1. Report No. 2. Government Accession No. SWUTC/13/600451-00020-1 A .
1. Report No.SWUTC/13/600451-00020-12. Government Accession No.4. Title and Subtitle3. Recipient's Catalog No.5. Report DateA MULTIVARIATE ANALYSIS OF FREEWAY SPEED ANDHEADWAY DATADecember 20137. Author(s)8. Performing Organization Report No.6. Performing Organization CodeYajie ZouReport 600451-00020-1Texas A&M Transportation InstituteThe Texas A&M University SystemCollege Station, Texas 77843-313511. Contract or Grant No.9. Performing Organization Name and Address10. Work Unit No. (TRAIS)DTRT 12-G-UTC0612. Sponsoring Agency Name and Address13. Type of Report and Period CoveredSouthwest Region University Transportation CenterTexas A&M Transportation InstituteThe Texas A&M University SystemCollege Station, Texas 77843-313514. Sponsoring Agency Code15. Supplementary NotesPartly supported by a grant from the U.S. Department of Transportation University Transportation CentersProgram.AbstractThe knowledge of speed and headway distributions is essential in microscopic traffic flow studies becausespeed and headway are both fundamental microscopic characteristics of traffic flow. For microscopic simulationmodels, one key process is the generation of entry vehicle speeds and vehicle arrival times. It is helpful to finddesirable mathematical distributions to model individual speed and headway values, because the individual vehiclespeed and arrival time in microscopic simulations are usually generated based on some form of mathematical models.Traditionally, distributions for speed and headway are investigated separately and independent of each other.However, this traditional approach ignores the possible dependence between speed and headway. To address thisissue, the research presents a methodology to construct bivariate distributions to describe the characteristics of speedand headway. Based on the investigation of freeway speed and headway data measured from the loop detector dataon IH-35 in Austin, it is shown that there exists a weak dependence between speed and headway.The research first proposes skew-t mixture models to capture the heterogeneity in speed distribution. Finitemixture of skew-t distributions can significantly improve the goodness of fit of speed data. To develop a bivariatedistribution to capture the dependence and describe the characteristics of speed and headway, this study proposes aFarlie-Gumbel-Morgenstern (FGM) approach to construct a bivariate distribution to simultaneously describe thecharacteristics of speed and headway. The bivariate model can provide a satisfactory fit to the multimodal speed andheadway distribution. Overall, the proposed methodologies in this research can be used to generate more accuratevehicle speeds and vehicle arrival times by considering their dependence on each other when developing microscopictraffic simulation models.16.17. Key WordsSpeed, headway, correlation, heterogeneity19. Security Classif.(of this report)UnclassifiedForm DOT F 1700.7 (8-72)18. Distribution StatementNo restrictions. This document is available to thepublic through NTIS:National Technical Information Service5285 Port Royal RoadSpringfield, Virginia 2216120. Security Classif.(of this page)Unclassified21. No. of Pages7022. PriceReproduction of completed page authorized
A MULTIVARIATE ANALYSIS OF FREEWAY SPEEDAND HEADWAY DATAByYajie ZouResearch assistantTexas A&M Transportation InstituteDecember, 2013
DISCLAIMERThe contents of this report reflect the views of the author, who is responsible for the facts and theaccuracy of the information presented herein.This document is disseminated under thesponsorship of the Department of Transportation, University Transportation Centers Program inthe interest of information exchange. The U.S. Government assumes no liability for the contentsor use thereof.ACKNOWLEDGEMENTThe author recognizes that support for this dissertation was provided by a grant from the U.S.Department of Transportation, University Transportation Centers Program to the SouthwestRegion University Transportation Center.i
ABSTRACTThe knowledge of speed and head way distributions is essent ial in m icroscopic trafficflow studies because speed and headway are bothfundamental microscopiccharacteristics of traffic flow. For microscopic simulation models, one key process is thegeneration of entry vehicle speeds and vehicle arrival times. It is helpful to find desirablemathematical distributions to m odel individual speed and headway values, because theindividual vehicle speed and arrival time in microscopic simulations are usu allygenerated based on som e form of mathematical models. Traditionally, distributions forspeed and headway are investigated separately and independent of each other. However,this traditional approach ignores the possible dependence between speed and headway.To address this issue, the research presents a m ethodology to construct bivariatedistributions to describe the characteristics of speed and headway. Based on theinvestigation of freeway speed and headway da ta measured from the loop detector dataon IH-35 in Austin, it is shown that there exists a weak dependence betw een speed andheadway.The research first proposes skew-t m ixture models to capture the he terogeneity in speeddistribution. Finite m ixture of skew-tdistributions can significantly im prove thegoodness of fit of speed data. To developa bivariate distribut ion to capture thedependence and describe the characteristics of speed and headway, this study proposes aii
Farlie-Gumbel-Morgenstern (FGM) approach to construct a bivariate distribution tosimultaneously describe the characteristics of speed and headway. The bivariate m odelcan provide a satisfactory fit to the multim odal speed and headway distribution. Overall,the proposed m ethodologies in th is research can be used to generate m ore accuratevehicle speeds and veh icle arrival times by considering th eir dependence on each otherwhen developing microscopic traffic simulation models.iii
TABLE OF CONTENTSPageABSTRACT . IITABLE OF CONTENTS .IVLIST OF FIGURES .VILIST OF TABLES . VIICHAPTER I INTRODUCTION . 11.1 Statement of the Problem . 21.2 Research Objectives . 31.3 Outline of the Research . 4CHAPTER II LITERATURE REVIEW . 62.1 Introduction . 62.2 Speed Distributions . 62.3 Headway Distributions . 72.4 Dependence between Speed and Headway . 82.5 Summary . 8CHAPTER III DATA INTRODUCTION AND PRELIMINARY ANALYSIS . 103.1 Introduction . 103.2 Data Description. 103.3 Preliminary Analysis . 113.4 Summary . 18CHAPTER IV METHODOLOGY I: MIXTURE MODELING OF FREEWAYSPEED DATA. 194.1 Introduction . 194.2 Finite Mixture Models. 194.3 Model Estimation Method. 224.4 Modeling Results . 234.5 Summary . 33iv
CHAPTER V METHODOLOGY II: MULTIVARIATE MIXTURE MODELINGOF FREEWAY SPEED AND HEADWAY DATA . 345.1 Introduction . 345.2 Methodology . 345.3 Results and discussions . 415.4 Conclusions . 55CHAPTER VI SUMMARY AND CONCLUSIONS . 566.1 Summary . 566.2 Conclusions . 566.3 Future Research . 57REFERENCES . 59v
LIST OF FIGURESPageFigure 3.1 (a) speed scatter plots by time of day; (b) headway scatter plots by time ofday; (c) vehicle length scatter plots by time of day; (d) hourly percentage oflong vehicles by time of day. . 12Figure 3.2 Scatter plot of speed and headway for peak period (T1). . 18Figure 4.1 The fitted mixture model for 2-component skew-t distribution. . 32Figure 4.2 The mixture model for 4-component normal distribution. . 33Figure 5.1 24-hour speed and headway histograms (a) speed (b) headway. . 43Figure 5.2 Bivariate speed and headway histogram and fitted distribution: (a)bivariate speed and headway histogram; (b) fitted bivariate distribution. 52vi
LIST OF TABLESPageTable 3.1 Summary statistics of speed and headway for different time periods . 15Table 4.1 Computed AIC, BIC and ICL values for three mixture models . 26Table 4.2 The K-S test results for three mixture models . 28Table 4.3 Parameter estimation results for the Skew-t mixture distribution . 30Table 5.1 Parameter Estimation Results, R2 and RMSE Values for Speed Models. 42Table 5.2 Parameter Estimation Results, R2 and RMSE Values for Headway Models . 44Table 5.3 Maximum Correlation Coefficients for Each Marginal Distribution . 45Table 5.4 Summary Statistics of Headway for Each Speed Group . 46Table 5.5 Seemingly Unrelated Regression Model of Individual Vehicle Speed . 49Table 5.6 Seemingly Unrelated Regression Model of Individual Vehicle Headway . 50Table 5.7 R2 and RMSE Values for Bivariate Distributions . 51vii
CHAPTER IINTRODUCTIONSpeed is a funda mental measure of traffi c performance of a highway system(May,1990). Most analytical and sim ulation models of traffic either produce speed as anoutput or use speed as an input for travel ti me, delay, and level of service determ ination(Park et al., 2010). It is desi rable to find an appropriate mathematical distribution todescribe the measured speeds, because in some microscopic simulations the individualvehicle speed needs to be de termined according to som e form of m athematical modelduring vehicle generation (Park et al., 2010).Headway is an important flow characteristic and headway distribut ion has applicationsin capacity estim ation, driver behavior st udies and safety analysis (May, 1990). Thedistribution of headway determ ines the re quirement and t he opportunity for passing,merging, and crossing (May, 1990). Theheadway distribution under capacity-flowconditions is also a primarily factor in determining the capacity of systems. Moreover, akey component in m any microscopic simulation models is to gene rate entry vehicleheadway in the sim ulation process. To generate accura te vehicle arrival tim es to thesimulated network, it is necessary to use appropriate mathematical distributions to modelheadway.1
As described above, th e knowledge of speed and headway is necessary because thesevariables are funda mental measures of tr affic performance of a hi ghway system.Therefore, developing reliable and innovative analytical techniques for analyzing thesevariables is very im portant. The primary goal of this resear ch is to develop som e newmethodologies for the analysis of microscopic freeway speed and headway data.1.1 Statement of the ProblemThis research consists of two parts. The firs t part concerns the heterogeneity problem i nfreeway vehicle speed d ata. If the characteristics of speed data are homogeneous, speedcan be generally m odeled by normal, log-normal and gamma distributions. However, ifthe speed data exhibit excess skewness and bimodality (or h eterogeneity), unimodaldistribution function does not give a satisfactory fit. Thus, the mixture model (compositemodel) has been considered by May (1990) for traffic stream that consists of two classesof vehicles or drivers. S o far, the mixture models used in previous studies to fit bimodaldistribution of speed data ctherefore, it is useful toonsidered normal density as the specified com ponent;investigate other types of com ponent density for the finitemixture model.The second parts concern the dependence between freeway speed and headway data.Traditionally, the dependence between speed an d headway is ignored in the m icroscopicsimulation models. As a result, th e same headway distribution m ay be assum ed fordifferent speed levels and this assu mption neglects the pos sible variability of headway2
distribution across speed values. Moreover, a num ber of developed m icroscopicsimulation models generate vehicle speedsand vehicle arrival tim es as independentinputs to the sim ulation process. Up to date , only a few st udies have been directed atexploring the dependence between speedand headway. Considering the potentialdependence between speed and headway, it is us eful to construct bi variate distributionmodels to describ e the characteris tics of speed and headway. Compared with onedimensional statistical models representi ng speed or headway separately, bivariatedistributions have the advantage thatthe possible correlation between speed andheadway is taken in to consideration. Given th is advantage, it is neces sary to cons tructbivariate distributions to improve the accuracy or validity of microscopic simulationmodels.1.2 Research ObjectivesThe primary goal of this research is todevelop new m ethodologies for analyzing thecharacteristics of speed and headway. To acco mplish this goal, followin g objectives areplanned to be addressed in this research.1.To address the heterogeneity problem in freeway vehicle sp eed data, w e applyskew-normal and skew -t mixture models to capture excess skewness, kurtos is andbimodality present in speed distribution. Sk ew-normal and skew-t distributionsareknown for their flexibility,andallowing for hea vy tails, high degree of kurtosisasymmetry. To investigate the applicability of mixture models with skew-norm al and3
skew-t component density, we fit a 24-hour speed data collected on IH-35 using skewnormal and skew-t mixture models with the Expectation Maximization type algorithm.2.To construct bivariate distribution of speed and headway, we examine thedependence structure between th e two variables. Three correlation coefficients (i.e.,Pearson correlation coefficient, Spearman’s rho and Kendall’s tau) are used to evaluatethe dependence between speed and headway.3.To develop a bivariate di stribution for capturing the dependence and describingthe characteristics of speed and headway simultaneously, the Farlie-Gumbel -Morgenstern (FGM) approach is proposed.1.3 Outline of the ResearchThe rest of this research is organized as follows:Chapter II o verviews various m athematical models that hav e been used for describingspeed and headway distributions. S ome studies that focused on the dependence betweenspeed and headway are also discussed.Chapter III provides the charac teristics of the traffic dataset used throughout in theresearch. A preliminary analysis is conducte d to investigate the dependence structurebetween speed and headway.4
Chapter IV applies skew-t mixture models to fit freeway sp eed data. This chapter showsthat finite mixture of skew-t distributions can significantly improve the goodness of fit ofspeed data and better account for heterogeneity in the data.Chapter V explores the applicability of the FGM approach to address the heterogeneityproblem in speed and headway data. This ch apter shows that the bivariate m odel canprovide a satisfactory fit to the speed and headway data.Chapter VI summarizes the m ajor results of in this research. General conclusions andrecommendations for future research are presented.5
CHAPTER IILITERATURE REVIEW2.1 IntroductionThis chapter first provides a review of mathematical models for speed and headway.Specifically, different speed and headway dist ributions proposed in the past studies areintroduced. Then, we discuss som e research focused on the dependence between speedand headway.2.2 Speed DistributionsPreviously, normal, log-normal and other form s of distribution have been used to fitfreeway speed data. L eong (1968) and McL ean (1979) proposed that speed dataapproximately follow a norm al distribution when flow rate is light. Haight and Mosher(1962) showed that the log-norm al distribution is proper for speed data. Gerlough andHuber (1976) and Haight (1965) have used nor mal, log-normal and gamma distributionsto model vehicular speed. Com pared with normal distribution, log-norm al and gammadistributions have the capacity toaccommodate the rig ht skewness and elim inatenegative speed values generated by norm al distribution. If the speed data exhibit excessskewness and bimodality, unimodal distribution function does not give a satisfactory fit;thus, several researchers used the m ixture model to fit the distribution of speed. W henthe traffic stream consists of two vehicletypes, the com posite distribution has beenproposed by May (1990). He also suggested that the vehicle sp eeds for subpopulations6
follow normal or lognormal distributions. Dey et al. (2006) introduced a new param eter,spread ratio to predict the shape of the speed curve. He stated that the bimodal speeddistribution curve consists of a mixture of two-speed fractions, lower fraction and upperfraction. Ko and Guensler (2005) did a si milar study by characterizing the speed datawith two different norm al components, one for congested and the other for noncongested speeds. The congestion characteri stics can be identified based on the speeddistribution. Recently, Park et al. (2010) explored the distribution of 24-hour speed datawith a g-component norm al mixture model. Jun (2010) investigated traffic congestiontrends by speed patterns during holiday travel periods using the normal mixture model.2.3 Headway DistributionsMany headway models have been proposed and these models can be classified into twotypes: single distribution m odels and m ixed models. For single distribution m odels,exponential (Cowan, 1975), norm al, gamma, lognormal and l og-logistic distributions(Yin et al., 2009) have been studied to m odel headway. The representatives of m ixedmodels are Cowan M3 model (Luttine n, 1999), M4 m odel (Hoogendoorn and Bovy,1998), the generalized queuing model and the semi-Poisson model (Wasielewski, 1979).Zhang et al. (2007) perfor med a c omprehensive study of the perform ance of typicalheadway models using the headway data recorded from general-purpose lanes.7
2.4 Dependence between Speed and HeadwayThere have been som e studies that focu sed on the dependence between speed andheadway. Luttinen (1992) found out thatspeed limit and road category have aconsiderable effect on the sta tistical properties of vehicle h eadways. WINSUM andHeino (1996) investigated the tim e headway and braking response during car-following.Taieb-Maimon and Shinar (2001) conducted astudy to investigate drivers’ followingheadways in car -following situation and th e results showed that drivers adjusted thedistance headways in relation to speed. Dey and Chandra (2009) proposed two statisticaldistributions for m odeling the gap and h eadway in the steady car-following state.Brackstone et al. (2009) found th at there is a limited depe ndence of following headwayon speed and the m ost successful relationship fit of headway and speed is an inverserelationship. Yin et al. (2009) also studied the dependence of headway distributions onthe traffic condition (speed pattern) and concluded that different headway models shouldbe used for distinct traffic conditions (speed patterns).2.5 SummaryFrom the above discussion, th ere are several current issu es existing in m odeling thespeed and headway data. First, when modeling multimodal distribution of speed data, themixture models used in previous studies ex tensively considered norm al density as thespecified component; therefore, other type s of com ponent density were not fullyinvestigated. Second, considering the possi ble dependence between speed and headway,8
there were very few studies focusing on cons tructing bivariate distribution m odels todescribe speed and headway simultaneously.9
CHAPTER IIIDATA INTRODUCTION AND PRELIMINARY ANALYSIS3.1 IntroductionAs discussed in Chap ter I, the main objective of this r esearch is to develop newmethodologies for analyzing the characteristics of freeway speed and headway data. Thetraffic data analyzed in this research are the microscopic traffic variables (i.e., individualspeed and headway observations) measured from the loop detector data. The study site ison IH-35 in Austin, Texas. This chapter introduces the c haracteristics of the tra fficdataset which is used throughout in the resear ch. A preliminary analysis is conducted toinvestigate the dependence structure between observed speed and headway data.3.2 Data DescriptionThe dataset was collected at a location on IH -35. IH-35 has four lanes in the southbounddirection and the free flow speed is 60 m ile/hour (or 96.56 kilometer/hour) for all typesof vehicles. Due to the heavy traffic dem and and a large volum e of heavy vehicles, thedata collection site is typi cally congested during the m orning and afternoon peak hours.The detector records vehicle arrival time, presence time, speed, length, and classificationfor each individual vehicle (Ye et al., 2006). This dataset was analyzed in some previousstudies (Ye and Zhang, 2009). The data have 27920 vehicles with recorded speed values,arrival times and vehicle lengths in a 24-hour period (from 00:00 to 24:00, December 11,2004), including 24011 (86%) passenger vehicl es and 3909 (14%) heavy vehicles. For10
this dataset, the headwa y value be tween two consecutive vehicles is the elaps ed timebetween the arrivals of a pair of vehicles. Th e arrival times were recorded in second (s);the observed speeds w ere recorded in m eter/second; and the vehicle lengths wererecorded in meter (m). To compare the result of this work with som e previous studies,we convert the m eter/second to kilom eter/hour (kph). W e also assume that 24-hourperiod (T) consists of two time periods: the p eak time period (T1) which contains twosub-periods 07:10-08:20 and 15:22-19:33; whil e the off peak period (T2) includes twosub-periods 08:20-15:22 and 19:33-07:10.3.3 Preliminary AnalysisFigure 3.1 (a), (b) and (c) display the scatte r plots of speed, headway and vehicle lengthby time of day for each tim e period. Because of large samples in th e dataset, sem itransparent points are used to alleviate som e of the over-plotting in Figure 3.1. Figure3.1 (c) indicates that the observed vehicles seem to consist of two sub-populations: oneat about 5 m eters, representing passenger ve hicles, and the other at about 22 m eters,representing trucks and buses. Previously,Zhang et al. (2008) estimated large truckvolume using loop detector data collected from IH-35, and they classified vehicles intotwo categories: short vehicles (smaller than 12.2 m (40 feet)) and long vehicles (largerthan or equal to 12.2 m (40 feet)). In order to see the changingpattern of vehiclecomposition over the tim e, we calculate the hour ly percentage of long vehicles (greaterthan or equal to 12.2 m ), which is shown in Figure 3.1 (d). It can be observed that the11
proportion of long vehicles is relatively high between 00:00 and 6:00 com pared withother time periods of the day.(a)Figure 3.1 (a) speed scatter plots by time of day; (b) headway scatter plots by timeof day; (c) vehicle length scatter plots by time of day; (d) hourly percentage of longvehicles by time of day.12
(b)(c)Figure 3.1 Continued13
(d)Figure 3.1 ContinuedFrom Figure 3.1 (a), we can see that the speed data exhibit hetero geneity and the m aincause for this heterog eneity is different traffic flow conditions over the 2 4-hour period.Since the characteristics of speed data are heterogeneous, the mixture models are used tocapture bimodality present in speed distribution. Then, we exam ine the correlationbetween speed and headway. Since the 24-hourtraffic data in the study consist ofdistinct traffic flow conditions, it is useful to evaluate the dependence between vehiclespeed and headway under different traffic c onditions. As discussed above, we dividedthe 24-hour traffic data into two tim e periods (i.e., the peak period T1 and the off-peakperiod T2) based on correspond ing traffic conditions. For each tim e period, threecorrelation coefficients are used to evaluate the dependence. These three m easures of14
dependence are Pearson correlation coeffi cient (PCC), Spearm an’s tau (SCC), andKendall’s pho (KCC). The summ ary statistics of speed and headway for different tim eperiods are given in Table 3.1.Table 3.1 Summary statistics of speed and headway for different time periodsT1 (07:10-08:20 andT2 (08:20-15:22 andT (24 dwaySpeedHeadwayMin.00a1.010001st Mean85.33.142.713.1597.243.083rd .6976Number 0.135SCC0.011-0.6350.186vehiclesNote: a Headway values are less than 0.5s.15
PCC measures the linear relationship between tw o continuous variables. It is defined asthe ratio of the covariance of the two variables to the product of their respective standarddeviations:PCC Cov( x, y )(3.1)σ xσ ywhere σ x and σ y are the standard deviations of variables x and y.SCC is a rank-based version of the PCC and it can be computed as:n (rank ( x ) rank ( x))(rank ( y ) rank ( y))SCC ii 1inn (rank ( x ) rank ( x)) (rank ( y ) rank ( y))2ii 1i 1(3.2)2iwhere rank ( xi ) and rank ( yi ) are the ranks of the observation xi and y i in the sample.Similar to SCC, KCC is designedto cap ture the asso ciation between two m easuredquantities. KCC quantifies th e discrepancy between the nu mber of concordant anddiscordant pairs. Its estimate can be expressed as follows:nKCC n sgn( x x ) sgn( y y )i 1 j 1ijij(3.3)1n(n 1)2 1 if ( xi x j ) 0 1 if ( yi y j ) 0 where sgn( xi x j ) 0 if ( xi x j ) 0 and sgn( yi y j ) 0 if ( yi y j ) 0 . 1 if ( x x ) 0 1 if ( y y ) 0ijij 16
Note that the PCC, KCC, and SCC are -0.469, -0.488 and -0.635 between speed andheadway for peak perio d T1, suggesting a moderate inverse relationship between thesetwo traffic variables. Since speed and headway values in peak period T 1 were observedunder congested traffic conditi ons, it is reason able to con sider most of the headwayvalues in tim e period T1 as following headwa ys. From Figure 3.2, it is observed thatheadway increases as speed decreases, and the relationship can be split into two regimes.The time headway is approxim ately stable when speed is above 20 kph in the firstregime. In the second regim e when speed is b elow 20 kph, the tim e headway increasessignificantly as sp eed decreases. The findings from Figure 3.2 are c onsistent with theresults reported in a study conducted byBrackstone et al. (2009 ). In their study, it isshown that there is a lim ited dependence of following headway on speed: the mostsuccessful relationship fit of headway and speed is an inverse relationship. Interestingly,KCC is 0.135 between speed and headway fo r o
Headway is an important flow characteristic and headway distribution has applications in capacity estimation, driver behavior studies and safety analysis (May, 1990). The distribution of headway determines the requirement and the opportunity for passing, merging, and crossing (May, 1990). The headway distribution under capacity-flow
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O ce of Local Government Contents FOREWORD 2 INTRODUCTION 4 5 key things every councillor needs to know 5 SECTION 1 - AN OVERVIEW OF LOCAL GOVERNMENT IN NSW 8 1.1 Local government 8 1.2 Local government reform 9 1.3 The principles for local government 10 1.4 A council's structure 11 1.5 Services and functions that councils provide 12
