On-line Calibration for Dynamic Traffic Assignment

On–line Calibration for Dynamic TrafficAssignment

Constantinos Antoniou

October 5, 2007

Seminar at Portland State University

Constantinos Antoniou October 5, 2007

Outline

• Introduction

• Formulation

• Solution approaches

• Case study

• Directions for further research

• A glimpse at ongoing research

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DynaMIT Evaluation

MITSIM

Surveillance System

DynaMIT

Information

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Information dissemination

DatabaseNetwork representationHistorical information

Real-Time inputsTraffic Surveillance and

Control

State Estimation

DemandSimulation

SupplySimulation

Prediction-based Information Generation

InformationGeneration

DemandSimulation

SupplySimulation


On–line DTA framework

Demand

simulator

Supply simulator

State estimation and model calibration

Network representation

Historical data

A-priori

parameter

values

Surveillance

information

Demand

simulator

Supply simulator

State prediction

Network

performance



Motivation• Models/components

– Demand–side! OD flows! Behavioral models (e.g. route choice)

– Supply–side

! Speed–density relationship (e.g. u = uf

[1 !

(max(0,K!Kmin)

Kjam

)!]"

· where u denotes the speed, uf is the free flow speed, K is thedensity, Kmin is the minimum density, Kjam is the jam density and" and ! are model parameters.)

! Segment capacities

• Existing state estimation and model calibration approaches include subsetof these models



Literature review

• Demand parameters

– OD estimation [Ashok and Ben–Akiva (1993, 2000)]

• Supply parameters

– Capacity estimation [van Arem and van der Vlist (1992)]

– Estimation of dynamic speed–density functions [Tavana andMahmassani (2000), Huynh et al. (2002), Qin and Mahmassani(2004)]

– Tra!c state estimation in freeway stretches [Wang and Papageorgiou(2005, 2007)]



Research objectives

• Develop an integrated approach for on–line calibration that

– Jointly estimates and predicts demand and supply parameters

– Is general and flexible! Not tied to a particular DTA system

! Applicable to any calibration parameters

! Can exploit any available information

– Is computationally feasible

• Demonstrate the feasibility of the approach



Inputs and outputs

On-line

calibration

A priori

parameter

values

Surveillance

data

Historical

data

On-line calibrated

parameters



Formulation

• State–space model

– State vector

– Measurement equations

– Transition equations

• Idea of deviations



State vector

• Deviations #!h of model inputs and parameters !h from availableestimates

#!h = !h ! !Hh

• State vector includes

– OD flows xijh: number of vehicles departing from origin i towardsdestination j during time interval h

– Model parameters, e.g.! route choice model parameters

! speed–density relationship parameters

! segment capacities



Measurement equations (I)

• Direct measurement equation

!ah = !h + vh

• In deviations’ form (subtracting !Hh from both sides)

!ah ! !H

h = !h ! !Hh + vh !

#!ah = #!h + vh

where

– a denotes a priori values, H indicates historical estimates, vh is arandom error vector.



Measurement equations (II)• Indirect measurement equation

Mh = S(!h

)+ "h

• In deviations’ form (subtracting MHh from both sides)

Mh ! MHh = S

(!h

)! MH

h + "h !

#Mh = S(!H

h + #!h

)! MH

h + "h

where

– Mh is a vector of measurements for interval h, and "h is a randomerror vector.

– S is the simulator function mapping the state to the measurements(no analytic expression).



Transition equations

!h+1 =h"

q=h!p

Fh+1q · !q + #h

• In deviations’ form (subtracting !H for appropriate interval from bothsides)

!h+1 ! !Hh+1 =

h"

q=h!p

Fh+1q

(!q ! !H

q

)+ #h !

#!h+1 =h"

q=h!p

Fh+1q · #!q + #h

where

– Fh+1q is a transition matrix of e"ects of #!q on #!h+1

– p is the degree of the autoregressive process, and– #h is a random error vector



The model at a glance

#!ah = #!h + vh

#Mh = S(!H

h + #!h

)! MH

h + "h

#!h+1 =h"

q=h!p

Fh+1q · #!q + #h



Solution approaches

• Linear models

– Kalman Filter (KF)

• Non–linear models

– Extended Kalman Filter (EKF)– Limiting Extended Kalman Filter (LimEKF)– Unscented Kalman Filter (UKF)



Kalman Filtering principles

• Prediction–correction approach

1. Predict a priori state for interval h (using information up to h ! 1)

2. Compute Kalman gain matrix

3. Combine Kalman gain and surveillance from interval h to correct stateprediction for interval h



Extended Kalman Filter

• Extension of Kalman Filter for non–linear models

– First order Taylor expansion approximation

• Can use numerical derivatives, if no analytical relationship exists (formeasurement equations)

– Large number of function evaluations is required– e.g. 2n evaluations if central di"erences are used (n is state dimension)

• Computation of Kalman gain most “expensive” operation



Limiting Extended Kalman Filter

• Use a pre–computed Kalman gain

– E.g.: (weighted) average of Kalman gains (computed o"–line)– No need to compute the Kalman gain on–line

• Only a single function evaluation is required on–line

• Can run EKF o"–line and periodically re–compute Kalman gain



Unscented Kalman Filter

• Uses Unscented Transformation

– The state mean and covariance are used to generate 2n + 1 sigmapoints (n is the dimension of the state)

– These points are propagated through the true nonlinearity

• Kalman gain is computed from the state and measurement covariances

• No limiting version is possible

– Mean measurement vector (from 2n + 1 replications) is required forthe correction stage



Case study — Objectives

• Demonstrate feasibility of the on–line calibration approach

• Verify importance of on–line calibration

– Impact of joint estimation and prediction of demand and supplyparameters

• Test candidate algorithms based on several criteria

– Performance/Fit (estimation and prediction)– Computational properties– Robustness



The DynaMIT–R system

• State–of–the–art, simulation–based DTA system

– Transportation network and supply characteristics– OD demand and travel behavior

• Operates in rolling horizon for:

– Real–time estimation of network performance– Short–term prediction of future network conditions– Generation of tra!c information

• Inputs

– A-priori OD flows– Route choice parameters– Tra!c dynamics models’ parameters– Segment capacities



Network (I)



Network (II)Southampton, U.K. (35km portion of freeway M27)

Direction of flow

Traffic sensor

• 45 segments (three types)

• 10 sensors

• 20 OD pairs

• PM data (peak direction)



Traffic flow characteristics



Experimental design

• Type of day: Dry/wet weather

• Scope of on–line calibration: demand only vs. joint demand and supply

Type of day

Scope Dry Wet

Demand only (base) KF/GLS KF/GLS

EKF EKF

Demand and supply LimEKF LimEKF

UKF UKF



Measures of effectiveness

1. Fit of estimated/predicted vs. observed speeds

2. Fit of estimated/predicted vs. observed sensor counts

• Aggregate statistic: Normalized root mean square error (RMSN)

RMSN =

√N"

#n=1:N(yn ! yn)2

#n=1:N yn

3. Computational performance



Summary results for estimated and predicted speeds(Dry)

0

0.05

0.1

0.15

0.2

0.25

Three-step PredictionTwo-step PredictionOne-step PredictionEstimation

RMSN

BaseEKF

LimEKFUKF



Summary results for estimated and predicted speeds(Wet)

0

0.05

0.1

0.15

0.2

0.25

Three-step PredictionTwo-step PredictionOne-step PredictionEstimation

RMSN

BaseEKF

LimEKFUKF



On–line calibrated speed–density relationshipsMainline segments – EKF

top: dry weather, bottom: wet weather

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10

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12

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! Observations

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! Observations



Computational performance (I)

• EKF/UKF

– Similar performance (2n vs. 2n + 1 function evaluations)

• Limiting Kalman Filter

– Order–of–magnitude improvement

– Single function evaluation required (independent of state dimension)



Computational performance (II)

Function Runtime

evaluations (min:sec)

EKF 2n 98:29

LimEKF 1 0:43

UKF 2n 97:44

• Pentium 4 2.8GHz computer with 512MB RAM



Additional analysis - LimEKF(top:speeds, bottom:counts)

0

0.05

0.1

0.15

0.2

0.25

0.3

Five-step pred.Four-step pred.Three-step pred.Two-step pred.One-step pred.Estimation

RMSN

BaseLimEKF

0

0.05

0.1

0.15

0.2

0.25

0.3

Five-step pred.Four-step pred.Three-step pred.Two-step pred.One-step pred.Estimation

RMSN

BaseLimEKF



Conclusions/Findings (I)

• Joint on–line calibration of demand and supply parameters can improveprediction accuracy

• Changes in parameters are consistent with tra!c engineering principlesand expectations

• Interactions between many parameters are captured

– Many degrees of freedom



Conclusions/Findings (II)

• EKF/UKF: “Real-time” performance in a small, but non-trivial realnetwork

• Limiting EKF

– Order(s) of magnitude improvement in computational performance

– Comparable accuracy to EKF/UKF

– Robustness



Conclusions/Findings (III)

• EKF has more desirable properties than UKF (in this application)

– EKF generally outperforms UKF! Full power of UKF is not exhibited, because transition equation is

already linear

– UKF is not as robust (perhaps due to the number of points used)

– EKF and UKF have similar computational properties

– Limiting EKF (while no similar UKF version exists)



Directions for further research

• Further experimental analysis

– Scalability– Robustness– Sensitivity

• Impact of grouping segments (for speed–density relationship estimation)

• Algorithms

– e.g. particle filters (generalization of UKF)



A glimpse at ongoing researchMesoscopic traffic simulation

Density

Calculate speed

Advance vehicles

Data-

base

Density and other

explanatory variables

Calculate speed

Advance vehicles

Typical approach Alternative approach

u=f(k)



More flexible functional forms

Plus: easily incorporate additional measurements



A comparison of data–driven approaches (I)

• Locally weighted regression

• Support vector regression

• Neural networks



A comparison of data–driven approaches (II)



A comparison of data–driven approaches (III)

Notes: Application runtimes. SVR is an order of magnitude slower.



An integrated machine learning approach

k-means,

Fuzzy c-means, etc

Flexible

relationships f(x)

K-nearest

neighbors

Clustering

Fitting

Classification

Flexible

relationships f(x)Estimation

Training data

(Speeds/Densities/flows)

Estimated

speeds

Training

Explanatory data

(Densities/flows)

Application



An integrated approach: loess and clustering (I)

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loess

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loess

(dsy+flw)

loess!clust

(dsy)

loess!clust

(dsy+flw)

!40

!20

020

40

B. Distribution of speed deviations by approach

Approach

Devia

tions in s

peeds (

kph)



An integrated approach: loess and clustering (II)

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050

100

150

A. Classic speed!density relationship

Observed speeds(kph)

Estim

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d s

peeds (

kph)

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B. Loess (density)

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100

150

E. Clustered Loess (density+flow)


Estim

ate

d s

peeds (

kph)

Reduction in RMSN:

Loess (density): -7%, Loess (density+flow): -35%

Loess/clust (density): -41%, Loess/clust (density+flow): -47%



An integrated approach: loess and clustering (III)

N/A: not applicable



Thank you

Questions?

Contact information:

Constantinos Antoniou ([email protected])


On-line Calibration for Dynamic Traffic Assignment

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