Tutorial Traffic state estimation
Transcript of Tutorial Traffic state estimation
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Challenge the future
DelftUniversity ofTechnology
TRAFFIC STATE ESTIMATION BASICS
Prof. Dr. Hans van Lint
Traffic simulation & Computing, Civil Engineering & Geosciences, TU Delft
Director of the (Interfaculty) Msc Transport, Infrastructuur & Logistics
Many thanks to: Dr. Tamara Djukic, Dr. Thomas Schreiter, Prof. Dr.
Serge Hoogendoorn
IPAM Tutorials
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2Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Contents two lectures
• Basic ingredients State estimation
• Idiosyncrasies of traffic data
o Bias due to time averaging
o Cumulative drift
• Adaptive Smoothing Method
• Travel time estimation
• The Kalman Filter• All assumptions matter (through
examples):o Tracking a bicycle tripo OD estimationo Traffic state estimation
Today Tomorrow
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3Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Overall story
• State estimation refers to estimating (partially unobservable) state
variables x from noisy, semantically different observations y using what
ever theories and models we have relating these two
• Recipe:• Address structural bias in observations first! (due to time mean averaging, cumulative
drift, etc). This can often be done using data-data consistency methods (use one
variable to correct the next)
• Optional (particularly for freeway data): use the ASM to smear / smooth the
observations over time and space. This removes high frequency noise and gives you
a rectangular grid of observations at a desired granularity to work with
• Then use (E)KF to estimate the state variables, but be aware of the pittfals:
• All assumptions matter and affect the estimation results
• Resort to KF alternatives if the assumptions (linearizability, Gaussianity) do not match the
problem: Ensemble KFs, Unscented KF, Particle filtering, etc
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4Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
The (ideal) traffic control cycle
Traffic system
Sensors / detectors
Actuators
(b) state prediction
ITS, DTM traffic control
Goals
(c) optimization
optimize
Historical Database
Data processing / data fusion
traffic flow model
Traffic demand, Network characteristics,
Parameters
current
state
(a) state estimation
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5Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Crucial component: estimating state variables /
inputs from data
Traffic system
Sensors / detectors
Actuators
(b) state prediction
ITS, DTM traffic control
Goals
(c) optimization
optimize
Historical Database
Data processing / data fusion
traffic flow model
current
state
(a) state estimation
Traffic demand, Network characteristics,
Parameters
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6Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
State estimationThe main idea is to translate the stuff that we measure (with sensors) into state
information on the basis of which we can apply control or do other useful things
x(t)
x(t 1)
u(t)
d(t)
y(t)
x(t)
u*(t)
u(t)
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7Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
x(t) y(t) u(t) d(t)
Coordinated ramp metering
density (+ speed, ramp queues)
Speeds, flows, occupancy
Greentimes(fraction ramp flow allowed in)
inflows, turns (or OD+splits), spillback from upstream, …
Route guidance Nr Vehicles queuing (for bottlenecks)
queue lengths, travel times, Speeds, flows, occupancy
provided information / guidance
Inflows, compliance / behavior, captives, …
Network management
Accumulation, spatial variability Accumulation, OD flows
Aggregate speeds, flows, travel times
Perimeter control
Inflow, outflow constraints from other networks, OD flows
Different applications: different traffic state
variables
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8Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Ingredients traffic state estimation
data y(t) (from
whatever source) related to our state
variables x(t)Theories, assumptions, models which describe the dynamics of x(t)
and the relationship dx/dt = g(x(t), …)
y(t)=h(x(t), …)Data Assimilation
Tools & Techniques
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9Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Recall traffic flow theory
Homogeneous &
stationary conditions
s = hvq=1/h ;r=1/s
¾ ®¾¾¾ q = ru
Instantaneous measurements
Local measurementsSpatiotemporal(average) data
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10Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
q=ρu, but which u ?
Red: 30 km/hYellow: 20 km/hBlue: 10 km/h
sensor
uL= 23 km/h ! (17% overestimation)
After 1 hour
Tot distance: 60 km
Total travel time: 3 hour
uS = 20 km/h
Length circle: 1 km
1, ,
1
1i i i
L i iiii
q z v
L i
i
i
z q zq
u vn
Local averaging
q= u (homogeneous & stationary conditions)
1 (i.e.: ), ,
1
1
1
1 1
i i i i i
S i iiii
q u v z v
S i i iii ii
S
ii
i
z z
z q u zq u
u
n v
Spatial averaging
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11Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Local
measurements
Instantaneous
measurements
Spatiotemporal
data (Edie)
VariableCross-section x
Period T
Road segment X
Time t
Road segment X
Period T
flow
q (veh/h)
density
ρ (veh/km)
Av. Speed
u (km/h)
Traffic variables
q =
n
T=
1
h q = ru
q =
diiå
XT
r =
q
u r =
n
X=
1
s r =
ttjjå
XT
1 / ii
nu
v
u =v
jjån
u =q
r
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12Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Ideal (the stuff we need)
Local
measurements
Instantaneous
measurements
Spatiotemporal
data (Edie)
VariableCross-section x
Period T
Road segment X
Time t
Road segment X
Period T
flow
q (veh/h)
density
ρ (veh/km)
Av. Speed
u (km/h)
q =
n
T=
1
h q = ru
q =
diiå
XT
r =
q
u r =
n
X=
1
s r =
ttjjå
XT
uL
=n
1/ vi( )
iå u =
vjjå
n
u =q
r
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13Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Local
measurements
Instantaneous
measurements
Spatiotemporal
data (Edie)
VariableCross-section x
Period T
Road segment X
Time t
Road segment X
Period T
flow
q (veh/h)
density
ρ (veh/km)
Av. Speed
u (km/h)
Practice (the stuff we have to work with)
q =
n
T=
1
h q = ru
q =
diiå
XT
r =
q
u r =
n
X=
1
s r =
ttjjå
XT
uL
=n
1/ vi( )
iå u =
vjjå
n
u =q
r
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14Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Space
Space
Time
Loop
detectors
Micro-wave
detector
Floating
car
1 2 3 4
1 3 42
Vehicles captured
by cameras
Vehicles captured
by cameras
Camera
Different traffic data sources tell (literally)
different stories
• Differences in
spatiotemporal
coverage, aggregation,
semantics • e.g. travel time (journey)
versus local speed
• Noise, Bias
• Observability
• Availability,
completeness
Challenges for traffic state estimation
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15Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
• Most important implication:
• Drift in cumulative balance
• Causes:
• (random) detector errors
• Errors in network description
• Errors in timestamps (did a vehicle
arrive in period t or t+1?)
Example measurement noise with actual data
Mass balance between different detector stations along the A13 freeway over the period 6:00-20:00 (data from 2008)
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16Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Example approach to solving structural errors in
traffic data
• Van Lint, H. and S.P. Hoogendoorn. A Generic Methodology to Estimate
Vehicle Accumulation on Urban Arterials by Fusing Vehicle Counts and
Travel Times. in Transportation Research Board Annual Meeting. 2015.
Washington D.C.: The National Academies.
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17Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
ta t (time)
n (vehicles)
N(t) = nb-na
na
nb
nath vehicle TT(t)= ta-td
td
Cross section x1
• Flow: q1 (veh/u)
Cross section x2
• Flow: q2 (veh/u)
Q
1(t)
d
dtQ
1(t) = q
1(t)
Q
2(t)
d
dtQ
2(t) = q
2(t)
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18Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
The cumulative drift problem
• Source errors (miscounts, double counts): • lane changes, power failure, etc.
• Errors may be random or structural (bias)
• Consequence:
With e.g.
1 2
1 2
( ) ( ) ( )ˆ( ) ( ) ( ) ( )
ˆ( ) ( ) ( )
t t
ti i i
N t q s ds q s dsN t N t s s ds
q t q t t
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19Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
The cumulative drift problem
• Source errors (miscounts, double counts): • lane changes, power failure, etc.
• Errors may be random or structural (bias)
• Consequence:
This is a random walk!(which means vehicle accumulation is practically unobservable using counts)
1 2
1 2
( ) ( ) ( )ˆ( ) ( ) ( ) ( )
ˆ( ) ( ) ( )
t t
ti i i
N t q s ds q s dsN t N t s s ds
q t q t t
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20Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
The cumulative drift problem
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21Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
ta t (time)
n (vehicles)
na
Q
1(t)
Q
2(t)
Solution
n0
t0
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td*
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22Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
t (time)
n (vehicles)
Q
1(t)
Q
2(t)
TT
r
obs(t2)
e
TT(t
2)Solution
Implies we either• Underestimated inflow• Overestimated outflow• Or both
n0
t0
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td ta
na
td*
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23Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
t (time)
n (vehicles)
Q
1(t)
Q
2(t)
Correction factor
is proportional to
εTT
( )r aTT t
( )obs
r aTT t
( )TT atSolution
n0
t0
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ta
na
td td*
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24Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
( )r aTT t
( )obs
r aTT t( )TT atan
n (vehicles)
Q
1(t)
Q
2(t)
t0
2 ( )aQ t
n
0t (time)
* 1
1 ( )d at Q n
1( )Q t
Full weight
Inflow
1 1( ) ( )d dQ t Q t
( )N at
2 ( )Q t
Full weight
outflow
2 2( ) ( )a aQ t Q t
( )obs
d a r at t TT t 9/11/2015
1
2 ( )a at Q n
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25Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Mathematically
• Correction factor can be
expressed as function of
known quantities
• You can choose which of
the two curves you want to
adjust by setting 𝛼
0 0
0 0
( ) ( )( ) ( )
( ) ( )
N a a i aN a TT a obs
TT a d a r a
t n n Q t nt t
t t t t TT t t
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1 1
2 2
( ) ( ) (1 ) ( )
( ) ( ) ( )
d d N a
a a N a
Q t Q t t
Q t Q t t
Correction cumulative curve Error travel time
Cum count downstream
• You can also use it recursively on a series of detectors starting from the most
upstream couple of detectors and work your way downstream (using 𝛼=1)
{t0,n0} is the reference point
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26Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management TutorialsRows (random errors): {1%, 5%, 10%}; Columns (bias): {-5%, 0, 5%};9/11/2015
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27Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Final note: there is also “noise” in the process
itself
Mass balance between two consecutive detectors at different time aggregation levels
The smaller the time aggregation the larger variance of course …
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28Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Ingredients traffic state estimation
data y(t) (from
whatever source) related to our state
variables x(t)Theories, assumptions, models which describe the dynamics of x(t)
and the relationship dx/dt = g(x(t), …)
y(t)=h(x(t), …)Data Assimilation
Tools & Techniques
9/11/2015
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29Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Usable theories / assumptions
• Basic (“physical”) relationships between variables:• ρ = q/u
• Travel time = L/u
• The fundamental diagram = behavioral (but statistical)
relationship• q = Qe(ρ)
• u = Ue(ρ)
• Shockwave / kinematic wave theory
• More complex/involved traffic flow simulation models – anything
that encapsulates our knowledge about traffic flow operations
Recall traffic flow theory
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30Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Queue at intersection = Queue on freeway?
x
t
maxx
maxL
roodT groenT
cyclusT
laatste voertuig dat even stilstaat
en nog door groen moet
BA C
D
/k vtg m
/q vtg s
kritq
vraagq
kritkvraagk
maxcc
maxV
stilstandk
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Recall traffic flow theory
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31Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Queue at intersection = Queue on freeway?
Queue traffic light Queue due to accident on freeway
t
x
• Yes, but scales differ hugely
– In space AND time
• Urban: more difficult!9/11/2015
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32Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Freeway traffic jam versus intersection queuing
x
t
Queue at intersection
Larger scale Jam (spill-back / overflow queues)
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33Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Ingredients traffic state estimation
data y(t) (from
whatever source) related to our state
variables x(t)Theories, assumptions, models which describe the dynamics of x(t)
and the relationship dx/dt = g(x(t), …)
y(t)=h(x(t), …)Data Assimilation
Tools & Techniques
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34Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
The simplest but still (traffic flow theoretically)
correct way of smoothing traffic data
1. Interpolation and smoothing (quick but flawed: for many years the
default method in practice)
2. Kinematic wave / shockwave theory: the adaptive smoothing method
(ASM)
• Treiber, M. and D. Helbing, Reconstructing the Spatio-Temporal Traffic Dynamics from
Stationary Detector Data. Cooper@tive Tr@nsport@tion Dyn@mics, 2002. 1: p. 3.1-
3.24.
• van Lint, J.W.C. and S.P. Hoogendoorn, A Robust and Efficient Method for Fusing
Heterogeneous Data from Traffic Sensors on Freeways. Computer-Aided Civil and
Infrastructure Engineering, 2010. 25(8): p. 596-612.
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35Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
?
Interpolation, smoothing,
time series
• Interpolation (space or time)
• Smoothing, forecasting (space or time)
• OK for “isolated” data with no spatial
dynamics involved (we know of)
?
1
1 1
1
1
,
, , ,
,
j a i
i
j i j i j a i K
a i
j i i K
z t x x x
xz t x z t x z t x x x x
x x
z t x x x
1 1 1
, ,
, , , ,
j i z j i
z j i z j i j i z j i
z t x f t x
f t x f t x z t x f t x
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36Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Interpolation, smoothing,
time series
• Interpolation (space or time)
• Smoothing, forecasting (space or time)
• OK for “isolated” data with no spatial
dynamics involved (we know of)
1
1 1
1
1
,
, , ,
,
j a i
i
j i j i j a i K
a i
j i i K
z t x x x
xz t x z t x z t x x x x
x x
z t x x x
1 1 1
, ,
, , , ,
j i z j i
z j i z j i j i z j i
z t x f t x
f t x f t x z t x f t x
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space
time
Missing value
M
space
time
M
M
M M
M
M
M
M
M
interpolation over
space
interpolation over
time
Legend
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37Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Empirical facts about traffic data
10:305:30 8:00
Direction of “information flow” in free conditions
Direction of “information flow” in congested conditions
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38Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
The adaptive smoothing method (ASM)
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• INPUT:• Any data zobs(tj,xi) from
fixed detectors, GPS
enabled vehicles, etc
• Associated speed
observation vobs(tj,xi) (or
another quantity with
which the regime can be
determined)
• OUTPUT• Discretised grid z(t,x) in
any desired granularity
Slides adapted from Martin Treiber’s lectures in Delft 2009
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39Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
The Three Components of the ASM
1. Linear kernel-based spatiotemporal filters (to smooth the data over space and time)
2. The direction of information flow of free and congested traffic is incorporated by skewing the time axes of the filters
3. Nonlinear weighting of the filters for free and congested traffic
cfree
ccong
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Slides adapted from Martin Treiber’s lectures in Delft 2009
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40Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
ASM (1): Linear Lowpass Filter
Orthogonal homogeneous kernel:
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Spatial smoothing σ ~ 400m,
temporal smoothing τ ~ 60s
Input Result
0
0
( , ) ( , ) ( , )
with ( , ) exp
i j i ju x t x x t t u x t
s rs r
Slides adapted from Martin Treiber’s lectures in Delft 2009
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41Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
ASM (2): Skewing the time axes
Two anisotropic kernels:
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Spatial smoothing σ ~ 400m,
temporal smoothing τ ~ 60s
0
0
0
( , ) ( , ,c ) ( , )
( , ) ( , ,c ) ( , )
with ( , , ) exp
free i j free i j
cong i j cong i j
u x t x x t t u x t
u x t x x t t u x t
r s css r c
cfree ccong
Slides adapted from Martin Treiber’s lectures in Delft 2009
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Effect on the Weighting of Data Points
x-xix-xi
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Slides adapted from Martin Treiber’s lectures in Delft 2009
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ASM (3): Adaptive Nonlinear Speed Filter
Superposition of free and congested speed regimes:
Nonlinear weight function (threshold Vc and width ΔV):
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( , ) ( , ) ( , ) 1 ( , ) ( , )cong congu x t w x t u x t w x t u x t
min[ , ]1( , ) 1 tanh
2
c free congV u uw x t
V
Slides adapted from Martin Treiber’s lectures in Delft 2009
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Validating the ASM: M 42 North (England)
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Original data
Slides adapted from Martin Treiber’s lectures in Delft 2009
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45Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Original data
ASM, =1 km
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Validating the ASM: M 42 North (England)Slides adapted from Martin Treiber’s lectures in Delft 2009
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46Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Original data
ASM, =2 km
X
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Validating the ASM: M 42 North (England)Slides adapted from Martin Treiber’s lectures in Delft 2009
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47Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Original data
ASM, =3 km
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48Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Original data
ASM, =4 km
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49Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Original data
ASM, =6 km
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50Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Application ASM: Travel time estimation
• Van Lint, J. W. C. (2010). "Empirical Evaluation of New Robust Travel
Time Estimation Algorithms." Transportation Research Record 2160: 50-
59.
• You’ll find many refs in that paper to other travel time estimation methods
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Trajectory methods
dx(n)(t)
dt= u(t, x(n)(t)),with x(n)(t0 ) = x0
x
t
Solve this differential equation:
Trajectory methods• use imaginary average vehicle
trajectories• to deduce average travel time
Trajectory methods differ• in discretization of space and
time (cells j,i)• in formulation of uji
TTk
t0
x0
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“Classic” Trajectory methods
x
t
• use grid of detector locations × time periods
• in cells: speed is function of up- and downstream detector speeds
(which (unfortunately) is the equivalent of orthogonal interpolation)
TTk
t0
x0
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dx(n)(t)
dt= u(t, x(n)(t)),with x(n)(t0 ) = x0
Solve this differential equation:
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53Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
“Classic” Trajectory methods ()PCSB: piece-wise constant speed-based method
t
x
Vj,i+1
Vj,i
u ji (t, x) =Vj ,i x <
1
2xi + xi+1( )
Vj ,i+1 otherwise
ì
íï
îï
xi
xi+1
tj tj+1
calculate consecutive cell exit locations until destination is reached
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t
x
“Classic” Trajectory methods
Vj,i+1
Vj,i
PLSB: piece-wise linear speed-based method
u ji (t, x) = Vj ,i +x - xi
xi+1 - xi
Vj+1,i - Vj ,i( )
xi
xi+1
tj tj+1
calculate consecutive cell exit locations until destination is reached
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55Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
“Classic” Trajectory methodsPQSB: piece-wise quadratic speed-based method, etc
u ji (t, x) = f Vj-1,i ,Vj ,i ,Vj+1,i( )
or any other polynomial combination of detector speeds in the same time period
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“Filtered” trajectory method
• filter (with ASM) speed
create equidistant grid with estimates uji (as coarse or fine as needed)
• Use PCSB with these filtered speeds
x
t
TTk
t0
x0
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dx(n)(t)
dt= u(t, x(n)(t)),with x(n)(t0 ) = x0
Solve this differential equation:
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57Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Differences classic (orthogonal) vs filtered
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“Inverse-Filtered” trajectory method
dx(n)(t)
dt= 1
w(t, x(n)(t)),with x(n)(t0 ) = x0
Solve this differential equation:
• filter (with ASM) 1/speed (pace) create equidistant grid with
estimates wji (as coarse or fine as needed)
• Use adjusted PCSB with filtered pace
• Result is an exponential (har-monically) averaged speed
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x
t
TTk
t0
x0
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Case
•A13 Z The Hague – R’dam
•Evaluated PCSB, PLSB, FSB
(filtered speed based) and FISB
(filtered inverse-speed based)
methods
•Three scenariosA. all detectors
B. 1 out of 2 detectors
C. 1 out of 4 detectors
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Case
• A13 Z The Hague – R’dam
• Evaluated PCSB, PLSB, FSB
(filtered speed based) and FISB
(filtered inverse speed based)
methods
• Three scenarios• all detectors
• 1 out of 2 detectors
• 1 out of 4 detectors
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Results
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Results
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Some lessons
• Orthogonal interpolation • Destroys shockwaves,
• results in biased travel time estimates
use the ASM method
• Moreover• Real traffic data looks very different
than predicted with first order traffic
flow theory (CTM)
• But that theory still is very powerful in
reconstructing real traffic patterns
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Challenge the future
DelftUniversity ofTechnology
INTRODUCTION KALMAN FILTERExample 1: dynamic state space systems
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65Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Rudolf Kalman
• Seminal paper:R.E. Kalman (1960). "A new approach to linear filtering and prediction problems". Journal of Basic Engineering 82 (1): 35–45
• Combine mathematical models with measurement information in a way that is efficient, optimal and elegant
• First important application: navigation of space-ships (Apollo program)• Data from radar, gyroscopes, visual observations• Knowledge regarding dynamic behavior of the space-ship
• Many other applications since then: location tracking, process control, oceanography, military applications, imaging, recursive parameter identification, etc
• Good resource / starting point: www.wikipedia.org/wiki/Kalman_filter
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66Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Kalman Filter Basics
• It’s a recursive method
(memoryless)
• Intuitive prediction <–>
correction structure
• Both are affected by noise
(i.e. both are uncertain)
Filter (blok) diagram
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Modelkxk 1x
k:=k-1
w
Filter
v
y
k+1
Observations
Prediction
Correction
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Kalman Filter Basics
• Measurement information is used to find and eliminate modeling errors, errors in the input, and errors in the parameters
• Model prediction is used to eliminate outliers in the measurements and to smooth the measurement
• But how do you combine the two?
Objectives and use
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Modelkxk 1x
k:=k-1
w
Filter
v
y
k+1
Observations
Prediction
Correction
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Example 1
• Suppose we have a model M for the
temperature x:
• We have a thermostat that makes noisy
observations y of x
• Assume error terms are independent
• Let’s for now choose as an estimator for
x the following linear combination of
model and measurement:
M
2
x x w
E(w) 0, E(w ) Q
T
2
y x v
E(v) 0, E(v ) R
M T
x (1 K)x Ky
Estimating the (CONSTANT) temperature in a room
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E[wv] 0
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Example 1
Mean and variance:
(bit of algebra)
For K=0 (no weight to measurement):
For K=1 (no weight to model):
2
2 2
ˆE(x) E(x)
ˆ ˆ ˆvar(x) E x E[x] ...
(1 K) Q K R
ˆvar(x) Q
ˆvar(x) R
Statistical properties (mean and variance) of this
estimator
M T
x (1 K)x Ky
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Example 1
ˆK argminvar(x)
dˆvar(x) 2(1 K)Q 2KR 0
dK
Optimal G in terms of minimum error variance?
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G
var[x]
0 1 G*
R
Q
KK
* QK
Q R
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G
var[x]
0 1 G*
R
Q
Example 1
• Basics of the general Kalman filter are the same!
• The Kalman Filter is a multi-dimensional generalization based on
a discrete-time dynamic system and measurement equation, i.e.
models which can be formulated in state-space form:
Relation to the Kalman Filter
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k 1 k k
k k k
F
H
x x w
y x v
KK
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Discrete state space models
• We consider linear stochastic discrete time systems
• Model and measurement noise are assumed Gaussian and mutually
independent; the following holds
k 1 k k k
k k k k
F
H
x x w
y x v
k
k k l
k
k k l
Q k lE( ) 0, E( )
0 k l
R k lE( ) 0, E( )
0 k l
w w w
v v v
The good news: most processes can be cast in that form!
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Examples of state-space formulation
• This is an ARMA(1,1) model
• Now consider ARMA(2,1) model
• Define new state
• then:
zk1
x
k1
xk
axk bx
k1 w
k
xk
a b
1 0
z
k 1
0
wk
z
k x
kx
k1
k 1 k kx ax w
k 1 k k 1 kx ax bx w
The good news: most processes can be cast in that form!
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74Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
The Kalman filter
• Initial conditions for the state x0
0 0
0 0 0 0 0 0
ˆE( )
ˆ ˆvar( ) E([ ][ ] ) P
x x
x x x x x
Initial conditions (discrete timestep k=0)
• KF is an estimator for the state that is
optimal in the following sense
• Maximum a-posteriori estimator
• Conditional mean estimator
• Minimum variance estimator
x
k argmax
xk
Pr xk|y
1,...,y
k
k
2
k k kˆ ˆ=argmin E
xx x x
k k 1 kˆ E | ,...,x x y y
Important property
If this thing is Gaussian,
then these three estimators
are identical !
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The Kalman filter
This describes the conditional mean
estimator (equal to the minimum variance
estimator, and the maximum a posteriori estimator!)
k 1|k k 1|k
k k|k k
k k|k k
k k|k
k k|k
ˆ E
E F
F E E
F E
ˆF
x x
x w
x w
x
x
Estimate of mean and error covariance at k:=k+1 …
k 1 k 1
k 1 k 1 k 1 k 1
k k k k k k k k k k
k k k k k k k k
k k k k k k k k
k k k k
P var
E ( E )( E )
E (F F E )(F F E )
E (F E )(F E )
F E E E F E
F P F Q
x
x x x x
x w x x w x
x x w x x w
x x x x w w
Recursive
Estimation:
ADDITIVE errors
Model prediction
without noise!
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What about the measurements?
• Reconsider our model in example 1:
• Then obviously we have
M T
x x w and y x v
w ~ N(0,Q) and v ~ N(0,R)
M Tx ~ N(x,Q) and y ~ N(x,R)
Remember we want to maximize p(xk|yk,yk-1,…)
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What about the measurements?
• The conditional p.d.f. for
x given y equals
• Some tedious
computation leaves us
with:
p(x | y) p(y | x)p(x)
p(y)
p(y | x)p(x)
p(y | x)p(x)dx
yxvp(x v| x)p(x)
p(x v| x)p(x)dx
p(v | x)p(x)
p(v | x)p(x)dx
m
R Q QRp(x | y) N x y,
Q R Q R Q R
Remember we want to maximize p(xk|yk,yk-1,…)
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And the optimal filter is …
• The maximum a-posteriori estimate in case of a Gaussian estimate is equal to the mean:
• Note • this is equal to the minimum variance estimator we derived earlier • we did not make any prior assumptions regarding the linear structure of the
estimator
• For general discrete linear processes with Gaussian noise:• Optimal filter is linear• Kalman filter is optimal in the sense of minimum variance
m T
R Qx x y
Q R Q R
Remember we want to maximize p(xk|yk,yk-1,…)
* *
m T(1 K )x K y !!!
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Recall: discrete state space models
• We consider linear stochastic discrete time systems
• Model and measurement noise are assumed Gaussian and mutually
independent; the following holds
xk1
Fkx
kG
kw
k
yk H
kx
k v
k
k
k k l
k
k k l
Q k lE( ) 0, E( )
0 k l
R k lE( ) 0, E( )
0 k l
w w w
v v v
Optimal filter estimate is indeed that nice linear combi of
model predictions and measurements!
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The Kalman Filter
1. Initial conditions
2. Time-propagation (prediction):
3. Measurement adaptation (correction):
Kalman gain:
k|k 1 k 1 k 1|k 1
k|k 1 k 1 k 1|k 1 k 1 k 1
ˆ ˆF
P F P F Q
x x
xk|k
I KkH
k
x
k|k1K
ky
k
Pk|k
I KkH
k
P
k|k1I K
kH
k
K
kR
kKk
K
k P
k|k1H
k H
kP
k|k1H
k R
k
1
The algorithm then becomes
x
0|0 x
0, P
0|0 P
0
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81Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
The Kalman Filter
1. Initial conditions
2. Time-propagation (prediction):
3. Measurement adaptation (correction):
Kalman gain:
k|k 1 k 1 k 1|k 1
k|k 1 k 1 k 1|k 1 k 1 k 1
ˆ ˆF
P F P F Q
x x
k|k k|k 1 k k k k|k 1
k|k k k k|k 1 k k k k k
ˆ ˆ ˆK H
P I K H P I K H K R K
x x y x
K
k P
k|k1H
k H
kP
k|k1H
k R
k
1
The algorithm then becomes (alternative formulation)
x
0|0 x
0, P
0|0 P
0
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The Kalman Filter
• Note the clear predictor – corrector structure
• The filter produces besides the optimal estimate, the covariance matrix
Pk|k which is an important estimate for the accuracy of the estimate (error
bars)
• The covariance Pk|k-1 and Pk|k and filter-gain Kk do not depend on the
measurements yk (only on assumed error terms!) and can thus be
computed off-line (drawbacks?)
properties
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A closer look at the filter gain
kK
H
kP
kH'
kR
k
P
kH
k
'
Looks familiar, right?
* PK
P RRemember Example 1:
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84Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
A closer look at the filter gain
Kalman Gain =
Variance in processequation
Variance in measurement
equation
×
Sensitivitymeasurement
equation
So balance between uncertainty in process model and uncertainty in measurements / observation model!
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Challenge the future
DelftUniversity ofTechnology
TRACKING A BICYCLE TRIP
All KF assumptions matter (largely):
Assumption 1. The process and observation models can be formulated as a linear dis-
crete state space system of equations.
Assumption 2. All noise terms in this system are independent, zero-mean Gaussian
noise processes of known covariance matrices.
Example 2
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86Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Tracking a bicycle
• Consider you’re on a bicycle trip
• Your driving with 10 km/h (approx. constant speed) , due to wind,
grades, i.e. some variation in the speeds are present.
• This is our process model:
• You measure both location and speed using your knowledge of the
terrain, physiology, celestial bodies. Per hour, we want to determine
location s and speed u of bicycle. (and you have a GPS in your pocket to
check how well you’ve done)
Using noisy data
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Tracking a bicycle
Per hour, we want to determine location s and speed u of bicycle
1. Using observations of the location
2. Using your obs on the speeds
3. Using both!
4. Same as 3, now adaptive
Process and observation models
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Tracking a bicycleAdditional scenarios: initial assumptions
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Just locations
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KF estimate improves RMSE locations but worsens speed estimates
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Just speeds
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KF estimate improves RMSE locations and slightly worsens speed estimates
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Both location
and speed
data
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KF estimate improves RMSE locations AND speed estimates
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location and
speed data +
adaptive filter
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KF estimate improves RMSE locations AND speed estimates (but CIC deteriorates)
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Tracking a bicycleResults
All scenarios with the wrong assumptions on Q and R perform worse than the “A” scenarios (even worse than no KF at all)
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Don’t put your money on the wrong data …
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Challenge the future
DelftUniversity ofTechnology
ESTIMATING OD FLOWS
All KF assumptions matter (largely):
Assumption 1. The process and observation models can be formulated as a linear dis-
crete state space system of equations.
Assumption 2. All noise terms in this system are independent, zero-mean Gaussian
noise processes of known covariance matrices.
Example 3
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95Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Crucial component: estimating inputs (off and
online) from data
Traffic system
Sensors / detectors
Actuators
(b) state prediction
ITS, DTM traffic control
Goals
(c) optimization
optimize
Historical Database
Data processing / data fusion
Online traffic flow
model
current
state
(a) state estimation
Traffic demand, Network
characteristics, Parameters
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Estimating OD matrices
General problem formulation
OD flows Prior OD flows Observations
Observations predicted using OD flows
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Estimating OD matrices
General problem formulation
This term (gently) forces solution in direction of prior, but no garantuees
(alternative is hard constraints)
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Estimating OD matrices
General problem formulation
State-space model: Solvable with a KalmanFilter!
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Estimating OD matrices
1st key problem: under-determinedness
3 4
1
2
5
6
100
100
200
50
150
OD matrix A OD matrix B
5 6 5 6
1 50 50 1 25 75
2 0 100 2 25 75
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15 16
3 3 1 225 26
4 4 1
15 16 25 26
15 25
16 26
15 16
5 5 3 2
25 26 4 4
16 25
16 26
1: 100
2 : 100
3: 200
4 : 50
5 : 150
1: 100
2 : 100
4 : 50
5 : 150
L L L L
L L L
L L L L
L L
L x x
L x x
L x x x x
L x x
L x x
L x x
L x x
L x x
L x x
15 16
25 26
16 25
1: 100
2 : 100
4 : 50
L x x
L x x
L x x
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Estimating OD matrices
• The simple observation model y = Ax is flawed in congestion
• There is mutual dependence between the assignment and the OD
matrices: A = A(x), but also x = x(A): fixed point problem• Route and departure time choice depend on prevailing conditions
• Gaussian assumption very unrealistic (OD flows < 0 ????)
But there are more (fundamental) problems
x
y ~ x y ~ C !!!
x
C = capacity bottleneck
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Estimating OD Matrix
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3 4
1
2
5
6
100
100
200
50
150
OD matrix A OD matrix B
5 6 5 6
1 50 50 1 25 75
2 0 100 2 25 75
We now only consider 3 unknowns:x15, x16 and x26
• By reformulating the problem it
becomes (a) much more linear and
(b) the noise terms become much
more Gaussian
• Given• Uncongested network
• Sufficient data
• Good prior assumptions
• The Kalman filter does a pretty good
job in reconstructing OD matrices …
Ashok & Ben-Akiva (2000)
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Example
• We perturb the prior OD• With random errors (OD scenario 1 )
• With structural errors (OD scenario 2)
• We perturb the measurements• With 20% random errors
• With 40% random erros
• With 60% random errors
• We restrict measurement availability• ALL data
• Only link 3
• Link 1 and 4
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Scenario’s and results
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Example shows
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(Based on work of Ashok and Ben-Akiva)
• By reformulating the problem it
does become (a) much more
linear and (b) the noise terms
become much more Gaussian
• Notes:
• Underdeterminedness is not
solved, BUT
• Filter divergence tell-tale sign
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Challenge the future
DelftUniversity ofTechnology
TRAFFIC STATE ESTIMATION
Non-linear KF approach: extended KF
Assumption 1. The process and observation models can be formulated as a linear dis-
crete state space system of equations.
Assumption 2. All noise terms in this system are independent, zero-mean Gaussian
noise processes of known covariance matrices.
Assumption 3: process & observation models can be sufficiently accurately approximated
through first order linearization around current state
Example 4
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105Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Traffic state estimation: traffic flow models +
Extended KF
Traffic system
Sensors / detectors
Actuators
(b) state prediction
ITS, DTM traffic control
Goals
(c) optimization
optimize
Historical Database
Data processing / data fusion
Online traffic flow
model
current
state
(a) state estimation
Traffic demand, Network characteristics,
Parameters
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Recall traffic flow theory
Process model: discrete solution pdf traffic flow
Observation model: fundamental diagram Qe(ρ)
Change in density (veh/km) over time
Change in flow (km/h) over space
Governed by the speed c (function of density) with
which perturbations move over space and time
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Recall traffic flow theoryReality check … only a first order approximation
Visit Martin Treiber & Arne Kesting www.traffic-states.com
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Prediction step Correction step
Discretized Traffic flow model + (Extended)
Kalman Filter
1 - estimate inputs & params2 - predict state, (e.g. withconservation of vehicles eqn)
x
k|k-1= f x
k-1|k-1,inputs,params,etc( )
Predictions of network state
3 - predict sensor data: e.g. withthe fundamental diagram
| 1 | 1ˆ ˆ , , ,
k k k kh inputs params etcy x
5 - correct state variables
k:=k+1
Prediction (output) errors
| | 1 | 1 | 1ˆ ˆ
k k k k k k k kKx x E
new data yk?
4 - calculate errors
E
k|k-1= y
k
obs - yk|k-1
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Here’s the
EKF math
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Assumption nr 3
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Traffic state estimation example
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(1) freeway stretch with bottleneck and sensors available
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
500
1000
1500
2000
2500
3000
Tra
ffic
dem
and
(ve
h/h
)
Time (hrs.)
(2) Assumed demand
(3) Resulting traffic state
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Traffic state estimation example
1. Distort the observation models (I: ok; II: wrong; III very wrong)
2. 3 scenario’s detector availability • ALL detectors; 1 out of 2 detectors; 1 out of 6 detectors
Scenarios: mess with the observation models (add bias)
0 50 100 1500
500
1000
1500
2000
2500
3000
(a) q = Qe(r)
Density (veh/km/lane)
Flo
w (
ve
h/h
/lane)
0 50 100 1500
20
40
60
80
100
120
(b) u = Ue(r)
Density (veh/km/lane)
Speed
(km
/h)
FD Scenario 1
FD Scenario 2
FD Scenario 3
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Traffic state estimation example
• These results may surprise you a bit (or not at all)• More data does not mean better estimation (particularly not with biased observation
models). Redundancy in some cases leads to larger errors
• Neither RMSE nor MAPE are very informative measures
Results (first numbers)
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Traffic state estimation example
• All detectors and reasonable observation model:• Congestion right place and time
• Largest errors at the interface free <-> congestion
• But also large errors within congestion (first order approximation cannot handle these
moving jams)
Results (qualitative)
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Traffic state estimation example
• 1/6 detectors and reasonable observation model:• Underestimation congestion (no capacity drop)
• Largest errors at the interface free <-> congestion
• But also large errors within congestion (first order approximation)
Results (qualitative)
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Traffic state estimation example
• 1/2 detectors and very biased observation model:• Congestion right place and time
• Largest errors at the interface free <-> congestion
• slightly smaller errors within congestion (biased observation model is actually of good
use here)
Results (qualitative)
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Traffic state estimation example
Lessons learned • Quality of the models critically important
(particularly systematic errors)
• Quality (right place, no bias) of the data is
important
• Choices KF (responsiveness) is important
• EKF does remarkable well (check the
literature), but there are better options
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Challenge the future
DelftUniversity ofTechnology
TRAFFIC STATE ESTIMATIONFinal Examples
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118Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Delft
University of
Technology
Regiolab-Delft
Server
Delft-Noord
Delft
Delft-Zuid
Petrol station
7505
8050
8510
9510
10000
10510
11510
12005
12305
12700
13180
13700
14500
15007
15535 Inductive
loop
detectors
Final example 1: ASM vs EKF
t
m
speeds (km/h) A13 (date: 20060411)
06:56 08:20 09:43 11:06 12:30 13:53 15:16 16:40 18:03 19:26
12000
11000
10000
9000
0
20
40
60
80
100
120
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Delft
University of
Technology
Regiolab-Delft
Server
Delft-Noord
Delft
Delft-Zuid
Petrol station
7505
8050
8510
9510
10000
10510
11510
12005
12305
12700
13180
13700
14500
15007
15535 Inductive
loop
detectors
State estimate ASM
t
m
speeds (km/h) A13 (date: 20060411)
06:56 08:20 09:43 11:06 12:30 13:53 15:16 16:40 18:03 19:26
12000
11000
10000
90000
20
40
60
80
100
120
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Delft
University of
Technology
Regiolab-Delft
Server
Delft-Noord
Delft
Delft-Zuid
Petrol station
7505
8050
8510
9510
10000
10510
11510
12005
12305
12700
13180
13700
14500
15007
15535 Inductive
loop
detectors
State estimate LWR model + Extended Kalman
Filter
t
m
speeds (km/h) A13 (date: 20060411)
06:56 08:20 09:43 11:06 12:30 13:53 15:16 16:40 18:03 19:26
12000
11000
10000
90000
20
40
60
80
100
120
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Think about this question: which of the two estimates is “right” and why?
HINT: I used quotation marks with a purpose
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121Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Final example 2: Data fusion
• Suppose a traffic manager wishes to install a monitoring system on this
freeway
• Van Lint, J.W.C. and S.P. Hoogendoorn. The technical and economic benefits of data
fusion for real-time monitoring of freeway traffic : preliminary results and implications of
a study with simulated data. in Proceedings of the 11th World Conference on Transport
Research. 2007. Berkeley: University of California.
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Data fusion: an example
• First question: Information needs, for example: • Speeds on every place and time (for incident detection)
• But there can be many other information needs:• Head and tale of queue
• Travel times, delay times
• Traffic volumes
• Total number of vehicles, total delay time
• Etcetera …
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Data fusion: an example
The (15) options:A. 1, 3 or 7 loops
costs: 20 k€ + 5k€/year/loop
• 0, 2, 6, 8, 10% probe vehicles (FCD) costs: 6 k€/1%
Extra difficulty: 20% of all measurements are missing or unreliable
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Data fusion: an example
Different approaches
A. Simple approach: averaging + some heuristics
B. LWR Model + Extended Kalman Filter
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Data fusion: an example
This is the Ground-truth
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2% FCD
Data fusion: an example
2% FCD + 3 loops
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Data fusion: an example
10% FCD
10% FCD +7 loops
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Data fusion: Costs and Benefits
PerformanceHow accurate is each
data fusion combination?
RobustnessHow well can each data fusion
combination handle missing data?
Costs“winning” combination: 3 loops
and 2% FCD
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A more elaborate evaluation (total 90 combinations scenario + state/parameters estimated + type of sensor
data used)
Tested monitoring/data fusion scenarios:
1. Speeds + interpolation/smoothing
2. Speeds + LWR/EKF state estimation
3. Speeds + LWR/EKF state+params estimation
4. Speeds & Flows + LWR/EKF state estimation
5. Speeds & Flows + LWR/EKF state+params estimation
Each with all possible combinations of 1 -7 loops and 0-10% FCD
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A more elaborate evaluation (total 90 combinations scenario + state/parameters estimated + sensor data used)
• Top 10 most and least efficient systems (compared to
lowest cost and best performance possible)
Rank Scenario % FCD No. Loops CB RMSE Total Cost
most efficient data fusion scenarios
1 4 2 1 96% 4,83 € 42.000
2 2 2 1 96% 4,84 € 42.000
3 3 2 1 96% 5,08 € 42.000
4 5 2 1 95% 5,32 € 42.000
5 4 0 1 95% 7,45 € 30.000
6 3 0 1 94% 7,94 € 30.000
7 4 4 1 94% 4,44 € 54.000
8 2 4 1 94% 4,48 € 54.000
9 3 4 1 94% 4,55 € 54.000
10 5 0 1 94% 8,24 € 30.000
least efficient data fusion scenarios
81 2 10 7 76% 4,10 € 150.000
82 4 10 7 76% 4,10 € 150.000
83 1 0 3 75% 12,40 € 50.000
84 1 10 3 75% 5,66 € 110.000
85 5 10 7 75% 4,20 € 150.000
86 3 10 7 75% 4,26 € 150.000
87 1 6 7 74% 5,12 € 126.000
88 1 8 7 70% 5,29 € 138.000
89 1 10 7 66% 5,44 € 150.000
90 1 0 7 52% 12,08 € 90.000
Interesting: current Dutch practice …
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131Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
Conclusions
• Use of a model (assumptions, knowledge) to enrich data IS ALWAYS A
GOOD IDEA
• Model-based data fusion offers benefits in terms of:• Accuracy
• Reliability
• Costs
• A few loops (at the right places) + FCD (+ cameras + … + …) = reliable
and accurate monitoring system
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132Traffic state estimation basics | Mathematical Approaches for Traffic Flow Management Tutorials
BUT …
• Not all traffic data can be fused with such a recursive state estimation
approach (e.g. traffic flow model + EKF)
• Bias (in speed, flows, etc) : better to deal with this first• filtering will “smear out” bias over space and time
• Underdetermined and or ill-posed relationships• speed-density relationship under free flow conditions
• travel times segment speeds
• Spatiotemporal alignment problem• observations y(t) (e.g. travel time) relate to different state variables x(t),
depending on the traffic conditions:
TTk = h uk ,uk-1,...,uk-TTk( )
(alternative / complementary data fusion and state estimation
approaches are needed)
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