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Data - Model Synchronization in Extended Kalman
Filters for Accurate Online Traffic State Estimation
Thomas Schreiter (corresponding author)
TRAIL Research School
Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands
t.schreiter@tudelft.nl, phone: +31 15 27 81723
Chris van Hinsbergen
TRAIL Research School
Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands
c.p.i.j.vanhinsbergen@tudelft.nl, phone: +31 15 27 86044
Frank Zuurbier
TRAIL Research School
Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands
f.s.zuurbier@tudelft.nl, phone: +31 15 27 83346
Hans van Lint
Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands
j.w.c.vanlint@tudelft.nl, phone: +31 15 27 85061
Serge Hoogendoorn
Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands,
s.p.hoogendoorn@tudelft.nl, phone: +31 15 27 85475
June 27, 2010
Abstract
Real-time freeway traffic state estimation plays an important role in Dynamic Traffic Management
(DTM) and Advanced Traveler Information Systems (ATIS). One of the model-driven estimation
techniques used in practice is based on the Extended Kalman Filter (EKF). It consists of two
components: the Predictor forecasts the traffic state over a short period of time, usually a few
seconds; the Corrector fuses the traffic state with online data, observed by induction loops, which
are usually aggregated over one minute. Currently, in many approaches, the traffic state is corrected
only once, namely as soon as the observation becomes available.
In this paper it is shown that this correction time scheme does not fully make use of all the
available data. A correction time scheme is proposed, which synchronizes the Correction step of
the EKF with the aggregation period of the data. Experiments with both synthetic and real data
show improvements in the estimation quality.
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 1
1 INTRODUCTION
Freeway traffic state estimation plays a fundamental role in controlling the traffic via Dynamic
Traffic Management (DTM) and Advanced Traveler Information Systems (ATIS). Often, DTM
and ATIS systems require the state estimates to be available in real-time.
For this purpose, sensors like dual-loop detectors provide data about traffic variables such
as flow and speed. These are used in traffic state estimators to fuse the data to an accurate image
of the current traffic state, usually represented by a density vector. The observations are usually
incomplete, noisy and local, so that interpolation between them is necessary.
There are two categories of parameterized traffic state estimators. Firstly, there are data-
driven methods that only use observations to obtain the current state. An example is the “Adaptive
Smoothing Method” [1] and its various extensions [2, 3] and calibrators [4].
Secondly, there are model-driven methods that use both observations and a traffic model
with predictive features to estimate the current state: the recursive Bayesian filters with the widely
used Kalman Filter approaches. Such a filter is divided into two components; the Prediction step
forecasts the traffic state over a short time; and the Correction step fuses real-time observations
with the current traffic state prediction. Examples of Kalman Filter approaches are Renaissance [5],
Fastlane combined with an Extended Kalman Filter [6], a multi-class realtime traffic surveillance
model based on the Unscented Kalman Filter [7], a filter approach based on the Markov Compart-
ment Model [8], a SCAAT Kalman Filter [9] and an Ensemble Kalman Filter [10].
Define xkto be the traffic state, for example a vector of densities. Define hto be the
observation function that maps the traffic state to a traffic measurement, for example a fundamental
diagram that maps density to flow. In Kalman Filter modeling, the observations yk=h(xk)are
based on one single time step konly. This means that the size of the time step should be equal
to the measurement interval. However, in the traffic state estimators cited above, this condition
is not met: loop detectors aggregate the speed and flow over a period of ∆tobs = 1 min, whereas
the system state represents the traffic over a significantly shorter period, for example ∆tpred =
3 s. An observation therefore contains information not just over one time step, but over η:=
∆tobs
∆tpred multiple time steps, namely from the current time step kto time step l:= k−η+ 1.
Due to the structure of Kalman Filters, only the current system state xkis corrected, whereas the
previous states xl,...,k−1remain unchanged. The data observed are therefore not synchronized with
the Kalman Filter model.
In this paper, a method to overcome this error by a different correction timing scheme is
presented. In this scheme, at all time steps that are observed by sensors, the Correction of the
EKF is applied. Since loop detectors observe the traffic state continuously, the Correction step
is applied at every time step (Section 3). This proposed correction timing scheme is tested with
both synthetic data and real data (Section 4). The results show an improvement of the filter quality
(Section 5) compared with the current approaches in practice.
2 THE EXTENDED KALMAN FILTER AND THE LWR MODEL
In this section, first the macroscopic LWR model is described. Second, the structure of the Ex-
tended Kalman Filter is defined.
2.1 The LWR Model
The traffic is described by macroscopic variables, such as density r, flow qand speed v. The traffic
model used in this paper is the LWR model [11, 12]. It is founded on the conservation law of
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 2
vehicles
∂r
∂t +∂q
∂x = 0 .(1)
In order to solve (1) numerically, the highway network is spatially discretized into cells iof length
∆xiand temporally discretized into time steps of length ∆tpred. Each cell iis characterized by an
equilibrium flow-density relation Qfd
i, often referred to as fundamental diagram.
The density ri,k of a cell iat time step kevolves to the next time step by the discretized
conservation law of vehicles:
ri,k+1 =ri,k +∆tpred
∆xi
·qin
i,k −qout
i,k .(2)
Here, an explicit time stepping is used; nevertheless, also different schemes are applicable. For
a discussion on this topic, see [13]. The inflow qin
i,k is the flow of vehicles entering cell iat time
step k; similarly the outflow qout
i,k is the flow of vehicles leaving cell iat time step k. Within a road
stretch with no merges or bifurcations,
qout
i,k =qin
i+1,k (3)
holds. In this contribution, these fluxes between cells are determined by the Godonov minimal-
supply-and-demand scheme [14]
qout
i,k = min(Di,k, Si+1,k ),(4)
with traffic flow demand Di,k of cell iat time step kand traffic flow supply Si+1,k of cell i+ 1 at
time step k. Those, in turn, depend on the current densities and the fundamental diagram of those
cells:
Di,k =(Qfd
i(ri,k)if ffi,k
qcap
iif congi,k
,(5)
Si,k =(qcap
iif ffi,k
Qfd
i(ri,k)if congi,k
,(6)
where ffi,k and congi,k indicate free-flow and congested traffic of cell iat time k, respectively, and
qcap
iis the capacity of cell i.
2.2 The Extended Kalman Filter (EKF)
This section describes the structure of the Extended Kalman Filter (EKF) [15]. In addition, the
EKF is applied to the LWR model and the loop detector model.
The traffic system state
˜xk=r1,k r2,k . . . rI,k T(7)
is modeled over discrete time steps kas the densities ri,k of the cells i. In the following, the two
components of the EKF, Prediction step and Correction step, are explained.
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 3
Prediction Step
In the Prediction step of the EKF, the LWR model (2) is used as system function fto model the
evolution of the system state and the error covariance over time:
xp
k+1 =f(xe
k)(8)
Cp
k+1 =FkCe
kFT
k+Qk,(9)
with xp
k+1 as the predicted mean system state at time k+ 1,xe
kas the estimated mean system state
at time k,Cp
k+1 the predicted covariance at time k+ 1,Ce
kthe estimated covariance at time k,Qk
at the system noise matrix, and Fkas the linearization of fat the current time step:
Fk=∂f
∂x x=xe
k
.(10)
Correction Step
In the Correction step of the EKF, the observation function hmodels the relation between the
system state and the sensor output. Here, dual-loop detectors are used as sensors. Such a detector
is modeled by the fundamental diagram: every dual-loop detector mis placed within a cell i,
observing time-mean speed yv
m,k and flow yq
m,k. Every observed cell is therefore related to the
observation by its fundamental diagram:
ym,k =yq
m,k
yv
m,k=Qfd
i(ri,k)
Vfd
i(ri,k),(11)
where the speed-density fundamental diagram Vfd
i(ri,k)is derived by the the flow-density funda-
mental diagram Qfd
i(ri,k)via the transportation equation:
Vfd
i(ri,k) = Qfd
i(ri,k)
ri,k
.(12)
The mean system state and the covariance matrix are updated by the Kalman Filter equa-
tions [16]:
xe
k=xp
k+Kk(yk−h(xp
k)) (13)
Ce
k=Cp
k−KkHkCp
k(14)
Kk=Cp
kHT
k
HkCp
kHT
k+Rk
,(15)
with Rkas the observation noise matrix, and Has the linearization of hat the current time step:
Hk=∂h
∂x x=xp
k
.(16)
As described in (8) - (16), the Prediction and Correction steps are recursively connected.
To start the loop in the recursive filter at k= 0, the system state xe
0and the error covariance Ce
0are
initialized.
The Prediction step in a recursive filter is applied exactly once per time step to evolve the
system over time. The Correction step, on the other hand, is applied dependent on the availability
of the observations. If there are multiple observations available, then multiple Correction steps are
applied in one time step. In contrast, if there is no observation available, then the Correction step
is not performed and the predicted state is directly fed into the next Prediction step.
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 4
3 CORRECTION TIMING SCHEMES
Two different correction timing schemes are compared in this paper, which are explained in the
following.
3.1 The classic correction time scheme
FIGURE 1 :Principle of the classic correction time scheme: the Correction step of the EKF
is applied once for every observation
In the classic correction time scheme, the observation ykis used in the Kalman Filter once,
namely to correct the predicted system state xk. Since the observations contain information over
more than one time step, this method induces an error in the corrected system state as the following
example explains.
Example 1: To illustrate how this error is induced, consider a road stretch as shown in
Figure 1. A platoon of vehicles is traveling at free-flow speed (1.). The filter is initialized with the
true state (2.). The first cell is in free-flow conditions, the other cells have zero density. The total
density in the filter state matches the true number of vehicles.
Dual-loop detectors are installed observing the flow aggregated over period ∆tobs. Let
detector µbe installed in cell ι. As soon as the platoon passes the detector, it observes a non-zero
flow (3.), yq
µ,k >0. This observation is aggregated over ∆tobs and used to correct the density of the
cell ι. However, at the end of the observation period the platoon has moved further downstream.
Cell ιnow contains zero density in the model, xp
ι,k. Since this cell has zero density but the flow
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 5
observed is greater than zero, the state is corrected to a non-zero density (4.), xe
ι,k >0. In other
words, a second platoon is generated in the estimate of the Extended Kalman Filter.
The total density estimated is now higher than the ground truth density (5.). To conclude, a
previously correct estimate of the state was fused with a correct observation and led to an incorrect
state. The reason is that the correction period ∆tcorr is not synchronized with the observation period
∆tobs.
Note that a similar error occurs when the platoon is coincidently located in the detector cell
during the Correction step. Then the density is of that cell is “corrected” to a lower value, since
the flow observed is lower than the flow of that cell. In this case, vehicles would be removed, also
leading an erroneous estimate.
The same error also occurs in congestion, when a stop-and-go wave is traveling over mul-
tiple cells during one observation period. Here, too, the state of only one cell is corrected, thereby
inducing an error.
3.2 The fully-recursive correction time scheme
FIGURE 2 :Principle of the fully-recursive correction time scheme: the Correction step of
the EKF is applied in every time step
To overcome the issue presented in the previous section, a fully-recursive correction time
scheme is proposed that avoids the errors described above. The observation ykis aggregated over
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 6
FIGURE 3 :Screenshot of microscopic simulator Fosim with the following simulation sce-
nario: a two-lane highway stretch with a one-lane bottleneck, causing congestion (red dots).
Dual-loop detectors (yellow boxes) observe the highway every 500 m and every 60 s.
period ∆tobs and is therefore influenced by multiple states xl,...,k. In the fully-recursive correction
time scheme, all these states are corrected by applying the Correction step of the EKF in every
time step. An observation ykis therefore fused sequentially with all contributing states xl,...,k. This
procedure leads to a scheme in the EKF, where the Prediction and the Correction step are applied
alternately.
Example 2: The principle of the fully-recursive correction time scheme is illustrated in
Figure 2. The setup is similar to the one of the classic scheme: a platoon of vehicles travels along
the road (1.), the filter is correctly initialized (2.) and the detector observes a non-zero flow (3.).
In the fully-recursive correction time scheme, however, the correction period is synchronized with
the observation period.
The observation is therefore used in every time step to correct the state of the detector cell
(4.). In the cases where the detector cell is empty, the state is corrected to a slightly higher density.
In contrast, in the one case where the detector cell contains a non-zero density, the state is strongly
corrected towards a lower density. Essentially, the platoon is smoothed out over the observation
period, whereas the total density is preserved (5.). To conclude, a previously correctly estimate of
the state was fused with a correct observation and preserved the total number of vehicles.
4 EXPERIMENTAL SETUP
The correction timing schemes are compared in two experiments. In the first experiment (Sec-
tion 4.1), a highway of ten kilometer length including a bottleneck is simulated, using the validated
microscopic traffic simulator Fosim [17]. In the second experiment (Section 4.2), real data from
NGSIM [18] are used.
The raw data sets of both experiments provide complete trajectory data of vehicles. These
are used to calculate the ground truth density by Edie’s definition [19]. The loop detector data of
flow and speed were calculated by aggregating vehicle passages over virtual detectors.
The remainder of this section explains the road network and the setup of the EKF param-
eters of both experiments. Finally, Section 4.3 defines the performance measurements used for
evaluating the filter quality of the correction timing schemes.
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 7
4.1 Experimental setup based on synthetic data
The scenario is a two-lane highway stretch with a bottleneck where one lane is closed. A screenshot
of this scenario in Fosim is shown in Figure 3. The total length of the road stretch is 10.5 km.
The lane drop has a length of half a kilometer, but since the vehicles merge earlier, the effective
bottleneck length is one kilometer. The detectors (yellow boxes) are placed 500 m apart, providing
time-mean speed and flow observations aggregated over 60 s.
The simulation time is one hour. At the beginning of the simulation, the traffic demand is
undersaturated, then increasing to oversaturated and maintaining this value for half an hour, so that
a jam emerges (red dots in Figure 3). Near the end of the simulation, the demand is undersaturated
again, so that the jam dissolves completely.
The EKF is based on a discretized LWR model (2) with cell length ∆x= 100 m and time
step length ∆tpred = 3 s, which leads to I= 105 cells and K= 1200 time steps. This discretization
complies with the Courant-Friedrichs-Levy condition [20] and a free speed of vfree = 120 km
/h. The
fundamental diagrams of the 2-lane and the 1-lane section were estimated in FOSIM; these are
shown in Figure 4. The specific values are: a critical speed of 100 km
/h; a jam density of 256 veh
/km
and 128 veh
/km, respectively; and a capacity of 4500 veh
/hand 2400 veh
/h, respectively. The inflow into
the filter model is set to a fixed value of 1000 veh
/h.
The system noise matrix Qwas estimated in FOSIM as well. Both the traffic prediction
model fand FOSIM were initialized with the same true state ˜xk. Then, a prediction step was
performed (8) and Fosim was run for a time step ∆tpred = 3 s, resulting in a predicted mean state
xp
k+1 =f(˜xk)and a true state ˜xk+1 , respectively. This procedure was repeated many times. Finally,
the system noise matrix
Q=Cov {˜xk+1 −f(˜xk)}(17)
is set to the covariance matrix of the errors between them. This calibration procedure showed that
the diagonal elements of Qare approximately 5veh2
/km2in free-flow conditions and 10 veh2
/km2
in congested conditions. Therefore, Qkwas set to a diagonal matrix with values of 5in case of
free-flow, and 10 in case of congestion.
The observation noise matrix Rwas obtained in a similar procedure. The variance for flow
observations was calibrated to 50000 veh2
/h2and for speed observations to 100 km2
/h2. The system
state is initialized with an empty road and a variance of 10 veh2
/km2of each cell.
In the classic time stepping approach, the Correction step is applied every ∆tcorr = ∆tobs =
60 s. In the fully-recursive approach, the Correction step is applied in every time step; therefore
∆tcorr = ∆tpred = 3 s.
Ten Monte Carlo simulations were performed, where the driving behavior and the vehicle
entrance into the network is randomized.
4.2 Experimental Setup based on real data
In this scenario, real data from NGSIM gathered at the American Interstate I-101 [18] are used.
This road stretch has a length of 640 m. It consist of five main lanes and it contains an on- and an
off-ramp including a weaving lane. Trajectories were gathered over 15min.
In practice, however, detectors are placed 500 m by 60 s apart on average, for example in the
Netherlands. Under these conditions, the the size of the NGSIM data set allows only two detectors
and 15 observation instants. Therefore, the setup of both the observation and the prediction model
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 8
0 100 200 300
0
2000
4000
6000
Density in veh
/km
Flow in veh
/h
(a) Flow-density fundamental diagram Qfd
0 100 200 300
0
50
100
150
Density in veh
/km
Speed in km
/h
(b) Speed-density fundamental dia-
gram Vfd
FIGURE 4 :Fundamental diagram used in the filter, estimated in Fosim; bold blue line: two
lanes; thin black line: bottleneck of one lane
are adjusted in order to enable more observations. The virtual detectors are placed at every 100 m
and aggregate flows and speeds over 12 s. In the traffic prediction model of the EKF, the cell length
is set to 20 m, and the time step length is set to 0.6 s.
The fundamental diagram was calibrated to a capacity of 10000 veh
/hand a jam density of
768 veh
/km in the 5-lane sections, and 10500 veh
/hand 640 veh
/km in the 6-lane section. The free speed
is set to 100 km
/hand the critical speed to 80km
/h. The system state is initialized with an empty
road and a variance of 10 veh2
/km2of each cell. The inflow into the prediction model of the EKF
is fixed to 5000 veh
/h. Since the noise parameters Rand Qare difficult to calibrate, several value
combinations are analyzed.
In the classic time stepping approach, the Correction step is applied every ∆tcorr = ∆tobs =
12 s. In the fully-recursive approach, the Correction step is applied in every time step; therefore
∆tcorr = ∆tpred = 0.6 s.
Ten Monte Carlo simulation have been performed, where the observations were subjected
to zero-mean white additive Gaussian noise with a variance of 100 veh2
/h2for flow observations and
5km2
/h2for speed observations.
4.3 Performance Measurements
The performance of the correction time schemes is evaluated by comparing the estimated states xe
k
against the ground truth ˜xk. Two quantities are compared. Firstly, in order to evaluate the spatio-
temporal estimation qualities of the filter (like the correct reconstruction of the time and position
of congestion), the spatio-temporal density maps are compared. Secondly, in order to compare the
total number of vehicles in the network, the densities averaged over time are also evaluated.
Those quantities are compared with the Mean Absolute Error
MAE =1
I·KX
i,k ˜xi,k −xe
i,k(18)
and the Root Mean Square Error
RMSE =s1
I·KX
i,k ˜xi,k −xe
i,k2.(19)
In addition, the execution times of the filters are compared.
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 9
5 RESULTS AND DISCUSSION
This section presents the results of the experiments based on synthetic data in Section 5.1 and
based on real data in Section 5.2.
5.1 Results of the experiments based on synthetic data
This section presents the results of the FOSIM experiment set up in Section 4.1.
(a) Ground-truth density map (b) Fully-recursive density map
(c) Classic method density map
FIGURE 5 :Density maps (in veh
/km) of a simulation run of the FOSIM experiment
Figure 5 contains the density map of the ground truth, the density map of the estimated
states xeof the fully-recursive method, and the density map of the classic method (xeif available,
otherwise xp) of a simulation run. The ground truth map (Figure 5(a)) clearly shows the congestion
at the bottleneck. Furthermore, platoons downstream of the bottleneck are identifiable.
The filter with the fully-recursive correction timing scheme (Figure 5(b)) reconstructed
this congestion to a great extend. Position and time of the congestion match with the ground truth.
Also, the stop-and-go waves were correctly estimated. This estimate is smoother than in the ground
truth, however, because the observations are spread out over one minute, effectively smoothing the
density over one minute.
In free flow, the fluctuation in the density caused by the platoon are also reconstructed to
an extent. Here, too, these features are not as sharply estimated as in the ground truth, because the
observations are aggregated over one minute.
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 10
12345678910
0
20
40
Simulation Run
RMSE
(a) RMSE spatio-
temporal density maps
1 2 3 4 5 6 7 8 910
0
5
10
Simulation Run
RMSE
(b) RMSE density aver-
aged over time
12345678910
0
10
20
Simulation Run
MAE
(c) MAE spatio-temporal
density maps
12345678910
0
5
10
Simulation Run
MAE
(d) MAE density aver-
aged over time
FIGURE 6 :Error measurements of the FOSIM experiments for each of the 10 Monte Carlo
simulation runs; blue: fully-recursive scheme, red: classic scheme
The filter with the classic correction timing scheme (Figure 5(c)) estimates the shape of the
congestion roughly. Position and time do not match as accurately as the fully-recursive method.
For example, the dissipation of the congestion is estimated too late. Furthermore, the border be-
tween free-flow traffic and congested traffic is serrated and does not correctly describe the true
transition of the jam.
fully-recursive classic
MAE spatio-temporal density in veh
/km 11.0 12.8
MAE density average over time in veh
/km 4.3 4.5
RMSE spatio-temporal density in veh
/km 21.9 22.9
RMSE density average over time in veh
/km 5.2 5.7
Execution time in s53.4 16.5
TABLE 1 :Error measurements averaged over 10 Monte Carlo simulation runs
Figure 6 shows the error measurements applied to each simulation run of both schemes; the
values averaged over all simulation runs are listed in Table 1. Comparing the MAE of the spatio-
temporal density map, the fully-recursive scheme (11 %) is more precise than the classic scheme
(12.8 %). This trend is also vissible in the other error measurements. The fully-recursive scheme
therefore outperforms the classic one.
Figure 7 shows a series of density vectors over time of ground truth (green, dashed), the
fully-recursive correction timing (blue, solid) and the classic correction timing (red, dotted). The
ground truth shows stop-and-go waves between Cells 30 and 45. At Time step 440, a new obser-
vation becomes available.
The filter with the fully-recursive scheme closely estimates this stop-and-go wave. Note
that the movement of this wave is correctly tracked over time. The filter with the classic correction
time scheme, on the other hand, estimates this stop-and-go wave only loosely.
Before an observation becomes available, the density vector of the classic scheme is a
smooth line (Figure 7(e)). After the Correction step, the density vector contains fluctuations with
local peaks at the observed cells (Figure 7(f)). After a couple of time steps, these local peaks are
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 11
20 40 60 80 100
0
100
200
(a) Time step 435
20 40 60 80 100
0
100
200
(b) Time step 436
20 40 60 80 100
0
100
200
(c) Time step 437
20 40 60 80 100
0
100
200
(d) Time step 438
20 40 60 80 100
0
100
200
(e) Time step 439
20 40 60 80 100
0
100
200
(f) Time step 440
20 40 60 80 100
0
100
200
(g) Time step 441
20 40 60 80 100
0
100
200
(h) Time step 442
20 40 60 80 100
0
100
200
(i) Time step 443
20 40 60 80 100
0
100
200
(j) Time step 445
20 40 60 80 100
0
100
200
(k) Time step 450
20 40 60 80 100
0
100
200
(l) Time step 455
20406080100
0
100
200
fully-recursive
classic
ground truth
FIGURE 7 :State vectors of twelve time steps of the fully-recursive correction timing scheme,
the classic correction timing scheme and the ground truth; horizontal axis: cell number,
vertical axis: density in veh
/km; at Time step 440 a new observation is available
smoothed out (Figure 7(g) et seqq.).
The inflow into the network is fixed in the prediction model f, as explained in Section 4.1.
In the first four cells, both correction time schemes estimate the densities incorrectly. However,
starting with the first observed cell, Cell 5, the fully-recursive correction scheme estimates the
density accurately. In contrast, the classic correction scheme miscalculated the density further
downstream as well.
Table 1 also presents the execution time of the filters. The filter with the fully-recursive
scheme runs within 53 s on average; the filter with the classic scheme runs in a third of this time.
The higher execution time of the fully-recursive method is caused, of course, by the higher number
of Correction steps that are performed.
5.2 Results of the experiments based on real data
This section presents the results of the simulations based on NGSIM data as described in Sec-
tion 4.2. Different combinations of error noise matrices Rand Qwere tested. For the sake of
brevity, the results of the combination leading to the highest performance are presented; this is the
case for both the system noise matrix Qand the observation noise matrix Rof tenfold compared
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 12
to the FOSIM setup.
(a) Ground-truth density map (b) fully-recursive density map
(c) classic method density map
FIGURE 8 :Density maps (in veh
/km) of a simulation run of the NGSIM experiment
In Figure 8, the ground truth and the simulation results the filters with the fully-recursive
time stepping scheme and the classic time-stepping scheme are presented. (Note that the road
length and simulation time are different than in Figure 5.) In the ground truth (Figure 8(a)), con-
gestion including typical stop-and-go waves can be seen.
The filter with the fully-recursive scheme reconstructs the congestion pattern to some extent
(Figure 8(b)). The estimation location and time of the congestion match with the ground truth,
starting at the first observed cell. In addition, the large stop-and-go wave near the end of the
experiment is reconstructed.
The filter with the classic scheme underestimates the congestion (Figure 8(c)). In particular,
even downstream of the first observed cell, the traffic state is erroneously estimated as undersatu-
rated, as the large blue area at the bottom of the plot shows. Also, the large stop-and-go wave near
the end of the experiment is reconstructed only partially.
The error measurements are presented in Table 2, averaged over all ten simulation runs.
Both MAE and RMSE are large for both schemes. A reason for this might be the unusual setup
of narrow detectors both in space and in time. Moreover, due the small data set, the calibration
of the parameters is difficult. The traffic is around capacity, which makes it hard to calibrate the
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 13
fully-recursive classic
MAE spatio-temporal density in veh
/km 70.1 85.2
MAE density average over time in veh
/km 20.5 30.1
RMSE spatio-temporal density in veh
/km 94.4 111.6
RMSE density average over time in veh
/km 26.3 36.7
Execution time in s33.2 17.1
TABLE 2 :Error measurements averaged over 10 Monte Carlo simulation runs of NGSIM
fundamental diagram. Furthermore, the noise parameters of the EKF are impossible to determine
with such a small data set.
Nevertheless, the filter with the fully-recursive scheme still performs significantly better
than the classic scheme, both in terms of the estimated spatio-temporal density and in terms of the
estimated number of vehicles in the network.
6 CONCLUSION
In this paper, two correction time schemes of a recursive filter like the Extended Kalman Filter
(EKF) were studied. In the classic scheme, the Correction step is applied exactly when a new ob-
servation becomes available. In the proposed full-recursive scheme, the Correction step is applied
during the whole aggregation time of the observation. In practical traffic applications, where loop
detectors continuously observe the flow and the speed, the Correction step is therefore applied in
every time step.
Experiments with both synthetic and real data indicate that the fully-recursive method out-
performs the classic scheme. Common macroscopic traffic features like stop-and-go waves in con-
gestion or platoons in free flow are estimated with a higher quality. In addition, the total number
of vehicles in the network at an instant is better estimated.
This result is intuitive, because every traffic state that contributes to the observation is
corrected. In other words, more of the available information is used to estimate the traffic state,
which leads to a higher filter quality.
This approach is not restricted to the Extended Kalman Filter, but furthermore applies to
recursive filtering in general, for example the Unscented Kalman Filter or Particle Filters.
These results suggest that the quality in online traffic management applications like Dy-
namic Traffic Management and Advanced Traveler Information Systems is improved by imple-
menting the fully-recursive correction time scheme.
For further research, a more complex network can be studied. Furthermore, this paper used
a single-class model; multi-class traffic can be analyzed to apply the results to multi-class models
like Fastlane [6], which are in use today.
ACKNOWLEDGMENT
This research work is sponsored under the Research Grant “The MultiModal Port Traffic Centre”
by the Port of Rotterdam Authority, Rijkswaterstaat Zuid-Holland and De Verkeersonderneming
Rotterdam.
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 14
REFERENCES
[1] Treiber, M. and D. Helbing, Reconstructing the Spatio-Temporal Traffic Dynamics from Sta-
tionary Detector Data. Cooper@tive Tr@nsport@tion Dyn@mics, Vol. 1, No. 3, 2002, pp.
3.1–3.21.
[2] Treiber, M., A. Kesting, and R. E. Wilson, Reconstructing the Traffic State by Fusion of
Heterogenous Data. accepted for publication in Computer-Aided Civil and Infrastructure
Engineering, Vol. preprint physics/0900.4467, 2010.
[3] Van Lint, J. and S. P. Hoogendoorn, A Robust and Efficient Method for Fusing Heterogeneous
Data from Traffic Sensors on Freeways. Computer-Aided Civil and Infrastructure Engineer-
ing, Vol. 24, 2009, pp. 1–17.
[4] Schreiter, T., H. van Lint, Y. Yuan, and S. Hoogendoorn, Propagation Wave Speed Estima-
tion of Freeway Traffic with Image Processing Tools. In in proceeding of the Transportation
Research Board, 89th Annual Meeting, Washington, D.C., DVD, 2010.
[5] Wang, Y., M. Papageorgiou, and A. Messmer, RENAISSANCE–A Unified Macroscopic
Model-Based Aapproach to Real-Time Freeway Network Traffic Surveillance. Transporta-
tion Research Part C, Vol. 14, No. 3, 2006, pp. 190–212.
[6] Van Lint, J., S. Hoogendoorn, and M. Schreuder, Fastlane: New Multiclass First-Order Traf-
fic Flow Model. Transportation Research Record: Journal of the Transportation Research
Board, Vol. 2088, No. -1, 2008, pp. 177–187.
[7] Ngoduy, D., Applicable Filtering Framework for Online Multiclass Freeway Network Esti-
mation. Physica A: Statistical Mechanics and its Applications, Vol. 387, No. 2-3, 2008, pp.
599 – 616.
[8] Xuan, D., H. X. Liu, and G. A. Davis, A Hybrid Extended Kalman Filtering Approach for
Traffic Density Estimation along Signalized Arterials Using GPS Data. In in proceeding of
the Transportation Research Board, 89th Annual Meeting, Washington, D.C., DVD, 2010.
[9] Byon, Y.-J., A. Shalaby, B. Abdulhai, and S. El-Tantawy, Traffic Data Fusion using SCAAT
Kalman Filters. In in proceeding of the Transportation Research Board, 89th Annual Meeting,
Washington, D.C., DVD, 2010.
[10] Work, D., O. Tossavainen, S. Blandin, A. Bayen, T. Iwuchukwu, and K. Tracton, An Ensem-
ble Kalman filtering Approach to Highway Traffic Estimation Using GPS Enabled Mobile
Devices. Proceeding of the 47th IEEE Conference on Decision and Control, Vol. 47, 2008,
pp. 5062–5068.
[11] Lighthill, M. and G. Whitham, On Kinematic Waves. II. A Theory of Traffic Flow on Long
Crowded Roads. Proceedings of the Royal Society of London. Series A, Mathematical and
Physical Sciences (1934-1990), Vol. 229, No. 1178, 1955, pp. 317–345.
[12] Richards, P., Shock Waves on the Highway. Operations research, Vol. 4, No. 1, 1956, pp.
42–51.
Schreiter, Van Hinsbergen, Zuurbier, Van Lint, Hoogendoorn 15
[13] van Wageningen-Kessels, F., J. van Lint, S. Hoogendoorn, and C. Vuik, Implicit and Ex-
plicit Numerical Methods for Macroscopic Traffic Flow Models. In TRB 88th Annual Meeting
Compendium of Papers DVD, 2009.
[14] Lebaque, J., The Godunov Scheme and What It Means for First Order Traffic Flow Models,
1996, pp. 647–677.
[15] Sorenson, H., Kalman filtering: theory and application. IEEE, 1985.
[16] Kalman, R., A new approach to linear filtering and prediction problems. Journal of basic
Engineering, Vol. 82, No. 1, 1960, pp. 35–45.
[17] Dijker, T., FOSIM (Freeway Operations SIMulation), 2002.
[18] FHWA, Next Generation Simulation. Website: http://ngsim-community.com/,
2008.
[19] Edie, L., Discussion of traffic stream measurements and definitions. In Proceedings, Organi-
sation for Economic Co-operation and Development, 1965, p. 139.
[20] Courant, R., K. Friedrichs, and H. Lewy, On the partial difference equations of mathematical
physics. IBM Journal, 1967, pp. 215–234.
