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Analysis of Kinematic Parameters and
Driver Behavior at Turbo Roundabouts
Marco Guerrieri1; Raffaele Mauro2; Giuseppe Parla, Ph.D.3; and Tomaz Tollazzi4
Abstract: This study focuses on both vehicle kinematic parameters (speed and acceleration) and behavior parameters (critical interval and
follow-up time) of drivers at turbo-roundabouts. Empirical evaluations of such parameters can be helpful in calibrating traffic microsimulation
models or assigning behavior parameters to closed-form capacity models in turbo-roundabouts (gap-acceptance capacity models) and are also
related to evaluation of vehicles pollutant emissions. The research was based on traffic process observed in the first turbo-roundabout imple-
mented in the city of Maribor in Slovenia. In 2016 a great number of traffic samples were taken with high-frame-rate video recordings [>50
frames per second (fps)]. All vehicle trajectories were obtained with the methods and algorithms typical of the digital imageprocessing
technique (DIP) by filtering the discrete signal of vehicle trajectories fðtÞiwith wavelet analysis. The research results showed that vehicle
speeds on entry lanes are rather moderate (below 25 km=h, 15 m prior to the Yield line), whereas accelerations usually have values inferior to
2m=s2on arm lanes and to 1.5m=s2on ring lanes. The critical intervals tc[distributed according to an Erlang probability distribution
function (PDF)] and follow-up headways tf(distributed according to an inverse Gaussian PDF) have, on the other hand, values in ranges
of tc¼4.03–5.48 s and tf¼2.52–2.71 s, respectively, according to the right- or left-turn lane and to the major or minor entry in question.
DOI: 10.1061/JTEPBS.0000129.© 2018 American Society of Civil Engineers.
Author keywords: Turbo-roundabout; Traffic processes; Driver behavior; Digital image processing technique.
Introduction
Road intersections are absolutely critical for road safety (Ahmed
et al. 2016;Kazazi et al. 2016). In the United States around 40%
of accidents are estimated to occur at intersections (Choi 2010).
When approaching or crossing intersections, the speed limitation
helps decisively to lower the number of fatal or serious accidents
(Tingvall and Haworth 1999). Therefore, in the last decade new
geometric layouts of signalized and nonsignalized intersections
have been designed to improve safety as a result of higher speed
limitation. Among signalized intersections are (Corben et al. 2010;
Stephens et al. 2017):
•Cut-through intersections; and
•Squircle intersections.
Among nonsignalized intersections are
•Turbo-roundabouts; and
•Flower-roundabouts.
The most widely implemented so far have been turbo-
roundabouts, studied especially in terms of capacity (Fortuijn
2009a,b;Mauro and Branco 2010;Tollazzi et al. 2011;
Vasconcelos et al. 2014), safety (Macioszek 2015;Mauro et al.
2015), and environmental performances (Fernandes et al. 2016;
Mauro and Guerrieri 2016). This study first examined the vehicle
kinematic parameters in turbo-roundabouts, in terms of speed and
acceleration, evaluated on each arm (at entry and at exit) and on
the ring lanes. To this end, it has been carried out a sampling of
traffic on a turbo-roundabout in Maribor (Slovenia), between the
roads Titova cesta and H. Bracica cesta, with a maximum radius
Rmax ¼25 m and 3.50-m-wide lanes (Fig. 1). The sampling of traf-
fic data was made after recording vehicle flows by means of a high-
frame-rate video camera (>50 fps) installed on a building adjacent
to the intersection, and able to zoom in on all the vehicle manouvers
in the turbo-roundabout area.
The kinematic parameters were measured with the digital image
processing technique—DIP (Richards 2013;Mitiche and Aggarwal
2014;Zalevsky et al. 2014;Hassannejad et al. 2015), which has,
however, required wavelet analysis in order to limit the noise of the
trajectory-time signal of every vehicle fðtÞi. The surveys made in
2016 were subdivided into intervals ΔT¼15 min. For each of
them the researchers have determined the O/D matrix (Table 1;
the arm numeration is given in Fig. 1), the vehicle arrival, and head-
way processes. Also, users’psycho-technical parameters in terms
of critical interval tcand follow-up time tfhave been estimated.
This analysis was necessary, in that there are still quite few studies
on users’real behaviors in turbo-roundabouts.
The results of this research can also have a practical use for
technical and scientific purposes [i.e., estimation of capacity, de-
lays, and queues with gap acceptance models or by traffic micro-
simulation models and vehicles pollutants emissions (Troutbeck and
Kako 1999;Pollatschek et al. 2002;Al-Madani 2003;Chevallier
and Leclercq 2007;Fernandes et al. 2017)].
Theory: Analysis of the Signal “Vehicle Trajectories
in Function of Time”
The following analyses were based on data obtained by examining
video sequences of road traffic. A digital video sequence, composed
1Adjunct Professor, Qualified as Full Professor, Polytechnic School,
Univ. of Palermo, 90144 Palermo, Italy (corresponding author). E-mail:
marco.guerrieri@tin.it
2Full Professor, Dept. of Civil, Environmental and Mechanical Engi-
neering, Univ. of Trento, 38123 Trento, Italy.
3Transportation Engineer and Consulting Engineer, via G. Pitrè n°3,
93100 Caltanissetta, Italy.
4Full Professor, Dept. for Roads and Traffic, Univ. of Maribor, 2000
Maribor, Slovenia.
Note. This manuscript was submitted on July 7, 2017; approved on
October 3, 2017; published online on March 28, 2018. Discussion period
open until August 28, 2018; separate discussions must be submitted for
individual papers. This paper is part of the Journal of Transportation
Engineering, Part A: Systems, © ASCE, ISSN 2473-2907.
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of a succession of images recorded with given discrete time inter-
vals (frames), contains all the space and time information which,
after specific processing, can be usefully used to rebuild the infra-
structural environment and/or vehicle movement. This information
is obtained automatically by implementing specific algorithms
which allow to detect and track objects and people of interest in
the recorded scene, and to analyze their movement from a kin-
ematic point of view.
A digital video sequence can be schematized as a series of bi-
dimensional matrices arranged according to the time t, in which
each matrix element called pel or pixel (picture element) has the
value of the function Fðx;y;tÞ, with 0≤x≤mand with 0≤y≤
nintegers. This value is obtained by sampling and digitally quantiz-
ing the measurement of the electromagnetic radiation intensity
(color) around every element defined by xand y.
According to the type and number of the used sensors, the
digital image acquisition system is characterized by a radiometric
resolution, also called color depth (bit depth), defined by the total
number of bits assigned to each pixel. For a color image in RGB
(red, green, blue) format, eight bits are generally reserved for each
color component, thus having 16,777,216 different tonalities. For a
color image represented in a matrix form, the value of the light
intensity of every pixel is synthetically expressed by: F½fRðx;y;tÞ;
fGðx;y;tÞ;fBðx;y;tÞ.
The vehicle instantaneous speed calculated through the image
processing technique can be subdivided into the following steps:
1. Detection: i.e., identification of pixels belonging to objects
(vehicles) in motion;
2. Classification or recognition: i.e., exclusive identification of
objects of interest (vehicles);
3. Labeling: i.e., attribution of a label to each object of interest
(vehicle); and
4. Tracking: i.e., when the same object (vehicle) can be seen in
different frames that temporally follow one another.
For details on this procedure and algorithms to be used, and
not included here for spatial reasons, see studies carried out by
Richards (2013), Mitiche and Aggarwal (2014), and Zalevsky et al.
(2014), and experiments to the traffic engineering by Guerrieri et al.
(2013) and Hassannejad et al. (2015).
This procedure allowed determination of the signals of vehicle
trajectories in function of the time fðtÞ.
The trajectory signal of each vehicle fiðtÞis affected by noise,
partly by small camera oscillations caused, for example, by the
wind. Consequently, we performed signal decomposition and
denoising through wavelet analysis (Gomes and Velho 2015;
Chatterjee 2015).
Wavelet Transform
Wavelets are families of real or complex functions, each coming
from the translation and dilation (or contraction) of a single func-
tion ψðtÞcalled the mother wavelet. The mother function must sat-
isfy a condition, called the admissibility condition,or
Zþ∞
−∞ j
ˆ
ψðωÞj2jωj−1dω<∞ð1Þ
where ˆ
ψðωÞ= Fourier transform function of ψðtÞ. Together with
the admissibility condition, also a regularity condition must be veri-
fied, that is, when the scale factor is reduced, the mother wavelet
function decreases rapidly or concentrates in time and frequency
domains. Moreover, the following relations must be valid:
Fig. 1. Turbo-roundabout under study: (a) photo (image by Tomaz Tollazzi and Giuseppe Parla); (b) layout (image courtesy of BPI d.o.o.)
Table 1. O/D Matrix (veh=15 min) on April 12, 2016
Time O/D N (3) E (4) S (1) O (2)
8:00–8:15 N (3) 0 78 210 10
E (4) 31 0 6 0
S (1) 101 30 0 2
W (2) 2 1 0 0
12:00–12:15 N (3) 0 103 267 12
E (4) 43 0 9 0
S (1) 134 39 2 4
W (2) 5 2 3 0
15:00–15:15 N (3) 0 90 243 11
E (4) 39 0 9 3
S (1) 118 35 0 6
W (2) 4 3 0 2
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Zþ∞
−∞ jψðtÞj2dt ¼1Zþ∞
−∞ jψðtÞjdt <∞
Zþ∞
−∞
ψðtÞdt ¼0Zþ∞
−∞
tmψðtÞdt ¼0ð2Þ
The condition that the the first order moment (mean) is zero
implies that ψðtÞhas at least a zero point determining an oscillatory
performance (usually damped) and giving the name to such a func-
tion type (from the English wave). Other typical wavelet proper-
ties are
1. Time-dependent spectral localization;
2. Linearity;
3. Time translation invariance;
4. Scale invariance; and
5. Invertibility.
The mother wavelet can be scaled (or dilated) by a factor aand
translated along the time axis by a factor bto generate daughter
wavelets, whose expressions are (Goupillaud et al. 1984):
Ψa;bðtÞ¼ 1
ffiffiffi
a
pψt−b
að3Þ
where ψa;bðtÞis generated by the mother wavelet; a∈ℜ−f0g=
scale parameter; b∈ℜ= translation parameter.
For the scale parameter athe following considerations can be
taken into account: elevated values of parameter adilate the wave-
let along the time axis and decrease wavelet frequency. Low values
of parameter acorrespond to wavelet compression and frequency
increase.
Continuous Wavelet Transform
The continuous wavelet transform (CWT) of a signal fðtÞis de-
fined as the convolution between the signal itself and the selected
wavelet family, and is analytically expressed by
CWTfða;bÞ¼Cða;bÞ¼ 1
ffiffiffi
a
pZþ∞
−∞
fðtÞψa;bðtÞdt ð4Þ
The wavelet transform provides the degree of similarity between
the function fðtÞand the wavelet function around the considered
signal with a fixed amplitude, in function of the scale factor aand
instant t. Therefore, the CWT establishes a relation between the
two-dimensional time/space function time-scale.
Every scale is provided with the similarity between the mother
wavelet and the signal at different times. Vice versa, every instant of
time is provided with the information on the similarity between the
signal in that period of time and the used mother wavelet. When a
scale factor ais established, the wavelet function is located at the
generic instant band the relevant coefficient Cða;bÞis calculated
with the relation in Eq. (4). Then, the researchers carried out suc-
cessive translations along the time axis until the observation period
was entirely covered (the whole original signal) by obtaining a co-
efficient Cða;biÞfor the scale aand the position bi, with the
generic index iranging between 1 and n(nbeing the number of
the performed translations).
An essential property of the CWT is the possibility of rebuilding
the original signal fðtÞ, if all the coefficients Cða;bÞare known, by
means of the following relation:
fðtÞ¼ 1
KψZZℜℜ
Cða;bÞ1
ffiffiffi
a
pψt−b
adadb
a2ð5Þ
where KΨ= constant that depends on the selected wavelet and,
given the admissibility condition, is equivalent to
kψ¼Zþ∞
−∞ j
ˆ
ψðωÞj2jωj−1dωð6Þ
The application of a continuous wavelet transform involves a
five-step process:
1. Comparison between the selected wavelet and the initial
“segment”of the signal under study;
2. Calculation of coefficient Crepresenting the value of the con-
volution product between the wavelet and the signal segment.
(In general, the greater is C, the more similar are the wavelet
function and the signal performance between each other in the
segment under study);
3. Wavelet time translation by repeating the first two steps until the
entire signal coverage;
4. Spatial wavelet scaling by repeating Steps 1–3; and
5. Iteration of Steps 1–4 for all scales.
Fig. 2. Example of distance-time curves fðtÞifor a 60 s interval (Arm 1, entry); on the left side it is an example of the original signal and filtered
signal for two vehicles (image by Tomaz Tollazzi and Giuseppe Parla)
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Wavelet Analysis for Discrete Signals
A procedure for the wavelet analysis of discrete signals or discrete
wavelet transforms (DWT), which allows a more efficient and
equally accurate analysis of the signals under examination, was
formalized by Mallat (1987). Such a procedure is based on the
discretization of the finite sets of the values assumed by coefficients
aand b, so that the discrete transform of the DWT signal can be
calculated for a number of pairs of values aand bnot higher than
that strictly necessary to decompose the original signal.
The following considerations support the choice for this
discretization:
•When parameter ais small, wavelets are characterized by high
frequency and consequently the translation of parameter bmust
be small; and
•Vice versa, when ais big, wavelets are characterized by reduced
frequency and therefore the translation parameter bmust be
high as well.
The most efficient discretization is given from the following
relations (Mallat 1987):
a¼a−j
0j∈Z;a0>1
b¼nan∈Zð7Þ
In case of a dyadic discretization (a0¼2), initially proposed by
Mallat and generally accepted, the complete wavelet family to be
used for j∈Zand n∈Zis given from
ψj
n¼1
ffiffiffi
a
pψt−a
b
a¼2j;b¼2−jn ¼2j=2ψð2jt−nÞð8Þ
Assuming integer values for jand n, we can easily infer the
coefficient numbers to be calculated in order to perform the DWT.
If then we consider the values in amin and amax as, respectively,
minimum and maximum scale values to be used in the analysis,
jassumes, consequently, all the integer values comprising between
jmax and jmin which can be obtained from relations amin ¼2−jmax
and amax ¼2−jmin. For each scale factor a¼2−J, a limited number
of translations n, depending on the avalue, needs to be carried out
to cover the whole domain of the signal to be analyzed.
Multiresolution Analysis and Signal Reconstruction
Given that the wavelet analysis provides a signal representation
which is more precise at low frequencies and less precise at high
frequencies, it is necessary to consider the low-frequency signal
content (approximation content) separately from the high-frequency
signal content (detail content). This is the assumption underlying the
multi-resolution analysis proposed by Mallat (1999). According to
Mallat, the separation can be obtained by using two different fam-
ilies of wavelet functions: the function family ðΦj
nðtÞÞðn;jÞ∈Z2, said
scale family which extracts the approximation content, and the func-
tion family ðΨj
nðtÞÞðn;jÞ∈Z2, linked to the former, extracting the detail
content. The DWT calculation, in a dyadic mode and for the func-
tion family ðΨj
nðtÞÞðn;jÞ∈Z2, means the same as calculating, when j
and nvary, the detail coefficients given from
Zþ∞
−∞
fðtÞ2j=2ψð2jt−nÞdt ð9Þ
From the function family ðΦj
nðtÞÞðn;jÞ∈Z2we can obtain the
approximation coefficients, when jand nvary, as follows:
Zþ∞
−∞
fðtÞ2j=2Φð2jt−nÞdt ð10Þ
When the resolution increases (with scale decrease), the set of
approximation coefficients progressively converges to the original
signal, while the set of detail coefficients provides a more and more
accurate representation of the details of the decomposed signal.
When the resolution decreases, on the other hand, both the approxi-
mation and detail coefficients become less and less significant.
The approximation coefficient calculation has been shown to
involve a convolution between the signal fðtÞand the function
Φ2jðtÞas well as a subsequent sampling operation every 2−jsam-
ples, the so-called subsampling operation of the original signal.
Table 2. Arrival Process
Section
¯
x
(veh=15 s)
σ2
ðveh=15 sÞ2PDF
Arm 1 (entry) 3.44 5.79 —
Arm 2 (entry) 0.56 0.83 Binomial
(p¼0.672;k¼1.146)
Arm 3 (entry) 4.97 10.10 —
Carriageway
(opposite Arm 2)
2.56 8.53 —
Carriageway
(opposite Arm 4)
3.68 5.00 —
Arm 1 (exit) 2.78 9.09 —
Fig. 3. Arm 1, time headways between vehicles coming from the arm (image by Tomaz Tollazzi and Giuseppe Parla)
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Signal Denoising
This section describes the computational steps reducing the un-
known noise from a discrete signal by means of the procedure pro-
posed by Donoho and Johnstone (1995). The technique for the
noise removal from signals can be synthetically expressed in the
following steps:
1. Signal decomposition at a given level L;
2. Application of a threshold level to detail coefficients, for each
decomposition level L. Some detail coefficients are practically
removed; and
3. Signal reconstruction (synthesis operation) by using, for each
level L, the original approximation coefficients and “thre-
sholded”detail coefficients.
The second step consists in defining a threshold value t,by
means of appropriate rules (soft or hard thresholding), to be applied
to detail coefficients at a given decomposition level L. Such a
threshold can be expressed as follows (Donoho 1995):
t¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2· logðNÞ
pð11Þ
As previously said, the rules for the application of the threshold
value tcan be divided into soft and hard thresholding. The soft
thresholding is defined as follows:
Sthresh ¼x;if jxj>t
0;if jxj≤tð12Þ
The hard thresholding is expressed with the following relation:
Hthresh ¼signðxÞ·ðjxj−tÞ;if jxj>t
0;if jxj≤tð13Þ
In agreement with the computational steps described earlier, the
denoised signal can be “rebuilt”by utilizing the original approxi-
mation coefficients and “thresholded”detail coefficients in the syn-
thesis process.
The procedure so far was implemented in MATLAB, with the
help of specific toolboxes designed for analyzing and solving
denoising problems of discrete signals affected by unknown noise.
Results and Discussions
The signal of the trajectories fðtÞiof the ith vehicle is a discrete
type (being conditioned to the video camera frame rates) and was
processed with the DWT analysis in a MATLAB environment.
For 15 min intervals such signals (fðtÞi) were diagrammed for
each lane at entry, exit,and ring of the intersection. Fig. 2shows, by
way of an example, the representation of fðtÞi,i¼1;2;:::;15,for
an interval ΔT¼60 s, of vehicles entering from Arm 1.
Moreover, different sections were identified for calculating traf-
fic flows (one for each entry, each exit, and on the ring opposite
each entry) and estimating vehicle flows. Therefore the probability
PðX¼x;tÞthat in such segments the random variable X“number
of vehicle transits observed in a time interval t”assumes the value x
(i.e., xvehicles in t) was investigated. It is widely known that in
flow stationarity conditions the random variable X(number of ve-
hicles transiting a section in a time interval t) can occur with differ-
ent probability laws.
A well-established criterion for choosing these laws of probabil-
ity PðXðtÞ¼x,t)—to be in compliance with statistical tests—is,
Fig. 4. Arm 1, entry—instantaneous speed, right lane (entry)
Fig. 5. Arm 1, entry—instantaneous speed, left lane Fig. 7. Arm 1, entry—mean acceleration on entry lanes
Fig. 6. Arm 1, entry—mean speed near the Yield line
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starting from the experimental count data, to determine the relation
between sampling variance σ2and mean ¯
x in arrivals; thus, the fol-
lowing probability laws are suggested (Gerlough and Huber 1975;
Mauro 2015):
•Negative binomial distribution if ðσ2=¯
xÞ>1;
•Poisson distribution if ðσ2=¯
xÞ≅1; and
•Binomial distribution or generalized Poisson distribution if
ðσ2=¯
xÞ<1.
For the turbo-roundabout in question, data sampling was made
by considering intervals Δt¼15 s. For every interval we have
calculated the frequencies recorded for each value of X¼
xiveh=15 s(i¼1;2; :::;10;>11).
On sampling days (year 2016), in each segment it was observed
that ðσ2=¯
xÞ>1; however, the statistical hypothesis test χ2allowed
to verify that the arrival process was a negative binomial type only
for vehicles entering from Arm 2 and for a time interval ΔT¼
8∶00–8∶15 on April 12, 2016 (see O/D matrix in Table 1):
PðX¼xÞ¼xþk−1
k−1pkð1−pÞxð14Þ
Fig. 8. Arm 1, time headways between vehicles exiting from the arm (image by Tomaz Tollazzi and Giuseppe Parla)
Fig. 9. Arm 1, exit—instantaneous speed, outer lane
Fig. 10. Arm 1, exit—instantaneous speed, inner lane
Fig. 11. Arm 1, exit—mean speed away from the intersection
Fig. 12. Arm 1, exit—mean acceleration away from the intersection
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With:
μ¼k·ð1−pÞ
pð15Þ
σ2¼k·ð1−pÞ
p2ð16Þ
Being ¯
x¼p0.56 and σ2¼0.8347, have been obtained p¼
0.672 and k¼1.146.
In all the other segments, although σ2>¯
x, the arrival process
does not follow this law (Table 2).
As a matter of fact, the identification of the probability density
function (PDF)—and therefore the correlated process of the vehicle
headways—is a prerequisite for the use of closed-form models to
estimate entry capacities in turbo-roundabouts, as well as all the
other at-grade intersections [e.g., Harders’formula and those deriv-
ing from it, implemented in the 2010 Highway Capacity Manual
(TRB 2010), need vehicle headways to be described with an ex-
ponential law].
The time headway process also allowed identification of the
arrival of platooned vehicles, which conventionally occurs when
Fig. 13. Time headways between vehicles opposite the west-side arm (image by Tomaz Tollazzi and Giuseppe Parla)
Fig. 14. Instantaneous vehicle speed on outer ring lane (opposite the
west-side arm)
Fig. 15. Instantaneous vehicle speed on inner ring lane (opposite the
west-side arm)
Fig. 16. Mean speed on the ring (opposite the west-side arm)
Fig. 17. Mean acceleration on the ring (opposite the west-side arm)
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their headway is τi<τm, with τmbeing the mean headway
(Mauro 2015).
Finally, the vehicle speeds and accelerations at entry, exit, and
on the ring lanes have been determined. The results are graphically
shown in Figs. 3–22.
When approaching the turbo-roundabout, speeds appear to be
always moderate, in that they were less than 25 km=h at 15 m from
the Yield line. On the ring, speeds were also observed to be always
less than 30 km=h. Such values are consistent with those measured
on turbo-roundabouts in Poland (Chodur and Radosław 2016) and
show turbo-roundabout efficiency in curbing speed.
On the ring, speed performances appeared to initially decrease
(up to the entry) and to increase toward the roundabout exits. As to
acceleration, the observed values were less than 2m=s2on the en-
try lanes and 1.5m=s2on the ring lanes; accelerations slightly
higher than 2m=s2were occasionally measured on the lanes exit-
ing from the intersection.
Estimation of the Critical and Follow-Up Headways
The critical interval (also called critical gap or critical headway)
tchas a crucial role in analyzing the performaces of nonsignalled
at-grade intersections, in conventional and turbo-roundabouts.
Fig. 18. Time headways between vehicles along the ring opposite the east-side arm (image by Tomaz Tollazzi and Giuseppe Parla)
Fig. 19. Instantaneous speeds opposite the east-side arm, outer ring
lane
Fig. 20. Instantaneous speeds opposite the east-side arm, inner ring
lane
Fig. 21. Mean speeds, ring (opposite the east-side arm)
Fig. 22. Mean accelerations, ring (opposite the east-side arm)
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Such a parameter, together with the follow-up time tf(or
follow-up headway), is used in the Highway Capacity Manual
(TRB 2010), based on Harders’formula.
The tcand tfvalues estimated in previous studies are shown
in Fig. 23.
Whereas tfcan be estimated directly from traffic samples, tc
can be determined only in an indirect way. For vehicle igetting into
an antagonist flow qj, the maximum gap rejected by a driver i,
τmax;iðqjÞand the gap accepted by a driver iτaccepted;iðqjÞ, can be
measured.
It follows:
τmax;iðqjÞ≤tc;i≤τaccepted;iðqjÞð17Þ
As pointed out by Brilon (2016), there are similar problems
in the survival theory (or lifetime analysis or reliability theory), in
that the extreme values of the interval—including the right value of
the required variable—are measurable and thus known (censored
data).
The likelihood function Lfor tccan be defined as (Lawless
2003;Brilon 2016):
L¼Y
n
i¼1
pðτmax;iðqjÞ≤tc≤τaccepted;iðqjÞjΘÞð18Þ
In which pðAÞ= probability of event A;τmax;iðqjÞ= maximum
gap rejected by a driver i(s); τaccepted;iðqjÞ= gap accepted by a
driver i(s); tc= critical gap (s); i= index for observed minor street
drivers; n= number of observed minor street drivers; and Θ=
parameter set of the assumed distribution function for tc.
The function Lgives the probability that the observed values
τmax;iðqjÞand τaccepted;iðqjÞare matched, by means of the measure-
ments from the hypothesized distribution function FcðtÞ, with the
parameter set Θ.
By maximizing the function L, we can find the parameters Θ
of the cumulative distribution function of critical gaps FcðtÞ, and
consequently the best fit to the observed sample.
Eq. (18) can be written as follows:
L¼Y
n
i¼1ðFcðτmax;iðqjÞÞ−Fcðτaccepted;iðqjÞÞjΘÞð19Þ
For computational reasons, Lis generally replaced with the
natural logarithm lnðLÞ:
L¼lnðLÞ¼X
n
i¼1Y
n
i¼1ðFcðτmax;iðqjÞÞ−Fcðτaccepted;iðqjÞÞjΘð20Þ
With L= log-likelihood function.
By maximizing Lthe same result for Θis achieved as if Lis
maximized because lnðLÞis a consistently increasing function.
An analysis comparing different statistical density functions
for some relevant types of distribution (gamma, lognormal, log-
logistic, Weibull) was carried out by Brilon (2016). The likelihood
analysis results were (Brilon 2016):
1. “From test applications the results do not significantly depend
on the applied type of distribution :::
Fig. 23. Critical and follow-up headways at turbo-roundabouts
Table 3. PDF for tc
Entry
Density
function
Sample
size kλ
Mean
μðtcÞ
(s)
Variance
σ2ðtcÞ
(s2)χ2
Minor—left Erlang 262 7 1.597 4.41 2.76 1.90
Major—left Erlang 280 13 2.471 5.48 2.22 14.68
Minor—right Erlang 160 6 1.519 4.03 2.65 8.40
Major—right Erlang 130 6 1.546 4.18 2.70 12.71
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2. It is reasonable to apply the type which is easier to be
handled :::
3. The variance of any type of distribution does not really contri-
bute to the precision of the result. Consequently, a simplified
method is useful where the standard deviation of the tcis reason-
ably assumed and only the scale parameter of the function is
optimized according to the measured gap data.
4. The results from this single-parameter method are rather con-
gruent with those from the more complicated two-parameter
method which is mainly a task for scientists.”
In this research, the random variable critical gap was assessed
with the method by Drew and Dawson (Drew 1968), and distin-
guished the entries of the main road (Arms 1–3, Fig. 1) from those
of the minor road (Arms 2–4, Fig. 1).
The variables tc;iwere estimated for both right-turn and left-turn
lanes.
By means of χ2test, we have verified that the random variable tc
is distributed according to an Erlang equation:
fðtcÞ¼λ·e−λ·tc;i ·ðλ·tc;iÞk−1
ðk−1Þ!ð21Þ
where fðtcÞ= probability density function of the critical gap; k=
shape parameter; and λ= rate parameter. The expected value μðtcÞ
and variance σ2ðtcÞof the distribution are given by
μðtcÞ¼k
λð22Þ
σ2ðtcÞ¼ k
λ2ð23Þ
The analysis results are shown in Table 3; the graphs in Fig. 24
show the statistical density functions.
Therefore, it has been observed that the tcvalues are systemati-
cally higher than those in conventional (single- and double-lane)
roundabouts (Vasconcelos et al. 2013;Giuffrè et al. 2016).
The mean values of tc(Table 3) are perfectly consistent with the
values evaluated by Brilon et al. (2014) and Macioszek (2017), but
higher than those obtained by Fortuijn (2009a). In this research
(Fig. 23), however, neither the estimation methodology of tcnor
the PDF (probability density function) are available.
Similarly, the research has continued to sample follow-up head-
ways tf. First, four different distributions (gamma, Weibull, Erlang,
and inverse Gaussian distribution) have been hypothesized and veri-
fied, by means of the chi-square test, that the best fit of data can be
obtained with the inverse Gaussian distribution, whose equation is
fðtfÞ¼λ
2π·t3
f1
2·e
−λ·ðtf−μÞ2
2μ2·tfð24Þ
With
Mean value ¼μð25Þ
σ2ðtfÞ¼μ3
λð26Þ
Table 4. PDF for tf
Entry Density function
Sample
size λ
Mean
μðtfÞ
(s)
Variance
σ2ðtfÞ
(s2)χ2
Minor—left Inverse Gaussian 456 11.821 2.53 0.77 10.75
Minor—right Inverse Gaussian 479 14.663 2.56 1.14 5.57
Major—left Inverse Gaussian 313 11.749 2.71 1.70 6.43
Major—right Inverse Gaussian 204 11.822 2.60 1.49 10.83
Fig. 24. tc, probability density function
© ASCE 04018020-10 J. Transp. Eng., Part A: Syst.
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Such tfdistribution is then the same as that obtained by Qu et al.
(2014). On the average, the tfvalues are between 2.53 and 2.71 s
depending on the entry and lane of the turbo-roundabout (Table 4
and Fig. 25). Therefore, the estimated values are in good agreement
with those obtained by Brilon et al. (2014) and Fortuijn (2009a)
(Fig. 23).
Conclusions
This article explores the kinematic parameters which are crucially
involved in safety and driving comfort in turbo-roundabouts,
i.e., vehicle speed and acceleration (which are also related to evalu-
ation of vehicles pollutant emissions). It also estimates the critical
intervals tcand follow-up headways tf.
To this end, many traffic samples were taken on a turbo-
roundabout in Maribor, a city in Slovenia. The analyses started in
the year 2016 and the traffic recordings were made with a high-
resolution video camera (>50 fps) installed on a building adjacent
to the turbo-roundabout.
The study asked for a proper assessment procedure for kin-
ematic parameters based on the image processing technique, fol-
lowed by wavelet analyses, in order to filter the trajectory-time
series signals fðtÞiof each vehicle and denoise them.
Thus the method allowed continuous measurement of thevehicle
kinematic parameters (speed and acceleration) on these elements:
•Entry lanes (both left- and right-turning);
•Exit lanes; and
•Inner and outer ring lanes.
First the researchers studied the process of vehicle headways
and arrivals.
In elementary intervals Δt¼15 min, the relationship between
the sample variance σ2and the sample mean ¯
x of arrivals was
observed to be always higher than 1. Nevertheless, the statistical
hypothesis tests (chi-square test) allowed the researchers to rule
out the possibility that arrivals followed a negative binomial distri-
bution, with the exception of one arm (Arm 2) and only in one of
the examined intervals ΔT¼15 min.
The study of the instantaneous vehicle speeds allowed the re-
searchers to observe that in the entry lanes speeds are rather mod-
erate (lower than 25 km=h, 15 m well ahead of the Yield line) and
accelerations are always lower than 2m=s2on both entry and ring
lanes (where they generally are below 1.5m=s2); only on exit lanes
accelerations proved to be slightly higher than 2m=s2.
Given that the critical interval tcand the follow-up headway tf
are two random variables, in the light of the sampled data, it was
observed by means of the statistical hypothesis test χ2that the for-
mer variable (tc) followed an Erlang distribution and the latter an
inverse Gaussian distribution.
The mean values ranges are tc¼4.03–5.48 s for critical inter-
vals, and tf¼2.52–2.71 s for follow-up headways, respectively,
according to the right- or left-turn lane and to the major ot minor
entry in question. Such values are very consistent with those in-
ferred in previous research.
These kinematic and users behavior values can be useful for the
estimation of the capacity, delays, and queues in turbo-roundabouts
by means of closed-form formulations, or for a better calibration of
traffic microsimulation models.
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