# Calibration of Dynamic Traffic Assignment Models

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1 Calibration of Dynamic Traffic Assignment Models Presented at DADDY 09, Salerno Italy, 2-4 December 2009

2 Outline 1 Statistical Inference as Part of the Modelling Process Model Calibration Toy Example 2 Some General Thoughts on Inference 3 Inference with Small Traffic Counts A Day-to-Day Assignment Model Likelihood Based Inference 4 Large Count Approximations Normal Approximations 5 Conclusions and Future Directions More Questions than Answers

3 Part 1: Statistical Inference as Part of the Modelling Process

4 Model Calibration Two Stages of Model Building 1 Development of mathematical description of process; 2 Calibration i.e. estimation of unknown model parameters.

5 Model Calibration Two Stages of Model Building 1 Development of mathematical description of process; 2 Calibration i.e. estimation of unknown model parameters. Thoughts on the Above Transport research literature has tended to focus more heavily on 1 than 2. Both stages are equally important.

6 Model Calibration Methods of Calibrating Assignment Models Stochastic Assignment Models Traffic flows modelled as random variables. Standard statistical methodologies can be applied (in theory). Maximum likelihood estimation Least squares estimation Method of moments Deterministic Assignment Models More difficult to apply principled methodology One approach is to embed in stochastic model and fit that E.g. SUE as approximate mean of Markov assignment process.

7 Model Calibration Methods of Calibrating Assignment Models Stochastic Assignment Models Traffic flows modelled as random variables. Standard statistical methodologies can be applied (in theory). Maximum likelihood estimation Least squares estimation Method of moments Deterministic Assignment Models More difficult to apply principled methodology One approach is to embed in stochastic model and fit that E.g. SUE as approximate mean of Markov assignment process. Assume henceforth that models are stochastic

8 Model Calibration Model Complexity Low Model Complexity High Variance Bias Examples User Equilibrium Markov day to day Microsimulation

9 Model Calibration Model Complexity Low Model Complexity High Variance Bias Examples User Equilibrium Markov day to day Microsimulation Bias-Variance Trade-Off MSE = bias 2 + var

10 Model Calibration The Dangers of Over-fitting Excessively complex models lead to over-fitting. Over-fitted models are deceptively realistic. Excellent at reproducing yesterday. Poor at forecasting tomorrow.

11 Model Calibration The Dangers of Over-fitting Excessively complex models lead to over-fitting. Over-fitted models are deceptively realistic. Excellent at reproducing yesterday. Poor at forecasting tomorrow. So... a Markov day-to-day model of traffic flows may be better in practice than a microsimulation.

12 Model Calibration Estimation, Reconstruction and Prediction Aims in Model Fitting Estimation of model parameters with minimum error. Forecasting future realized flows.

13 Model Calibration Estimation, Reconstruction and Prediction Aims in Model Fitting Estimation of model parameters with minimum error. Forecasting future realized flows. Reconstruction of historical realized flows much less important.

14 Model Calibration Preparatory Notation u = (u 1,..., u L ) T y = (y 1,..., y M ) T x = (x 1,..., x N ) T Random Variables OD flows route (path) flows link (arc) flows Model Parameters µ = (µ 1,..., µ L ) T mean OD flows λ = (λ 1,..., λ M ) T mean route flows p 1,..., p L route choice probability vectors by OD pair

15 Toy Example An Illustrative Toy Example 1 O D 2 Aim To model hourly traffic flow from O to D by paths 1 and 2. Model Structure OD demand model: u Pois(µ). Route choice: travellers take route 1 with prob. p 1. Hence y 1 u Bin(u, p 1 ).

16 Toy Example Toy Example: Model Parameters Known Parameter µ = 10 travellers per hour. Unknown Parameters (need estimating) Route choice prob. p t 1 varies from hour to hour. Available Data Hourly counts: y t = (y t 1, yt 2 )T for t = 1, 2,..., 24 hours.

17 Toy Example Toy Example: Models to be Calibrated Hour-to-Hour Model Model correctly specified. Model complex: 24 unknown parameters to be estimated. Day-to-Day Model Just model aggregate path flows over whole day. Assume that p t 1 = p 1, a constant (counterfactual). Model mis-specified. Model simple: just 1 parameter to be estimated.

18 Toy Example Toy Example: Fitted Model Comparison Parameter Estimation Estimate route choice probability by maximum likelihood: ˆp t 1. For day-to-day model, ˆp t 1 = ˆp 1. Fitted Model Comparison Today s reconstruction errors: y1 t µˆpt 1 Tomorrow s predictive errors: y 1 t µˆpt 1 Week average predictive errors: ȳ1 t µˆpt 1

19 Toy Example Toy Example: Hourly Errors in Reconstruction Hourly errors in fitted model Hour hour model Day to day model

20 Toy Example Toy Example: Aggregate Error in Reconstruction RMSE in fitted model Hour hour model Day to day model

21 Toy Example Toy Example: Hourly Errors for Tomorrow Hourly forecasting error Hour hour model Day to day model

22 Toy Example Toy Example: Aggregate Error for Tomorrow RMSE of forecast Hour hour model Day to day model

23 Toy Example Toy Example: Hourly Errors for Next Week Hourly forecasting error Hour hour model Day to day model

24 Toy Example Toy Example: Aggregate Error for Next Week RMSE of forecast Hour hour model Day to day model

25 Toy Example Toy Example: Summary of Results Complex (hour-to-hour) model is great at forecasting yesterday. Simple (day-to-day) model is much better at predicting tomorrow. A General Conclusion Model design should account for feasibility of good calibration.

26 Part 2: Some General Thoughts on Inference

27 Some General Thoughts on Inference Data sources Model parameterization Link counts and indeterminism The Importance of second order properties Linear inverse framework

28 Data Link count data Widely available Typically unbiased Vehicle routing information Availablility varies Can be biased Other Surveys (bias? coverage?) Experiments

29 Data Link count data Widely available Typically unbiased Vehicle routing information Availablility varies Can be biased Other Surveys (bias? coverage?) Experiments We will focus primarily on inference from link count data.

30 Model Parameterization Some parameters can be estimated directly from link counts E.g. cost (delay) functions Behavioural parameters control route choice, hence route counts provide direct information

31 Model Parameterization Some parameters can be estimated directly from link counts E.g. cost (delay) functions Behavioural parameters control route choice, hence route counts provide direct information Example (logit route choice) p i = exp( θc i) j exp( θc j) Parameter θ a behavioural parameter.

32 Link counts and indeterminism Fundamental equation x = Ay A = (a ij ) is routing matrix. a ij = 1 if link i on route j, 0 otherwise. Number links = N = dim(x). Number routes = M = dim(y). Typically N << M so equations hugely underdetermined. Feasible route set Y x = {y : x = Ay} can defy enumeration.

33 The Importance of second order properties Data x 1, x 2,..., x n sequence of link counts First Order Statistical Properties x = Aȳ Mean link counts provide just N pieces of information.

34 The Importance of second order properties Second Order Statistical Properties S x = A T S y A Sample variance provides N(N 1)/2 pieces of information.

35 The Importance of second order properties Second Order Statistical Properties S x = A T S y A Sample variance provides N(N 1)/2 pieces of information. Conclusion Second order properties provide lots of additional information.

36 Linear Inverse Framework Statistical Linear Inverse Problem q(x) = h(x, y) dp (y) P is probability measure for latent variables y h is blurring function q is density/mass function for observed variables x. Examples: Image deblurring Decomposition of chemical spectra

37 Linear Inverse Problems in Transport q(x) = h(x, y) dp (y) P (y) probability measure for route flows possibly over multiple days q(x) probability density/mass function for link flows. h(x, y) = 1 y Yx for error-free counts. E.g. h(x, y) = f(x Ay) for counts with measurement error.

38 Statistical Linear Inverse Problems (SLIPs) Puts inference for transport networks in wider context. Lots known about these problems... SLIPs are hard Regularization typically necessary Bayesian framework attractive Each problem is different... but much remains to be done.

39 Part 3: Inference with Small Traffic Counts

40 A Day-to-Day Assignment Model A Day-to-Day Assignment Model Markov Process Model Assume traffic pattern evolves as Markov process from day-to-day. Route (link) flows on day t are y t (x t ). Route travel costs experienced on day t are c t = c t (x t ). Transition Probabilities Probability distribution of y t specified in term of ν previous travel costs: c t 1,..., c t ν ; Parameter vector θ, requiring estimation. Denote by p(y t c t 1,..., c t ν, θ).

41 A Day-to-Day Assignment Model Figure-of-Eight Example 1 3 O D 2 4 Route Constituent links Route cost 1 1,3 c 1 = k 1 + k 3 2 1,4 c 2 = k 1 + k 4 3 2,3 c 3 = k 2 + k 3 4 2,4 c 4 = k 2 + k 4

42 A Day-to-Day Assignment Model Figure-of-Eight Example OD Demand u t Pois(µ)

43 A Day-to-Day Assignment Model Figure-of-Eight Example OD Demand u t Pois(µ) Route Choice depends on yesterday s costs (first order Markov) p t i p t i(ζ) = e ζct 1 i N j=1 e ζct 1 j

44 A Day-to-Day Assignment Model Figure-of-Eight Example OD Demand u t Pois(µ) Route Choice depends on yesterday s costs (first order Markov) p t i p t i(ζ) = e ζct 1 i N j=1 e ζct 1 j Parameter vector θ = (µ, ζ) T

45 Likelihood Based Inference Likelihood Based Inference Likelihood L(θ) = f(x θ) f generically denotes probability mass/density function X = (x 1,..., x n ) is all link data. Parameter θ describes route flows. Decomposition f(x θ) = f(x Y )f(y θ) = f(y θ) Y Y Y X Y X = {x t = Ay t : t = 1,..., n} is feasible route set.

46 Likelihood Based Inference Application to Figure-of-Eight Example x 1 =? k 1 = 4 x 3 =? k 3 = 4 O D k 2 = 5 x 2 = 2 k 4 = 5 x 4 = 2 Simple example (for clarity) Link count data from just one day. Counts available on links 2 and 4 only: x = (NA, 2, NA, 2) T. Link costs fixed: k = (4, 5, 4, 5) T Is this sufficient information to estimate µ and ζ?

47 Likelihood Based Inference Likelihood for Figure-of-Eight Example L(µ, ζ) = y Y 4 i=1 e µ i µ y i i y i! Feasible set Y = {(y 1, y 2, y 3, y 4 ) T : y 2 + y 4 = 2, y 3 + y 4 = 2}. Can sum out unobserved y 1. Then Y = {y : y 2 = 2 y 4, y 3 = 2 y 4, y 4 = 0, 1, 2} L(µ, ζ) = y 4 {2,3,4} e (µ 2+µ 3 +µ 4 ) µ 2 y 4 2 µ 2 y 4 3 µ y 4 4 (2 y 4 )!(2 y 4 )!y 4!

48 Likelihood Based Inference Normalized Likelihood for Example Dashed line is set of GLS estimates. Likelihood has unique maximum at (µ, ζ) = (4, 0). µ ζ

49 Likelihood Based Inference Computational Problems Likelihood based inference desirable (see example). Evaluation of full likelihood requires enumeration of all feasible routes. Only feasible for very small examples. In general, direct likelihood approach is impractical.

50 Likelihood Based Inference Bayesian Approach In Bayesian paradigm, parameters are random variables. Distribution of parameter represents current knowledge about it. Before data collected, knowledge given by prior distribution f(θ). After data X observed, knowledge given by posterior distribution f(θ X). Calculating the Bayesian Posterior f(θ X) = f(x θ)f(θ) f(x) L(θ)f(θ)

51 Likelihood Based Inference Bayesian MCMC Bayesian inference cannot proceed directly without likelihood L(θ). Computationally feasible alternative is to sample from posterior. Can do this using Markov chain Monte Carlo (MCMC) methods.

52 Likelihood Based Inference Implementing MCMC Must jointly sample parameters θ and route flows Y conditional on X. Sampling Y given X is challenging since Y X not enumerable. Working in progress. See presentation by Katharina Parry.

53 Part 4: Large Count Approximations

54 Normal Approximations Normal Approximations u Pois(µ) y i u i Mult(p i ) y i Pois(µ i p i ) Define: λ i = µ i p i, λ = (λ T 1,..., λ T M )T. Then approximately for large λ: y N (λ, diag(λ))

55 Normal Approximations Normal Approximation Magic? Large counts f(y θ) N (λ, diag(λ)) f(x θ) N (Aλ, Adiag(λ)A T) where λ = λ(θ). Small counts f(x θ) = y Y x f(y θ)

56 Normal Approximations Normal Approximation Magic? Large counts f(y θ) N (λ, diag(λ)) f(x θ) N (Aλ, Adiag(λ)A T) where λ = λ(θ). Small counts f(x θ) = y Y x f(y θ) Large sample distribution looks much more tractable. Is this magic??

57 Normal Approximations Smoke and Mirrors Smoke and Mirrors The apparent advantage of the large count likelihood is partly an illusion. Even if link counts large, what about path flows? Complexity is hidden in mean Aλ and covariance matrix Adiag(λ)A T.

58 Normal Approximations Application to Figure-of-Eight Example l(µ, ζ) = log(l(µ, ζ)) = 1 2 log Σ 1 2 (x m)t Σ 1 (x m)+const. where x = [ x2 x 4 ] [ p3 + p m = µ 4 p 2 + p 4 ] [ p3 + p Σ = µ 4 p 4 p 4 p 2 + p 4 ] and p i (ζ) = e ζc i N j=1 e ζc j

59 Normal Approximations Lessons from the Example Likelihood is a complex function of µ, ζ even in simple example. Even if mean and variance estimated well, may be difficult to draw conclusions about canonical parameters.

60 Normal Approximations Normal Approximations for Day-to-Day Assignment Theorem (from Davis and Nihan (1993) 1 ) For fixed demand µ, Markov assignment process x 1, x 2,... can be approximated by a normal vector autoregressive (VAR) process. In other words: x t x t 1,..., x t ν N(m t, Σ t ) where m t, Σ t functions of θ and x t 1,..., x t ν. 1 Davis G. and Nihan N. (1993). Op Res

61 Normal Approximations Inference Using VAR Approximation Earlier comments notwithstanding, VAR approximation provides best current hope for inference for day-to-day models. Computation of VAR process mean vector and covariance matrix is challenging. Dealing with terms without full history (x 1,..., x ν ) difficult. Does VAR approximation work for Poisson (etc.) demand?

62 Part 5: Conclusions and Future Directions

63 More Questions than Answers Conclusions Parameter estimation is a critical step in modelling day-to-day traffic patterns. Statistics inference is challenging. Problems inevitable dealing with large scale linear-inverse problems. Methods for small counts and large counts differ markedly. There remain many more questions than answers.

64 More Questions than Answers Future Directions MCMC seems best hope for inference for small count models. Good sampler for route flows is crucial. Try VAR approximation for large flows. Need better understanding of VAR model properties.

65 More Questions than Answers Acknowledgement Support from the Royal Society of New Zealand (Marsden fund) gratefully acknowledged.

66 More Questions than Answers For a copy of these slides... ~ mhazelto/seminars

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