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This article presents a methodology that integrates cumulative plots with probe vehicle data for estimation of travel time statistics (average, quartile) on urban networks. The integration reduces relative deviation among the cumulative plots so that the classical analytical procedure of defining the area between the plots as the total travel time can be applied. For quartile estimation, a slicing technique is proposed. The methodology is validated with real data from Lucerne, Switzerland and it is concluded that the travel time estimates from the proposed methodology are statistically equivalent to the observed values.
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for publication in the following source:
Bhaskar, Ashish,Chung, Edward, & Dumont, Andre-Gilles
(2011)
Fusing loop detector and probe vehicle data to estimate travel time statis-
tics on signalized urban networks.
Computer-Aided Civil and Infrastructure Engineering,26(6), pp. 433-450.
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https://doi.org/10.1111/j.1467-8667.2010.00697.x
Page 1 of 38
Fusing loop detector and probe vehicle
data to estimate travel time statistics on
signalized urban networks
Ashish Bhaskar*
Laboratory of Traffic Facility
Swiss Federal Institute of Technology
Lausanne, Switzerland
Ph: +41 21 693 2341;
Fax: +41 21 693 63 49
ashish.bhaskar@epfl.ch
Prof. Edward Chung
School of Urban Development
Queensland University of Technology
Brisbane, Australia
Ph: +61 731381143
Fax: +61 3138 1827
edward.chung@qut.edu.au
Prof. André-Gilles Dumont
Laboratory of Traffic Facility
Swiss Federal Institute of Technology
Lausanne, Switzerland
Ph: +41 21 693 2345
Fax: +41 21 693 63 49
andre-gilles.dumont@epfl.ch
First submission: 22nd June 2009
Final submission: 29th April 2010
*Corresponding author
Abstract
This paper presents a methodology that integrates cumulative plots with probe vehicle data for
estimation of travel time statistics (average, quartile) on urban networks. The integration reduces
relative deviation amongst the cumulative plots so that the classical analytical procedure of defining
the area between the plots as the total travel time can be applied. For quartile estimation, a slicing
technique is proposed. The methodology is validated with real data from Lucerne, Switzerland and it is
concluded that the travel time estimates from the proposed methodology are statistically equivalent to
the observed values.
Page 2 of 38
1. Introduction
Travel time is defined as the time needed to travel from a point upstream (u/s) to a point downstream
(d/s) on the network. It quantifies congestion and is easily perceived by all road users and operators. It
is an important network performance measure and a decision making variable. For instance, travel
time information is utilized by the operators to develop traffic control strategies for reducing congestion
on both spatial and temporal scales.
Literature is abundant with models for obtaining travel time values. Majority of the literature is limited to
freeways (Nam and Drew, 1996; Dharia and Adeli, 2003; Jintanakul et al., 2009) and cannot be
applied on urban networks. The development of a travel time model on urban networks is more
challenging than freeways due to number of reasons. For instance, interruption in traffic flow due to
traffic signals; non conservation of traffic flow on urban link due to mid-link sources and sinks (e.g.
parking, side street) etc. Travel time models can be differentiated into estimation models and
prediction models. Estimation models provide experienced travel time whereas, prediction models
(Park et al., 1999; You and Kim, 2000; Zhang and Rice, 2003; Vlahogianni et al., 2004; Vlahogianni et
al., 2008; Hamad et al., 2009) provide expected (forecasted) travel time in future. The future can be
immediate (i.e., for trips just starting) to several minutes (say 30 minutes) ahead.
Traffic flow on an urban network is in stop-and-go running condition i.e., vehicles have to the stop at
the intersection during signal red phase and queue of vehicles is formed. The individual vehicle travel
time on urban link depends on number of factors. For instance, traffic demand; signal parameters;
vehicles’ entry time on the link relative to downstream intersection signal red phase; and number of
vehicles queued in front of it when it reaches the downstream signal etc. The distribution of travel time
from different vehicles departing within an estimation interval (Here, referred as within interval
distribution.) is bi-modal (or even multi-modal) with modes corresponding to the vehicles that can pass
the link without stopping at intersection and for vehicles that experience delay at intersection.
Average travel time is an important indicator for network performance measure and generally most of
the in-practice models are applicable for average travel time estimation (Review provided in Section 2).
Due to the above mentioned bi-modal distribution; none of the vehicle can encounter average travel
time. Hence, for better understanding of the network performance it is important to estimate other
statistics, such as upper quartile of travel time in addition to the average travel time. Little research is
performed on the estimation of within interval distribution mainly due to unavailability of individual
vehicle travel time data. Robinson (2005) has applied k-NN approach, on the Automatic Vehicle
Identification (AVI) data from central London, to estimate travel time variance for within interval
distribution. He has observed around 23% variance in travel time between vehicles traversing the link
during 15 minutes estimation interval. This is more than three times of the variance observed by Part
et al., (1999) on Houston freeways.
The objective of this paper is to develop and validate a methodology for estimation of travel time
statistics (average and quartiles) on signalized urban networks. The estimates are for certain time
period that is integral multiple of signal control cycle. For instance, average travel time during five
Page 3 of 38
signal control cycles. The develop methodology should be robust with respect to urban complexities,
such as mid-link sources and sinks and detector counting error. The proposed methodology is named,
CUmulative plots and PRobe Integration for Travel timE estimation (CUPRITE) (Bhaskar, 2009). The
methodology is based on classical analytical procedure for travel time estimation using cumulative
plots. Analytical modeling is performed through integrating or fusing cumulative plots with probe
vehicle data for accurate estimation of travel time statistics (average travel time and quartile of travel
time). The proposed methodology can be applied for real time application, where the estimated travel
time is the experienced travel in the last estimation interval.
The paper is organized as follows: section 2 provides the literature review for the average travel time
estimation on urban networks. The classical analytical procedure for travel time estimation and its
vulnerability for application on urban environment are introduced in section 3. Thereafter, the
proposed methodology is developed in section 4, followed by its validation with real data in section 5.
Finally, the conclusions are presented in section 6.
2. Literature review
Advancement of technology has resulted in different traffic data retrieval systems from traditional loop
detectors to advanced electronic systems onboard a vehicle, such as Vehicle Information and
Communications Systems (VICS). Fixed sensors, such as loop detectors provide traffic flow and
occupancy at the specific location on the network whereas; mobile sensors, such as probe vehicle
provide data for the entire journey of the vehicle. Based on the type of data available, different models
are proposed to estimate average travel time for all the vehicles traversing the road. Moreover, the
availability of different data systems provide avenue for application of data fusion techniques for more
reliable and robust travel time estimation. Thereafter, here we classify the literature into: i) fixed
sensor; ii) mobile sensor; and iii) data fusion based models for average travel time estimation.
Fixed sensor based:
The initial motivation for the development of travel time estimation models was to consider effect of
congestion in conventional traffic assignment step used in four-step transportation modeling. Several
travel time functions (or volume delay functions) are proposed that define relationship between link
travel time and traffic intensity (flow/capacity ratio). These include: Bureau of Public Roads (BPR,
1964); Davidson’s function (Akçelik, 1978; Tisato, 1991); conical-volume delay function (Spiess, 1990)
etc. Webster delay model (Webster and Cobbe, 1966) is a pioneer model for estimating average
deterministic delay at undersaturated signalized intersection. Researchers have followed Webster’s
work to suit different field conditions and modified models are proposed, such as Akçelik (Akçelik,
1988; Akçelik, 1991; Akcelik and Rouphail, 1993; Akcelik and Rouphail, 1994) and Highway Capacity
Manual’s procedure for delay estimation (TRB, 1998; TRB, 2000).
The simplicity of these travel time function make them favorable candidate for transport planning and
policy analysis. They are not suited for ITS applications where more accurate and reliable analysis
especially for variable traffic conditions in real time is required.
Page 4 of 38
Regression analysis based models to estimate link travel time, as a function of site characteristics and
detector data are also proposed. Wardrop (1968) has defined regression relationship between
average journey speed in central urban area as a function of average traffic flow, width of the road, the
number of controlled intersections per miles and average proportion of green time. Researchers (Gault,
1981; Young, 1988; Sisiopiku and Rouphail, 1994; Sisiopiku et al., 1994) have observed a regression
relationship between average travel time and certain ranges of detector occupancy, for mid-link
detectors, with queue that does not persist over detector location. Zhang (1999) has proposed a
regression equation for journey speed as a function of volume to capacity ratio and mid-link detector
flow and occupancy.
Generally, the regression models are site specific and require calibration to suite different environment.
These models should not be generalized and the effect of parameters, such as detector location,
effective green time, progression quality, link length, opposing flow from permissive phasing, traffic
composition etc. should not be overlooked.
Researchers have also applied machine learning algorithm, such as k-Nearest Neighbor (Robinson
and Polak, 2005) and Artificial Neural Networks (Palacharla and Nelson, 1999; Liu et al., 2005), for
travel time estimation and prediction (Hamed et al., 1995). Such models require measured link travel
time values and the corresponding detector data for a training period. The training data set should
properly represent the required extend of the solution space and the model should be applied well
within the limits for which it is trained.
Skabardonis and Geroliminis (2005) have proposed a model for travel time estimation based on
upstream detector data and signal parameters. The queueing at the signal is considered by applying
kinematic wave theory. The required detector should be sufficient upstream from the intersection
stopline, so that the flow and occupancy measures from the detector are not affected by the presence
of queue at the signals. The model involves calibrating the fundamental diagram (flow-density
relationship) for links using detector data.
Recent advancement in sensor technology has produced Advance Inductive Loop Detectors (AILD)
that can provide magnetic vehicle signatures. The signatures from upstream and downstream
detectors can be correlated to reidentify the vehicles (or platoon) at downstream location for travel
time estimation. Ritchie et al., (2002; 2005) have demonstrated the potential application of vehicle
signature for travel time estimation on urban arterial. However, this approach is still in initial research
stage, and further study is needed to increase the accuracy, reliability and reidentification rate.
Moreover, for the implementation of reidentification technique, existing infrastructure should be
upgraded with AILD and a high bandwidth in the data communication channel.
Mobile sensor based:
Mobile sensors, such as probe vehicle is a vehicle equipped with vehicle tracking equipment (e.g.,
GPS) and can provide data for the vehicles’ trajectory (time stamp and position coordinates) and
hence its travel time. In practice, only a small fraction of all the vehicles traversing the link are probe
vehicles. Average travel time for all the vehicles traversing the link can be estimated by applying
Page 5 of 38
statistical sampling techniques on the travel time obtained from the probe vehicles (Hellinga and Fu,
2002; Long Cheu et al., 2002). Researchers (Srinivasan and Jovanis, 1996; Long Cheu, Xie and Lee,
2002) have shown interest to determine minimum number of probes required for statistically significant
travel time estimation.
Data fusion based:
Researchers have also applied data fusion techniques to fuse data from different sources, specifically
detector and probe vehicles data for travel time estimation. El Faouzi (2004) provides an overview of
the application of data fusion techniques in road traffic engineering. Dailey et al., (1996) summaries
ITS data fusion projects.
Berka et al. (1995) has applied weighted average based fusion technique where the fused average
travel time is the weighted average of the estimated average travel time from detectors and estimated
average travel time from probe vehicles. The weights are defined by considering variables, such as
the standard deviation of the travel time from detector data and from probe data, respectively; weights
assigned to detector travel time in data screening; the sum of weights of reasonable probe reports etc.
A similar weighted average based data fusion approach for travel time estimation is proposed by El
Faouzi (2004).
Choi and Chung (2002) have applied the data fusion technique for 5 minutes average travel time
estimates using detector and probe vehicle data. The algorithm first estimates space-mean speed
from detector counts and occupancy using Dailey (1999) equation, which provides travel time
estimates for each minute. Each minute travel time estimates are aggregated using Voting Technique
for 5 minutes average travel time (TTd). Average 5 minutes travel time (TTg) from GPS probes is
obtained using Fuzzy regression. Finally, fused link travel time is obtained by applying Bayesian
Pooling Method on TTd and TTg. The algorithm is tested for undersaturated traffic condition and should
be tested for oversaturated traffic condition too. They quote that “a different level of service might
produce totally different weights of each data collection mechanism. In such cases, a different data
fusion method and/or a revision of the proposed algorithm may be needed”.
Xie et al., (2004) have applied two independent neural network methodologies: Multi-Layer Perception
(MLP) and Multi-Layer regression (MLR) models to fuse average travel time estimates from detector
data and probe vehicles. The average travel time from detector is the sum of the free flow travel time
and signal delay. The signal delay is estimated using Singapore model (Xie et al., 2001). Average
travel time from probe samples are considered only if the sample size during estimation interval is
more than 10 vehicles or is more than the minimum required sample size determined by central limit
theorem.
Page 6 of 38
3. The classical analytical procedure for travel time estimation
3.1 The procedure
Cumulative plot is a plot of cumulative count of vehicles versus time at a specific location on the
network. The classical analytical procedure for travel time estimation considers cumulative plots U(t)
and D(t) at upstream (u/s) and downstream (d/s) locations, respectively (Daganzo, 1997). It defines
the total travel time from u/s to d/s as the area (Refer to Figure 1a) between the plots. Say, time t1 and
time t2 correspond to the start and end of the U(t) represented in the area, respectively. Similarly t3 and
t4 are time corresponding to the start and end of D(t) represented in the area, respectively. Then,
mathematically, the average travel time
TT
is represented as follows:
1 1 1 1
1 1 1
( ) ( ) ( ) ( )
N N N
i i i
D i U i D i U i
TT NN




(1)
2 1 4 3
( ) ( ) ( ) ( )N U t U t D t D t
(2)
Here N is the number of vehicles that depart downstream (arrives upstream) during the time interval
from t3 to t4 (t1 to t2).
3.2 The Relative Deviation (RD) issue with the procedure
The area between the plots is the total travel time from upstream to downstream as long as all the
vehicles represented in U(t), from time t1 to t2, and in D(t), from time t3 to t4, are same. Therefore, the
plots should be based on only those vehicles that traverse from upstream to downstream.
Cumulative plots are defined based on the detector counts at a specific location. Practically, detectors
are not perfect and one can easily observe 5% error in detector counting. Moreover, urban network
has mid-link sources and sinks, such as parking or side-street. This results in non conservation of
vehicles (loss or gain of vehicles) between upstream and downstream locations. Due to detector
counting error; non conservation of vehicles between plots location; and any such combinations over
time, there is relative deviation (RD) amongst the plots (also termed as “drift”).
Let us explain RD with the help of an example. Consider a scenario where upstream detector is
overcounting. In Figure 1b: U(t) is the cumulative plot observed from the overcounting upstream
detector; U’(t) is from a perfect detector. U(t) deviates from U’(t), or there is a relative deviation
between U(t) and D(t). The observed cumulative plots are U(t) and D(t) and if the classical procedure
is applied then the error in the estimation of travel time, during TEI travel time estimation interval, is
represented as the shaded region in the figure. If RD is left unchecked then the error can exponentially
grow with time. Hence, the RD issue is critical in the application of the classical procedure.
Note: U(t) and D(t) will eventually diverge from each other if: upstream detector is overcounting; or
downstream detector is undercounting; or there is mid-link sink. U(t) and D(t) will eventually cuteach
Page 7 of 38
other if: upstream detector is undercounting; or downstream detector is overcounting; or there is mid-
link source. If the plots diverge then the travel time is highly overestimated and if the plots cut then
travel time estimates are negative. In practice, there is complex combination of detector errors, mid-
link sources and mid-link sinks over time, which defines the relative deviation for each estimation
interval.
Figure 1: Classical analytical procedure and its vulnerability to relative deviation (RD) amongst
the plots.
Traffic flow direction
Area (A)
Average
Travel Time
= A/N
D(
t
)
U(
t
)
Time
Cumulative counts
u/s
d/s
Road
t3
t4
t1
t2
D(
t
)
Time
Cumulative counts
U(
t
): from perfect detector
U(
t
): from
overcounting
detector
U(
t
) deviated from U(
t
)
Relative deviation
between U(
t
) and D(
t
)
(a)
(b)
Example: Upstream detector is overcounting
TEI
Error in travel
time estimation
i
t
tti- Horizontal distance
(temporal separation)
between the curves for
rank i.
n- Vertical distance
(counts separation)
between the curves at
time t.
Page 8 of 38
3.3 How RD issue can be addressed?
Refer to Figure 1a: The vertical distance (counts separation) between the plots (at time t) is the
number of vehicles (n) between upstream and downstream locations. The horizontal distance
(temporal separation) between the plots (for rank i) is an estimate of travel time, tti, for the ith vehicle
under FIFO assumption. Therefore, the knowledge about the number of vehicles between upstream
and downstream locations; or the travel time of individual vehicle can be applied to address the RD
issue.
The number of vehicles between upstream and downstream locations is difficult to obtain. Alternatively,
we can consider the knowledge about the queue length. Theoretically, the queue length can be
defined as follows:
( ) ( )
ff
Queueattimet U t t D t
(3)
Here, tff is the free flow travel time of the link. This is further discussed in section 4.4.
There is an increasing use of probe vehicle, which can provide its travel time. Hence, in this paper we
propose a methodology that- integrates probe vehicle with cumulative plots to resolve the RD issue;
and applies slicing technique for estimation of travel time quartiles.
4. The proposed methodology
4.1 Probe vehicle data
Here, probe vehicle is a vehicle equipped with vehicle tracking equipment. There are issues related to
the probe vehicle data, such as map matching, frequency of probe data etc. To address these issues
is beyond the scope of this paper. We assume that the time when the probe vehicle is at upstream (tu)
and downstream (td) locations is accurately obtained and its travel time is td tu.
4.2 Architecture of CUPRITE
The proposed methodology integrates cumulative plots with probes. The basic concept for the
integration is introduced in Section 4.3. The architecture (see Figure 2) of the proposed algorithm is as
follows (the details for which are provided in the following sections):
Step 1 Cumulative plots U(t) and D(t) are defined. Here, if the detector data is individual
vehicle data (pulse data), then the cumulative counts can be obtained by
cumulative the vehicles. However, if detector data is not a pulse data but an
aggregated traffic counts during certain detection interval (for instance counts per
60 seconds), then cumulating the counts for each detection interval will not reflect
the actual traffic fluctuations within the detection interval. These fluctuations can
be captured by integrating the detector counts with signal timings, where the
counts during the signal red phase is assigned to be zero, and counts during the
signal green phase is segregated into counts from the saturation flow and counts
Page 9 of 38
from non saturation flow. Refer to Bhaskar et al., (2008) for the methodology for
integration of signal timings with aggregated traffic counts from detector data for
accurate representation of cumulative plots.
Step 2 Probe vehicle data, list of [tu] and [td], is defined (Refer to Sections 4.1 and 4.3).
Moreover, if the conditions for virtual probe (Section 4.4.1) are satisfied then the
list [tu] and [td] is appended with additional elements corresponding to the virtual
probe i.e., tu= tGE-tff; td = tGE (where tGE and tff are time corresponding to the end of
signal green phase and link free flow travel time, respectively). Else, only real
probes are considered.
Step 3 Points through which U(t) should pass are defined (Section 4.5).
Step 4 U(t) is redefined by first vertical scaling and shifting the plots (Section 4.6) so that
it passes through the above defined points (Step 3); and
Step 5 Finally, for each estimation interval: a) average travel time (Section 4.7) is
estimated using classical analytical procedure; and b) quartile for travel time
(Section 4.8) is estimated using slicing technique.
Page 10 of 38
Figure 2: CUPRITE architecture for estimation of travel time statistics.
4.3 Integration of cumulative plots and probes
Suppose, there is no RD and both U(t) and D(t) are perfect. Due to non-FIFO traffic behavior, even in
the absence of RD, the rank of a vehicle in upstream and downstream cumulative plots may not be
same i.e, U(tu) D(td) (Figure 3a). We fix the vehicle to downstream cumulative plot, i.e., we fix the
rank of the vehicle in the cumulative plots as D(td) (Refer to Figure 3b) and define a parameter Δt :
1( ( ))
du
t U D t t
(4)
Where U-1(D(td)) is the time when the vehicle is represented at U(t) given that we fix its rank to D(td).
Signal data
U(t) D(t)
Probe
vehicle
Probe data
Lists of [tu] & [td]
Redefine upstream
cumulative plot
Refefined
U(t)
Classical analytical procedure
No
Points through
which U(t)
should pass
Define points through which
U(t) should pass
Virtual probe
conditions
satisfied*
Yes
*If the virtual probe
conditions are not satisfied
then only real probes are
considered
Slicing technique
Quartile of
travel time
Average
travel time
Detector
data
Is pulse
data?
Yes
Integrate signal timings
with detector data
Page 11 of 38
Figure 3: a) Illustration for the relationship between probe data and cumulative plots; b) Fixing
of probe data to D(t).
If all the vehicles in U(t) and D(t) are same then
t
from all the vehicles should be zero (5). This is
an important property and is the explanation for the area between the plots to be the total travel time.
0
i
i
t

(5)
If there is presence of RD then the equation (5) is not satisfied. Therefore, RD issue can be addressed
by correcting the cumulative plots such that equation (5) is satisfied.
Equation (5) is satisfied when we are considering all the vehicles. Each vehicle has an equal
probability of being a probe vehicle and only a fraction of vehicles are randomly selected as probes.
We make a hypothesis that we can remove or at least reduce the RD by redefining U(t) such that
t
from all the probes is zero.
D(
t
)
U(
t
)
Time
Cumulative counts
td
tu
(a)
U(tu)
D(td)
D(
t
)
U(
t
)
Time
Cumulative counts
td
tu
(a)
U(tu)
D(td)
U-1(D(td))
Δt
Probe Data
tu
td
Probe fixed
to D(t)
U(tu) may not be
equal to D(td)
Page 12 of 38
In practice, we do not know which plot is responsible for RD issue. It can be U(t), D(t) or both. It is
complicated to correct both U(t) and D(t) simultaneously. As the deviation amongst the plots is relative
therefore, we can correct either U(t) or D(t). Here, we redefine U(t) because we fix the rank of the
probe considering D(t). Alternatively, we can redefine D(t), if we fix the rank of the probe considering
U(t).
4.4 Virtual probe
Virtual probe (Figure 4) is defined as a virtual vehicle that, during undersaturated traffic flow, departs
from the downstream at the end of signal green phase (at time tGE) and its travel time is free-flow travel
time (tff) of the link. The probe is not real and is defined with the aim to reduce RD.
For undersaturated traffic condition, the vehicle queue formed during the signal red phase should be
completely served during the signal green phase i.e., the queue length at time tGE should be zero.
Considering equation (3): U(tGE - tff) = D(tGE) i.e., the travel time of the vehicle that enters the
intersection at time tGE should be close to tff. Therefore, under such conditions we can define virtual
probe (see Figure 4) such that it is observed at upstream and downstream locations at time tGE - tff and
time tGE, respectively (i.e. for virtual probe tu = tGE - tff and td = tGE.).
Note: Virtual probe is only defined if the following conditions for virtual probe are satisfied.
4.4.1 Conditions for virtual probe
i. As the travel time of a virtual probe is defined as free-flow travel time of the link,
therefore on the link the sources for significant mid-link delay, such as mid-link
intersections and on-street bus stop should be absent.
ii. Virtual probes are defined only for undersaturated condition with logic of zero queue
length at the end of signal green phase. Traffic condition is defined as undersaturated if
counts during the signal cycle (or more specifically during signal green time) are less
than the corresponding capacity (Figure 4) i.e.,
( ) ( ) *
GE GE
D t D t c s g
(6)
Where: s and g are saturation flow rate and effective signal green time, respectively;
s*g is the capacity and is a calibration parameter to take into account the error in the
estimation of capacity.
To define equation (6) it is assumed that there is no spill-over from downstream link. If
there is spill-over, then vehicles are restricted to flow resulting in low counts at stop-line
detector. Capacity is generally not corrected to account for the spill-over from
downstream link. Due to which equation (6) is satisfied and system can falsely indicate
undersaturated situation for spill-over cases. Though under such situation the queue
may not vanish and hence virtual probe should not be defined.
Page 13 of 38
iii. Virtual probe is defined with the aim to reduce RD. Hence, it should only be defined if
there is presence of RD i.e., the following equation should be satisfied:
1( ( )) [ , ]
GE GE ff ff
U D t t t t

(7)
Where: δ is a calibration parameter taking into account the variation in the estimation of
tff. It can be considered equal to the standard deviation of the estimate of tff.
Figure 4: Illustration of a virtual probe fixed to D(t).
4.5 How to define the points through which U(t) should pass?
Say, we have n probe vehicles and the database for the probe is defined as list of [tu] and list of [td],
where the size of each list is n. The value of jth element in the list represents the data from the jth
probe. These lists are appended with additional elements satisfying the conditions for virtual probe
(Section 4.4.1). If the conditions are satisfied, then time tGE is appended to the list [td]; and time
(tGE - tff) is appended to the list [tu].
4.5.1 Grid technique
Consider an example, in Figure 5a, where we have four probes fixed to D(t). The U(t) should pass
within the region satisfying the following constrain (Refer to the rectangular region in Figure 5a)
min[ ] max[ ]
uu
probes probes
t t t


(8)
min[ ( )] max[ ( )]
dd
probes probes
D t counts D t


(9)
Saturation flow rate (s)
tGE
Green (g)
Red (r)
Cycle (c)
(tRS ,D(tRS))
(
tGE
, D(
tGE
))
(
tGE
-
tff
, D(
tGE
))
Virtual probe
(tff)
D(t)
Time
Cumulative counts
D(tGE)-D(tGE - c)
s*g
D(tGE)-D(tGE - c) < s*g
Queue is served
tGE - c
Page 14 of 38
We can define a grid with rows corresponding to D(td) and columns corresponding to tu within the
above region (Refer to Figure 5b). If U(t) passes through the diagonal nodes of the above grid then
∑∆t = 0 is satisfied. Therefore, the required points to pass are the diagonal nodes of the grid and can
be obtained from the following algorithm:
Step 1 Sort list [td] in ascending order of its values. This is required as the rank of the
probe is defined considering D(t).
Step 2 Sort list [tu] in ascending order of its values. This is required to make sure that the
redefined U(t) is monotonically increasing and satisfies the property of ∑∆t = 0.
Step 3 The required points through which U(t) should pass are (tuj, D(tdj)); where tuj and tdj
are jth value in the sorted list of [tu] and [td], respectively.
Figure 5: Points through which the U(t) should pass.
Time
Cumulative counts
Time
tu2 tu1 tu4 tu3
D(t)
td1 td2 td3 td4
tu2 tu1 tu4 tu3 td1 td2 td3 td4
D(t)
Point through which U(t) should pass
t1
∑∆ti=0
t2
t3
∆t4
∆t1=-∆t2; ∆t3=-∆t4
Region
(b)
Cumulative counts
(a)
Grid
Page 15 of 38
4.6 How to redefine U(t)?
4.6.1 Reference point
We define reference point as the point in which we have confidence that it is a correct point on the
cumulative plot. Initially, U(t) and D(t) are two independent cumulative plots. When the traffic condition
is free-flow (for instance during night) then counts for cumulative plots can be initialized to zero. This is
the initial reference point (P0). Say [P1, P2, P3, …, Pn] is the list of n points from where U(t) should
pass, then for redefining U(t) for point Pi, the reference point is Pi-1 and so on.
4.6.2 Vertical scaling and shifting technique
The RD issue is the result of: a) the error in the cumulative counts- due to error in detector counting;
and b) inconsistency between the cumulative count at upstream and downstream locations - due to
mid-link sources and sinks. Therefore, to address the issue, the cumulative counts (vertical axis)
should be corrected. For this, we apply the following vertical scaling and shifting technique.
Say, a) point (tRef, U(tRef)) is a reference point; and b) point (tp, Yp) is a point through which U(t) should
pass (Section 4.5). The redefined U(t) should pass through both these points.
Refer to Figure 6, we define
p
t
Y
= U(tp) - U(tRef) and
p
t
y
= Yp - U(tRef).
Figure 6: Illustration of the abbreviations for vertical scaling.
a) For time tRef
Time
Reference Point
(tRef, U(tRef))
Point to pass
(tp, Yp)
tp
tRef
Cumulative counts
(tp, U(tp))
t
p
t
y
p
t
Y
p
t
Ref
Ref
Ref
()
( ) ( )
()
( ) ( )
p p p
p
p
t t t p p
tp
tp
t
Y y U t Y
Y U t U t
y Y U t
Y U t U t



t
Y
U(t)
Page 16 of 38
The reference point is the correct point on the cumulative plot; therefore no correction on the
cumulative plot is required for time less than and equal to tRef.
b) For tRef < time tp
We perform vertical scaling on U(t) such that it passes through the point (tp, Yp). The scale is defined
as follows:
Ref Ref
Ref
Ref
() ( ) ( )
( ) ( )
1 ( ) ( )
p
p
tp
p
tp
p
yY U t if U t U t
Y U t U t
scale
if U t U t

(10)
The value of the scale reflects the net effect of the RD on the cumulative plots:
i. scale >1: The plots are diverging. For diverging plots, the error can exponentially grow
with time. For instance, there is a mid-link sink. The vehicles from the sink are
observed at upstream and not at downstream.
ii. scale <1: The plots are converging. For converging plots, U(t) can cut D(t) resulting in
negative travel time estimation. For instance, there is a mid-link source. The vehicles
from the source are observed at downstream but not at upstream.
iii. scale =1: RD is absent.
Here, the relative deviation in the cumulative count at time t, (
t
) is the result of the accumulation of
the relative deviation since time tRef (Refer to Figure 6). Equation (11) defines the RD (
p
t
) at time tp.
()
p p p
t t t p p
Y y U t Y
(11)
The proportion of this relative deviation to the cumulative counts (
p
p
t
t
Y
) is assumed to be constant(12).
;
p
p
ttRef p
tt
t t t
YY
(12)
Where:
t
is the relative deviation at time t; Yt = U(t) - U(tRef).
The above equations can be rearranged to define
t
as follows:
(1 )* ;
t t Ref p
scale Y t t t
(13)
c) For time > tp
All the points on U(t) beyond time tp are shifted vertically so that the redefined U(t) is continuous. The
magnitude of the shift is the relative deviation observed at time tp (Eq (11)).
Page 17 of 38
The above is summarized as follows: we redefine U(t) (14) by applying correction (15) on it such that
all points on the plot: i) before time tRef have no correction; ii) between tRef to tp are scaled vertically;
and iii) beyond tp are shifted vertically.
( ) ( ) t
U t U t

(14)
Ref
0
(1 )*( ( ) ( ))
()
Ref
t Ref p
p p p
tt
scale U t U t t t t
U t Y t t

(15)
Figure 7 represents an example, where P0 is the initial reference point; and points P1 and P2 are two
points through which U(t) should pass (refer to section 4.5). First, the correction for point P1 is
performed with P0 as the reference point (Refer to Figure 7b). Thereafter, P1 becomes the reference
point for P2 and correction for P2 is performed (Refer to Figure 7c). The redefined U(t) considering
points P1 and P2 is represented in Figure 7c.
Page 18 of 38
Figure 7: Example for redefining U(t) based on vertical scaling and shifting technique.
Time
Cumulative counts
Time
Cumulative counts
Time
Cumulative counts
P0
P1
P2
P0:Reference point
P1
P1:Reference point
P2
(a)
(b)
(c)
Vertical
Scaling
Vertical
Shifting
No
Correction
Vertical
Scaling
Vertical
Shifting
No
Correction
Correction for P1
with P0 as
reference
Correction for P2
with P1 as
reference
U(t)
Parallel
curves
Parallel
curves
Redefined U(t)
Page 19 of 38
4.7 Average travel time estimation
The classical procedure (see section 3) is applied between redefined U(t) and D(t) to estimate average
travel time.
4.8 Quartiles of travel time estimation
By definition, quartile is any value that divides the sorted data into equal parts:
i. Q1: the first quartile is the 25th percentile and 25% of the data is lower than Q1.
ii. Q2: the second quartile is the 50th percentile (or median) and it divides the data into two
equal parts.
iii. Q3: the third quartile is the 75th percentile and 75% of the data is lower than Q3.
To obtain quartiles of travel time, we need either individual vehicle travel time data or grouped vehicle
travel time data. The later is the data consisting of representative travel time from different group of
vehicles. For FIFO systems, the horizontal distance (temporal separation) between the cumulative
plots is an estimate for individual vehicle travel time. For both FIFO and non FIFO systems, the area
between the plots is an estimate for total travel time for a group of vehicles represented within the
plots. We propose the following slicing technique where, within an estimation interval, we slice the
area between the cumulative plots to obtain the grouped vehicle travel time data.
4.8.1 Slicing technique
Cumulative plot is a two dimensional piece wise linear graph with coordinates (ti, i) as its nodes; where
ti is the time when the ith vehicle is observed. In Figure 8, we illustrate a study link between u/s and d/s
locations. In the figure, links L1, L2 and L3 are three upstream links that contribute to the flow on the
study link. Each of these links has a signal phase Phase1, Phase2 and Phase3 that permit the flow
towards the study link, respectively. Time tgs1, tgs2 and tgs3 correspond to the start of signal green time
for Phase1, Phase2 and Phase3, respectively.
Upstream cumulative plot, U(t), is defined by the flow contributions from different upstream links (L1, L2
and L3). We define a “cut node” as the node corresponding to the start of each signal green time (tgs1,
tgs2 and tgs3) for the upstream signal phases that permits the flow towards the study link. The cut
nodes” in the U(t) is marked in the figure. Here a node is a “cut node” if the following is satisfied:
1 2 3
1 1 1
(( ) ( ) ( ))
( , ) " "
i gs i i gs i i gs i
i
if t t t or t t t or t t t
thennode t i is a cut node
(16)
Similarly, for downstream cumulative plot, we define a “cut node” as the node corresponding to the
start of the signal green time for the downstream signal phases that permits the departure of the
vehicles from the study link.
Page 20 of 38
Figure 8: Simplified illustration of how a “cut node” is defined for upstream cumulative plot
with flow contribution from different upstream links.
We define, Mu as a matrix of the “cut nodes” for the upstream cumulative plot and similarly MD for
downstream cumulative plot. Within an estimation interval, the total area A between the cumulative
plots, is fragmented into different areas (Ai) (see Figure 9), by horizontal cuts corresponding to the
nodes at MU (“cut node” matrix for U(t)), MD (“cut node” matrix for D(t)) and with the following
constraint:
For each fragmented area Ai, if the counts Ni, are above a certain threshold number, Nthreshold, then the
area (Ai) is further fragmented by a horizontal cut into two fragmented areas: Ai1 and Ai2 with counts
Nthreshold and Ni-Nthreshold, respectively. The threshold value provides an upper limit on the number of
vehicles for each fragmented area.
The process is repeated until each fragmented area satisfies the constraint. Finally, each fragmented
area (Ai) represents the total travel time for the Ni number of vehicles. We assume that Ni number of
u/s
L2
L3
L1
d/s
Study link
tgs1
tgs2
tgs3
Phase1
Phase2
Phase3
Time
u/s Signal phases associated with the flow towards the study link
Time
Cumulative counts
tgs1
tgs2
tgs3
cut nodes
U(t)
For simplicity of presentation, the green time illustrated here is effective green.
Page 21 of 38
vehicles experience similar travel time (
i
TT
) equal to the Ai/Ni. This defines a grouped vehicle travel
time data.
Figure 9: Illustration for slicing the area between cumulative plots for defining travel time for
different group of vehicles within an estimation interval.
Note: if we define Nthreshold =1 vehicle, then the travel time estimates from the above procedure is
equivalent to obtaining individual vehicle estimates as the horizontal distance between the plots.
An estimate for the quartiles of travel time is obtained by sorting the travel time values obtained from
all the sliced areas and corresponding number of vehicles as follows:
For an estimation interval, say we have a two dimensional array with first column as list of Ai/Ni (i.e.,
list LA/N) and second column as list of Ni (i.e., list LN). Following steps are followed:
Step 1 Sort the array with respect to the values in the list LA/N;
Step 2 Define a cumulative frequency list (Lf) by cumulating the values in the list LN;
Step 3 Define N, as total number of vehicles in the estimation interval. This is the last
element of the above cumulative frequency list;
Time
Cumulative counts
N
TEI
Time
Cumulative counts
Ni
A
MU
MD
Ai
For each fragmented area Ai
Ni
Ai
Ni -Nthreshold
Ai1
if Ni > Nthreshold
Ai2
Nthreshold
Horizontal cut
New Horizontal cut
For Ni vehicles travel time is:
i
i
i
A
TT N
Page 22 of 38
Step 4 Define the index for the quartiles as follows:
Q1_index = 0.25*N
Q2_index = 0.5*N
Q3_index = 0.75*N
Step 5 Quartiles are defined as the value corresponding to the jth element of the sorted
list LA/N where j is the rank of Lf such that:
if(j=0) and ( Lf[j] ≥ Q3_index),
then Q3 = LA/N[0]
if(j>0) and ( Lf[j-1] < Q3_index) and (Lf[j] ≥ Q3_index),
then Q3 = LA/N[j]
Similarly, for Q2 and Q3;
Note: here the elements of the list start from rank 0.
For better understanding of the above algorithm a self explaining example is presented in the Figure
10, where Table A is the original list of LA/N and LN ; Step 1 and Step 2 are executed in Table B; Step 3
and Step 4 are performed in Table C; and finally, the quartiles are defined by performing Step 5 in
Table B.
Page 23 of 38
Figure 10: An example for quartile estimation using slicing technique.
5. Validation of the methodology with real data
The methodology is validated using real data from Lucerne, Switzerland. The traffic at the site is
controlled by a fully actuated signal controller named VS-PLUS (VS-PLUS) that provides the detector
counts and signal timings. Ground truth individual vehicle travel time, is obtained from manual number
Table A
Table B
Original List
Step 1
Step 2
Sorted w.r.t Ai/Ni
[Cumulative Ni]
LA/N
[Ai/Ni]
LN
[Ni]
LA/N
[Ai/Ni]
LN
[Ni]
Lf
122.14
2.0
122.13
2.0
2.0
192.84
2.0
122.14
2.0
4.0
176.64
4.0
130.96
5.0
9.0
130.96
5.0
154.90
1.0
10.0
122.13
2.0
164.88
2.0
12.0
< Q1_index
198.54
5.0
Q1
166.08
5.0
17.0
≥ Q1_index
200.68
4.0
176.64
4.0
21.0
191.27
1.0
177.60
2.0
23.0
164.88
2.0
188.31
5.0
28.0
< Q2_index
234.54
1.0
Q2
191.27
1.0
29.0
≥ Q2_index
217.51
5.0
192.84
2.0
31.0
166.08
5.0
198.54
5.0
36.0
154.90
1.0
200.68
4.0
40.0
< Q3_index
228.88
3.0
Q3
213.28
4.0
44.0
≥ Q3_index
188.31
5.0
217.51
5.0
49.0
177.60
2.0
228.88
3.0
52.0
253.28
4.0
234.54
1.0
53.0
213.28
4.0
253.28
4.0
57.0
Table C
Step 3
N=
57.0
Step 4
Q1_index =0.25*N =
14.25
Q2_index =0.5*N =
28.50
Q3_index =0.75*N =
42.75
Page 24 of 38
plate survey performed from 3:00 pm to 6:00 pm on 15th April, 2008 (working day). The probes are
randomly selected from the survey individual vehicle data.
CUPRITE is applied on the link (see Figure 11) from Intersection A (Kaserneplatz) to intersection D
(Pilatusplatz). From A to B, there is minor side street acting as both source and sink; from B to C
there is on-street bus stop; and from C to D, there are two different movements (left and through)
associated with the link. Around 15% of the vehicles are lost in the side street in between intersection
C to D. The detectors at the site are also not perfect.
The four stop-line detectors at A (as1,as2, as3, as4) provide total cumulative plot at the upstream (UT).
The downstream cumulative plots for through movement (DThru) and left movement (DLft) are obtained
from stop-line detectors (ds1,ds2) and detector (ds3), respectively. UT is scaled vertically using the
average turning ratio of 55% for through movement and 30% for left movement to define the initial
arrival cumulative plot for each movement.
Figure 11: Illustration of the link characteristics between intersections A and D.
5.1 Average Travel Time estimation
5.1.1 Ground truth travel time
The number plate survey captures the sample of vehicles traversing the link. We are interested in
actual average travel time for all the vehicles departing the link during travel time estimation interval.
Say the mean and standard deviation of the travel time obtained from the survey be
s
X
and Ss,
respectively. We define the confidence bounds in the actual average travel time (µs) of the vehicles as:
Loss
A
C
Figure not to scale
B
Bus stop
Detectors
DThru
DLft
Bus Lane
as1
as2
as3
as4
ds1
ds2
ds3
D
Page 25 of 38
/2, 1 /2, 1
ss
ss
s n s s n
ss
SS
X t X t
nn


(17)
Where:
/2, 1
s
n
t
is the t-statistic with α level of significance and ns-1 degrees of freedom; ns is number
of survey vehicles in an estimation interval.
5.1.2 CUPRITE application
As the survey vehicle data is available for a fixed time period and the probe data required for
CUPRITE application is randomly selected from the survey vehicle data. Therefore, for each
estimation interval CUPRITE is applied for nC times with different values of the seed for random
number generator to randomly select probe vehicles. Hence, the application of CUPRITE provides
different travel time estimates for a given estimation interval. Say for an estimation interval the mean
and standard deviation of the estimates be
C
X
and SC, respectively. Then we define the confidence
bounds for the travel time estimate by CUPRITE as:
/2, 1 /2, 1
CC
CC
C n C C n
CC
SS
X t X t
nn
(18)
Where:
µC is the mean of the population of estimates from CUPRITE application;
/2, 1
C
n
t
is the t-statistic at α level of significance and nC-1 degrees of freedom.
Figure 12 illustrates an example for the presentation of results. For each estimation interval, the black
box represents the confidence bounds for the ground truth average travel time (see Figure 12a) and
the orange box represents the confidence bounds for the travel time estimates from the CUPRITE
(Figure 12b). Note: In the results, if the mean from the CUPRITE is in within the confidence bounds
from the survey then we can say that the travel time estimates from CUPRITE are statistically
equivalent to that from the survey.
Accuracy of the estimates from CUPRITE is defined as following:
(%) (100 )Accuracy MAPE
(19)
1
ni
i
Error
MAPE n
(20)
( )*100
ii
i
sC
is
XX
Error X
(21)
Page 26 of 38
Where: Errori is the absolute percentage error for ith estimation interval;
i
s
X
and
i
C
X
are the mean of
survey travel time and mean of travel time estimates from CUPRITE application during ith estimation
interval, respectively; n is the number of estimation intervals; and MAPE is the Mean Absolute
Percentage Error obtained from the CUPRITE application for different estimation intervals during
survey period.
Here, the estimation interval is five continuous signal cycles. During the analysis period, the cycle time
varies between 96 s to 116 s. The level of significance for t-statistics considered is 0.05 (=α).
The results presented here are from for ADLft. Figure 13, Figure 14 and Figure 15 illustrate results
with one, two and three probes per estimation interval (Sn). The orange box overlaps with black box,
indicating that the CUPRITE can estimate the true actual travel time. It can be seen that even the
short term oversaturation in the system can be accurately estimated. For instance, in Figure 13: fourth,
fifth, sixth and seventh estimation intervals (time from 15:30 hr to 16:00 hr) have significant variation in
average travel time between the periods. This fluctuation is also accurately captured by CUPRITE.
For A→DLft: the accuracy (19) of the CUPRITE model increases from 92.3% to 94.6% with increase in
number of probes from one probe per estimation interval (see Figure 13) to three probes per
estimation interval (see Figure 15), respectively.
Figure 12: Systematic representation of the results for CUPRITE validation.
Time
Travel time
Estimation interval
From Survey data
Time
Travel time
Estimation interval
From CUPRITE application
(b)
(a)
/2, 1
s
s
sn
s
S
Xt n
/2, 1
C
C
Cn
C
S
Xt n
/2, 1
s
s
sn
s
S
Xt n
/2, 1
C
C
Cn
C
S
Xt n
s
X
C
X
Page 27 of 38
Figure 13: Results for ADLft with Sn = 1.
Figure 14: Results for A→DLft with Sn = 2.
0
50
100
150
200
250
300
350
15:00 15:30 16:00 16:30 17:00 17:30 18:00
Travel time (seconds)
Time (hr:mm)
Survey
CUPRITE
Individual vehicle
A →DLft (Sn=1)
Accuracy (%) = 1-MAPE= 92.2%
0
50
100
150
200
250
300
350
15:00 15:30 16:00 16:30 17:00 17:30 18:00
Travel time (seconds)
Time (hr:mm)
Survey
CUPRITE
Individual vehicle
A →DLft (Sn=2)
Accuracy (%) = 1-MAPE= 93.9%
Page 28 of 38
Figure 15: Results for A→DLft with Sn = 3.
5.2 Quartile for travel time estimation
Here slicing technique is applied and quartile Q3 (75th percentile) is estimated. The results are
presented in Figure 16, Figure 17 and Figure 18 for one, two and three probes per estimation interval,
respectively. Here the accuracy of the estimates from CUPRITE is defined as following:
(%) (100 )Accuracy MAPE
(22)
1
ni
i
Error
MAPE n
(23)
( )*100
ii
i
sC
i
s
QQ
Error Q
(24)
Where: Errori is the absolute percentage error for ith estimation interval;
i
s
Q
is the Q3 of survey travel
time during ith estimation interval (As CUPRITE is applied nc number of times (Refer to section 5.1.2),
therefore
i
C
Q
is the 75th percentile of the different values of Q3 of travel time obtained from CUPRITE
application during ith estimation interval.); n is the number of estimation intervals.
0
50
100
150
200
250
300
350
15:00 15:30 16:00 16:30 17:00 17:30 18:00
Travel time (seconds)
Time (hr:mm)
Survey
CUPRITE
Individual vehicle
A →DLft (Sn=1)
Accuracy (%) = 1-MAPE= 94.6%
Page 29 of 38
It is observed that the accuracy increases from 92.4% to 94.7% for increase in Sn from one to three
probes per estimation interval. The results are similar to what we have observed for application of
CUPRITE for average travel time estimation
The above analysis indicates the potential of CUPRITE for quartile travel time estimation in addition to
the average travel time estimation.
Figure 16: Q3 estimation using CUPRITE for route from A→DLft (Sn=1).
0
50
100
150
200
250
300
350
15:00 15:30 16:00 16:30 17:00 17:30 18:00
Travel time (secomds)
Time (hr:mm)
A→DLft (Sn=1)
Accuracy (%) = 1-MAPE= 92.4%
Survey
CUPRITE
Individual vehicle
Page 30 of 38
Figure 17: Q3 estimation using CUPRITE for route from A→DLft (Sn=2).
Figure 18: Q3 estimation using CUPRITE for route from A→DLft (Sn=3).
0
50
100
150
200
250
300
350
15:00 15:30 16:00 16:30 17:00 17:30 18:00
Travel time (secomds)
Time (hr:mm)
A→DLft (Sn=2)
Accuracy (%) = 1-MAPE= 94%
Survey
CUPRITE
Individual vehicle
0
50
100
150
200
250
300
350
15:00 15:30 16:00 16:30 17:00 17:30 18:00
Travel time (secomds)
Time (hr:mm:ss)
A→DLft (Sn=3)
Accuracy (%) = 1-MAPE= 94.7%
Survey
CUPRITE
Individual vehicle
Page 31 of 38
5.3 Discussion on probe as percentage of vehicles traversing the link (Sp)
The results presented in the previous section are with fixed number (Sn) of probes per estimation
interval. In order, to capture the effect of probe market penetration we apply the model with probes as
percentage (Sp) of vehicles traversing the route during three hour of survey period. Here during an
estimation period there can be no probe (Sn=0) or at least one probe (Sn>0) (Refer to Figure 19). In
the present analysis around 60% of the estimation intervals have no probe (Sn=0) for Sp=1% and the
percentage of estimation intervals with Sn=0 decreases with increase in Sp.
Figure 20, Figure 21 and Figure 22, illustrates results for average travel time estimation for Sp equals
1%, 2% and 3%, respectively. The results indicate that even with 1% of the probes CUPRITE can
capture the fluctuations in time series of travel time. The accuracy of the estimation increased from
83.5% to 92.3% with increase in Sp from 1% to 3%, respectively. Similar, Figure 23, Figure 24 and
Figure 25 illustrate results for Q3 estimates for Sp equals 1%, 2% and 3%, respectively. The accuracy
of the Q3 estimation is close to 90%.
Figure 19: Percentage of estimation intervals versus Sn for route A→DLft for different Sp.
0%
10%
20%
30%
40%
50%
60%
0
1
2
3
4
5
6
7
8
Percentage of estimation intervals
Number of probes per estimation interval (Sn)
SP=1
SP=2
SP=3
Sp=1% : Around 60% of the estimation periods
with no probe (Sn=0)
Sp=1%
Sp=2%
Sp=3%
Page 32 of 38
Figure 20: Results for A→DLft with Sp = 1%.
Figure 21: Results for A→DLft with Sp = 2%.
0
50
100
150
200
250
300
350
15:00 15:30 16:00 16:30 17:00 17:30 18:00
Travel time (seconds)
Time (hr:mm)
Survey
CUPRITE
Individual vehicle
A →DLft (Sp=1%)
Accuracy (%) = 1-MAPE= 86.7%
0
50
100
150
200
250
300
350
15:00 15:30 16:00 16:30 17:00 17:30 18:00
Travel time (seconds)
Time (hr:mm)
Survey
CUPRITE
Individual vehicle
A →DLft (Sp=2%)
Accuracy (%) = 1-MAPE= 89.24%
Page 33 of 38
Figure 22: Results for A→DLft with Sp = 3%.
Figure 23: Q3 estimation using CUPRITE for route from A→DLft (Sp=1%).
0
50
100
150
200
250
300
350
15:00 15:30 16:00 16:30 17:00 17:30 18:00
Travel time (seconds)
Time (hr:mm)
Survey
CUPRITE
Individual vehicle
A →DLft (Sp=3%)
Accuracy (%) = 1-MAPE= 91.9%
0
50
100
150
200
250
300
350
15:00 15:30 16:00 16:30 17:00 17:30 18:00
Travel time (secomds)
Time (hr:mm)
A→DLft (Sp=1%)
Accuracy (%) = 1-MAPE= 89.9%
Survey
CUPRITE
Individual vehicle
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Figure 24: Q3 estimation using CUPRITE for route from A→DLft (Sp=2%).
Figure 25: Q3 estimation using CUPRITE for route from A→DLft (Sp=3%).
0
50
100
150
200
250
300
350
15:00 15:30 16:00 16:30 17:00 17:30 18:00
Travel time (secomds)
Time (hr:mm)
A→DLft (Sp=2%)
Accuracy (%) = 1-MAPE= 90.2%
Survey
CUPRITE
Individual vehicle
0
50
100
150
200
250
300
350
15:00 15:30 16:00 16:30 17:00 17:30 18:00
Travel time (secomds)
Time (hr:mm)
A→DLft (Sp=3%)
Accuracy (%) = 1-MAPE= 90.5%
Survey
CUPRITE
Individual vehicle
Page 35 of 38
6. Conclusions
A majority of research on travel time estimation provide average travel time. The distribution of travel
time from different vehicles departing within an estimation interval is generally bi-modal and hence it
can happen that none of the vehicle can experience average travel time. For better understanding of
the network performance statistics, such as quartiles should be explored.
The methodology proposed and validated in this paper provides travel time statistics (average and
quartiles). The methodology is based on the classical analytical procedure for travel time estimation.
The procedure is vulnerable to the relative deviation issue. This issue is addressed by integrating
cumulative plots with probe vehicle data. For this, the probe data is fixed to the downstream
cumulative plot (D(t)) and upstream cumulative plot (U(t)) is redefined: First, the points through which
U(t) should pass are defined by applying grid technique thereafter, the U(t) is redefined by applying
vertical scaling and shifting technique. The average travel time is estimated by applying classical
procedure between redefined U(t) and D(t). For estimation of quartiles, slicing technique is proposed.
The methodology is validated using real data from Luzern city, Switzerland. The application site
represents a typical urban network with: a) mixed traffic (with buses); b) on-street bus stops;
c) mid-link sinks and sources; and d) detector counting error. The validation of the methodology on
real network demonstrates its potential for practical application. The methodology requires few probes
per estimation interval for accurate estimation. The current market penetration of probe is low, and
with limited number of probes per estimation interval, it can considerably enhance the accuracy of
travel time estimation on urban networks.
Though, the development of methodology is based on urban networks, but it can be equally applied to
freeway facilities. It can be easily integrated with traffic monitoring system to simultaneously monitor
both urban and freeway networks.
The probe vehicle data for the methodology is the time when the probe is at upstream and
downstream locations. Advanced loop detectors with the capacity to provide vehicle signatures can be
explored for vehicle re-identification. The re-identified vehicle can be a proxy for probe vehicle data.
For this, even with low re-identification rate, the methodology can accurately estimate travel time.
The proposed methodology accurately estimates travel time, which is the experienced travel time. It
should be extended further, for short-term travel time prediction, by exploring forecasting techniques,
such as time series analysis, Artificial Neural Network applications etc.
The methodology integrates cumulative plots with probe vehicles. The cumulative plot considered is
two dimensional i.e., both U(t) and D(t) are represented in the same figure. An avenue for extension of
this research is to consider three three-dimensional (3D) representations of cumulative plots i.e., to
consider cumulative counts; time; and location from upstream to downstream as three different axis,
respectively. The integration of 3D representation of cumulative plots with the trajectory of a probe
vehicle should be explored for detailed modeling of individual vehicle trajectories.
Page 36 of 38
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... However, transport researchers have identified another advantage and the extended its utility from in-car communication to a complementary data source in transportation . Bluetooth data in transport has been used for estimation of travel time (Bhaskar et al., 2009, Bhaskar et al., 2010, Bhaskar et al., 2011, Kieu et al., 2012, travel patterns analysis (Behara et al., 2018b), vehicle trajectories (Michau et al., 2017a), human mobility (Abedi et al., 2013, Abedi et al., 2014, OD demand estimation (Michau et al., 2015), etc. Fundamentally, the communication range of an in-built Bluetooth device ranges from 100 to 250m radius. ...
Thesis
Lately, new big datasets are generated in the transportation sector, providing prospects for transportation engineers to design innovative ways to understand and comprehend transport-related problems and propose the solutions accordingly. Broadly, this study utilizes big datasets, i.e., smart card data and Bluetooth data, to identify public transit improvement areas. For this purpose, the aim of this study is to advance knowledge on the gap identification between supply and demand of public transit using big transit and traffic data. To achieve this aim, the research objectives of this study are to determine total transit demand using advanced data sources such as smart card data; to determine public transit supply index by employing smart card data and identify the transit gaps (non-conformity between supply and demand); to determine the transit spatial gap using smart card and Bluetooth data. Firstly, one of the smart card data applications is the estimation of the public transit origin-destination matrix (tOD). A detailed review on this topic is conducted to identify the current research gaps in the literature. Here, the primary focus is to critically analyse the contemporary literature on essential steps involved in the tOD estimation process. The steps include processes of data cleansing, estimation of unknowns, transfer detection, validation of developed algorithms, and ultimately estimation of zone level transit OD (ztOD). Estimation of unknowns includes boarding and alighting information estimation of passengers. Transfer detection algorithms distinguish between a transfer or an activity between two consecutive boarding and alighting. The findings reveal many unanswered critical research questions which need to be addressed. The research questions are primarily related to the conversion of stop level OD (stOD) to ztOD, transfer detection, and a few miscellaneous problems. Further, it is projected that the observed demand (tOD) is in fact, the served demand and not the total transit demand. In literature, attempts are made to quantify potential transit demand by employing regression equations. However, none of the studies used the observed demand from smart card data to model the potential demand or total transit demand. Here, the smart card data is used to estimate the observed demand (served demand) of a zone, based upon which high and low trip zones are segregated. An assumption is made stating that the served and total transit demand for high demand zones is approximately equal, while for low demand zones, total demand is the summation of served demand and potential demand. An ensemble tree-based Gradient Boosting model is trained and validated using observed trips from high demand zones as the dependent variable. A group of demographic, socio-economic, and geographic variables are employed as a dependent variable. The total and, subsequently, potential demand is predicted for all the low-demand zones. From the analysis, zones with high and low potential demand can be identified. Furthermore, the PD is normalized by the total area of zones to determine per unit area potential demand. Further, all the zones are clustered into four groups identifying the areas with the lowest, low, medium, and high transit improvement requirements. Secondly, this study proposed a transit Supply Index (SI), which is an effective way to estimate the transit resource provision to a zone. However, in literature, SI is modelled considering the transit services allocated to a zone and ignores the quality and quantity of the supply provided between two zones. Addressing this need, this study proposes a novel origin-destination based supply index (odSI) that incorporates a) zonal characteristics such as transit routes operate in origin zone, number of trips made by those routes, walk buffer areas of transit stops, and the total area of the zone; b) the quality of the service provided in terms of the average speed and route straightness ratio; and c) origin-destination dependent variables such as the minimum number of transfers needed to travel between two zones, the average transit speed between those zones, straightness of transit routes, and total available capacity. As proof of concept, the proposed odSI model is applied on the real network from Brisbane, Australia, at Statistical Area-2 level. The odSI results for 136×136 zonal pairs are presented for four different time periods (two peak and two off-peak periods). Furthermore, detailed analysis is carried for a zone, Grange, as an origin and the remaining 135 zones as destinations. The applicability of the proposed odSI model on Brisbane demonstrates its capability to provide valuable insights for an extensive network, helping planners visualise and improve the complex and cumbersome networks in a more effective manner. Thirdly, a comparative analysis of multi-modal travel demand is proposed that can help transport planners to improve the sustainability of a transport system. This research proposes a framework to qualitatively compare and analyse multi-modal travel demand (of constrained and unconstrained transport modes) to identify opportunities and prioritize areas for improvement of constrained mode usage. The framework includes two methods. First method, CLAN is a Coarser Level ANalysis, comparing travel patterns and demand ratios of the two modes. Second method, FLAN is a Finer Level ANalysis based on density distributions of zones and OD pairs. The proposed framework is applied to the car (relatively unconstrained mode) and transit (constrained mode) OD matrices developed from observed Bluetooth and smart card data, respectively, for the Brisbane City Council region, Australia. The gaps in transit service usage are identified at different sections of the network using both methods. This study's findings reveal that CLAN can be effectively used to prioritise OD pairs for transit improvement at a coarser level. The OD pairs with the highest and least priority from CLAN are further tested using FLAN. The results from FLAN further confirmed the findings from CLAN. The study showed that both methods have their own advantages and disadvantages if applied independently. However, when applied together, the two methods (CLAN and FLAN) can help prioritise zones and OD pairs for wider benefits of transport systems such as improvement in transit patronage. The methodological framework proposed in this study is generic and can be applied to compare other multi-modal combinations. This research suggests that the spatial gap identification can be identified by employing only smart card data or both smart card and Bluetooth data. Both the proposed methods are capable of identifying the disparities between the transit supply and demand. The results of this research can be utilised to understand the transit-related resource provision throughout the network, plan the transit resources accordingly and prioritise zones for forthcoming transit funding. The future research directions are associated to three areas: tOD estimation, odSI, and CLAN and FLAN. tOD estimation can be further improved by incorporating first and last mile, and modes not integrated in public transit system, enhancing transfer detection algorithm, and aggregation of stop-level tOD to zone-level tOD. To determine the potential demand, future research includes use of finer level of census data so that high number of zones can be included in the model calibration phase, and exploration of other machine learning and econometric models for this purpose. The proposed odSI does not consider the intrazonal supply index; therefore, it is required to accommodate the intrazonal supply index, and scrutinise its transferability to other regions of the world. Furthermore, CLAN and FLAN methods are based on the single day data only. The accuracy of results is highly dependent on the accuracy of OD matrices supplied. This research does not build on the OD estimation of Bluetooth data, hence Bluetooth OD enhancement, sensitivity analysis of various parameters used, and employment of different clustering techniques for transit gap identification is suggested for future work.
... Yao et al. (2020) estimated traffic volume and queue length utilizing sampled trajectory data at intersections. Bhaskar et al. (2011) estimated travel time using loop detector data and probe vehicle trajectory data on signalized urban network. However, such proposed approaches are mainly based on fixed detector or sampled trajectory data, which suffer from the detection failure and limited coverage. ...
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Trajectory data of road users play a crucial role in transportation engineering and traffic safety research. Previous vehicle trajectory data, including naturalistic driving data, trajectory data captured by drones and traffic cameras, have greatly promoted the research of traffic safety, traffic simulation, and other transportation engineering fields. However, existing technologies in collecting trajectory data are limited their detection range. With the development of the sensing and tracking technologies, it is possible to track the trajectory information of road users at the wide-area level, i.e., the real-time trajectory data of all road users on the entire road. Recently, we published our data sharing platform, Tongji Road Trajectory Sharing (TJRD TS, available online at https://tjrdts.com) Platform, providing our open wide-area vehicle trajectory data for researchers, academics, and road traffic professionals. Advanced vehicle tracking and trajectory splicing technologies, such as millimeter wave radar and edge computing technology, were employed to obtain such wide-area vehicle trajectory data. In this short communication article, we brainstorm ideas of using wide-area vehicle trajectory data in traffic safety research, traffic efficiency research, and connected and autonomous vehicle (CAV) application. We hope this short communication and our published data sharing platform can serve as the steppingstone for those who are interested in road transportation engineering and road safety research using such wide-area vehicle trajectory data.
... Besides the aforementioned works, research also focuses on the fusion of different data sources to achieve accurate travel time estimates, e.g., LD and probe vehicle data (Bhaskar et al., 2011) or drone data to derive naturalistic vehicle trajectories (Bock et al., 2019). In addition, the travel time reliability is highlighted by several kinds of research works as an essential measure to evaluate not only empirical measurements but also estimated/predicted outputs (see, Lyman andBertini, 2008 or Chen andFan, 2019). ...
Thesis
Full-text available
Urban transport infrastructure is under increasing pressure as more and more people want to live in cities and surrounding metropolitan areas. This development leads to increasing congestion, which deteriorates the level of service of networks and results in externalities. Targeted traffic management is an important component in counteracting these effects and meeting the mobility challenges in urban areas. Since transportation networks are complex systems, machine learning has gained importance in transportation research. This dissertation aims to solve traffic engineering problems by exploring the potential of machine learning in the field of traffic management. Approaches for challenges at three levels in a network are proposed: at an intersection, a perimeter/corridor, and at a large-scale network level. The first part of this thesis focuses on signal phase timing prediction at an isolated urban intersection. A framework is developed for multi-modal intersection systems to predict the duration of the next signal phase of a fully actuated signal control system. Based on an empirical data set from the city of Zurich, physically informed feature engineering is performed to evaluate state-of-the-art machine learning techniques (linear regression, random forest, long short-term memory neural network). We find that applying machine learning with feature engineering based on traffic flow theory is a promising tool for predicting signal phase timings when the traffic signal control system is proprietary. The second part focuses on inferring traffic conditions in an urban area with novel sensor technologies. Measurements of traffic flow and travel times are obtained from various sensors and compared to a ground truth data set. In this first part of the study, we find that thermal cameras and a license plate algorithm can accurately determine traffic flow. However, the analysis of travel time measurements shows that the WiFi module of the thermal cameras and Google distance matrix data cannot accurately replicate the current traffic conditions in the area. Moreover, we propose a simple but efficient linear regression model that performs data fusion of moving sensors (a vehicle fleet fraction of 5%), static loop detectors, and traffic signal data. We find that (a) 5% of ground truth is insufficient to represent travel time for a given route, and (b) data fusion with a simple linear model can significantly reduce travel time prediction error. Finally, the third part of this thesis presents a novel approach to traffic management with dynamic congestion pricing. We focus on developing a dynamic pricing system for multi-regional urban networks with machine learning. For this purpose, the optimal routing decisions of the system are derived first by solving a linearized optimization problem. Based on the optimal solution and the user's route choice, neural network models that capture the user's travel behavior are trained. Consequently, these models can be used to determine the dynamic price functions. We show that a linear solution of the model's system optimum is accurate and can be computed in real-time. We also find that applying the dynamic price functions significantly reduces the total time spent and the total distance traveled. The research presented in this thesis has applications for research and practice. First, sophisticated feature engineering and the application of machine learning improve (a) prediction of signal phase timings, (b) traffic state estimation, and (c) traffic conditions by applying dynamic pricing based on users' (optimal) travel behavior. Second, although data availability continues to increase, appropriate data processing and the ability to deal with aggregation and scarcity are of great importance depending on the network's level of detail. Third, the study results show that data-driven traffic management enables models to capture complex and highly non-linear traffic dynamics, which is of great importance to provide efficient and sustainable solutions in the future.
... Further, the delay estimates are sensitive to the information about the initial number of vehicles present within the section. To overcome these limitations, researchers have proposed different modifications, varying from manual adjustment [8] to probe databased adjustment [9] of the cumulative count curves. Also, other analytical methods like Incremental Queue Accumulation (IQA) have been reported in the literature for delay estimation [10]. ...
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In developing countries with limited or no availability of traffic sensors, theoretical delay models are the most commonly used tool to estimate control delay at intersections. The traffic conditions in such countries are characterised by a large mix of vehicle types and limited or no lane discipline (Heterogeneous, Less Lane-Disciplined (HLLD) traffic conditions), resulting in significantly different traffic dynamics. This research develops a queueing theory-based theoretical delay model that explicitly incorporates HLLD traffic conditions’ characteristic features like lack of lane discipline, violation of the First-In-First-Out rule, and a large mix of vehicle types. A new saturation flow-based Passenger Car Equivalent (PCE) estimation methodology to address heterogeneity and a virtual lane estimation approach to address lack of lane-discipline are proposed. The developed model shows 64% lesser error in average control delay estimation compared to the in-practice delay estimation models under HLLD traffic conditions. The developed model is used for signal optimisation under HLLD traffic conditions and reductions of up to 24% in control delay in comparison to the in-practice signal timing approach are observed. The study also highlights the significance of knowing the variation of delay in addition to average delay and presents a simple approach to capture the variation in delay.
... Most existing research on travel-time estimation focuses on using non-vision sensors, such as GPS [11,12] and loop detectors [13]. Due to the simplicity of implementation and low computational cost, historical-data-based travel-time estimation methods have been widely used in practice [14,15]. ...
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Travel-time estimation of traffic flow is an important problem with critical implications for traffic congestion analysis. We developed techniques for using intersection videos to identify vehicle trajectories across multiple cameras and analyze corridor travel time. Our approach consists of (1) multi-object single-camera tracking, (2) vehicle re-identification among different cameras, (3) multi-object multi-camera tracking, and (4) travel-time estimation. We evaluated the proposed framework on real intersections in Florida with pan and fisheye cameras. The experimental results demonstrate the viability and effectiveness of our method.
... The mobile-location sensors collect traffic data with mobile sensors like GPS or cellphones. Several studies have proved that important traffic phenomena cannot be captured properly with a penetration rate of 3%-5% level of information (Bhaskar et al. 2014(Bhaskar et al. , 2011Jenelius and Koutsopoulos 2013;Liu et al. 2009;Ramezani and Geroliminis 2015). ...
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Estimation of passenger car unit (PCU) values is very important for traffic capacity analysis and other relevant applications such as determining the saturation flow rate and signal design. However, the usual methods are not completely analogous for mixed traffic due to the presence of a wide variety of vehicle types, poor lane discipline, variability of intraclass vehicles. The parameters required for PCU estimation under mixed traffic are relatively more complex to measure, causing a severe data limitation. The accurate estimation of PCUs relies on the collection of sufficient amounts of data, and unmanned aerial vehicles (UAV) like drones are used in recent years by researchers as they significantly reduce efforts of data collection and extraction. This paper illustrates three methods for PCU estimation under mixed traffic on two signalized intersections of China using UAV monitoring data. The first method is based on the travel time of a vehicle while clearing the intersection. The second method is based on the average speed estimated in terms of different vehicle categories. The lagging headway ratio is used as the basis for PCU estimation in the third method. The author explores the effect of the driving direction and compared the PCU values at different time periods. The results show that the PCU value of vehicles turning left is always greater than that of going straight. The usage of a combined parameter is more appropriate than a single parameter in mixed traffic. The values are similar to those of India but are quite different from the United States. The values obtained in this study can be used as a guideline in the traffic analysis at signalized intersections in China and generally in other developing countries.
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Traffic density is an essential parameter for assessing the performance of uninterrupted facilities during planning, design, and operations. This study reviews and evaluates existing density estimation methods based on 87 publications. The density estimation methods are classified into two categories: definition of concept (DOC) methods and automated methods. The traditional approach includes aerial photographic, fundamental traffic flow equation, Edie's, and cumulative input-output methods. The automated approach includes the occupancy method, filtering method, and integrated method. In addition, the strengths, weaknesses, opportunities, and threats (SWOT) analysis for the preceding methods is conducted. The SWOT analysis will be helpful to researchers and practitioners in selecting the appropriate density estimation method based on its relative merits. Finally, guidelines for using various density methods for heterogeneous and homogeneous traffic are presented.
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The ability to estimate travel time on-line for use in signal timing optimization and route guidance and vehicle navigation applications is becoming necessary. Detector data are considered a valuable source of information on traffic conditions in transportation facilities. The development of models that use detector information to estimate travel time in urban networks is traced. Existing research efforts are briefly described and interrelated. A comparative discussion of the alternatives is aimed at providing a qualitative evaluation of a range of options available today for developing arterial travel time functions. The need for further calibration and validation of existing models is identified, and enhancements to improve their quality and applicability are recommended.