Research Article Crossing Reliability of Electric Bike...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 108636 8 pageshttpdxdoiorg1011552013108636

Research ArticleCrossing Reliability of Electric Bike Ridersat Urban Intersections

Huan Mei Yang Xiaobao and Jia Bin

MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology Beijing Jiaotong UniversityBeijing 100044 China

Correspondence should be addressed to Yang Xiaobao yangxbbjtueducn

Received 13 June 2013 Accepted 31 July 2013

Academic Editor Wuhong Wang

Copyright copy 2013 Huan Mei et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a crossing reliabilitymodel of electric bike riders at urban intersections using survival analysis approach Ridersrsquocrossing behavior was collected by video camerasWaiting times in the red-light phase weremodeled by reliability-basedmodel thatrecognizes the covariate effectsThree parametric models by the exponential Weibull and log-logistic distributions were proposedto analyzewhen andwhy electric bike riders cross against the red lightThe results indicate thatmovement information and situationfactors have significant effects on ridersrsquo crossing reliability The findings of this paper provide an important demonstration ofmethod and an empirical basis to assess crossing reliability of electric bike riders at the intersection

1 Introduction

Nonmotorized vehicles (ie mainly regular bicycles and elec-tric bikes) are one of the most popular means of transporta-tion in some Asian developing countries such as VietnamCambodia and China Even in developed countries cyclingtravel is recognized as low energy consumption healthy to theusers and do not damage the health of others For examplepeople who go to work by bike would obtain material rewardfrom the government in Korea [1] In London in order toencourage people to travel by bike the local governmentcarried out the cycling revolution [2]

In recent years the electric bike has entered peoplersquoslife Because of electric bike is labor-saving and speedy ithas emerged as a popular mode of transportation in manylarge cities in China [3] The number of Chinese electricbikes was about 140 million in 2012 [4] Electric bikes inChina are defined as electric two wheelers with relatively lowspeeds and weights compared to a motorcycle Both bicycle-style electric bikes (with functioning pedals) and scooter-style electric bikes (with many of the features of gasolinescooters) are classified as bicycles and are given access tobicycle infrastructure (see Figure 1)

However the growing popularity of cycling traffic alsoentails safety concerns as observed in accident and injury

statistics Traffic accident proportion of cyclists has alwaysbeen high because cyclists are vulnerable groups in the trafficconflict In 2010 4616 regular bicyclists were killed and 14283were seriously injured in road accidents representing 71 ofall traffic fatalities and 56 of injuries [5] With the rapidlyincreasing number of electric bikes more and more peoplepaymuch concerned about traffic security problems involvedelectric bikes In 2004 589 electric bike riders were diedand 5295 were seriously injured in road accidents [6] In2010 the corresponding figures increased to 4029 and 20311respectively representing 62 of all traffic fatalities and 80of injuries [5]

Accident analysis reveals that over 60 of cyclist fatalcrashes were caused by violation of traffic rules [5] Red-light crossing is a typical type of rule violation behaviorBecause of the poor law enforcement and peoplesrsquo low safetyawareness violation behavior in the red-light period is ratherprevalent and represents a substantial safety problem inChinese urban intersections [7] Particularly electric bikeriders with relatively high speed are much more likely toincrease the risk of traffic incident

So far many scholars have studied red-light crossingbehaviors but many focused onmotorized vehicles [8 9] andpedestrians [10ndash13] Unfortunately only a few studies haveinvestigated bicyclistsrsquo red-light crossing behavior much less

2 Mathematical Problems in Engineering

Figure 1 Bicycle-style electric bike at the left and scooter-style electric bike at the right

about electric bike riders Johnson et al analyzed the bicycleviolation behavior at intersections in Melbourne [14] Theyfound out that the rate of red-light crossing before noonwas 3 and 11 in the afternoon Johnson et al collected4225 bicyclists who arrived at the intersections at red-lighttime and their result indicated that 69 bicyclists violatedtraffic rules [15] Wu et al used logistic model to analyze howgender age and conformity behavior affected the cyclistrsquosviolation and classify red-light crossing behavior to threetypes obey the rules risk taking opportunistic [7] Johnsonet al obtained the data through the Internet and used TPB(theory of planned behavior) to investigate the bicyclistsrsquored-light crossing behavior and attitude and the influencingfactors in Australia [16]Their result showed that 373 of theriders violated traffic rules and the violation rate ofmales washigher than females the young was higher than the old anda rider who was accident-free may violate them easier

Most of the existing researches on cyclistsrsquo crossingbehavior used logistic model to analyze the violating prob-ability and its influence factors And little effort was focusedon the red-light crossing behavior of electric bike riders Inthis paper we use survival analysis method to study electricbike ridersrsquo crossing reliability at the signalized intersectionSurvival analysis is a common topic in many areas includingbiomedical engineering and social sciences In the trans-portation field it has been applied to describe a number oftime-related issues including activity duration [17 18] trafficaccidents [19] and vehicle travel time [20 21] Hazard-basedduration models of survival analysis have an advantage inthat it allows the explicit study of the relationship betweenduration time and the explanatory variables [22ndash24] Moreimportantly survival analysis models can deal with not onlyuncensored data but also censored data For example theexact waiting duration reflecting cyclist endurance cannot beobserved if cyclists wait until the permission of traffic rulesThis is the very reasonwhy survival analysismethod is chosento analyze ridersrsquo crossing behaviorThe empirical data whichwere obtained by video cameras are modeled by three typicalparametric hazard functions Both crossing reliability andwaiting times of electric bike riders under various conditionsare calculated and the covariate effects are quantified Thefinding of this paper can explain when and why electric bikeriders violate traffic light at the intersection It is hoped thatthe results can help to improve the planning and designing ofsignalized intersections in developing countries

2 Method

Reliability is the probability that a system or component willperform its required function under stated conditions fora specified period of time Accordingly crossing reliabilityof electric bike riders can be defined as the probability thatan electric bike rider obey the traffic light after a specifiedwaiting time at the signalized intersection

Let 119879 denote a nonnegative random variable represent-ing the failure time or time-to-failure Mathematically thereliability function 119877(119905) is the probability that a system willbe successfully operating without failure in the interval fromtime 0 to time 119905

119877 (119905) = 119875 (119879 gt 119905) 119905 gt 0 (1)

In this paper it is assumed that an electric bike rider violatethe traffic light when hisher waiting duration time is 119879119877(119905) is the crossing reliability of electric bike riders It is theprobability that a rider who arrives at the intersection in thered-light period will not violate the traffic light in hisherwaiting duration time from 0 to 119905

The failure probability or unreliability of ridersrsquo crossingbehavior is then

119865 (119905) = 1 minus 119877 (119905) = 119875 (119879 le 119905) (2)

which is known as the distribution of 119879If the random variable 119879 has a density function119891(119905) then

119877 (119905) = int

infin

119905

119891 (119909) 119889119909 (3)

The density function can be mathematically described aslimΔ119905rarr0

119875(119905 le 119879 lt 119905 + Δ119905) This can be interpreted as theprobability that the failureviolation time will occur betweentime 119905 and the next interval of waiting time 119905 + Δ119905 Thethree functions 119877(119905) 119865(119905) and 119891(119905) are closely related toone another If any of them is known all the others can bedetermined

The waiting time of an electric bike rider in the red-lightperiod can be regarded as the waiting duration that startswhen a cyclist arrives at the intersection in the red periodand ends when the rider starts to cross the intersectionTherefore hazard-based duration model in survival analysiscan be adopted to calculate the probability of rider violation

Mathematical Problems in Engineering 3

with different waiting durations In survival analysis 119877(119905)is also called survival function or survivor probability Thesurvival function is defined to be the probability that thewaiting time of a rider in a red-light is longer than a specifictime 119905

Another important function in survival analysis is thehazard function ℎ(119905)The hazard function in this paper is theinstantaneous rate at which the waiting duration will end inan infinitesimally small time period Δ119905 after time 119905 giventhat the duration time has lasted to time 119905

ℎ (119905) = limΔ119905rarr0

119875 (119905 lt 119879 le 119905 + Δ119905 | 119879 gt 119905)

Δ119905

= limΔ119905rarr0

119875 (119905 lt 119879 le 119905 + Δ119905)

Δ119905 times 119875 (119879 gt 119905)=119891 (119905)

119877 (119905)

(4)

The importance of the hazard function is that it indicatesthe changing rate in the red-light violating behavior over thewaiting period of a rider For example two riders may havethe same crossing reliability at a specific point in time butthe hazard rate curves can be very different

Note that crossing reliability of electric bike riders isinfluenced by various factors The influential factors canbe defined as a vector of explanatory variables x =

(1199091 1199092 119909

119898)1015840 To accommodate the effects of these influ-

ential factors a parametric hazardmodel or a semiparametrichazardmodel may be adopted If little or no knowledge of thefunctional form of the hazard is available a semiparametricapproach for modeling the hazard function is convenientIf theoretical support for a parametric shape is availableusing a particular parametric distributional form for thehazard function will be appropriate Both semiparametricand fully parametric hazard-based models have been widelycited in the literature For detailed discussion of differentapproaches for duration models see Bhat [17] Lee and Wang[25] Here the primary objective of this paper is to examinethe crossing reliability and consider the application of themodel A parametric hazard approach is adopted becauseits hazard function can be chosen flexibly Several typicallyparametric distributions are exponential Weibull and log-logistic These common distributions are summarized below

The exponential distribution is suitable for modeling datawith constant hazard The exponential hazard and survivorfunctions are

ℎ (119905) = 120582 (constant hazard)

119877 (119905) = exp (minus120582119905) (5)

The model is implemented by parameterizing 120582119895= exp(minus119883

119895

120573) when the influenced covariates are considered Let ℎ119894(119905 120582119894)

and ℎ119895(119905 120582119895) be the hazards of individuals 119894 and 119895 the hazard

ratio of these two individuals is

ℎ119894(119905 120582119894)

ℎ119895(119905 120582119895)

=120582119894

120582119895

= exp[minussum119896

120573119896(119909119896119894minus 119909119896119895)] (6)

This ratio is dependent only on the differences of thecovariates of the two individuals and the coefficients It doesnot depend on the time 119905

The Weibull distribution is suitable for modeling datawith monotone hazard rates that either increase or decreaseexponentially with time The Weibull hazard and survivorfunctions are

ℎ (119905) = 120582120574(120582119905)120574minus1

119877 (119905) = exp (minus(120582119905)120574) (7)


119895

120573) where 120574 is an ancillary parameter to be estimated fromthe data Note that the hazard increases with duration timeif 120574 gt 1 decreases if 120574 lt 1 and reduces to exponentialif 120574 = 1 Similar to the exponential distribution model thehazard ratio of theWeibull distributionmodel is also not timedependent

The log-logistic model assumed that the natural loga-rithm of time follows a logistic distribution The log-logistichazard and survivor functions are

ℎ (119905) =120582120574(120582119905)

120574minus1

1 + (120582119905)120574

119877 (119905) = 1 + (120582119905)120574

minus1

(8)


119895

120573) and treating the scale parameter 120574 as an ancillary param-eter to be estimated from the data Unlike the exponentialand Weibull distributions the log-logistic distribution isindicated for data exhibiting non-monotonic hazard ratesWhen 120574 gt 1 the hazard initially increases and then decreaseswith duration time When 0 lt 120574 le 1 the hazard decreaseswith duration time The hazard ratio of the log-logisticdistribution model is time dependent

In order to keep the linear form of the covariates threenew terms can be defined

119910119894= log 119905

119894= 120573119883 + 120576

119894120590 (9)

120576 =(log 119905 minus 120573119883)

120590 (10)

120574 =1

120590 (11)

The likelihood and log-likelihood functions for 119873 observa-tions on 119910

119894and right censoring indicator 120575

119894are

119871 = prod

119894

[120590minus1

119891 (120576119894)]120575119894

[119877 (120576119894)]1minus120575119894 (12)

log 119871 = sum119894

[120575119894(minus log120590 + log119891 (120576

119894)) + (1 minus 120575

119894) log119877 (120576

119894)]

(13)

Log-likelihood functions can be maximized using any stan-dard method (eg Newton BFGS)

3 Data

The field observation approach was used in this study It hasbeen widely used to study pedestrian and cyclist behaviors


Table 1 Covariates selection and explanation

Covariate Type Explanation

AGE (age group) Continuous variable 1 if under 20 2 if 20ndash29 3 if 30ndash39 4 if 40ndash49 5 if50ndash59 and 6 otherwise

GEN (gender) Binary indicator 1 if male 0 female

WN (waiting number) Continuous variable The number of other cyclists that are waiting for a greenlight when arrives

CN (crossing number) Continuous variable The number of other cyclists that are crossing againstthe red light when arrives

TC (twice crossing) Binary indicator 1 if twice crossing behavior 0 otherwise

MV (motor vehicle volume) Continuous variable Motor vehicle volume per lane per min in red-lightphase when the rider arrives

LT (left turn phase) Binary indicator 1 if an electric bike rider violates traffic rules in theleft-turn period of motorized vehicles 0 otherwise

YL (yellow light) Binary indicator 1 if an electric bike rider violates traffic rules in theyellow light period 0 otherwise

in actual traffic situations [7 13 26 27] For example Tiwariet al used video recording to analyze pedestriansrsquo violatingbehavior in signalized intersections in India [13] Yao et alalso used video cameras to study the behavior characteristicsof pedestrian crowd weaving flow in transport terminal [27]Wu et al used video recordings to examine urban commutercyclistsrsquo red-light running behaviors in Beijing [7] Wang etal used video data to study pedestriansrsquo crossing behavior inred-light period in Beijing [26]

31 Site Characteristics A cross-sectional observationalstudy was conducted at five signalized intersections in Bei-jing Three criteria were used to select the observationalsites First the selected sites should represent the typicalintersection design characteristics and traffic conditions ofurban areas in Beijing Second the selected intersectionsshould have similar characteristics involved geometrics traf-fic conditions traffic control and the absence of pointsmenIn addition there have to be a reasonably high number ofelectric bike traffic during the observation period

32 Data Collection and Processing Video cameras were usedto collect data of electric bike ridersrsquo crossing behaviors atsignalized intersections The cameras were carefully placedso that the road users were unaware that they were beingobserved The data collection was conducted on weekdaysduring daylight hours (ie 800 am to 530 pm) in goodweather conditions

All road users who entered the intersectionwere recordedon video but only the riders arriving in red-light phases werecoded In addition only the straight-going riders crossingthrough the intersection were recorded Left-turners andright-turners of electric bike were excluded because of thelimited field of view of the camerasThe waiting duration wasfrom the time a rider arrived at the stop line to the time heshebegan to cross It can be classified into two kinds uncensoreddata and censored data Uncensored data is defined as thewaiting duration which ends within the red-light period(violating crossing) Otherwise the waiting duration is called

censored data as long as it ends within the green light period(normal crossing) For censored data it is unknown abouthisher exact maximum endurance of waiting time

Considering the previous researches and intuitive argu-ments regarding the influential factors of cyclist crossingbehavior three sets of variables were coded (see Table 1)The first set described the ridersrsquo individual characteristicsincluding gender and ageThe second set of variables focusedon the ridersrsquo movement information including twice cross-ing behavior the times of arrival at and departure from thestop line the time when crossing is completed the statusof the traffic light at each of three times and whether theviolating behavior occurs in the left-turn period ofmotorizedvehicles or the yellow-light period The last set of variables ofconcern were situation factors including the red-light cyclelength the number of riders waiting upon arrival the numberof riders crossing against the red-light and crossing trafficvolume (ie the number of motorized vehicles that crossedthe intersection from either direction during the time of thered-light cycle when the rider arrives)

4 Results

A total of 8 h of video recordings (approximately 15 h for eachsite) were collected and 1946 crossing events of cyclists wereobserved Demographics and behavioral data were codedonly for electric bike riders during the red-light phasesThusa total of 312 (16) valid observations were obtained (seeTable 2)

41 Descriptive Statistics The proportions of red-light cross-ing of electric bike riders in each sub-groupwere presented inTable 2Of all the electric bike riderswe observed the numberof males was larger than females (843 versus 157) Andmost of the electric bike riders are young and middle-agedpersons the respective rates are 265 and 676 The oldriders only occupied 61

The overall proportion of electric bike riders who crossedagainst the red-light was 754 Male electric bike riders


Table 2 Red-light crossing rates and waiting times by each subcategory

Violating crossing Normal crossing OverallRates Waiting time (s) Rates Waiting time (s) Rates Waiting time (s)

GenderMale 769 2611 231 6488 843 3506Female 673 2211 327 6914 157 3746

Age groupYoung (lt30) 807 2483 193 8257 265 3596Middle (30ndash50) 744 2667 256 5946 676 3512Old (gt50) 632 1378 368 7597 61 3669

Overall 754 2555 246 6576 100 3544

Table 3 Parameter regression values of the influence variables

Parameter Exponential model Weibull model Log-logistics modelCoef Haz ratio 119875 value Coef Haz ratio 119875 value Coef 119875 value

Constant 2960 0052 lt0001 1009 0364 0084 1449 0359SITE 0145 0865 0077 0044 0957 0543 0012 0951AGE minus0065 1067 0424 minus0055 1057 0485 minus0220 0329GEN 0012 0988 0954 0040 0961 0841 0197 0737WN 0041 0960 0018 0064 0938 lt0001 0266 lt0001CN minus0134 1143 lt0001 minus0089 1094 lt0001 minus0382 lt0001TC minus0206 1229 0254 minus0331 1392 0049 minus1151 0017MV 0189 0828 0001 0146 0864 0006 0429 0006LT minus1166 3209 lt0001 minus095 2585 lt0001 minus2431 lt0001YL minus0737 2090 0002 minus0882 2415 lt0001 minus3065 lt0001120574 1000 0408 0514log 119871 minus8316 minus6566 minus6712

were found to be more likely to violate traffic rules at signalintersections than females (769 versus 673) Red-lightviolation rates also differed by age group Young- andmiddle-aged riders were more likely to run against a red-light thanthe old ones (807 and 744 versus 632) The result isconsistent with the research in pedestrian crossing behavior[13 23 26]

The averagewaiting time of all sampleswas 3544 secondsThe average waiting time of the violating crossing was2555 seconds while the average waiting time of the normalcrossing is 6576 seconds The maximum waiting durationwas 161 seconds while the minimum was 0 second The lattermeans people cross the street without any waiting time Asshown in the last column of Table 2 the overall waiting timesof electric bike riders vary from 3506 to 3746 seconds indifferent genders and from 3512 to 3699 seconds in differentage groupsTherefore there is no significant difference of theoverall waiting times in gender and age group

This descriptive statistic cannot reflect the exact waitingbehavior due to the neglect of the censored data Theestimation of the waiting duration with censored data will bediscussed later

42 Parameter Regression Analysis Table 3 shows the esti-mated parameters of the parametric hazard models in pre-dicting red-light crossing behavior of electric bike ridersThree models show perfect overall goodness-of-fit indicatedby the log-likelihood values Among them the Weibulldistribution is considered the best to describe ridersrsquo crossingbehavior The hazard ratios of both the exponential andWeibullmodels are not time-dependent According to (6) thehazard ratio is exp(minus120573

119894) times if the 119894th covariate increases by

one unit However the risk of the log-logistic model cannotbe calculated by this way since it is time dependent

From the estimated results the site variable is insignif-icant which means the selected intersections have similartraffic and geometric characteristics Besides gender and agegroup have low significant level It is partly because of the lowproportions of female (157) and old people (61) in thesample Most of other covariates are statistically significant atthe 010 level of significanceThese covariates are significantlyrelated to violation behavior

For the specific parameter estimation in Table 2 based on(9) the positive parameters have a positive effect on waitingtime of riders (eg WN and MV) Therefore an increase


00

02

04

06

08

10

0 20 40 60 80 100

Cros

sing

relia

bilit

y

Waiting duration (s)

ExponentialWeibullLog-logistic

Figure 2 Crossing reliability with waiting duration in differentparametric models

in the corresponding covariates can increase the waitingduration or decrease the risk rate For example an electricbike rider is willing to wait longer time if many other cyclistsare waiting when heshe arrives at the intersection Alsoa rider is likely to wait to reduce the collision risk in thepresence of heavy traffic

On the contrary the negative effectmeans that an increasein the corresponding covariates can decease the waitingduration or increase the risk rate (eg CN TC LT and YL)For example an electric bike rider may follow to violate ruleswhen other cyclists are crossing against the red-light A riderof twice crossing has higher hazard and shorter waiting timeBesides a rider is more likely to violate rules during the left-turn period ofmotorized vehicles and the yellow-light period

43 Crossing Reliability Analysis of Electric Bike RidersCrossing reliability of electric bike riders calculated by threeparametric distributions are shown in Figure 2 All of theprobability curves present a monotonically decreasing trendwhich means the crossing reliability decreases with theincreasingwaiting time In addition the curves of theWeibulland log-logistic distributions have a rapidly decreasing trendin the early part of the waiting time It is consistent with thefact that about 281 of electric bike riders are at high riskof violation and low waiting time to cross against the redright in our sample But the exponential distribution has alow adaptability to describe the waiting time of electric bikeriders

Figure 3 gives the risk rate curve calculated by threeparametric models The constant risk in the exponentialmodel reflects the fact that the risk of violating behaviorwould not change with the elapse of waiting time Thereforethe exponential model is not appropriate for describing thecrossing behavior of electric bike riders On the other hand

000

002

004

006

008

010

0 20 40 60 80 100

Haz

ard

rate



Figure 3 Hazard rate with waiting duration in different parametricmodels

the curves of theWeibull and log-logistic distributions show aclear downward trend over waiting time reflecting a negativedependence It means that the likelihood of terminatingthe waiting time decreases with elapsed waiting time It isconsistent with the fact that a part of riders are generallynonrisk takers who can obey the traffic rules after waiting along time

44 Reliability Function of Crossing Behavior Once electricbike riders violate traffic light such a crossing behavior isdefined as the occurrence of a risk event Crossing reliabilityof electric bike riders is the probability that an electric bikerider obey the traffic light after a specified waiting time at theintersection The Weibull distribution model gives the bestdescription of ridersrsquo crossing behavior Taken the Weibulldistribution model crossing reliability of electric bike ridersusing our sample data can be written as

119877 (119905) = exp (minus(120582119905)120574) = exp minus[119905 exp (minus120573119883)]120574

= exp minus [119905 exp (minus1009 minus 0064 lowast119882119873 + 0089 lowast 119862119873

+ 0331 lowast 119879119862 minus 0146 lowast119872119881

+ 095 lowast 119871119879 + 0880 lowast 119884119871)]0408

(14)

The proposed reliability model can capture the covariateeffects of crossing behavior of electric bike riders at theintersection Before the applications however it is noted thatthe model should be estimated using the specified field dataAdditionally the explanatory variables should be chosenflexibly according to the specified traffic circumstance


5 Conclusions

The paper uses the concept of crossing reliability and themethodology of survival analysis to examine violating behav-ior of electric bike riders at the intersectionThree parametricdurationmodels by the exponentialWeibull and log-logisticdistributions are proposed to analyze the covariate effects oncrossing reliability of electric bike riders in red-light phaseThemethodology uses a durationmodel structure that is ableto predict crossing reliability at any given traffic conditionsMore importantly three sets of variables are selected asthe covariates individual characteristics movement infor-mation and situation factors It is proved that the Weibulldistribution is the most appropriate for describing crossingbehavior of electric bike riders

The paper provides several important insights into thedeterminants of crossing reliability of electric bike ridersFirstly from the methodological standpoint this study hasprovided the empirical evidence that hazard-based durationapproach is appropriate for crossing reliability analysis Astatistical test of covariate effects on crossing reliabilityclearly indicates the goodness-of-fit of the used parametrichazard models Secondly the results indicate that movementinformation and situation factors are significantly relatedto ridersrsquo crossing reliability Finally various factors canmodify crossing reliability of electric bike riders The moreriders waiting at the stop line the less other riders crossingagainst the red-light and themoremotorized vehicle volumeelectric bike riders would endure longer waiting times andhave higher crossing reliability While electric bike ridershave lower crossing reliability during the left-turn period ofmotorized vehicles and the yellow-light period

In terms of the future work research with more datasetsis required Also more parameters under different situationsshould be taken into account In addition it is necessaryto study the comparison between electric bikes and com-mon bikes It is expected that the findings will help trafficengineers urban planners and policy makers to understandelectric bike ridersrsquo behavior at signalized intersections

Acknowledgments

This work was supported by the National Basic ResearchProgram of China (Grant no 2012CB725401) the NationalNatural Science Foundation of China (Grant nos 7090100571131001) and Fundamental Research Funds for the CentralUniversities (Grant no 2013JBM043)

References

[1] httpzhaokuang541020blog163comblogstatic1279301062-01021931336293

[2] httpwwwneagovcn2012-0620c 131664583htm[3] Q Zhou H Lu andW Xu ldquoComparison and analysis of urban

resident trip characteristicsrdquo Journal of Central South HighwayEngineering vol 32 no 2 pp 145ndash149 2007 (Chinese)

[4] httpwwwcebikecomnewshtml2012042012042908414362htm

[5] CRTASR China Road Traffic Accidents Statistics Report TrafficAdministration Bureau of China State Security Ministry Bei-jing China 2010 (Chinese)


[7] C XWu L Yao andK Zhang ldquoThe red-light running behaviorof electric bike riders and cyclists at urban intersections inChina an observational studyrdquo Accident Analysis and Preven-tion vol 49 no 11 pp 186ndash192 2012

[8] K L H Martinez and B E Porter ldquoCharacterizing red lightrunners following implementation of a photo enforcementprogramrdquo Accident Analysis and Prevention vol 38 no 5 pp862ndash870 2006

[9] B E Porter K L Johnson and J F Bland ldquoTurning offthe cameras red light running characteristics and rates afterphoto enforcement legislation expiredrdquo Accident Analysis andPrevention vol 50 pp 1104ndash1111 2013

[10] M M Ishaque and R B Noland ldquoBehavioural issues in pedes-trian speed choice and street crossing behaviour a reviewrdquoTransport Reviews vol 28 no 1 pp 61ndash85 2008

[11] E Papadimitriou G Yannis and J Golias ldquoA critical assessmentof pedestrian behaviourmodelsrdquoTransportationResearch F vol12 no 3 pp 242ndash255 2009

[12] R G ZhouW J Horrey and R F Yu ldquoThe effect of conformitytendency on pedestriansrsquo road-crossing intentions in China anapplication of the theory of planned behaviorrdquoAccident Analysisand Prevention vol 41 no 3 pp 491ndash497 2009

[13] G Tiwari S Bangdiwala A Saraswat and S Gaurav ldquoSurvivalanalysis pedestrian risk exposure at signalized intersectionsrdquoTransportation Research F vol 10 no 2 pp 77ndash89 2007

[14] M Johnson J Charlton and J Oxley ldquoCyclists and red light-a study of behaviour of commuter cyclists in Melbournerdquo inProceedings of the Australasian Road Safety Research Policingand EducationConference Adelaide Australia November 2008

[15] M Johnson S Newstead J Charlton and J Oxley ldquoRidingthrough red lights the rate characteristics and risk factors ofnon-compliant urban commuter cyclistsrdquoAccident Analysis andPrevention vol 43 no 1 pp 323ndash328 2011

[16] M Johnson J Charlton J Oxley and S Newstead ldquoWhy docyclists infringe at red lights An investigation of Australiancyclistsrsquo reasons for red light infringementrdquo Accident Analysisand Prevention vol 50 no 1 pp 840ndash847 2013

[17] C R Bhat ldquoDuration modelingrdquo in Handbook of TransportModelling D A Hensher D A Hensher and K J Button Edspp 91ndash111 Elsevier Science Oxford UK 2000

[18] K M N Habib ldquoModeling commuting mode choice jointlywith work start time and work durationrdquo TransportationResearch A vol 46 no 1 pp 33ndash47 2012

[19] Y Chung ldquoDevelopment of an accident duration predictionmodel on the Korean freeway systemsrdquo Accident Analysis andPrevention vol 42 no 1 pp 282ndash289 2010

[20] X B Yang Z Y Gao H W Guo and M Huan ldquoSurvivalanalysis of car travel time near a bus stop in developingcountriesrdquo Science in China E vol 55 no 8 pp 2355ndash2361 2012

[21] X B Yang M Huan H W Guo and L Gao ldquoCar traveltime estimation near a bus stop with non-motorized vehiclesrdquoInternational Journal of Computational Intelligence Systems vol6 no 4 pp 1350ndash1357 2011

[22] H W Guo W H Wang W W Guo and F C Zhao ldquoModel-ing lane-keeping behavior of bicyclists using survival analysis


approachrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 197518 6 pages 2013

[23] H W Guo W H Wang W W Guo X B Jiang and H BubbldquoReliability analysis of pedestrian safety crossing in urban trafficenvironmentrdquo Safety Science vol 50 no 4 pp 968ndash973 2012

[24] X B YangMHuan B F Si L Gao andHWGuo ldquoCrossing ata red light behavior of cyclists at urban Intersectionsrdquo DiscreteDynamics in Nature and Society vol 2012 Article ID 490810 12pages 2012

[25] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis John Wiley amp Sons New York NY USA 2003

[26] W H Wang H W Guo Z Y Gao and H Bubb ldquoIndividualdifferences of pedestrian behaviour in midblock crosswalk andintersectionrdquo International Journal of Crashworthiness vol 16no 1 pp 1ndash9 2011

[27] L Yao L S Sun Z Y Zhang S W Wang and J RongldquoResearch on the behavior characteristics of pedestrian crowdweaving flow in transport terminalrdquoMathematical Problems inEngineering vol 2012 Article ID 264295 9 pages 2012

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Stochastic AnalysisInternational Journal of


Figure 1 Bicycle-style electric bike at the left and scooter-style electric bike at the right

about electric bike riders Johnson et al analyzed the bicycleviolation behavior at intersections in Melbourne [14] Theyfound out that the rate of red-light crossing before noonwas 3 and 11 in the afternoon Johnson et al collected4225 bicyclists who arrived at the intersections at red-lighttime and their result indicated that 69 bicyclists violatedtraffic rules [15] Wu et al used logistic model to analyze howgender age and conformity behavior affected the cyclistrsquosviolation and classify red-light crossing behavior to threetypes obey the rules risk taking opportunistic [7] Johnsonet al obtained the data through the Internet and used TPB(theory of planned behavior) to investigate the bicyclistsrsquored-light crossing behavior and attitude and the influencingfactors in Australia [16]Their result showed that 373 of theriders violated traffic rules and the violation rate ofmales washigher than females the young was higher than the old anda rider who was accident-free may violate them easier

Most of the existing researches on cyclistsrsquo crossingbehavior used logistic model to analyze the violating prob-ability and its influence factors And little effort was focusedon the red-light crossing behavior of electric bike riders Inthis paper we use survival analysis method to study electricbike ridersrsquo crossing reliability at the signalized intersectionSurvival analysis is a common topic in many areas includingbiomedical engineering and social sciences In the trans-portation field it has been applied to describe a number oftime-related issues including activity duration [17 18] trafficaccidents [19] and vehicle travel time [20 21] Hazard-basedduration models of survival analysis have an advantage inthat it allows the explicit study of the relationship betweenduration time and the explanatory variables [22ndash24] Moreimportantly survival analysis models can deal with not onlyuncensored data but also censored data For example theexact waiting duration reflecting cyclist endurance cannot beobserved if cyclists wait until the permission of traffic rulesThis is the very reasonwhy survival analysismethod is chosento analyze ridersrsquo crossing behaviorThe empirical data whichwere obtained by video cameras are modeled by three typicalparametric hazard functions Both crossing reliability andwaiting times of electric bike riders under various conditionsare calculated and the covariate effects are quantified Thefinding of this paper can explain when and why electric bikeriders violate traffic light at the intersection It is hoped thatthe results can help to improve the planning and designing ofsignalized intersections in developing countries

2 Method

Reliability is the probability that a system or component willperform its required function under stated conditions fora specified period of time Accordingly crossing reliabilityof electric bike riders can be defined as the probability thatan electric bike rider obey the traffic light after a specifiedwaiting time at the signalized intersection

Let 119879 denote a nonnegative random variable represent-ing the failure time or time-to-failure Mathematically thereliability function 119877(119905) is the probability that a system willbe successfully operating without failure in the interval fromtime 0 to time 119905

119877 (119905) = 119875 (119879 gt 119905) 119905 gt 0 (1)

In this paper it is assumed that an electric bike rider violatethe traffic light when hisher waiting duration time is 119879119877(119905) is the crossing reliability of electric bike riders It is theprobability that a rider who arrives at the intersection in thered-light period will not violate the traffic light in hisherwaiting duration time from 0 to 119905

The failure probability or unreliability of ridersrsquo crossingbehavior is then

119865 (119905) = 1 minus 119877 (119905) = 119875 (119879 le 119905) (2)

which is known as the distribution of 119879If the random variable 119879 has a density function119891(119905) then

119877 (119905) = int

infin

119905

119891 (119909) 119889119909 (3)

The density function can be mathematically described aslimΔ119905rarr0

119875(119905 le 119879 lt 119905 + Δ119905) This can be interpreted as theprobability that the failureviolation time will occur betweentime 119905 and the next interval of waiting time 119905 + Δ119905 Thethree functions 119877(119905) 119865(119905) and 119891(119905) are closely related toone another If any of them is known all the others can bedetermined

The waiting time of an electric bike rider in the red-lightperiod can be regarded as the waiting duration that startswhen a cyclist arrives at the intersection in the red periodand ends when the rider starts to cross the intersectionTherefore hazard-based duration model in survival analysiscan be adopted to calculate the probability of rider violation




ℎ (119905) = limΔ119905rarr0

119875 (119905 lt 119879 le 119905 + Δ119905 | 119879 gt 119905)

Δ119905

= limΔ119905rarr0

119875 (119905 lt 119879 le 119905 + Δ119905)

Δ119905 times 119875 (119879 gt 119905)=119891 (119905)

119877 (119905)

(4)



(1199091 1199092 119909





119877 (119905) = exp (minus120582119905) (5)


119895




ℎ119894(119905 120582119894)

ℎ119895(119905 120582119895)

=120582119894

120582119895


120573119896(119909119896119894minus 119909119896119895)] (6)



ℎ (119905) = 120582120574(120582119905)120574minus1

119877 (119905) = exp (minus(120582119905)120574) (7)


119895



ℎ (119905) =120582120574(120582119905)

120574minus1

1 + (120582119905)120574

119877 (119905) = 1 + (120582119905)120574

minus1

(8)


119895



119910119894= log 119905

119894= 120573119883 + 120576

119894120590 (9)

120576 =(log 119905 minus 120573119883)

120590 (10)

120574 =1

120590 (11)



119894are

119871 = prod

119894

[120590minus1

119891 (120576119894)]120575119894

[119877 (120576119894)]1minus120575119894 (12)

log 119871 = sum119894

[120575119894(minus log120590 + log119891 (120576

119894)) + (1 minus 120575

119894) log119877 (120576

119894)]

(13)


3 Data



















4 Results









Overall 754 2555 246 6576 100 3544













00

02

04

06

08

10

0 20 40 60 80 100

Cros

sing

relia

bilit

y








000

002

004

006

008

010

0 20 40 60 80 100

Haz

ard

rate









+ 095 lowast 119871119879 + 0880 lowast 119884119871)]0408

(14)



5 Conclusions




Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












ℎ (119905) = limΔ119905rarr0

119875 (119905 lt 119879 le 119905 + Δ119905 | 119879 gt 119905)

Δ119905

= limΔ119905rarr0

119875 (119905 lt 119879 le 119905 + Δ119905)

Δ119905 times 119875 (119879 gt 119905)=119891 (119905)

119877 (119905)

(4)



(1199091 1199092 119909





119877 (119905) = exp (minus120582119905) (5)


119895




ℎ119894(119905 120582119894)

ℎ119895(119905 120582119895)

=120582119894

120582119895


120573119896(119909119896119894minus 119909119896119895)] (6)



ℎ (119905) = 120582120574(120582119905)120574minus1

119877 (119905) = exp (minus(120582119905)120574) (7)


119895



ℎ (119905) =120582120574(120582119905)

120574minus1

1 + (120582119905)120574

119877 (119905) = 1 + (120582119905)120574

minus1

(8)


119895



119910119894= log 119905

119894= 120573119883 + 120576

119894120590 (9)

120576 =(log 119905 minus 120573119883)

120590 (10)

120574 =1

120590 (11)



119894are

119871 = prod

119894

[120590minus1

119891 (120576119894)]120575119894

[119877 (120576119894)]1minus120575119894 (12)

log 119871 = sum119894

[120575119894(minus log120590 + log119891 (120576

119894)) + (1 minus 120575

119894) log119877 (120576

119894)]

(13)


3 Data



















4 Results









Overall 754 2555 246 6576 100 3544













00

02

04

06

08

10

0 20 40 60 80 100

Cros

sing

relia

bilit

y








000

002

004

006

008

010

0 20 40 60 80 100

Haz

ard

rate









+ 095 lowast 119871119879 + 0880 lowast 119884119871)]0408

(14)



5 Conclusions




Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























4 Results









Overall 754 2555 246 6576 100 3544













00

02

04

06

08

10

0 20 40 60 80 100

Cros

sing

relia

bilit

y








000

002

004

006

008

010

0 20 40 60 80 100

Haz

ard

rate









+ 095 lowast 119871119879 + 0880 lowast 119884119871)]0408

(14)



5 Conclusions




Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Overall 754 2555 246 6576 100 3544













00

02

04

06

08

10

0 20 40 60 80 100

Cros

sing

relia

bilit

y








000

002

004

006

008

010

0 20 40 60 80 100

Haz

ard

rate









+ 095 lowast 119871119879 + 0880 lowast 119884119871)]0408

(14)



5 Conclusions




Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

02

04

06

08

10

0 20 40 60 80 100

Cros

sing

relia

bilit

y








000

002

004

006

008

010

0 20 40 60 80 100

Haz

ard

rate









+ 095 lowast 119871119879 + 0880 lowast 119884119871)]0408

(14)



5 Conclusions




Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 Conclusions




Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Crossing Reliability of Electric Bike...

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