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On–line Calibration for Dynamic Traffic 
Assignment 
Constantinos Antoniou 
October 5, 2007 
Seminar at Portland State Uni...
Constantinos Antoniou October 5, 2007 
Outline 
• Introduction 
• Formulation 
• Solution approaches 
• Case study 
• Dire...
!"#$%&' 
DYnamic Network Assignment 
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Information to Travelers
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Estimati...
Broadcasting forecasts does not affect the weather 
But with traffic…
DynaMIT Evaluation 
MITSIM 
Surveillance System 
DynaMIT 
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Database 
Network representation 
Historical information 
Real-Time inputs 
Traffic Surveillanc...
Constantinos Antoniou October 5, 2007 
On–line DTA framework 
Demand 
simulator 
Network representation 
Historical data 
...
Constantinos Antoniou October 5, 2007 
Motivation 
• Models/components 
– Demand–side 
! OD flows 
! Behavioral models (e....
Constantinos Antoniou October 5, 2007 
Literature review 
• Demand parameters 
– OD estimation [Ashok and Ben–Akiva (1993,...
Constantinos Antoniou October 5, 2007 
Research objectives 
• Develop an integrated approach for on–line calibration that ...
Constantinos Antoniou October 5, 2007 
Inputs and outputs 
On-line 
calibration 
A priori 
parameter 
values 
Surveillance...
Constantinos Antoniou October 5, 2007 
Formulation 
• State–space model 
– State vector 
– Measurement equations 
– Transi...
Constantinos Antoniou October 5, 2007 
State vector 
• Deviations #!h of model inputs and parameters !h from available 
es...
Constantinos Antoniou October 5, 2007 
Measurement equations (I) 
• Direct measurement equation 
!a 
h = !h + vh 
• In dev...
Constantinos Antoniou October 5, 2007 
Measurement equations (II) 
• Indirect measurement equation 
Mh = S 
% 
!h 
& 
+ "h...
Constantinos Antoniou October 5, 2007 
Transition equations 
!h+1 = 
"h 
q=h−p 
Fh+1 
q · !q + #h 
• In deviations’ form (...
Constantinos Antoniou October 5, 2007 
The model at a glance 
h = #!h + vh 
#!a 
#Mh = S 
% 
!Hh+ #!h 
& 
−MHh 
+ "h 
#!h+...
Constantinos Antoniou October 5, 2007 
Solution approaches 
• Linear models 
– Kalman Filter (KF) 
• Non–linear models 
– ...
Constantinos Antoniou October 5, 2007 
Kalman Filtering principles 
• Prediction–correction approach 
1. Predict a priori ...
Constantinos Antoniou October 5, 2007 
Extended Kalman Filter 
• Extension of Kalman Filter for non–linear models 
– First...
Constantinos Antoniou October 5, 2007 
Limiting Extended Kalman Filter 
• Use a pre–computed Kalman gain 
– E.g.: (weighte...
Constantinos Antoniou October 5, 2007 
Unscented Kalman Filter 
• Uses Unscented Transformation 
– The state mean and cova...
Constantinos Antoniou October 5, 2007 
Case study — Objectives 
• Demonstrate feasibility of the on–line calibration appro...
Constantinos Antoniou October 5, 2007 
The DynaMIT–R system 
• State–of–the–art, simulation–based DTA system 
– Transporta...
Constantinos Antoniou October 5, 2007 
Network (I) 
Seminar at Portland State University 20
Constantinos Antoniou October 5, 2007 
Network (II) 
Southampton, U.K. (35km portion of freeway M27) 
Direction of flow 
T...
Constantinos Antoniou October 5, 2007 
Traffic flow characteristics 
Seminar at Portland State University 22
Constantinos Antoniou October 5, 2007 
Experimental design 
• Type of day: Dry/wet weather 
• Scope of on–line calibration...
Constantinos Antoniou October 5, 2007 
Measures of effectiveness 
1. Fit of estimated/predicted vs. observed speeds 
2. Fi...
Constantinos Antoniou October 5, 2007 
Summary results for estimated and predicted speeds 
(Dry) 
0.25 
0.2 
0.15 
0.1 
0....
Constantinos Antoniou October 5, 2007 
Summary results for estimated and predicted speeds 
(Wet) 
0.25 
0.2 
0.15 
0.1 
0....
Constantinos Antoniou October 5, 2007 
On–line calibrated speed–density relationships 
Mainline segments – EKF 
top: dry w...
Constantinos Antoniou October 5, 2007 
Computational performance (I) 
• EKF/UKF 
– Similar performance (2n vs. 2n + 1 func...
Constantinos Antoniou October 5, 2007 
Computational performance (II) 
Function Runtime 
evaluations (min:sec) 
EKF 2n 98:...
Constantinos Antoniou October 5, 2007 
Additional analysis - LimEKF 
(top:speeds, bottom:counts) 
0.3 
0.25 
0.2 
0.15 
0....
Constantinos Antoniou October 5, 2007 
Conclusions/Findings (I) 
• Joint on–line calibration of demand and supply paramete...
Constantinos Antoniou October 5, 2007 
Conclusions/Findings (II) 
• EKF/UKF: “Real-time” performance in a small, but non-t...
Constantinos Antoniou October 5, 2007 
Conclusions/Findings (III) 
• EKF has more desirable properties than UKF (in this a...
Constantinos Antoniou October 5, 2007 
Directions for further research 
• Further experimental analysis 
– Scalability 
– ...
Constantinos Antoniou October 5, 2007 
A glimpse at ongoing research 
Mesoscopic traffic simulation 
Typical approach Alte...
Constantinos Antoniou October 5, 2007 
More flexible functional forms 
Plus: easily incorporate additional measurements 
S...
Constantinos Antoniou October 5, 2007 
A comparison of data–driven approaches (I) 
• Locally weighted regression 
• Suppor...
Constantinos Antoniou October 5, 2007 
A comparison of data–driven approaches (II) 
Seminar at Portland State University 3...
Constantinos Antoniou October 5, 2007 
A comparison of data–driven approaches (III) 
Notes: Application runtimes. SVR is a...
Constantinos Antoniou October 5, 2007 
An integrated machine learning approach 
k-means, 
Fuzzy c-means, etc 
Flexible 
re...
Constantinos Antoniou October 5, 2007 
An integrated approach: loess and clustering (I) 
! 
! 
! 
!! 
! 
! 
! 
! 
! 
! 
! ...
! ! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!! 
!!!!!!!!!!!!!!!!!!!! ! 
!!!!!!!!!!!! !!!!!!!!!! !!!!!! ! 
! 
! ! !!! ! ! !!! 
!!!!!!!!...
Constantinos Antoniou October 5, 2007 
An integrated approach: loess and clustering (III) 
N/A: not applicable 
Seminar at...
Constantinos Antoniou October 5, 2007 
Thank you 
Questions? 
Contact information: 
Constantinos Antoniou (antoniou@centra...
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On-line Calibration for Dynamic Traffic Assignment

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Constantinos Antoniou, National Technical University of Athens, Greece

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On-line Calibration for Dynamic Traffic Assignment

  1. 1. On–line Calibration for Dynamic Traffic Assignment Constantinos Antoniou October 5, 2007 Seminar at Portland State University
  2. 2. Constantinos Antoniou October 5, 2007 Outline • Introduction • Formulation • Solution approaches • Case study • Directions for further research • A glimpse at ongoing research Seminar at Portland State University 1
  3. 3. !"#$%&' DYnamic Network Assignment for the Management of Information to Travelers
  4. 4. !"#$%&' ()*$ )(+, $.(/#01$)23 42$ 0.(+2*)").2+ 5423(6.(#7*.4$88(6 54/9(3(#7*.4$92 *(#8/4+$.(/#
  5. 5. DynaMIT GUI
  6. 6. :(+, $.(/# !!2+$#3 ; %(64/*)(+, $./4 ; !25$4.,42*.(+2*6</(62 ; =/,.2*6</(62 ; =2)5/#)2*./*&': ; >4(7(#032).(#$.(/#*8 /?) !:,55 " ; %2)/)6/5(6 '4$88(6*)(+, $./4
  7. 7. =2$ 0'(+2 !@/#.(#,/,)*3$.$*6/ 26.(/# !A$).*54/62))(#7 ; B$).C >4(7(#$ "*32)(7#23*$)*3().4(1,.23* )/8.?$42 ; B42)2#.C D+/#/ (.<(6E ; A,.,42C 5$4$ 2 (F$.(/#*288/4.)*,#324?$" "Processors no longer getting faster, but adding cores !&++23($.2*3())2+(#$.(/#
  8. 8. '4$88(6*5423(6.(/# ! =/ (#7*</4(F/# ! @/#)().2#6"
  9. 9. =/ (#7*G/4(F/# !"#$%&' 6/#).$#. "*5423(6.)*.4$88(6* 6/#3(.(/#)*8/4*$*5420)526(8(23*.(+2*524(/3 7:55 8:00 9:00 Estimation Prediction Running time 7:55 8:00 9:00 Estimation Prediction Running time 8:07 8:07 9:07 At 8:00 At 8:07
  10. 10. Broadcasting forecasts does not affect the weather But with traffic…
  11. 11. DynaMIT Evaluation MITSIM Surveillance System DynaMIT %2$),42) Information /8 28826.(92#2))
  12. 12. !"#$%&'C*H2.?/4I*=2542)2#.$.(/#
  13. 13. !"#$%&' >924$ A4$+2?/4I Database Network representation Historical information Real-Time inputs Traffic Surveillance and Control State Estimation Demand Simulation Supply Simulation Prediction-based Information Generation Demand Simulation Supply Simulation Information Generation Information dissemination
  14. 14. Constantinos Antoniou October 5, 2007 On–line DTA framework Demand simulator Network representation Historical data Supply simulator A-priori parameter values State estimation and model calibration Surveillance information Demand simulator Supply simulator State prediction Network performance Seminar at Portland State University 2
  15. 15. Constantinos Antoniou October 5, 2007 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 the density, 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 subset of these models Seminar at Portland State University 3
  16. 16. Constantinos Antoniou October 5, 2007 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 and Mahmassani (2000), Huynh et al. (2002), Qin and Mahmassani (2004)] – Traffic state estimation in freeway stretches [Wang and Papageorgiou (2005, 2007)] Seminar at Portland State University 4
  17. 17. Constantinos Antoniou October 5, 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 Seminar at Portland State University 5
  18. 18. Constantinos Antoniou October 5, 2007 Inputs and outputs On-line calibration A priori parameter values Surveillance data Historical data On-line calibrated parameters Seminar at Portland State University 6
  19. 19. Constantinos Antoniou October 5, 2007 Formulation • State–space model – State vector – Measurement equations – Transition equations • Idea of deviations Seminar at Portland State University 7
  20. 20. Constantinos Antoniou October 5, 2007 State vector • Deviations #!h of model inputs and parameters !h from available estimates #!h = !h − !Hh • State vector includes – OD flows xijh: number of vehicles departing from origin i towards destination j during time interval h – Model parameters, e.g. ! route choice model parameters ! speed–density relationship parameters ! segment capacities Seminar at Portland State University 8
  21. 21. Constantinos Antoniou October 5, 2007 Measurement equations (I) • Direct measurement equation !a h = !h + vh • In deviations’ form (subtracting !Hh from both sides) !a h − !Hh = !h − !Hh + vh ! h = #!h + vh #!a where – a denotes a priori values, H indicates historical estimates, vh is a random error vector. Seminar at Portland State University 9
  22. 22. Constantinos Antoniou October 5, 2007 Measurement equations (II) • Indirect measurement equation Mh = S % !h & + "h • In deviations’ form (subtracting MHh from both sides) Mh −MHh = S % !h & −MHh + "h ! #Mh = S % !Hh + #!h & −MHh + "h where – Mh is a vector of measurements for interval h, and "h is a random error vector. – S is the simulator function mapping the state to the measurements (no analytic expression). Seminar at Portland State University 10
  23. 23. Constantinos Antoniou October 5, 2007 Transition equations !h+1 = "h q=h−p Fh+1 q · !q + #h • In deviations’ form (subtracting !H for appropriate interval from both sides) !h+1 − !H h+1 = "h q=h−p Fh+1 q % !q − !Hq & + #h ! #!h+1 = "h q=h−p Fh+1 q · #!q + #h where – Fh+1 q is a transition matrix of effects of #!q on #!h+1 – p is the degree of the autoregressive process, and – #h is a random error vector Seminar at Portland State University 11
  24. 24. Constantinos Antoniou October 5, 2007 The model at a glance h = #!h + vh #!a #Mh = S % !Hh+ #!h & −MHh + "h #!h+1 = "h q=h−p Fh+1 q · #!q + #h Seminar at Portland State University 12
  25. 25. Constantinos Antoniou October 5, 2007 Solution approaches • Linear models – Kalman Filter (KF) • Non–linear models – Extended Kalman Filter (EKF) – Limiting Extended Kalman Filter (LimEKF) – Unscented Kalman Filter (UKF) Seminar at Portland State University 13
  26. 26. Constantinos Antoniou October 5, 2007 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 state prediction for interval h Seminar at Portland State University 14
  27. 27. Constantinos Antoniou October 5, 2007 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 (for measurement equations) – Large number of function evaluations is required – e.g. 2n evaluations if central differences are used (n is state dimension) • Computation of Kalman gain most “expensive” operation Seminar at Portland State University 15
  28. 28. Constantinos Antoniou October 5, 2007 Limiting Extended Kalman Filter • Use a pre–computed Kalman gain – E.g.: (weighted) average of Kalman gains (computed off–line) – No need to compute the Kalman gain on–line • Only a single function evaluation is required on–line • Can run EKF off–line and periodically re–compute Kalman gain Seminar at Portland State University 16
  29. 29. Constantinos Antoniou October 5, 2007 Unscented Kalman Filter • Uses Unscented Transformation – The state mean and covariance are used to generate 2n + 1 sigma points (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 for the correction stage Seminar at Portland State University 17
  30. 30. Constantinos Antoniou October 5, 2007 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 supply parameters • Test candidate algorithms based on several criteria – Performance/Fit (estimation and prediction) – Computational properties – Robustness Seminar at Portland State University 18
  31. 31. Constantinos Antoniou October 5, 2007 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 traffic information • Inputs – A-priori OD flows – Route choice parameters – Traffic dynamics models’ parameters – Segment capacities Seminar at Portland State University 19
  32. 32. Constantinos Antoniou October 5, 2007 Network (I) Seminar at Portland State University 20
  33. 33. Constantinos Antoniou October 5, 2007 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) Seminar at Portland State University 21
  34. 34. Constantinos Antoniou October 5, 2007 Traffic flow characteristics Seminar at Portland State University 22
  35. 35. Constantinos Antoniou October 5, 2007 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 Seminar at Portland State University 23
  36. 36. Constantinos Antoniou October 5, 2007 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 Seminar at Portland State University 24
  37. 37. Constantinos Antoniou October 5, 2007 Summary results for estimated and predicted speeds (Dry) 0.25 0.2 0.15 0.1 0.05 0 Estimation One-step Prediction Two-step Prediction Three-step Prediction RMSN Base EKF LimEKF UKF Seminar at Portland State University 25
  38. 38. Constantinos Antoniou October 5, 2007 Summary results for estimated and predicted speeds (Wet) 0.25 0.2 0.15 0.1 0.05 0 Estimation One-step Prediction Two-step Prediction Three-step Prediction RMSN Base EKF LimEKF UKF Seminar at Portland State University 26
  39. 39. Constantinos Antoniou October 5, 2007 On–line calibrated speed–density relationships Mainline segments – EKF top: dry weather, bottom: wet weather ! ! ! ! !! ! ! ! ! ! ! ! 10 20 30 40 50 60 70 80 90 100 110 120 Density (veh/km/lane) Speed (kph) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Off!line On!line ! Observations ! ! ! ! ! ! ! ! ! ! ! ! ! ! 10 20 30 40 50 60 70 80 90 100 110 120 Density (veh/km/lane) Speed (kph) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Off!line On!line ! Observations Seminar at Portland State University 27
  40. 40. Constantinos Antoniou October 5, 2007 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) Seminar at Portland State University 28
  41. 41. Constantinos Antoniou October 5, 2007 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 Seminar at Portland State University 29
  42. 42. Constantinos Antoniou October 5, 2007 Additional analysis - LimEKF (top:speeds, bottom:counts) 0.3 0.25 0.2 0.15 0.1 0.05 0 Estimation One-step pred. Two-step pred. Three-step pred. Four-step pred. Five-step pred. RMSN Base LimEKF 0.3 0.25 0.2 0.15 0.1 0.05 0 Estimation One-step pred. Two-step pred. Three-step pred. Four-step pred. Five-step pred. RMSN Base LimEKF Seminar at Portland State University 30
  43. 43. Constantinos Antoniou October 5, 2007 Conclusions/Findings (I) • Joint on–line calibration of demand and supply parameters can improve prediction accuracy • Changes in parameters are consistent with traffic engineering principles and expectations • Interactions between many parameters are captured – Many degrees of freedom Seminar at Portland State University 31
  44. 44. Constantinos Antoniou October 5, 2007 Conclusions/Findings (II) • EKF/UKF: “Real-time” performance in a small, but non-trivial real network • Limiting EKF – Order(s) of magnitude improvement in computational performance – Comparable accuracy to EKF/UKF – Robustness Seminar at Portland State University 32
  45. 45. Constantinos Antoniou October 5, 2007 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) Seminar at Portland State University 33
  46. 46. Constantinos Antoniou October 5, 2007 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) Seminar at Portland State University 34
  47. 47. Constantinos Antoniou October 5, 2007 A glimpse at ongoing research Mesoscopic traffic simulation Typical approach Alternative approach Density Calculate speed Advance vehicles Data-base Density and other explanatory variables Calculate speed Advance vehicles u=f(k) Seminar at Portland State University 35
  48. 48. Constantinos Antoniou October 5, 2007 More flexible functional forms Plus: easily incorporate additional measurements Seminar at Portland State University 36
  49. 49. Constantinos Antoniou October 5, 2007 A comparison of data–driven approaches (I) • Locally weighted regression • Support vector regression • Neural networks Seminar at Portland State University 37
  50. 50. Constantinos Antoniou October 5, 2007 A comparison of data–driven approaches (II) Seminar at Portland State University 38
  51. 51. Constantinos Antoniou October 5, 2007 A comparison of data–driven approaches (III) Notes: Application runtimes. SVR is an order of magnitude slower. Seminar at Portland State University 39
  52. 52. Constantinos Antoniou October 5, 2007 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 Seminar at Portland State University 40
  53. 53. Constantinos Antoniou October 5, 2007 An integrated approach: loess and clustering (I) ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! !! ! !! !!! !! ! !!! ! ! ! ! ! ! ! Classic loess (dsy) loess (dsy+flw) loess!clust (dsy) loess!clust (dsy+flw) !40 !20 0 20 40 B. Distribution of speed deviations by approach Approach Deviations in speeds (kph) Seminar at Portland State University 41
  54. 54. ! ! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!! ! !!!!!!!!!!!! !!!!!!!!!! !!!!!! ! ! ! ! !!! ! ! !!! !!!!!!!!!!!!!!!!!! ! ! ! ! !!!!!!!!! ! ! !!!!! !! ! ! ! ! ! ! ! ! !! ! ! !! ! !! ! !! ! ! ! ! !!! ! ! ! !! ! ! ! ! !! ! ! !!! ! !! ! ! !! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !! Constantinos Antoniou October 5, 2007 An integrated approach: loess and clustering (II) ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! ! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!! ! !!!!!!!!!!!! !!!!!!!!!! !!!!!! ! ! ! ! !!! ! ! !!! !!!!!!!!!!!!!!!!!! ! ! ! ! !!!!!!!!! ! ! !!!!! !! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! !!! !! ! ! ! ! !! ! ! ! ! ! !!! ! ! ! !! ! ! ! ! ! ! ! !! ! ! !!! ! ! ! ! ! !! ! !! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !!!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! !! ! ! ! ! 0 50 100 150 0 50 100 150 A. Classic speed!density relationship Observed speeds(kph) Estimated speeds (kph) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! !!!!! !!!!!! !!!!!! !!! ! ! ! !! ! !!!!!!!!! !!! ! ! !!!!!! ! ! !!!!! ! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!! ! ! ! ! !!!!!!!!!!!!!!! !!! ! ! ! ! ! !!!! ! !! ! ! !! !! !! !!!!! !!!!! ! ! !!!!! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! !!! ! ! ! ! !!!!! ! !! !!!!! !!! ! ! ! ! ! ! !!!!! !!! ! ! ! ! ! !! !! ! ! ! ! !!!!!!! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! !!! ! !! ! !! ! ! ! ! ! ! ! ! ! ! ! !!!!! !!!! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !!! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !!!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! !!! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! !! 0 50 100 150 0 50 100 150 B. 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Clustered Loess (density+flow) Estimated speeds (kph) ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !!! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !!!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! !! ! ! ! ! 0 50 100 150 0 50 100 150 Observed speeds(kph) Estimated speeds (kph) ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!! ! ! ! ! !!!!!!!!!!!!!!! !!! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !!!!! !!!! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !!!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! !!! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! !! 0 50 100 150 0 50 100 150 B. 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Loess (density+flow) Observed speeds (kph) Estimated speeds (kph) !! ! ! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! !!! ! ! ! ! ! ! ! ! ! ! !! ! ! !!! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! ! ! ! ! ! ! !!! ! !! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! !!! !! ! !! ! ! ! ! !!!!! ! ! !! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! !!!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !!! !! ! !! ! ! !! ! ! ! ! ! !!!!! ! ! ! !! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !!! ! ! ! ! ! ! !! ! !!!! ! ! !!! ! ! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! 0 50 100 150 0 50 100 150 D. 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Clustered Loess (density+flow) Observed speeds (kph) Estimated speeds (kph) Reduction in RMSN: Loess (density): -7%, Loess (density+flow): -35% Loess/clust (density): -41%, Loess/clust (density+flow): -47% Seminar at Portland State University 42
  55. 55. Constantinos Antoniou October 5, 2007 An integrated approach: loess and clustering (III) N/A: not applicable Seminar at Portland State University 43
  56. 56. Constantinos Antoniou October 5, 2007 Thank you Questions? Contact information: Constantinos Antoniou (antoniou@central.ntua.gr) Seminar at Portland State University 44

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