsored by an insurance company?

This is an example of a problem involving a proportion (p = 5720/11,000). We state the Hypotheses, compute a test statistic and use it for comparison on the normal curve.

Mario Triola, Elementary Statistics, (Addison Wesley,10th ed.)415.

5720.5

ˆ 11,000 4.199(.52)(.48)11,000

p pz

pqn

H0: p=.5H1: p>.5

Testing a Claim About a Proportion

The test statistic is determined by the formula

With the test statistic z = 4.199 deep into the rejection region, we have sufficient evidence to reject the null hypothesis at the .01 significance level and support alternative hypothesis that more than 50% of accidents occur within 5 miles of the home.

Normal distribution example with hypothesis testing applied to traffic on the 405.

A section of Highway 405 in Los Angeles has a speed limit of 65 mi/h, and recorded speeds are listed below for randomly selected cars traveling on northbound and southbound lanes. Using all the speeds, test the claim that the

mean speed is greater than the posted speed limit of 65 mi/h.

Ho: u=65Ha: u>65

Hypothesis Testing of Traffic Speeds on the 405 Freeway

68.375 653.765

5.66940

N

N

xt

sn

The critical value corresponding to a 99% confidence level is t=2.429.

The area of the reject region is .01. Our test statistic of t = 3.765 is to the right of t = 2.429. This puts us in the rejection region and corresponds to an area smaller than .01.That means there is less than a 1% chance that the actual Mean speed is not greater than 65mph.

Test statistic t = 3.765. Critical value for 95% confidence is approximately 1.686. For 99% confidence it is 2.429. In either case we reject the null hypothesis. We can be 99% sure that the average speed driven on this section of the 405 is greater than 65 mph, at least for this time of day.

Hypothesis Testing of Two Independent Samples

Here we test the claim that the mean speed on the northbound lane is equal to the mean speed on the southbound lane.

If we assume the data comes from a normally distributed population, we can use a version of the student t-distribution for two independent samples.

1 2 1 2

2 21 2

1 2

x xt

s sn n

69.5 67.25 01.265

3.41 7.1920 20

t

The critical values for a .05 significance level are t = +/-2.093. Our test statistic is 1.265.

With t = 1.265 we fail to reject the null hypothesis.

Do Airbags save lives? The National Highway Transportation Safety Ad ministration reported that for a recent year, 3,448 lives were saved because of air bags. It was reported that for car drivers involved in frontal crashes, the fatality rate was reduced 31%; for passengers, there was a 27% reduction. It was noted that "calculating lives saved is done with a mathematical analysis of the real-world fatality experience of vehicles with air bags compared with vehicles with out air bags. These are called double-pair comparison studies, and are widely accepted methods of statistical analysis." (Triola p487)

IV Queue Theory

• Developed by French mathematician S.D. Poisson (1781-1840)

• A statistical approach applied to any situation where excessive demands are made on a limited resource.

• Early applications in telephone traffic.

• Applications in road traffic build-up at intersections or in congestion

• A discrete probability distribution

• Expresses the probability of a number of discrete independent events occurring in a fixed period of time

• The discrete events are called "arrivals"

• Events take place during a time-interval of given length.

The Poisson Distribution

( )!

xeP x

x

(1.18)

• x = number of occurrences of an event over some interval (time or space).

• Mean where p is the probability of the event. • Standard deviation

np

Probability of n arrivals during one service time period has Poisson distribution with parameter (number of arrivals) where v is service period and is the mean. The mean is calculated by number of arrivals during service period. One challenge is to evaluate the probability of queue length changes.

v

• Chaos theory studies how complexity emerges from simple events. The butterfly effect is a classic example.

• Chaos theory was formulated during the 1960s. The name chaos was coined by Jim Yorke, an applied mathematician at the University of Maryland.

• Applied to traffic: Small changes in one part of traffic result in large changes “down the road.” Traffic also has a self-replicating characteristics.

V. Chaos

• Fractals

• Hilbert’s Curve

• Computing fractional dimension

Is it new?

» From late 1980s: talk about road rage and aggressive driving increased.

» At the same time, the number of deaths due to crashes gradually decreased.

» Increase in vigilante behavior, driven by examples in movies and TV.

What IS Road Rage?

"Aggressive driving" - an incident in which an angry or impatient motorist or passenger intentionally injures or kills another person or attempts to injure or kill another in response to a traffic dispute, altercation, or grievance or intentionally drives his or her vehicle into a building or other structure or property.

Prevalence

» Estimates of the number of aggressive driving incidents reach as high as 1.8 billion per year.

» 25% of drivers surveyed admitted that they have driven aggressively

What IS Road Rage?

Frustration leads to angerAnger can lead to aggression, but not in

everyoneIn road rage, aggression escalates as a

result of repetition.

”Aggressive drivers become angry when someone blocks them from achieving goals they have set for themselves. They believe their goals to be virtuous, and their self-esteem is at stake if they can’t achieve them.”

Causes of Road Rage

» Psychological issues

» A lack of responsible driving behavior, driven by psychological issues

» Environmental issues

» Reduced levels of traffic violation enforcement

» More traffic congestion, especially in urban areas.

Psychological Issues

» Personality or environment?» Self-esteem

» Cars are an extension of the self.» Insult or injury to our cars is a threat to our self-esteem.

» Cars provide anonymity.» Cars are powerful and obedient.» Fundamental error of attribution


» Certain personalities are predisposed to act aggressively.» Less control of hostility» Less tolerance of tension» Less maturity» Tendency to take risks


» Actual pathologies may be involved» Higher incidence in aggressive drivers than in the general population

Psychological AnalysisTypes Beliefs

Speeders I’m making good time

Competitors I need to be Number One

Passive-Aggressive Try and make me!

Narcissists They shouldn’t be allowed on the road

BurbankTraffic Command

Center

Traffic Command Center

Group ProjectsGroup Projects

» Requirements» The paper should contain an analysis of the

components that are discussed in the class and an analysis of your contribution to the topic TRAFFIC. The relations between the components and their relation to the TRAFFIC should clearly be discussed.


» Requirements (continued)» The paper also should contain a discussion of the

impact of your solution and how it compares with other solutions if any

» A conclusion/solution to the topic TRAFFIC supported by reasoning should be presented in the paper.


» Traffic Control Systems for the Traffic Control Systems for the 21st Century21st Century» Traffic signal communicationTraffic signal communication» Automation of driving systems Automation of driving systems » Smart/hybrid carsSmart/hybrid cars» And their impact on pollution, congestion, and And their impact on pollution, congestion, and

psychological behaviors.psychological behaviors.


» Reorganizing Los Angeles: Reorganizing Los Angeles: A A transportation plan for Los Angeles to transportation plan for Los Angeles to be Re-Routedbe Re-Routed» Complexity of traffic issuesComplexity of traffic issues» Impact of urban planningImpact of urban planning» Large mass transit systemsLarge mass transit systems» Stackable concept carsStackable concept cars» Alternate energy sourcesAlternate energy sources» All parts synthesized into a group of suggestionsAll parts synthesized into a group of suggestions


» PollutionPollution» Pollution as a societal problemPollution as a societal problem» Integrated information from science, psychology, and Integrated information from science, psychology, and

experiential knowledge.experiential knowledge.


» Los Angeles IntegratedLos Angeles Integrated» Traffic as an urban landscaping and infrastructure Traffic as an urban landscaping and infrastructure

issueissue» Residential areas near business complexesResidential areas near business complexes» Flexible freewaysFlexible freeways» Density of traffic flowDensity of traffic flow

IntegrationIntegrationTraffic

Urban Planing Infrastructure Education

The SyllabusThe Syllabus

Determining course Determining course effectiveness througheffectiveness through

» Criteria AnalysisCriteria Analysis

» SALG Data AnalysisSALG Data Analysis

8 Criteria

» Recognizing the issues (C1)

» Realizing the knowledge components (C2)

» Analyzing and synthesizing (C3)

» New ideas (C4)

» Interpreting and evaluating solution/s (C5)

» Addressing society’s problems in an informed manner (C6)

» Concept of the common good (C7)

» Participation and practice (C8)

SALG Data Analysis

» Visualizing results with bar graphs

» Wilcoxon Hypothesis Test to determine significance of results

Pre and Post SALG Results

Wilcoxon Hypothesis Test

Statement of Hypothesis to be tested.

H0: the median responses do not differ in the

pre and post SALG surveys

Ha: the median responses differ in the pre and

post SALG surveys

SPSS Wilcoxon Results

Things we’d do differently

Documents

SENCER Summer Institute 2008 Traffic!. Woodbury University »Small »Professional focus »Architecture »Professional Design »Business »Liberal Arts »Burbank