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Accident
Analysis
and
Prevention
jo u r n al ho me p a g e :w w w . e l s e v i e r . c o m / l o c a t e / a a p
The
effect
of
Operation
24
Hours
on
reducing
collision
in
the
City
of
Edmonton
Siana
Halim
a,∗,
Heming
Jiang
b,1aIndustrialEngineeringDepartment,PetraChristianUniversity,Jl.Siwalankerto21-131,Surabaya,Indonesia
bCityofEdmontonOfficeofTrafficSafety,9304–41Ave,EdmontonT6E6G8,Alberta,Canada
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received21February2013
Receivedinrevisedform21April2013 Accepted25April2013
Keywords:
Negativebinomialtimeseries Segmentedregression Changepoint
a
b
s
t
r
a
c
t
IntheCityofEdmonton,inordertoreducetheprevalenceofcollisions,theOperation24Hoursprogram (OPS24)wasdevelopedbyusingexistingpoliceandtransportationservicesresources.Theprogramuses traditionalmannedpolicespeedenforcementmethod,whicharesupplementedbytrafficsafetymessages displayedonpermanentandmobiledynamicmessagingsigns(DMS).Inthispaper,collisiondataanalysis wasperformedbylookingatthedailynumberofcollisionsfrom2008to2011thatcovers28Operation 24Hours(OPS24)events.Theobjectiveofthecollisiondataanalysisistoanalyzeifthereisareduction incollisionfrequenciesafterOPS24washeldandexaminedhowlongthecollisionreductioneffectlast. Weatherfactorssuchastemperature,thicknessofsnow,andwindgusthavebeenconsideredbymanyas agreatinfluenceoncollisionoccurrences,especiallyinacitywithlongandcoldwintersuchasEdmonton. Therefore,collisionmodelingwasperformedbyconsideringtheseexternalweatherfactors.Toanalyze thelinearandperiodictrendofdifferentcollisiontypes(injury,fatal,andpropertydamageonly(PDO)) andexaminetheinfluenceofweatherfactorsoncollisions,negativebinomialtimeseriesmodelthat accountsforseasonalityandweatherfactorswasusedtomodeldailycollisiondata.Themodelingalso consideredcollisionproportiontoaccountformissingtrafficvolumedata;theGaussiantimeseriesmodel thataccountsforseasonalityandweatherfactorswasusedtomodelcollisionproportion.Toestimatethe collisiontrendandtestforchangesincollisionlevelsbefore/afterOPS24,interruptedtimeseriesmodel withsegmentedregressionwasused.WhileforestimatinghowlongtheeffectoftheOPS24last,change pointmethodwasapplied.
© 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Manyresearchershaveconsideredweatherfactorsasagreat influenceoncollisionoccurrences.Poorweather-relateddriving conditionsare associated with7000 fatalities, 800,000 injuries, and more than 1.5million vehicular collisions annually in the UnitedStates(NationalResearchCouncil,2004).Adverseweather ispresent in 28%of total collisionsand nearly20% ofhighway fatalities(WeatherandHighways,2004).Analystsestimatethe eco-nomictollofweather-relatedcollisionsat$42billion(Lombardo, 2000).InCanada,weather-relatedcollisionshavebeenestimatedto costCanadiansanaverageofapproximatelyCDN$1billionperyear
(Andreyetal.,2001;Usmanetal.,2011).Understandingtheeffects
ofadverseweather onmotor vehiclecollisionsmattersbecause expertshaveidentifiedanumberofcommunicationsand
engineer-∗Correspondingauthor.Tel.:+62312983425.
E-mailaddresses:halim@peter.petra.ac.id,sianahalim@yahoo.com(S.Halim), heming.jiang@edmonton.ca(H.Jiang).
1 Tel.:+17804959906.
inginnovations(largelytechnologiestocollectandcommunicate realtimeroadconditioninformation,suchassensorsanddynamic messagesigns)that couldsignificantlyreducethecollision and injuryrates,butatapotentiallysubstantialcost(NationalResearch
Council,2004;WeatherandHighways,2004).
Itisa knownfactthatcollisionscanbeinfluencedby exter-nalenvironmentalvariablessuchastemperature,snowfalllevel, andwindgust.Previousstudieshaveassociatedprecipitationwith markedlyincreasedcollisionsrates(Eisenberg,2004;Brodskyand
Hakkert,1988; AndreyandYagar, 1993;Fridstrometal.,1995).
Recentworkalsoshowsthattheriskposedbyprecipitationrises dramatically with the time since last precipitation (Eisenberg, 2004).Therehasnotbeenanystudyontheeffectofsnowfall.
Numberofcollisionsisnotinevitablyhigherinsnowyweather thanindryweather.Ontheonehand,snowmakesdrivingmore dangerous;byreducingtireadherenceandimpairingvisibility.On theotherhand,experienceddriverstypicallydrivemoreslowly andcarefullyinsnowyweather,andmanypeopleavoidor post-poneunnecessarytravel.Perhapsasareflectionoftheseoffsetting factors,thehandfulofpublishedstudiesaddressingthecollision consequencesofsnowhasproducedsomeconflictingresults.The
0001-4575/$–seefrontmatter© 2013 Elsevier Ltd. All rights reserved.
S.Halim,H.Jiang/AccidentAnalysisandPrevention58 (2013) 106–114 107
Table1
2008–2011OPS24schedules.
2008 2009 2010 2011
Thursday,January29 Wednesday,January20 Wednesday,January19 Tuesday,March10 Friday,February05 Friday,February04 Tuesday,April07 Monday,March29 Monday,March28 Tuesday,May12 Tuesday,April13 Thursday,April21 Tuesday,June09 Monday,June28 Thursday,May19 Friday,June26 Tuesday,June07 Monday,September15 Tuesday,September29 Monday,September13
Tuesday,October21 Friday,October16 Monday,November15 Tuesday,November15 Tuesday,December16 Tuesday,December15 Tuesday,December14 Friday,December16
weightoftheevidencesuggeststhatless severecollisions(e.g., thoseproducingonlypropertydamage)increaseduringsnowfall, whilemoreseverecollisions(thoseresultinginmajorinjuriesor fatalities)decrease.
Significantlyincreasedcollisionrateshavebeendocumentedin snowymonthsinCanada(AndreescuandFrost,1998),onsnowy daysintheUnitedKingdom(Perryetal.,1991),andduring snow-storms in Iowa, UnitedStates (Knappet al., 2000).Perryet al.
(1991)foundincreasedratesofcollisionsinvolvinginjuries and
fatalitiesonsnowydaysintheUnitedKingdom,butBrownand
Baass(1997)notedfewercollisionsinvolvinginjuriesinthewinter
monthsinCanada,asdidFridstrometal.(1995)insnowymonths inDenmarkandFinland.Eisenberg(2004)founddecreasedrates offatal collisionsonsnowydaysintheUnitedStates,afinding echoedinanalysisofwintermonthsinCanada(BrownandBaass, 1997)andsnowymonthsinScandinavia.Todate,onlytwoprevious studieshaveexaminedtheeffectsofthefirstsnowfallofthe sea-son.Definingfirstsnowfallasthefirstsnowinamonthfollowing amonthwithoutsnow,FridstromandIngebrigtsen(1991)found significantincreasesinbothinjuryandfatalcollisionsinNorway. Subsequently,however,researchbyFridstrometal.(1995) pro-ducedmixedfindings:Injurycollisionsrosesignificantlyduring thewinter’sfirstmonthwithsnow(comparedwithothermonths withsnow)inDenmarkbutnotineitherFinlandorNorway. Fatal-ityrateswerenodifferentinthefirstsnowymonththaninother snowymonths.
1.1. TheOperation24Hours
IntheCityofEdmonton,inordertoreducetheprevalenceof speedrelatedcollisions,theOperation24Hoursprogram(OPS24) wasdevelopedbyusingexistingpoliceandtransportationservices resources.Theprogramusestraditionalpolicespeedenforcement methodssuchasmannedenforcement,whicharesupplemented by traffic safety messages displayed onpermanent and mobile dynamicmessagingsigns(DMS).Theprogramutilizesan alternat-ingtwo-messageapproach“BigTicketEvent–Don’tSpeed”.The messageswererunconsecutivelyforfourdayspriortothe opera-tiondayandalsoduringtheoperationday(thefifthday).During theOPS24,theentireCityofEdmontonbecameanenforcement zone. Twenty-eight OPS24events have been conducted during 2008–2011(seeTable1).
TheobjectivesofthisstudyaretoexaminetheimpactofOPS24 oncollisionsingeneralaswellasondifferentcollisiontypes(injury, fatal,andpropertydamageonly(PDO)).Thecollisionanalysisdid not limitto speed related collisionssince theEdmontonPolice
Service collected speed information of vehicles involved in a collision only starting from2011 and only ifthe police visited thecollisionscene.Thecollectedspeedinformationis alsoonly categorical: unsafe speed, no unsafe speed, unknown, and not applicable.Generalcollisiontrendandtheinfluenceofweather fac-torssuchastemperature,thicknessofsnow,andwindgustwere alsoinvestigated.Inaddition,weaimtofindhowlongthecollision reductionlasts.TheimpactofOPS24onvehiclespeeddistribution isbeyondthescopeofthispaperandcanbefoundinHalimetal.
(2012).
2. Data
Thisstudyused3 yearsand4monthsofdailycollisiondata rangingfromSeptember2008toDecember2011.Thecollisiondata usedinthis analysiswasobtainedfromtheCityofEdmonton’s MotorVehicleCollisionInformationSystem(MVCIS),adatabaseof motorvehiclecollisionsthatoccuronpublicroadsintheCityof Edmonton.Theinformationinthedatabaseiscollectedfromthe provincialCollisionReportForm,whichiscompletedbymembers oftheEdmontonPoliceService(EPS)eitheronpaperatthesceneof thecollisionorelectronicallyatthefrontcounterofadivisionalor communitypolicestation.Thecollisionreportsreceivedfromthe EPSwerereviewedandcorrectedifnecessary(inconsultationwith theEPS)beforeenteredintoMVCIS.
Thetotalnumberofcollisionswas109,459duringthisperiod. Thedailycollisiondatasetcontainscollisionfrequencies,collision types(injury,fatal,andPDO),collisionlocations,roadwayportions, andcollisioncauses.Wecombinedfatalandinjurycollisionsand namedit‘severecollisions’whichcontributes14.6%tothetotal col-lisions.ThestatisticalsummaryofthedataispresentedinTable2.
2.1. Weatherdata
Theweather data usedinthis analysisincludes 3years and 4 months of dailyweather dataof the Cityof Edmonton from EnvironmentCanadawithdatesrangingfromSeptember2008to December2011.Thedailyweatherdatacontainsvariablessuchas dailymaximum,minimumandmeantemperaturesindegree Cel-sius,thethicknessofsnowontheground(incm),andthespeed ofmaximumwindgustduringtheday(inkm/h).Thestatistical summaryoftheweatherdataisshowninTable3.
BothMVCISandEnvironmentCanadaprovidereliabledatafor purposeofthisstudy.Thecompletenessandaccuracyofdataare significantlyhigh,forexample,nounknowninformationaboutthe
Table2
Summaryofcollisiondata.
Minimum Maximum Mean Stddeviation Numberofcollisionsperday 13 305 75.13 35.24
Severecollisionsperday 1 36 10.98 4.69
Table3
Summaryofweatherdata.
Minimum Maximum Mean Stddeviation Mintemperature(◦C) −46.1 14.3 −5.49 11.99
Maxtemperature(◦C) −33.2 34.0 8.46 13.21
Meantemperature(◦C) −39.7 22.8 1.48 12.34
Groundsnow(depth,cm) 0 52 7.52 12.31
Speedofmaxgust(km/h) 4 95 37.88 10.95
collisiontypeandthepercentageofunknowncollisioncausesis onlylessthan1%.
Fig.1showsseasonalvariationofcollisionsandtheir relation-shiptoweather.Thecollisionsinautumnandwinterarehigher thaninspringandsummer,ithadaperiodictrend.Wecanalsosee thatwhenthemeantemperaturewaslow,thecollisionswerehigh andviceversa.Fig.1alsoshowsthatwhenthesnowontheground waslow,thecollisionswerelow.Therewasnoclearrelationship betweenthespeedofmaximumwindgustandthecollisions.
3. Methods
Wedividedcollisionandweatherdataanalysisintothree sec-tions.First,wemodeledthecollisiontrendsandanalyzedtheeffects ofenvironmentalvariablessuchastemperature,thicknessofsnow, andspeed of maximumwind gust oncollision frequencies and collisionproportions.Forthecollisionfrequencies,weused neg-ativebinomialtimeseriesmodel.Sincetrafficvolumedatawere notavailable,wealsolookedattheproportionsofdailyinjuryand fatalorPDOcollisions.Thoseareexpectedtobeindependentof thetrafficvolume.For thesecollision proportions,we usedthe Gaussiantimeseriesmodelthatwasdesignedtohandlecontinuous dependentvariables.
Somemodelsfor collisions have beenconstructed basedon timeseriesanalysis.Vanlaaretal.(2011)modeledthelog trans-formof monthlycollisionsfrom1994–2008at 48 intersections usingARIMA(p,d,q)models.Quddus(2008)modeledtraffic col-lisionsinGreatBritainusingclassofINARmodelsfortimeseries. Heusedtwosetsofdatatotesthismodels,thatis,annualtraffic collisionsinGreatBritainfrom1950to2005andmonthlytraffic collisioncasualtieswithintheLondoncongestion-chargingzone betweenJanuary1991andOctober2005.Anotherapproach pro-posedbyCastroetal.(2012)wasbasedonpartitioningofasingle latentcontinuousvariableintomutuallyexclusiveintervals.Bythis
way,Castro et al.(2012) reformulateddailycount modelsas a
specialcaseofgeneralizedordered-responsemodels.Thismodel wasappliedtopredictcollisionfrequenciesaturbanintersections inArlington,Texas.ModelsofferedbyVanlaaretal.(2008)and
Quddus(2008)areonlyforunivariatedataandthemodelfittings
inthesepaperswereappliedonlyforshortseriesoraggregated series,whereasCastroetal.(2012)introducedspatialandtemporal
dependenciesthroughthelatentcontinuousvariables. Areview andassessmentofmethodologiesinthestatisticalanalysisof colli-sionsisgivenbyLordandMannering(2010).Thenegativebinomial modelwasalsochosenoverthetimeseriesanalysismodelssince itisamultivariatemodelratherthanaunivariateone.
Moreover,thenegativebinomialtimeseriesischosensincefor timeseriesconsistingofcounts,classicalGaussianmodelsare inap-propriate.Countdataarenon-negative,integer-valuedandoften overdispersed,i.e.,thevarianceislargerthanthemean.Poisson modelsarenotrobustformodelingcountdata,since theyhave over dispersion problems (Cameron and Trivedi, 1998). There-fore,manyresearchersusedoverdispersedPoissonandbinomial regressionmodeltohandlethisproblem.Asanaturalextensionof Poissondistribution,itiswellknownthatnegativebinomial dis-tributionismoreflexibleandallowsforoverdispersion(Davisand
Wu,2009).El-BasyounyandKwon(2012)modeledthetimeand
weathereffects oncollisionfrequenciesusingCityofEdmonton data from 2000to 2010 by using multivariatePoisson lognor-malmodels.ThismodelismorerobustthenthePoissonmodels. However,thePoisson-lognormal haslimitations,i.e., themodel estimationismorecomplexbecausethePoisson-lognormal dis-tributiondoesnothaveaclosedform.Therefore,manyresearchers usedtheMarkovChainMonteCarlosimulationforestimatingthe parametersofthosemodels,sincetheclassicalestimationofthose parameterscannotbecalculatedinastraightforwardway. More-over,thePoisson-lognormalcanstillbeadverselyaffectedbysmall samplesizesand low sample meanvalues (Miaouand Mallick, 2003).Thenegativebinomialtimeseriescanbesolvedusing clas-sicalestimation(DavisandWu,2009).
Secondly, we estimated the collision trend and tested for changesincollisionlevelsbefore/afterOPS24byusinginterrupted timeserieswithsegmentedregression(Perrin,2009).
Finally,wemeasuredthestatisticalproperties,i.e.,asequenceof collisionmeansafterOPS24endedandestimatedapointatwhich thestatisticalpropertyofthatsequenceischanged.Thatpointis calledachangepoint(Killicketal.,2011).Fromthelengthofdays betweentheendsof OPS24tothatchangepoint, wecan mea-surehowlongthebehaviorlastsintermofcollisionsreduction. Moreover,thechangepointdetectionmodelsareusuallyusedin thestatisticalprocesscontrolfordetectingachangeinthe distri-butionfromthetargetintheprocess(HawkinsandDeng,2010).
S.Halim,H.Jiang/AccidentAnalysisandPrevention58 (2013) 106–114 109
Thedetectionofasinglechangepointcanbasicallyberegardedas ahypothesistest;withnullhypothesisbeingthereisnochange pointversusthealternativehypothesisbeingthereexistsachange point. Somemodelsthat assume normalityin theobservations aregivenbyHawkinsetal.(2003),whichdetectedthechangein mean,andHawkinsandZamba(2005),whichdetectedchangein meanandvariance.Here,weappliedtheHawkinsandDeng(2010)
nonparametricchangepointdetectionmodel.Thenon-parametric approachwaschosenbecausewedonotneedtoassumeany dis-tributionofthedatasetasinaparametricmodel.
3.1. Negativebinomialmodel(DavisandWu,2009)
Let Yt be a time series of counts and Xt
be an observed l-dimensional covariate, Xt=
1,t N,cos
2t360
,sin
2t360
,Tempt,Snowt,Gustt
, t is thedis-crete time, s is the seasonal term, and the link function is:
log
r(1−pt) pt=xT
tˇ+˛t (1)
where{˛t}latentprocesswithGaussianinnovations.Hereˇisthe vectorofregressioncoefficients.Theusedofharmonicfunctions inthecovariateXtisforcapturingtheseasonaleffectsinthedata
(Davisetal.,2000).
TheconditionaldensityfunctionofYtis
p(Yt=yt|˛t)=
yt+r−1
r−1
prt(1−pt)yt (2)
Foryt=0,1,...Lettheprocess{εt=e˛t},whichisastrictly
sta-tionarynonnegative timeserieswhen {˛t} isstrictlystationary,
center{˛t}suchthat{εt}hasmeanone.Theconditionalmeanof Ytcanbewrittenintermsofεt,
E(Yt|˛t)= r
(1−pt)
pt
=rexp
xTtˇ+˛t=rexp(xTtˇ)εt∞ (3)withE(εt)=1,wehaveE(Yt)=rexp(xTtˇ)theformforapure
gen-eralizedlinearmodelinthenegativebinomialcase.When{˛t}is,
inordertosatisfytheconditionofE(e˛t)=1,itisrequiredthat
˛t∼N(−˛2/2,˛2).Astandardnegativebinomialgeneralizedlinear
modelwasfittedtothedata.Theestimate ˆrwasdeterminedbythe rvaluethatyieldedthesmallestAIC.The{˛t}wasestimatedfrom
theresidualsof(3).Whenthe{˛t}wasastationaryGaussian lin-earprocesswithinnovationsZtwhichareindependentidentically
distributedN(0,2)variables.Thenthe2wasestimatedusingthe
methodofmoments.
var(Yt)=t+2t
(r+1)2
ε+1
r (4)
andtheestimateofˆ2
ε andˆε(k) wereobtainedusinganordinary
leastsquaretypeestimators(BrännäsandJohansson,1994)
ˆ 2ε=
1 ˆ r+1
⎡
⎣
ˆ r
t=1n ˆ
2
t
(Yt−ˆt)2−ˆt
t=1n ˆ4t
−1
⎤
⎦
(5)ˆ ε(k)=
ˆ ε−2
t=k+1
n ˆtˆt−k(Yt−ˆt)(Yt−k−ˆt−k)
t=k+1n ˆ2tˆ2t−k
(6)
Thelatentprocessmodels’parameterscanbeestimatedusing
(6)andthestandardmethodsas,YuleWalkerEstimate,algorithm forestimatingmovingaverageparameters(seeBoxandJenkins, 1970).Togetthesolutionofthismodelforpredictingthecollisions inEdmonton,weusedR-programming(CRAN,2012).
3.2. Gaussiantimeseriesmodel
Toaccountformissingtrafficvolumedata,themodelingalso consideredcollisionproportions.SincetheNegativeBinomialtime seriesmodelisdesignedforcountdependentvariables,itwillnot beappropriatetohandlethedailyinjuryandfatalorPDOcollision proportions.Therefore,wechangedthelinkfunctiontoGaussian, whichisappropriatetohandlecontinuousdependentvariables.
3.3. Interruptedtimeserieswithsegmentedregressionmodel
Studyofinterruptedtimeserieshasbeenappliedinmanytraffic conditions;Hamedetal.(1999)usedadiffusionmodelwithjumps toinvestigatetheeffectofGulfcrisistotrafficcollisionsinJordan. Here,weusedasegmentedregression(Perrin,2009)forestimating thetrendandlevelofcollisionsbefore/afterOPS24.Thismodelis moreclassicalthanthediffusionmodel.However,itcanbeusedto examinetheimpactofexternalvariables,suchastemperatureand snowonthegroundtothechangeoftrendandlevelofcollisions before/afterOPS24.Thediffusionmodelisappropriateformodeling aunivariatevariable.Thesegmentedregressioncanbeformulated asfollow:
Yt=b0+b1T+b2D+b3P+b4Tempt+b5Snowt+et, (7)
whereYtistheproportionofinjuryandfatalcollisions;Tistime,
Disadummyvariableforpreorpostintervention,Pistimeafter theintervention,etistherandomvariationattimetnotexplained
bythemodel.Temptisthemeanoftemperatureattheobservation
timet,bi,i=0,...,5arecoefficientsthatshouldbeestimated.
3.4. Changepointdetectionmodel
Hawkins and Deng developed non-parametric approach of changepointmodels,byassumingY1,Y2,...,Y,Y+1,...,Ynas
independent,continuousrandomvariableswithstatistical distri-bution
Yi∼F(y)fori=1,2,...,
Yi∼F
y−
fori=+1,...,n,(8)
whereisashiftinlocationoccurringafterthechangepointand nisthenumberofdailyspeeddataforagivenlocation.Inthiscase, YiisthenumberofcollisionsstartedaftertheOPS24.
Inthenon-parametricapproach,thelikelihoodratiotestforthe nullhypothesisthat;H0:1=2(themeanofbeforeandafterthe
changepointisthesame)isthetwo-samplet-statistic.Itisdefined as:
Tk,n=
Uk,n
k(n−k)(n+1)/3
; (9)
withUk,n=2
ki=1Ri−k(n+1);1≤k≤n−1andRiistherankof
YiwithinY1,...,Yn,Inthenullcase,weareassumingconstant
vari-ance,theTk,nfollowsat-distributionwithn−2degreesoffreedom.
TheisgivenbyfindingTmax,n,themaximumof
Tk,n acrossallpossiblekvalues.
4. Results
Table4
Parametersofnegativebinomialmodel.
Allcollisions Injuryandfatalcollisions PDO
Explanatoryvariables Coeff. Pr(>|t|) Coeff Pr(>|t|) Coeff Pr(>|t|) Intercep 4.3131 <2e-16* 2.4201 <2e-16* 8.473e-01 <2e-16*
Slope −0.2159 2.39e-13* −0.2252 1.55e-11* −7.777e-04 0.8489
Cosinus 0.1709 9.22e-10* 0.1844 7.95e-09* −8.535e-03 0.027+
Sinus 0.0448 0.0240+ 0.0354 0.11713 3.286e-03 0.2361
Temperature -0.018 <2e-16* −0.0037 0.05635# −1.623e-03 3.13e-11*
Snow 0.0054 2.13e-05* 0.0045 0.0021− 2.966e-05 0.8682
Gust 0.0022 0.0226+ 0.0009 0.40756 1.265e-04 0.3461
Nulldeviance 1842.5 1331.9 3.7317
Residualdeviance 1238.5 1228.9 2.9317
AIC 11179 6893.6 −3867.
Signif.codes: * <0.01. +0.01. − <0.05. #<0.10.
4.1. Generalcollisiontrendandweatherinfluenceanalysis
Firstly,wefittedthecollisiondatausinga standardnegative binomialgeneralizedlinearmodelandthe ˆrwasdeterminedbythe rvaluethatyieldedthesmallestAIC.AICstandsfor‘Akaike Infor-mationCriterion’,itisameasureoftherelativegoodnessoffitof astatisticalmodel.TheAICisgroundedintheconceptof informa-tionentropy,ineffectofferingarelativemeasureoftheinformation lostwhenagivenmodelisusedtodescribereality.Itcanbesaid todescribethetradeoffbetweenbiasandvarianceinmodel con-struction,orlooselyspeakingbetweenaccuracyand complexity ofthemodel.AICvaluesprovideameansformodelselection.The smallertheAICvalue,thebetterthemodelfit.Theparametersfrom thismodelweresummarizedinTable4.
FromTable4,wecanseethatforalldailycollisions,therewas
asignificantdecliningannualtrend;theperiodictrendfolloweda cosinefunction,meaningthatcollisionsdecreasedfromJanuaryto June,andincreasedtoDecember;therewasasignificantinverse relationshipwithmeantemperature;significantpositive relation-shipwiththicknessofsnowandwindgust.Fordailyinjuryand fatalcollisions,therewasalsoasignificantdecliningannualtrend; theperiodictrendalsofollowsacosinefunction;non-significant inverserelationshipwithmeantemperature;significantpositive relationshipwiththicknessofsnow;andnon-significantpositive relationshipwithwindgust.PDOcollisionsdidnothaveatrend; italsofollowedacosinefunction.Ithadasignificantinverse rela-tionshipwiththemeantemperature,sowecansaythatthePDO collisionsincreaseduringthewintertime.Therelationshipwiththe thicknessofsnowandwindgustwaspositivebutnotstatistically significant.
Now,usingtheaboveparameters,wethencalculatedthelatent process ˛t using Eqs.(7) and (8). Since thelatent processis a
randomseries,wedidaMonteCarlosimulationforthelatent pro-cessbygeneratingthatprocessB=1000times.Further,wefitted themeanoftheestimatedseries,i.e., ˆYt intothedataand
con-structedtheconfidenceintervalbytakingthe2.5%and97.5%of theorderedestimatedseries.Thefittedvaluesandtheconfidence
intervalsfordaily,injuryandfatalandPDOcollisionsaredepicted inFig.2(a–c)respectively.Fromthosefigureswecanseethatthe modelsfollowedtheseasonalitytrend.ExceptfortheInjuryand Fatalcollisions,theconfidenceintervalofthefittedvaluecovered thedatawell.
Thefittedmodelmeasurementsaregiveninfivetypes,i.e.,mean squareerror(MSE),rootmeansquareerror(RMSE),meanof abso-lutedeviation(MAD),symmetricmeanabsolutepercentageerror (SMAPE)andTheil’sU1.SMAPEisanaccuracymeasurebasedon percentageerrors.SMAPEcanbeformulatedas
Nt=1|Yt−Yˆt|/(Yt+
ˆ
Yt),whereYtisthetruevalueofcollisionsand ˆYtistheestimated
oneattimet.TheTheil’sU1statisticisboundbetween0and1, withvaluescloserto0indicatinggreaterforecastingaccuracy.The Theil’sUstatisticsmeasurewhetherthemodelbeingusedis pro-vidingvaluableinformation.
Thesummaryofthosemeasurementsforthemeanofthe esti-matedseriesisgiveninTable5.InwhichwecanseethattheSMAPE oftheinjuryandfatalcollisionsonlyreach15.9%,whileforthedaily collisionsandPDOcollisionsitcanreach13.5%and14.2%, respec-tively.SincetheTheil’sU1statisticiscloserto0,wecandeducethat themodelsareprovidingvaluableinformation.Amongfour mea-surements,i.e.,MSE,RMSE,MAD,SMAPE,thesmallerthevalue, thebetterthefittedmodel.However,MSE,RMSE,and MADare notrobusttotheoutliersinthedata.SMAPEismorerobusttothe outliersthanthoseotherthreemeasurements(Flores,1986).
4.2. Collisionproportionanalysis
Toanalyzethedailyproportionofinjuryandfatalcollisionsor PDOcollisionstototalcollisions,weusedtheGLMwithGaussian densityfunctionasthelinkfunctionsincethedependentvariableis continuous.AsshowninTable6,fordailyPDOandinjuryandfatal collisionproportions,therewasanon-significantdecliningannual trend;fairlysignificantcosineperiodictrend;significantnegative relationshipwithmeantemperature;andnon-significantpositive relationshipwiththicknessofsnowandwindgust.Thefittedmodel
Table5
Thefittedmodelmeasurements.
Negativebinomialtimeseries
Measurement Dailycollision Injuryandfatalcollision PDO
MSE 824.6126 17.9288 689.3258
RMSE 28.7161 4.2343 26.2550
SMAPE 13.505 15.87 14.15
MAD 20.3815 3.3675 18.1344
S.Halim,H.Jiang/AccidentAnalysisandPrevention58 (2013) 106–114 111
Fig.2. Fittedmodelintothedatafor(a)dailycollisions,(b)injuryandfatalcollisions,(c)PDOcollisions.
totheactualproportionoftheinjuryandfatalcollisionproportions andthePDOcollisionproportionsaredepictedinFig.3.Inthose figureswecanseethatthe95%confidenceintervalofbothfitted modelscancovertheactualproportions.
Ingeneral,thePDOcollision frequenciesand theproportion ofPDOshaveasimilarpropertyintermofrelationshipwiththe weather. The collision frequencies and proportion of PDOs are higherinthewintertimethaninthesummertime.However,this relationshipisoppositeforinjuryandfatalcollisionfrequencies andproportions.DuringthesummertimeinEdmonton,drivers tendtospeedmorewhentheroadconditionisgoodandtrafficis notcongested.
4.3. ImpactofOPS24oncollisions
Inthissection,wepresentthegeneralimpactofOPS24on col-lisionsbyperformingtrend,interceptandweatherfactoranalysis usinginterruptedtimeserieswithsegmentedregressionmodel. In2008,theOPS24wereheldduringthewintertime(September, October,andDecember).However,duetotheweatherdata avail-ability,westartedtheanalysisfromOctober2008.
Twoexamples of analysisare discussed here.For OPS24 on October21,2008,Fig.4(left)andTable7showedthattherewasno significantchangeinthelevelofinjuryandfatalcollision propor-tionsafterOPS24,althoughthetrendbecamesignificantlysteeper.
Table6
ParametersofGaussianModels.
Injuryandfatalcollisionsproportion PDOproportion
Explanatoryvariables Coeff Pr(>|t|) Coeff Pr(>|t|) Intercept 1.529e−01 <2e−16* 8.473e−01 <2e−16*
Slope 1.625e−03 0.6911 −7.777e−04 0.8489
Cosinus 7.635e−03 0.0494− −8.535e−03 0.0279#
Sinus −2.843e−03 0.3055 3.286e−03 0.2361
Temperature 1.647e−03 1.62e−11* −1.623e−03 3.13e−11*
Snow −5.751e−05 0.7478 2.966e−05 0.8682
Gust −1.360e−04 0.3111 1.265e−04 0.3461
Nulldeviance 3.7213 0.7478 3.7317
Residualdeviance 2.9278 2.9317
AIC −3864.5 −3867.1
Fig.3.Fittedmodelintotheproportionfor(left)injuryandfatalproportion,(right)PDOproportion.
Fig.4.Trendandinterceptbefore/afterOPS24in2008(blue:intercept;green:trend;red:theOPS24date;magenta:thechangepoint).(Forinterpretationofthereferences tocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)
Table7
Trendandinterceptanalysisresults2008–2011.
Date Changepoint Meanbefore Trendbefore Meanafter Trendafter Temperature Snow 10/21/2008 40 0.16842* −0.00027 −0.00176 −0.00085+ −0.00215 0.00323+
12/16/2008 17 0.15978* −0.00085* −0.04473+ −0.00009 −0.00295* 0.00384*
1/29/2009 25 0.12336* −0.00004 0.00303 −0.00022 −0.00192 0.00292
3/10/2009 8 0.11768* −0.00022 −0.01913 0.00176 −0.00364 0.00492
4/7/2009 20 0.11185* 0.00162 0.03539 −0.00080 −0.00201 0.00299
5/12/2009 4 0.12836* −0.00074 −0.00339 −0.00044 −0.00313 0.00541+
6/9/2009 9 0.11946* −0.00015 −0.02880 0.00317 −0.00466 0.00546
6/26/2009 7 0.14802* 0.00340 0.01317 0.00103* 0.00291 −0.00170
9/29/2009 13 0.23054* −0.00024 −0.04045 −0.00459+ 0.00401 −0.00329
10/16/2009 35 0.14672* −0.00212 −0.00688 −0.00040 0.00067 0.00060
12/15/2009 27 0.13867* −0.00051 −0.00881 −0.00024 −0.00013 0.00141
1/20/2010 13 0.11886* −0.00010 −0.04137 0.00486 −0.00171 0.00308
2/5/2010 17 0.06027+ 0.00634* 0.06988* 0.00056 0.00217 −0.00248
3/29/2010 1 0.12452* 0.00067 0.02840 0.00345 0.00196 −0.00272
4/13/2010 6 0.13046* 0.00282 0.01170 −0.00008 −0.00189 0.00207
6/28/2010 40 0.12986* 0.00034 0.01254 −0.00036 −0.00076 0.00134
9/13/2010 42 0.14103* 0.00053 0.02260 −0.00006 0.00152 −0.00119
11/15/2010 26 0.17917* −0.00017 −0.05020+ 0.00001 −0.00337* 0.00256
12/14/2010 28 0.11547* 0.00013 0.01362 −0.00020 −0.00186 0.00373*
1/19/2011 12 0.11140* 0.00002 −0.02274 0.00203 −0.00238 0.00369+
2/4/2011 39 0.09215* 0.00181 0.01366 0.00060 −0.00157 0.00231
3/28/2011 21 0.12205* 0.00041 0.05790 −0.00110 −0.00041 0.00140
4/21/2011 14 0.18725* 0.00144 −0.05663 0.00135 0.00008 0.00184
5/19/2011 13 0.21358* 0.00068 0.03262 −0.00254 0.00752+ −0.00625
6/7/2011 45 0.22788* 0.00504+ −0.03288 0.00008 −0.00239 0.00094
11/15/2011 5 0.13324* −0.00015 −0.03703 0.00226* −0.00405* 0.00632*
12/16/2011 13 0.10786* 0.00214+ 0.02011 0.00177 −0.00266 0.00458
S.Halim,H.Jiang/AccidentAnalysisandPrevention58 (2013) 106–114 113
Table8
Summaryofcollisiontrendandweatherinfluenceanalysisresults.
Daily collisions
Dailyinjury andfatal collisions
Dailyinjuryand fatalcollision proportions DailyPDO collisions DailyPDO collision proportions
Lineartrend − − + − −
Periodictrend Cosine Cosine Cosine Cosine Cosine
MeanTemperature − − + − −
Thichnessofsnow + + − + +
Windgust + + − + +
+,statisticallysignificantpositiveassociationbetweendependentandindependentvariable.−,statisticallysignificantnegativeassociationbetweendependentand indepen-dentvariable,+/−,non-statisticallysignificantpositive/negativeassociationbetweendependentandindependentvariable.
Snowonthegroundshowedsignificantinfluenceonthepattern change.Themeanofcollisionsproportionslastedupto40days aftertheOPS24ended.Theanalysisof OPS24onDecember 16, 2008showedtheoppositeresult:Fig.4(right)andTable7showed thatthelevelofinjuryandfatalcollisionproportionsafterOPS24 changedsignificantlybutnotthetrend.Bothmeantemperature andsnowonthegroundsignificantlyinfluencedthefatalandinjury collisionproportions.Themeanofcollisionsproportionslastedup to17daysaftertheOPS24ended.
Overall,OPS24eventsaswellastemperatureandsnowonthe grounddidnotinfluencetheinjuryandfatalcollisionproportions. ThiscouldhappensincemostoftheOPS24swereheldwhensnow waspresent,inwhichdriverstendtoslowdown.Theschedulesof OPS24eventssometimeswereclosetoeachother.Theseschedules weredeterminedbytheEdmontonPoliceServicebasedonseveral factors,includingresourcesavailability.
5. Conclusions
The negative binomial time series model that accounts for seasonalityandweatherfactorsshowedthatthemean tempera-tureisnegativelyassociated(withstatisticalsignificance)withall collisiontypes,i.e.,thehigherthemeantemperature,thelower the number of daily collisions. Thickness of snow is positively associatedwithinjuryandfatalcollisions(withstatistical signifi-cance),butisnotsignificantlyassociatedwithPDOcollisions.Wind gust is not associated witheither injury and fatal collisionsor PDOcollisions.Furthermore,dailycollisions,injuryandfatal col-lisionsshowedadecreasingtrendandtheperiodictrendfollowed thecosine function,which meansthatthecollisionswere high inJanuary(winter),slowlydecreasedtoJune(summer),andthen startedtoincreaseagaininSeptember(autumn).Theresultsfrom thisanalysisweremainlysimilartoEl-BasyounyandKwon(2012)’s paper,althoughoneofresultswascompletelyopposite.The simi-laritiesoccurredinthefollowingterms:inbothstudies,theinjury andfatalcollisionshadasignificantdecliningannualtrend, signif-icantinverserelationshipwithmeantemperature,andsignificant positiverelationshipwiththicknessofsnow.Also,inboth stud-ies,PDOcollisionshadsignificantinverserelationshipwithmean temperatureandpositiverelationshipwiththicknessofsnow.The contradictionoccurredinannual trendofPDO collisions:in
El-BasyounyandKwon(2012)’spaper,PDOcollisionshadsignificant
growingannualtrend,whileinthisstudythePDOcollisionshada significantdecliningannualtrend.Onepossiblereasonisthat El-BasyounyandKwonused2000–2010data,wherethetrendswere notuniformoverthiscourseof11yearsthatcouldbeduetoseveral reasons,includingchangesinpopulation,roadwaystructure,and trafficenforcementstrategies
TheGaussian timeseriesmodelthataccountsforseasonality andweather factorsshowedthatonlymeantemperaturehad a significantpositiveassociationwiththedailyinjuryandfatal colli-sionproportions.Thelineartrendisnotstatisticallysignificantfor thedailyinjuryandfatalcollisionsorPDOcollisionproportions.
Table 8 summarizes the association between collision and
weatherfactors.El-BasyounyandKwon(2012)didnotanalyzethe relationshipbetweentheweatherfactorsandthedailyinjuryand fatalcollisionproportions.
Theinterruptedtimeseriesmodelwithsegmentedregression modelshowedthattherewasnostatisticalevidencethatOPS24 hadimpactoncollisionreduction.Thechangeincollisionpattern afterOPS24wasduetothechangeinweatherfactors.
GiventhefactthattheCityofEdmontonischaracterizedby itssometimesextremewinterweatherwhichtypicallyconstitutes fivetosixmonthoftheyear(El-BasyounyandKwon,2012)and the magnitude of the problemcaused by the adverse weather conditions,itisimperativetohaveasolidunderstandingofthe associationbetweencollisionandweatherfactorstocomeupwith anoptimumproactivetrafficsafetycountermeasures.Detail analy-sisofvariousrelevantcountermeasuresisbeyondthescopeofthis study.However,alistofrelevantcountermeasuresandtheir corre-spondingcollisionmodificationfactors(CMFs)basedonprevious studies,suchasinstallingicycurvewarningsystem(CMF=0.82for totalcollisions)andincreasingpavementfriction(CMF=0.76for totalcollisions)canbereviewedfromCrashModificationFactors Clearinghouse(ClearingHouse,2013).
Acknowledgements
Theauthorsarevery gratefultoDr.StevanusA. Tjandar,Dr. YongshengChenandtwoanonymousrefereesfortheirvaluable commentsandsuggestions,whichcontributedtoimprovethe qual-ityofthepaper.
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