Hierarchical Capacity Analysis of Freeways via ... · 77 Lorenz and Elefteriadou defined the...

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Hierarchical Capacity Analysis of Freeways via Nonparametric Bayesian 4

Estimation with Censored Data 5

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7

8 Eren Erman Ozguven, M.Sc. (Corresponding Author) 9

Graduate Assistant, 10

Department of Civil and Environmental Engineering, 11

Rutgers University 12

623 Bowser Road, Piscataway, NJ 08854 USA, 13

Tel: (732)445-4012 14

Fax: (732)445-0577 15

e-mail: [email protected] 16

17

18

Kaan Ozbay, Ph.D. 19 Professor, 20

Department of Civil and Environmental Engineering, 21

Rutgers University 22

623 Bowser Road, Piscataway, NJ 08854 USA, 23

Tel: (732)445-2792 24

Fax: (732)445-0577 25

e-mail: [email protected] 26

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29 30 31 32 33 34 35 36 37

38

Word count: 5498 + 3 Figures + 5 Tables = 7498 39

Abstract: 195 40

Re-submission Date: November 15, 2010 41

42

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44 Paper Submitted for Presentation and Publication at the 45

Transportation Research Board’s 90th

Annual Meeting, Washington, D.C., 2011 46

47

TRB 2011 Annual Meeting Paper revised from original submittal.

ABSTRACT 48 49

Traffic capacity of a freeway differs depending on its distinct sections with different 50

spatial characteristics such as the number and width of lanes, existence and type of 51

shoulders and/ or medians, traffic characteristics (such as the number of breakdowns 52

defined using the sudden changes in the speed and density values that occur during the 53

flow phase transition), and population characteristics (rural and urban areas). To account 54

for these spatial differences, this paper investigates the hierarchical estimation of the 55

traffic capacity distribution on a highway using a nonparametric Bayesian approach 56

assuming two prior distributions, namely Dirichlet and Gamma process priors under the 57

minimization of a squared-error loss function. This approach addresses the difficult 58

problem of the censored observations while treating the model parameters as random 59

variables represented by a probability distribution. The methodology is applied on the 60

highway sections with different spatial characteristics. An application of the method for 61

on and off-ramps of a highway is also presented. Finally, the results are discussed 62

hierarchically with presenting a methodology to simulate censored and breakdown 63

observations and to analyze them statistically using a bootstrap approach in order to 64

obtain the capacity distributions for sections without sufficient data. 65

66


Ozguven E. E., Ozbay K.

3

INTRODUCTION 67

Traffic capacity has utmost importance in every phase of the planning, 68

maintenance and operations for various sections of a roadway. The overestimation of the 69

capacity leads to delays and traffic incidents due to the congestion and excessive volumes 70

whereas the underestimation can lead to high costs and unnecessarily large 71

ramp/mainline delays. Thus, there is a need to study the spatial characteristics of the 72

probabilistic distribution of traffic capacity utilizing the traffic state information 73

including free flow and breakdown observations. Although they have not explicitly 74

considered a concept of capacity distribution, several researchers have recognized the 75

necessity of characterizing roadway capacity as a stochastic phenomenon. 76

Lorenz and Elefteriadou defined the traffic capacity by including probabilistic 77

concepts (2), (3). Kim and Elefteriadou (4) performed a simulation analysis to estimate 78

the capacity of 2-lane/2-way highways focusing on the difference from the deterministic 79

capacity values. Persaud et al. (5) studied the probabilistic breakdown phenomenon in 80

freeway traffic and confirmed that breakdown is stochastic in nature. Chung et al. (6), 81

and Hall and Agyemang-Duah (7) worked on the freeway capacity drop and the 82

definition of the capacity whereas Cassidy and Bertini (8), and Smilowitz et al. (9) 83

studied some important traffic features with respect to the bottleneck flows. A 84

deterministic method was proposed by Cassidy and May (10) for estimating the traffic 85

capacity of major freeway weaving sections. Wang et al. (11), and Lertworawnich and 86

Elefteriadou (12) also worked on the capacity of weaving sections. Another approach 87

employing neural network techniques for estimating traffic capacity of weaving segments 88

has been proposed by Awad (13). Yeon et al. (14), and Kuhne et al. (15) worked on the 89

development of a stochastic theory of freeway traffic. Kuhne and Mahnke (16) derived a 90

formulation for obtaining the cumulative distribution for traffic breakdowns defining the 91

breakdown as a car cluster formation process. Kuhne (17) used Weibull curves to match a 92

broad variety of cumulative distribution functions. Uchida and Muniero (18) proposed a 93

traffic capacity and covariance estimation methodology using a macroscopic traffic 94

model based on a generalized car following model. Jun et al. (19) used a simultaneous-95

spline regression model for the joint estimation of traffic variables. Similarly, for the 96

estimation of the key traffic stream parameters, a dual regime model was proposed by 97

Yao et al. (20) whereas Banks (21), (22), (23), (24) conducted several important studies 98

to develop a new approach to bottleneck capacity analysis. Polus and Pollatschek (25) 99

explored the stochastic nature of freeway capacity by fitting speed-flow diagrams. Laval 100

(26) came up with approximate formulas for highway capacity using the presence of slow 101

vehicles. 102

However, none of these studies focus on the use of statistically robust 103

methodologies to estimate the probabilistic distribution of the freeway capacity spatially. 104

Brilon et al. (27), (28) used a practical estimation method based on an analogy to the 105

statistics of lifetime analysis to obtain the distribution of freeway capacity. They used the 106

Kaplan-Meier estimator to obtain the capacity distribution function, and estimated the 107

parameters of this distribution with various functions, where Weibull function appeared 108

to be the best fit. This lifetime method had been first proposed by van Toorenburg (29) 109

and discussed by Minderhoud et al. (30). Regler (31), applied the capacity analysis in 110

(28) to other freeway sections. Recently, Geistefeldt and Brilon (1) and Geistefeldt (32) 111



4

made a comparative assessment between the direct estimation of breakdown probabilities 112

and estimation of capacity distribution functions via Kaplan-Meier method. They 113

concluded that Kaplan-Meier method performs better than the direct breakdown 114

probability estimation techniques. Ozbay and Ozguven (33), studied the effects of 115

spatial/temporal differences of freeway sections on the capacity distribution using a non-116

informative Bayesian prior distribution called Jeffrey's prior. Ozguven and Ozbay (34), 117

improving the method given in (27), introduced a nonparametric Bayesian estimation 118

under an informative Dirichlet process prior, for the first time, to robustly estimate the 119

traffic capacity model parameters in the presence of insufficient and unreliable data. 120

In this article, we make the following contributions: 121

Improving the Bayesian estimation performed by Ozguven and Ozbay (34), we 122

propose a hierarchical nonparametric Bayesian approach to estimate the traffic 123

capacity distribution where the prior knowledge about the traffic parameters specific 124

to a freeway section is represented by a probabilistic function. 125

We use two different probability distributions, namely Dirichlet and Gamma 126

processes as the priors to compare the traffic capacity distributions for different 127

sections of the same highway, and to test the location specific differences among the 128

estimated distributions. 129

We apply this estimation methodology to the ramp sections using the occupancy 130

data which can be used to enhance freeway control strategies such as ramp metering. 131

With the proposed model, a freeway can be studied hierarchically with respect to 132

the following factors: 133

Number of lanes, 134

Population affects (rural and urban areas), 135

Traffic features (number of breakdowns, free flows and flow phase 136

transition). 137

HIERARCHICAL BAYESIAN ESTIMATION WITH CENSORED DATA 138

The use of Bayesian estimation to obtain the traffic capacity is useful when the 139

following conditions are met: 140

There is substantial amount of censored data in the traffic flow observations. 141

There is insufficient data to study the behavior of the traffic capacity accurately 142

and efficiently. 143

Traffic observations are not 100% reliable. 144

In the context of the probabilistic traffic capacity modeling mentioned in previous 145

studies, these conditions are present. Therefore, there is definitely a need for using 146

hierarchical Bayesian estimators. 147

We first consider the following model. Let 1 2( , ,..., )nq q q q be the traffic flow 148

data and 1 2( , ,..., )nV V V V be the speed data of n observations obtained from the sensors 149

on a freeway section through a specific time period. Data is censored from the right by a 150

threshold speed value of Y where q consists of breakdown and censored flows. 151

Censored flows are the ones where the speed values are higher than the threshold speed 152

so that they do not cause breakdown. Therefore, the data is in the form of: 153



5

, 1,...,

1 if

0 if

j

j

j

j

q j n

V Y

V Y

(1) 154

If 1j , there is a traffic breakdown at volume jq , and, if 0j , traffic is 155

fluent above the threshold speed without any breakdown (censored). Therefore, the 156

important point is to make use of the traffic observations, including the censored 157

volumes, to calculate the probability distribution of the capacity. With this information, 158

we obtain the capacity distribution function ( )F q as: 159

( ) 1 ( ) ( )F q S q P C q (2) 160

Here, C represents the section capacity, and ( ),S q the survival function. The 161

Kaplan-Meier estimation approach to find this capacity distribution was presented in 162

(27). The difference of Bayesian statistics from this approach is that it incorporates the 163

prior knowledge along with a given set of current observations to make statistical 164

inferences. The prior information could come from operational/observational data, from 165

previous experiments or engineering knowledge. The pioneering studies on the Bayesian 166

estimation with right-censored data were conducted by Susarla and Ryzin (35), and 167

Ferguson and Phadia (36). The idea is to obtain the nonparametric estimation of the 168

capacity distribution function using a Bayesian approach with right-censored 169

observations. As a base decision rule to obtain an estimate of the traffic capacity function, 170

a loss function is specified as 171

2

0

( , ) ( ( ) ( )) ( )L F F F q F q dw q

(3) 172

where w is a nonnegative weight function. This loss function simply presents the 173

weighted integrated difference between the actual and estimated values of the capacity 174

function. 175

Kaplan-Meier Estimation (Product Limit Method) 176

The standard estimator of the survival function is the Kaplan-Meier estimator 177

(Product-Limit Estimator) proposed by Kaplan and Meier (37), and applied into traffic 178

engineering by Brilon et al. (27) to obtain the distribution of freeway capacity. Using this 179

approach, traffic capacity distribution function, ( )KF q is obtained as: 180

:

( ) 1 ( ) 1j

j j

K K

j q q j

nF q S q

n

(4) 181

where 182

( )KF q : estimated Kaplan-Meier capacity distribution function, 183

( )KS q : estimated Kaplan-Meier survival function, 184

jn : number of intervals with a flow rate jq , 185

jq : traffic volume in interval j , 186

j : number of breakdowns at a volume of jq . 187



6

This product is calculated over all time intervals j with a traffic volume jq q 188

that were followed by a traffic breakdown. With a comparative assessment, Geistefeldt 189

and Brilon (1) concluded that Kaplan-Meier method performs better than the direct 190

breakdown probability estimation techniques. 191

Hierarchical Bayesian Approach 192

In this paper, a nonparametric hierarchical Bayesian estimation of the traffic 193

capacity analysis is given under the squared-error loss notion with stochastic process 194

priors. We consider a nonparametric approach in which 0 ( )F q , the prior guess on the 195

estimated capacity function, is drawn from a stochastic process ( , )f H based on the 196

observations ( , )q . Here, 0 and H are the baseline probability measures for 197

( , )f H . With this information, the posterior distribution of the capacity distribution 198

function, ( )F q is derived. Therefore, the hierarchical stochastic process becomes (see 199

Teh et al. (38) for details): 200

0

0 0

( ) , ( , )

( ) , ( ) ( , ( ))

F q H f H

F q F q f F q

(5) 201

where is a parameter of the prior distribution 0 ( )F q . The selection of the prior guess 202

has utmost importance. This prior guess based on ( , )f H can be obtained from previous 203

empirical work, data, or researcher's subjective beliefs. In the proceeding sections, we try 204

two different priors, namely Dirichlet and Gamma process priors, to obtain the posterior 205

distribution of the traffic capacity function. 206

Dirichlet process prior 207

Dirichlet distribution is the first prior distribution selected to lead to closed-form 208

estimates for the survival function using the loss function given in Equation (3). This 209

prior is chosen because it is a conjugate prior (the one which produces a posterior 210

distribution of the same type as the prior) for the survival function. This makes the 211

computation of the posterior particularly simple. 212

When we use a Dirichlet process (DP) in Equation (5), the general hierarchical 213

model becomes as follows: 214

0

0 0

( ) (0, ) ( (0, ))

( ) , ( ) ( , ( ))D

F q DP

F q F q DP F q

(6) 215

To assign a prior distribution to the capacity function, the parameter function is 216

taken to be of the form 0( , ) (1 ( ))q F q where 0( ) 1 (0, )F q is the prior 217

guess at the traffic capacity function and is a kind of measure that indicates how much 218

weight to put on the prior guess. Here, we take the prior guess at the survival function 219

0 0( ) 1 ( )S q F q with a Dirichlet prior given as (0, ) qe and our problem becomes: 220

0 ( ) 1 (0, ) 1

( ) , ( , ) ( , ( , ))

q

D

F q e

F q q DP q

(7) 221



7

As observed, there is a precise relationship between the prior and posterior 222

distributions; therefore, we obtain the posterior distribution as: 223

0( , ) (1 ( )) (0, ) qq F q e (8) 224

where is heuristically calculated using the Kaplan-Meier data of breakdown intervals. 225

To find the Bayesian estimator with Dirichlet prior, the conditional pth

moment of 226

the breakdown probability ( ) 1 ( )D DF q S q given ( , )q is calculated as follows, where 227

( , )q is distributed as a Dirichlet process on (0, )R : 228

1

0

( , )(1 ( )) ( , ) ( )

(0, )

pqp

D

s

q s nE F q q j

s n

(9)

229

which is adapted from (35) where 230

( , ) :

( , )( )

un : 1

j j

j

j j j j

j

q nwhile q censored

q nj

while q censored

231

jn : number of intervals with a flow rate greater than or equal to jq , 232

n : total number of observations, 233

qn : total number of observations with a flow rate greater than q , 234

j : number of censored observations at a volume of jq . 235


p , s : nonnegative integers. 237

Using Equation (9), the Bayesian estimator of the survival function with the 238

Dirichlet prior is calculated taking 1p . That is, we are focusing on the conditional 239

mean of 1 ( )DF q given ( , )q . Then, our Bayesian estimator becomes: 240

( , )( ) 1 ( ) 1 ( )

(0, )

q

D D

q nF q S q j

n

(10) 241

where 242

( , ) :

( , )( )

un : 1

j j

j

j j j j

j

q nwhile q censored

q nj

while q censored

243

( )BS q : estimated Bayesian survival function. 244



8

Gamma process prior 245

To compare the Dirichlet results with another distribution, we choose the Gamma 246

prior, namely ( )GF q . With the Gamma prior having a continuous shape parameter ( )q 247

and reciprocal scale parameter , the posterior expectation is calculated for the traffic 248

capacity as follows (see (36) for the original formulation): 249

( )( )

1

1 1

1 ( 1, )(1 ( )) ( , )

1 ( , )1

jq

qqn n

j jq G j j j

G

jq G j j jj j

n nn nE F q q

n nn n

250

(11)

251

where 252

jn : number of intervals with a flow rate jq , 253

n : total number of observations, 254

qn : total number of observations with a flow rate greater than q , 255

1 if

0 if

j

j

j

V Y

V Y

256

j : number of censored observations at a volume of jq . 257


: scale parameter of the Gamma prior, 259

( )q : shape parameter of the Gamma prior, 260

1

0

1 1( , ) 1 log( )

i

G

i

i

i i

. 261

Here, the prior Gamma guess at the capacity function is selected as: 262

( )

0 0( ) 1 ( ) 11

q

F q S q

(12) 263

Then, for all q , we have ( )q for fixed as: 264

0ln(1 ( ))( )

ln1

F qq

(13) 265

266

Finally, the Bayesian capacity function estimator with the Gamma prior is 267

calculated as in the Dirichlet prior case, and our Bayesian estimator becomes: 268

( )( )

1

1 1

1 ( 1, )( ) 1 ( ) 1

1 ( , )1

jq

qqn n

j jq G j j j

G G

jq G j j jj j

n nn nF q S q

n nn n

269

(14) 270

where ( )q is given as in Equation (13) and ( )G as in Equation (11). 271



9

The Bayesian estimator, using the Dirichlet or Gamma process priors, estimates 272

all possible probability values between the breakdown and censored observations 273

(achieving continuity) and therefore smoothes the posterior distribution curve at all 274

discontinuities. Suzarla and Ryzin (39) state that "the Bayesian estimator is a better 275

admissible estimator smoothing the resulting posterior on the discrete survival data using 276

the expectation and variances of the cumulative distribution given in Equation (9) under 277

the loss function used and under the weak convergence topology". Survival models are 278

based on data that measure time to some event such as death, transition or failure. In our 279

case, this time period is replaced by the traffic volume and the discrete failure event 280

becomes the discrete traffic breakdown event. Therefore, assuming the breakdown as a 281

failure event, the statistical methods developed for the survival data analysis can be 282

successfully used to estimate the traffic capacity, which is the analog of the lifetime in 283

this context. 284

COMPARATIVE MODEL APPLICATION 285

Basic Assumptions 286

The concept of determining how to define the breakdown is very crucial in terms 287

of the accuracy and efficiency of the proposed methodology. There are several 288

approaches available to define the freeway breakdown. One approach is to determine the 289

breakdown value by using the speed difference between consecutive intervals whereas 290

another approach is including the average space mean speed to detect the breakdown 291

point. Recently, Geistefeldt and Brilon (1) used a more sophisticated set of criteria to 292

determine the breakdown value. In this study, the flow values are classified, using a 5-293

minute interval denoted as i , and shown in Table 1 where Y is a threshold speed, and 294

is a small speed value selected for comparison to determine the breakdown. This 295

approach is logical as the traffic breakdown is usually followed by a significant amount 296

of reduction in speed, however every interval followed by a speed lower than the 297

threshold value cannot be considered as a breakdown interval (B). If each congested 298

volume was considered as a B-value, we would be considering all the breakdown flows 299

again and again until the end of the analysis after the failure of a specific road segment. 300

Hence, the intervals that causes breakdown with satisfying the conditions in Table 1 is 301

considered as a B-value. 302

The traffic congestion usually leads to a breakdown, which is usually determined 303

by a sudden decrease in the speed level accompanied with a simultaneous steep increase 304

of the traffic density level that occurs during the phase transition from free-flow to 305

breakdown flow. When the traffic breakdown ends, the speed and density both return to 306

the normal level of the uncongested conditions. Therefore, while considering the land 307

usage for the proposed spatial analysis in this study, the phase transition is also 308

considered during the determination of the breakdown points used in Table 1. 309

310

311

312

313

314 315



10

TABLE 1 Breakdown Interval Definitions (Based on (1)

) 316

Interval Definition

B

Traffic is fluent in time interval i , but the observed volume causes a

breakdown, satisfying the following conditions:

1iv Y ,

iv Y ,

1iv Y ,

2iv Y ,

1 1 2

1 1( ) ( )

2 2i i i iv v v v .

C

Traffic is fluent in the intervals i and 1i . This interval i contains a

censored value. Its information is that the actual capacity in interval i is

greater than the observed volume iq . C-values represent the right-censored

data in the analysis. That is, C-values are accompanied by speed values

higher than the threshold speed and they do not cause breakdown.

D1

Traffic is congested in interval i , i.e. the average speed is below the

threshold value. As this interval i provides no information about the

capacity, it is disregarded. These are mainly those intervals representing

congested flow conditions, which do not contain any information about the

traffic capacity.

D2

D2-values are similar to B-values; however they are due to a breakdown in

a downstream cross-section. That is, the reason that the speed goes below

the threshold value is not the interval itself, but rather downstream

congestion. Traffic is fluent in interval i , but the observed volume causes a

breakdown. However, unlike classification B, traffic is congested at a

downstream cross section during interval i or 1i , this case, the

breakdown at the observation point is supposed to be due to a tailback from

downstream. As this interval i does not contain any information for the

capacity assessment at the observation point, it is disregarded.

D3

Traffic data may include low traffic flow values having a wide range of

corresponding speed values. As they have nothing to do with the

breakdown, they are disregarded within a certain threshold for the data. 317

Data 318

Data obtained from different sections of the I-5 Highway, California, USA from 319

the PEMS database (40) is used to test the location specific differences among the 320

estimated distributions. The information about the sections studied are given in Table 2. 321

322

323

324

325

326



11

TABLE 2 I-5 Highway Section Information 327

Vehicle

Detection

System (VDS)

Number

Type Number of

Lanes Location

Maximum

Observed

Volume per

lane (PEMS(40)

)

1015010 Mainline 2 Rural (Stanislaus

County, CA) 1524 veh/h

1108613 Mainline 4

Urban

(Washington St,

San Diego, CA)

1914 veh/h

1205157 Mainline 6 Urban (Orange

County, CA) 1732 veh/h

1214241 On Ramp 2 Urban (Orange

County, CA) -

1211850 Off Ramp 1 Urban (Orange

County, CA) -

328

The important point of the study while using the data is to obtain plausible 329

cumulative distribution functions for different sections of the freeway (which results in 330

obtaining data from several single loop detectors on different lanes of the freeway 331

sections) so that this can be used to make it possible for a traffic engineer/planner to 332

choose the proper pairs of the breakdown probability and traffic capacity for the whole 333

freeway. After analyzing one-year (2009) traffic data obtained for these highway 334

sections, threshold speeds for the mainline sections are selected as 70 m/h , whereas it 335

was taken as 40 m/h for the ramp sections, and is selected 5 m/h for all sections. 336

We have also checked the validity of the breakdown speed values from 337

consecutive detectors. For instance, as observed from Table 2, Vehicle Detection System 338

(VDS) No. 1108613 is one of the detection systems used in the analysis. However, one of 339

the consecutive systems namely, VDS-1108603 is also tested using the model to 340

determine if the breakdowns and the corresponding traffic flow per lane values are 341

reasonably similar. This is simply a way of validating the data. 342

Example Calculation 343

Firstly, we work on an example to illustrate the concept of the Bayesian 344

estimation. Kaplan-Meier estimator is indeed a limit of the Bayesian estimator given in 345

Equation (6) when ( ) 0R . This reduction and better performance of Bayesian 346

estimation using censored data is clearly seen on an example with the following data (8 347

observations obtained from VDS-1015010): 348

: 40,70,90,120 veh/5 min

:50,60,100,130 veh/5 min

Breakdown flows

Censored flows 349

Given the data and selecting the prior guess of the survival function as 350

0 0( ) 1 ( ) qS q F q e , the Bayesian estimators and Kaplan-Meier estimates are 351

calculated in Table 3. Based on the prior guess of the survival function, and taking 1 , 352

we obtain ( ) 1.443q q from Equation (13). 353



12

TABLE 3 Bayesian and Kaplan-Meier Estimates of ( )S q for the Example 354

Flow

(veh/5

min) in

the

interval

Bayesian

Estimate with Dirichlet prior

ˆ ( )DS q

Bayesian

Estimate with Gamma prior

ˆ ( )GS q

Kaplan-

Meier

Estimate

ˆ ( )KS q

[0,40) 8

8

qe

1.449

10

q

1.0

[40,50) 7

8

qe

1.44 57.7210

ln( )8 81 9* *

99 80ln( )

8

q

7/8

[50,60)

50

50

6 7*

8 6

qe e

e

1.44 57.72 72.1510

ln( )7 81 649* * *

98 80 63ln( )

8

q

7/8

[60,70)

50 60

50 60

3 7 6* *

8 6 5

qe e e

e e

1.44 86.586 49

*......*7 48

q

7/8

[70,90)

50 60

50 60

3 7 6* *

8 6 5

qe e e

e e

1.44 1017

ln( )5 36 6*......*

66 35ln( )

5

q

7/10

[90,100)

50 60

50 60

3 7 6* *

8 6 5

qe e e

e e

1.44 129.876

ln( )4 25 5*......*

55 24ln( )

4

q

21/40

[100,120)

50 60

50 60

100

100

2 7 6* *

8 6 5

3*

2

qe e e

e e

e

e

1.44 144.3

3 16*......*

4 15

q

21/40

[120,130)

50 60

50 60

100

100

2 7 6* *

8 6 5

3*

2

qe e e

e e

e

e

1.44 173.16

4ln( )

2 9 3*......*33 8

ln( )2

q

21/80

[130, )

50 60

50 60

100 130

100 130

7 6* *

8 6 5

3 1* *

2

qe e e

e e

e e

e e

187.59

4*......*

3

qe

Not

defined

355



13

As ( ) 0Rq

R e

, the ratios coming from the censored data naturally 356

vanish as the values common to both products cancel out reducing the Bayesian estimator 357

with Dirichlet prior to the Kaplan-Meier. However, the Gamma prior estimate does not 358

converge to the Kaplan-Meier. It converges to an estimate which gives more weight to 359

the right tails than the Kaplan-Meier estimate. This takes the Bayesian estimator with 360

Gamma prior away from the other two estimators. 361

Comparative Spatial Results 362

Distributions for the Dirichlet and Gamma priors, namely (0, ) and

( )

1

q

363

are calculated based on the following heuristic argument given in (35) for each freeway 364

section separately. The idea is to force the estimators to satisfy the following equation: 365

,0.5M

Ke quantile P (15) 366

where : ,0.5KM quantile P for the estimated Kaplan-Meier breakdown probability 367

vector KP . Therefore, M represents the 0.5 quartile of the Kaplan-Meier estimated flow 368

vector. Using this method, and by doubly using the observations, is obtained for each 369

section. This approach is found to be useful in (35) as other values of tend to pull the 370

Bayesian estimator away from the Product-Limit estimator. 371

The selected parameters for the Gamma and Dirichlet distributions are given in 372

Table 4. The parameters for Dirichlet prior and for Gamma prior define the prior 373

mass on the traffic volume data. For instance, when the data sample size is n , the prior 374

mass on (0, ) becomes n

. The extreme case is to let in which case the 375

Bayesian estimator reduces to qe . values are optimally obtained for each section 376

iteratively tending not to pull the Bayesian estimator from the Kaplan-Meier curve. On 377

the other hand, the intensity parameter is selected as 1 according to the argument given 378

in (36) and by choosing that value, we correspond reasonably well to the prior sample 379

size for the Gamma process given in this study. With this information, ( )q is calculated 380

with Equation (13). 381

382

383

384

385

386

387

388

389

390

391

392

393

394



14

TABLE 4 Distribution Parameters for the Highway Sections Studied 395

Section ( )q Sample Size

( )q

VDS-1205157 (6 Lanes) 0.1071 0.1071qe 0.0050

0.1071ln( )

ln1

qe

VDS-1108613 (4 Lanes) 0.0498 0.0498qe 0.0040

0.0498ln( )

ln1

qe

VDS-1015010 (2 Lanes) 0.0126 0.0126qe 0.0045

0.0126ln( )

ln1

qe

VDS-1211850 (1 Lane) 0.0344 0.0344qe 0.0035

0.0344ln( )

ln1

qe

VDS-1214241 (2 Lanes) 0.0536 0.0536qe 0.0048

0.0536ln( )

ln1

qe

396

Figure 1 represents the estimated capacity distribution curves for the 2-lane, 4-397

lane and 6-lane sections of I-5 Highway and the spatial comparative plot for the 398

estimators obtained using these parameters. 399



15

400 (a) 401

402 (b) 403



16

404 (c) 405

406 (d) 407

FIGURE 1 Capacity estimators of I-5 highway (FIGURE 1a) for section 1, 408

(FIGURE 1b) for section 2, (FIGURE 1c) for section 3, and (FIGURE 1d) capacity 409

comparison with Bayesian estimation for all sections 410



17

The maximum value of the Kaplan-Meier curve only reaches 1 if the maximum 411

observed volume q is followed by a breakdown. This is encountered in (27), where 412

despite the used large sample, no complete distribution function could be obtained since 413

the highest flow values observed were not followed by breakdown. Bayesian estimators 414

solve this problem and give more complete curves for the estimated capacity 415

distributions. To sum up, the lower and upper tails of the capacity distribution cannot be 416

estimated by non-Bayesian estimation techniques. 417

Moreover, Gamma prior does not work as well as the Dirichlet prior. The 418

Dirichlet prior estimate is closer to the Kaplan-Meier estimate than the Gamma prior 419

estimate. This is because the estimator with Dirichlet prior assigns more mass on the 420

breakdown flows and less mass between these flows and at the tails than Gamma prior 421

estimate. Unlike for the Dirichlet process where the relative change due to a breakdown 422

in the estimator occurs only at the observation itself, for the Gamma prior, some of the 423

information obtained from a breakdown is used to change the relative weight of the 424

points of the estimator to the left of it. Therefore, based on the observations obtained 425

from empirical data, Dirichlet distribution is chosen as a reasonable prior for our model 426

that represents the distribution of traffic capacity. 427

For the Kaplan-Meier estimator, the breakdown probability is 0 up-to the flow 428

with first breakdown, which is not realistic for every case. For instance, consider the 429

section with 4-lanes (VDS-1108613) of I-5 Highway. With a flow value of 1959 veh/h, 430

the Kaplan-Meier curve has a breakdown probability of 0.33 roughly whereas the 431

breakdown probability is 0 for 1515 veh/h. This means that the traffic is almost free 432

flowing (no chance of breakdown) at that volume which is merely 444 veh/h less than 433

1959 veh/h. This unrealistic situation is addressed by the Bayesian estimator calculating 434

the breakdown probabilities for all traffic flow values possible including the relatively 435

lower flows for which breakdowns are not actually observed. Therefore, Bayesian 436

estimation becomes more effective while defining the capacity profile of a freeway 437

section. 438

Bayesian estimators are smoother than the Kaplan-Meier estimator in the sense 439

that the jumps at censored observations are not as large for the Bayesian estimators. 440

Actually, both the conventional and Bayesian estimate are discontinuous at uncensored 441

observations. However, Bayesian approach smoothes the estimator at censored 442

observations by achieving continuity. This smoothness depends on the value of , and 443

the percentage of censored observations which differs from section to section. For 444

example, for the rural section (VDS-1015010), the breakdowns are very limited, and at 445

the same time, the censored observations for higher traffic volumes are not present. This 446

makes it even difficult for the Bayesian estimation to obtain the upper part of the curve. 447

Therefore, possible trajectories (trend lines) for the sections without enough data are 448

plotted based on the shape of the section with VDS-1205157 which has the satisfactory 449

data. A procedure to obtain a trend line mathematically is given in the proceeding 450

sections. 451

Application to Ramps 452

PEMS ramp data (40) do not include the density and volume observations. It only 453

has the number of vehicles passing through the detector, and their occupancies. 454



18

Therefore, we apply the following formula to calculate the speed values from the single-455

loop detector data (41): 456

ˆ( )DL LQV

O

(16) 457

where 458

Q : number of vehicles per 5 minutes, 459

O : occupancy, percentage of the time the loop is occupied by vehicles 460

during the 5 minutes interval, 461

L̂ : average vehicle length, 462

DL : length of detector, 463

: conversion constant. 464

Choosing the average vehicle length as 16 ft, and the length of detectors obtained 465

from the PEMS data, we obtain the speed observations for the ramp sections where Y is 466

selected as 40 m/h , and as 5 m/h . Despite the large data size, it is difficult to come up 467

with a reasonable amount of traffic observations (both breakdowns and censored 468

observations) for the off-ramp section using Table 1. The breakdown values are very 469

limited, and the largest flow value is followed by a breakdown. Hence, Kaplan-Meier and 470

Bayesian estimator curves are almost the same. For VDS-1214241 (on-ramp with 2-471

lanes), we have better data to obtain the capacity distribution. This is possibly because the 472

section with 2-lanes is more suitable to apply a breakdown analysis suggested in this 473

paper. Actually, the section can be viewed as a short mainline freeway section. Moreover, 474

the comparison figure indicates that the capacity of the urban on-ramp is significantly 475

higher than the urban off-ramp section. 476



19

477

478 (a) 479

480 (b) 481



20

482 (c) 483

FIGURE 2 Capacity estimators of two ramp sections of I-5 highway (FIGURE 2a) 484

for section 1, (FIGURE 2b) for section 2, and (FIGURE 2c) capacity comparison 485

with Bayesian estimation for all sections 486 487

Ramp metering strategies that heavily depend on the accurate estimation of the 488

section capacities can be significantly improved using this methodology. For example, 489

for the VDS-1214241 section, rather than having just one probability value for the 490

capacity, a capacity range from 150 to 250 veh/h/lane can be used as (Figure 2): 491

Capacity interval: 150 185 Breakdown probability: 0.25 492



These intervals indicate that the breakdown probability can occur for any traffic 495

flow value between the boundaries of these intervals. For instance, knowing that VDS-496

1214241 section has a 0.50 probability of breakdown for 185 to 210 veh/h/lane, it is 497

possible to adjust the traffic signaling system at the end of the ramp accounting for the 498

higher flows. 499

Discussion of the Hierarchical Modeling Approach 500

The choice of the traffic capacity depends on the engineer’s judgment/purposes at 501

the specific period s/he is working in. In general, the main idea is to choose a strategy 502

that ensures the maximum flow with minimum breakdown probability. Moreover, the 503

breakdown probability initially selected at the design stage may change due to 504

operational characteristics, external conditions, or even a new construction. Using the 505



21

presented methodology, the operating agency has the opportunity to revise the breakdown 506

probability. 507

Spatially, the steepness of the curve is related to the quality and quantity of the 508

data, which changes the parameters of the posterior distributions. For instance, 509

parameter directly depends on the number of breakdown and censored observations. As a 510

rule of thumb, the larger it is, a better data set (better representative information) we have 511

for the roadway section. The parameter for Dirichlet prior, on the other hand, defines 512

the prior mass on the traffic volume data. When the sample size is n , the prior mass on 513

(0, ) becomes n

. The larger it is, the smoother the Bayesian estimator becomes when 514

compared to the Kaplan-Meier estimator. 515

To calculate the Kaplan-Meier estimator, we do not have to know the actual 516

censored observations, but the number of censored observations between two uncensored 517

traffic volumes is sufficient. The Bayesian estimators, however, are calculated with using 518

all the data including both uncensored and censored observations. The fact that Bayesian 519

estimates use all the data makes it preferable to the other estimates. In this sense, the 520

Bayesian estimators serve as a function of the sufficient statistic by fully utilizing the 521

data. 522

The spatial results presented indicate the need for a hierarchical sampling 523

approach to obtain the freeway capacity curves incorporating the stochastic nature of the 524

traffic volume and the section characteristics. For instance, for the rural section of I-5 525

highway (VDS-1015010), it is difficult to obtain a full curve because data obtained do 526

not include reasonable amount of breakdown and even censored observations (Figure 1). 527

On the other hand, for the urban section of I-5 highway (VDS-1205157), we have a better 528

capacity distribution estimate where substantial amount of data including breakdowns are 529

present. Although these two sections differ in terms of driver population characteristics 530

and number of lanes, the lanes were actually built with similar design principles, and 531

therefore the actual capacity distribution curves of the roadways should be similar in 532

shape. Therefore, given the sufficient amount of data, it can be possible to obtain a 533

capacity estimator curve for VDS-1015010 by changing the parameters of the capacity 534

estimating distribution. With this idea, data obtained for the urban section (with sufficient 535

amount of breakdown and censored observations) is used as the base to simulate the prior 536

distribution of the hierarchical model presented. This is also a way of assessing the 537

robustness of the estimation results by analyzing the subsets of the traffic data samples. 538

Geistefeldt and Brilon (1) chose half of their data set as the subset sample, and obtained 539

the estimator curves calculating the parameters of a Weibull distribution. Similarly, we 540

focus on the data obtained from VDS-1205171, and we take a sample of 3 -months from 541

the data set to obtain reliable parameters for the Dirichlet estimator. 542

To recall, the data should be in the form of: 543

, 1,...,

1 if

0 if

j

j

j

j

q j n

V Y

V Y

544

If 1j , there is a traffic breakdown at volume jq , and, if 0j , traffic is fluent above 545

the threshold speed without any breakdown (censored). From Equation (10), the Bayesian 546



22

estimate of the capacity curve for any breakdown Bq value with the Dirichlet prior, 547

ˆ( ( ))BF q is given as follows: 548

( , )ˆ ( ) 1

(0, )

q

B

q nF q

n

(17) 549

whereas the estimate for the censored observations ( ˆ ( )CF q ) for any censored value Cq is 550

as follows: 551

:

( , ) ( , )ˆ ( ) 1

(0, ) ( , )j

q j j

C

j q censored j j j

q n q nF q

n q n

(18) 552

We obtain these breakdown and censored observations, namely Bq and

Cq , where 0 553

or 1 as Bq q or Cq q , respectively. Then, we estimate the parameter values , to 554

obtain ˆ ( )CF q and ˆ ( )BF q based on the observations ( , )q of the VDS-1205157 section 555

where ( , )q is the empirical distribution function assigning the prior mass on each 556

observed value of traffic volume in (0, ) as n

. To perform that, we employ the well-557

known bootstrap method (Efron (42)) and assign a measure of accuracy to the estimated 558

parameters. The bootstrap methodology used for this model can be described as follows: 559

560

We randomly obtain a 3-months data sample of size l from the VDS-1205157. In 561

this data set, ˆ ( )BF q puts mass only at those observations Bq q where 1j , 562

and ˆ ( )CF q puts mass only at those observations Cq q where 0j . 563

This gives a bootstrap empirical distribution function *( , )q , the empirical 564

sample distribution , 1,...,jq j l , and the corresponding bootstrap value 565

* *ˆ ( )F F q . 566

These steps are repeated independently for a large number of iterations, say N , 567

obtaining bootstrap values *1 *2 *ˆ ˆ ˆ, ,..., NF F F . 568

The value of the measure of accuracy ˆˆ ( )BOOT F is approximately calculated 569

using the sample standard deviation of the *F̂ values: 570

2

*

1* 2

1

ˆ( )

ˆ( )

ˆ1

Nj

Njj

j

BOOT

F

FN

N

(19) 571



23

With this approach, we obtain the bootstrap distribution percentiles *F̂ , and 572

standard deviation estimates based on those percentiles. We select N as 1000, and use 573

the Monte Carlo simulation procedure to obtain values in Table 5. The results indicate 574

that the sampling estimator performs sufficient enough in terms of representing the 575

original Bayesian capacity distribution using the VDS-1205157 data. However, it is 576

important to note that the reduced data samples still do contain a considerable amount of 577

at least 95 breakdown observations. That is actually why this section is selected as an 578

input for the analysis. For the data samples including only a few traffic breakdowns, it 579

will be harder to maintain a reasonable sampling bootstrap distribution. 580

581

TABLE 5 Bootstrap Distribution Results 582

q 278q 548q 818q 1097q 1371q 1643q 1918q 2250q

2465q

260 514 769 1042 1297 1550 1803 2125 2340

ˆ ( )F q 0.053 0.056 0.059 0.062 0.071 0.097 0.218 0.449 0.995

ˆBOOT 0.088 0.070 0.065 0.058 0.051 0.046 0.042 0.037 0.031

Percentiles of the bootstrap distribution of *ˆ ( )F q

10% 0.048 0.052 0.056 0.058 0.068 0.082 0.153 0.367 0.876

25% 0.049 0.054 0.058 0.060 0.070 0.089 0.193 0.402 0.914

50% 0.052 0.056 0.060 0.062 0.072 0.095 0.224 0.443 0.995

75% 0.055 0.057 0.062 0.065 0.075 0.102 0.301 0.496 0.996

90% 0.057 0.061 0.064 0.068 0.077 0.107 0.367 0.537 0.999

583

Now, we change the parameters , of the sampling distribution to obtain a 584

reasonable capacity distribution curve for VDS-1015010. The selected values for the 585

parameters , are given in Table 3 for VDS-1015010 roughly representing the 586

sectional and traffic characteristics of the section. As observed in Table 5, the simulated 587

distribution has breakdown probability values similar to those of the input data set. The 588

issue is to which extent the sampled capacity distribution can estimate the capacities for 589

VDS-1015010. Figure 3 shows the actual Bayesian estimator and Kaplan-Meier curve 590

obtained for VDS-1205157, the actual Kaplan-Meier curve for VDS-1015010, and its 591

estimated bootstrap curve obtained via sampling from the input data of VDS-1205157. 592

The result shows that the new sampled curve gives a reasonably good fit for VDS-593

1015010 curve. This can be interpreted as if we had sufficient data for that section to 594

create a capacity distribution curve. 595

596



24

597 FIGURE 3 Capacity estimator for VDS-1015010 via Sampling 598

CONCLUSIONS AND FUTURE RESEARCH 599

This paper proposes a hierarchical estimation methodology to determine roadway 600

traffic capacity distribution using nonparametric Bayesian estimation techniques based on 601

two different priors, namely Dirichlet and Gamma process priors. The distribution of 602

traffic capacity in the presence of censored observations is studied incorporating the 603

spatial features of the roadway sections. Then, Kaplan-Meier estimator is compared with 604

the Bayesian estimators. The analysis is based on five-minute interval observations from 605

I-5 Highway in California, USA. The results indicate that, especially for urban roadways 606

and ramp sections where breakdowns are most likely to occur, the use of a stochastic 607

approach is necessary and crucial. 608

It is mostly impossible to obtain the actual capacity distribution as the flow 609

observations have right-censored data points. However, several non-parametric 610

estimation methods that can handle data with censored observations are available in the 611

literature. Kaplan-Meier method is the most well-known estimation technique to obtain 612

the capacity distribution (Geistefeldt and Brilon (1)). In the presence of sufficient amount 613

of data, these techniques can work efficiently. The comparative results presented in this 614

paper suggest that, for a section with sufficient amount of breakdown values, and without 615

too many censored volume values, it is possible to use the Kaplan-Meier estimator. 616

However, in cases where there are substantial amount of censored traffic volumes in the 617

data, it is necessary to use a hierarchical Bayesian estimation approach to obtain a better 618

capacity estimation curve. As shown before, Kaplan-Meier estimator is indeed a limit of 619

the Bayesian estimator with the Dirichlet prior. Thus, we can state that the estimators 620



25

complement each other, and the traffic engineer should be careful to decide which one to 621

use for the specific case on hand. Moreover, the Kaplan-Meier estimator is not defined 622

for all possible values of traffic flow values ranging from zero to infinity. Since a 623

complete curve cannot be obtained, parametric distribution fitting techniques have to be 624

applied on the Kaplan-Meier estimator. On the other hand, the proposed non-parametric 625

Bayesian estimators serve as a function of the sufficient statistic by fully utilizing the 626

data. 627

Moreover, it is found that Gamma prior performs worse than Dirichlet prior. This 628

is basically because Dirichlet prior estimator assigns more weight on the breakdown 629

flows and less weight between these flows and at the tails than the one with Gamma 630

prior. Unlike for the Dirichlet process where the relative change due to a breakdown in 631

the estimator occurs only at the observation itself, for the Gamma prior, some of the 632

information obtained from a breakdown is used to change the relative weight of the 633

points of the estimator to the left of it. Therefore, based on the observations obtained 634

from empirical data, Dirichlet distribution is chosen as a reasonable prior for our model 635

that estimates the capacity distribution. 636

The effects of the distribution parameters on the shape of estimated capacity 637

curves have shown to have the utmost importance. For instance, parameter directly 638

depends on the number of breakdown and censored observations. As a rule of thumb, the 639

larger it is, a better data set (better representative information) we have for the given 640

section. The parameter for the Dirichlet prior, on the other hand, defines the prior 641

weight on the traffic volume data. When the sample size of the observations is n , the 642

prior mass on (0, ) becomes n

. The larger it is, the smoother the Bayesian estimator 643

becomes compared to the Kaplan-Meier estimator. 644

Finally, the capacity intervals for the breakdown probabilities of ramp sections 645

make it possible for a traffic engineer/planner to choose the proper pairs of the 646

breakdown probability and traffic capacity. Freeway control strategies such as ramp 647

metering that heavily depend on the accurate estimation of the section capacities can be 648

improved by using this methodology. The temporal changes in the freeway capacity can 649

also be studied regarding the weather conditions, seasonal demand, and daylight/darkness 650

conditions. 651

652

ACKNOWLEDGMENTS 653

654 We would like to acknowledge California DOT for providing us with the PEMS 655

data. 656

657

REFERENCES 658

1. Geistefeldt, J., and Brilon, W., “A Comparative Assessment of Stochastic Capacity 659

Estimation Methods”, Transportation and Traffic Theory 2009, Edited by W.H.K. 660

Lam, S.C. Wong and H. K. Lo, pp. 583-602, Springer, 2009. 661

2. Lorenz, M. and Elefteriadou, L., “A Probabilistic Approach to Defining Freeway 662

Capacity and Breakdown”, Proceedings of the 4th International Symposium on 663

Highway Capacity, pp. 84-95, TRB-Circular-E-C018, Transportation Research 664

Board, Washington D. C, 2000. 665



26

3. Lorenz, M., and Elefteriadou, L., “Defining Highway Capacity as a Function of the 666

Breakdown Probability”, Transportation Research Record, Volume 1776, pp. 43-51, 667

2001. 668

4. Kim, J., Elefteriadou, L., “Estimation of Capacity of Two-Lane Two-Way Highways 669

Using Simulation Model”, ASCE Journal of Transportation Engineering, Volume 670

136-1, 2010. 671

5. Persaud, B., Yagar, S., and Brownlee, R., “Exploration of the Breakdown 672

Phenomenon in Freeway Traffic”, Transportation Research Record, Volume 1634, 673

pp. 64-69, 1998. 674

6. Chung, K., Rudjanakanoknad, J., and Cassidy, M.J., “Relation between Traffic 675

Density and Capacity Drop at Three Freeway Bottlenecks”, Transportation Research 676

Part B, Volume 41-1, p. 82-95, 2007. 677

7. Hall, F.L. and Agyemang-Duah, K., “Freeway Capacity Drop and the Definition of 678

Capacity”, Transportation Research Record, Volume 1320, pp. 91-98, 1991. 679

8. Cassidy, M.J., and Bertini, R.L., “Some Traffic Features at Freeway Bottlenecks”, 680

Transportation Research Part B, Volume 33, pp. 25-42, 1999. 681

9. Smilowitz, K., Daganzo, C.F., Cassidy, M.J. and Bertini, R.L., “Some Observations 682

of Highway Traffic in Long Queues”, Transportation Research Record, Volume 683

1678, pp. 225-233, 1999. 684

10. Cassidy, M.J, and May, A.D., “Proposed Analytical Technique for Estimating 685

Capacity and Level of Service of Major Freeway Weaving Sections”, Transportation 686

Research Record, Volume 1320, pp. 99-109, 1991. 687

11. Wang, M., Cassidy, M.J., Chang, P. and May, A.D., “Evaluating the Capacity of 688

Freeway Weaving Sections”, ASCE Journal of Transportation Engineering, Volume 689

119-3, pp. 360-384, 1993. 690

12. Lertworawnich, P. and Elefteriadou, L., “Capacity Estimation for Type B Weaving 691

Areas Based on Gap Acceptance”, Transportation Research Record, Volume 1776, 692

pp. 24-34, 2001. 693

13. Awad, W.H., “Neural Networks Model to Estimate Traffic Capacity for Weaving 694

Segments”, Proceedings of the 4th

International Symposium on Uncertainty 695

Modelling and Analysis, 2003. 696

14. Yeon J., Hernandez S., and Elefteriadou, L., “Differences in Freeway Capacity by 697

Day of the Week, Time of Day, and Segment Type”, Presented at the 86th

Meeting of 698

the Transportation Research Board, Washington D.C., 2007. 699

15. Kuhne, R., Mahnke, R., Lubashevsky, I., and Kaupuzs, J., “Probabilistic Description 700

of Traffic Breakdowns”, The American Physical Society, Physical Review E, Volume 701

65, 2002. 702

16. Kuhne, R., and Mahnke, R., “Controlling Traffic Breakdowns”, Proceedings of the 703

16th International Symposium on Transportation and Traffic Theory, pp. 229-244, 704

2005. 705

17. Kuhne, R., “Application of Probabilistic Traffic Pattern Analysis to Capacity”, Siam 706

Conference, 2001. 707

18. Uchida, K., and Munehiro K., “Impact of Stochastic Traffic Capacity on Travel 708

Time in Road Network”, Presented at the 89th Meeting of the Transportation 709

Research Board, Washington D.C., 2010. 710



27

19. Jun, Y., Hualiang, T., Heng, W., and Siji, H., “Estimating Roadway Capacity Using 711

the Simultaneous Spline Regression Model”, Journal of Transportation Systems 712

Engineering and Information Technology, Volume 9-1, pp. 87-98. 713

20. Yao, J., Rakha, H., Hualiang, T., Kwigizile, V., and Kaseko, M., “Estimating 714

Highway Capacity Considering Two-Regime Models”, ASCE Journal of 715

Transportation Engineering, Volume 135-9, 2009. 716

21. Banks, J.H., “New Approach to Bottleneck Capacity Analysis: Final Report, 717

California PATH Research Report, 1055-1425, UCB-ITS-PRR-2006-13, 2006. 718

22. Banks, J.H., “Effect of Time Gaps and Lane Flow Distributions on Freeway 719

Bottleneck Capacity”, Presented at the 85th

Annual Meeting of the Transportation 720

Research Board, 2006. 721

23. Banks, J.H., “Effect of Site and Population Characteristics on Freeway Bottleneck 722

Capacity”, Presented at the 86th

Annual Meeting of Transportation Research Board, 723

2007. 724

24. Banks, J.H., “Review of Empirical Research on Congested Freeway Flow”, 725

Transportation Research Record, Volume 1802, pp. 225-232, 2002. 726

25. Polus, A., and Pollatschek, M.A., “Stochastic Nature of Freeway Capacity and Its 727

Estimation”, Canadian Journal of Civil Engineering, Volume 29, pp. 842-852, 2002. 728

26. Laval, J.A., “Stochastic Processes of Moving Bottlenecks: Approximate Formulas for 729

Highway Capacity”, Transportation Research Record, Volume 1988, pp. 86-91, 2006. 730

27. Brilon, W., Geistefeldt, J. and Regler, M., “Reliability of Freeway Traffic Flow: A 731

Stochastic Concept of Capacity”, Proceedings of the 16th

International Symposium on 732

Transportation and Traffic Theory, pp. 125-144, 2005. 733

28. Brilon, W., Geistefeldt, J., and Zurlinden, H., “Implementing the Concept of 734

Reliability for Highway Capacity Analysis”, Transportation Research Record, 735

Volume 2027, pp. 1-8, 2008. 736

29. van Toorenburg, J., “Praktijwaarden voor de capaciteit”, Rijkswaterstaatdienst 737

Verkeerskunde, Rotterdam, 1986. 738

30. Minderhoud, M.M., Botma, H., and Bovy, P.H.L., “Assessment of Roadway Capacity 739

Estimation Methods”, Transportation Research Record, Volume 1572, pp. 59-67, 740

1997. 741

31. Regler, M., “Freeway Traffic Flow and Capacity”, Master Thesis, Institute for 742

Transportation and Traffic Engineering, Bochum University, 2004. 743

32. Geistefeldt, J., “Consistency of Stochastic Capacity Estimations”, Presented at the 744

89th

Meeting of the Transportation Research Board, Washington D.C., 2010. 745

33. Ozbay K, and Ozguven E.E., “A Comparative Methodology for Estimating the 746

Capacity of a Freeway Section”, Proceedings of the 10th International IEEE 747

Conference on Intelligent Transportation Systems, Seattle, WA, USA, Sept. 30 - Oct. 748

3, 2007. 749

34. Ozguven, E.E., and Ozbay, K., “A Nonparametric Bayesian Estimation of Freeway 750

Capacity Distribution from Censored Observations”, Transportation Research 751

Record, Volume 2061, pp. 20-29, 2008. 752

35. Susarla, V., and Ryzin, J.V., “Nonparametric Bayesian Estimation of Survival Curves 753

from Incomplete Observations”, Journal of the American Statistical Association, 754

Volume 71-356, pp. 897-902, 1976. 755



28

36. Ferguson, T.S., and Phadia, E.G., “Bayesian Nonparametric Estimation Based on 756

Censored Data”, The Annals of Statistics, Volume 7, pp. 163-186, 1979. 757

37. Kaplan, E.L., and Meier, P, “Nonparametric Estimation from Incomplete 758

Observations”, Journal of the American Statistical Association, Volume 53, pp. 457-759

481, 1958. 760

38. Teh, Y.W., Jordan, M.I., Beal, M.J., and Blei, D.M., “Hierarchical Dirichlet 761

Processes”, Journal of the American Statistical Association, Theory and Methods, 762

Volume 101-476, 2006. 763

39. Susarla, V., and Ryzin, J.V., “Large Sample Theory for a Bayesian Nonparametric 764

Survival Curve Estimator Based on Censored Samples”, The Annals of Statistics, 765

Volume 6-4, pp. 755-768, 1978. 766

40. PEMS Database, http://pems.dot.ca.gov. 767

41. Kutz, M., Handbook of Transportation Engineering, McGraw-Hill, 2004. 768

42. Efron, B., “Censored Data and the Bootstrap”, Journal of the American Statistical 769

Association, Volume 76, Issue 374, 1981. 770


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