Research ArticleBayesian Dynamic Linear Model with Adaptive ParameterEstimation for Short-Term Travel Speed Prediction

Tai-Yu Ma 1 and Yoann Pigneacute 2

1Luxembourg Institute of Socio-Economic Research (LISER) 11 Porte des Sciences 4366 Esch-Sur-Alzette Luxembourg2Normandie Univ UNIHAVRE UNIROUEN INSA Rouen LITIS 76600 Le Havre France

Correspondence should be addressed to Tai-Yu Ma tai-yumaliserlu

Received 27 November 2018 Accepted 5 March 2019 Published 23 June 2019

Academic Editor Emanuele Crisostomi

Copyright copy 2019 Tai-Yu Ma and Yoann Pigne This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Bayesian dynamic linear model is a promising method for time series data analysis and short-term forecasting One research issueconcerns how the predictivemodel adapts to changes in the system especiallywhen shocks impact systembehavior In this study wepropose an adaptive dynamic linear model to adaptively updatemodel parameters for online system state predictionThe proposedmethod is an automatic approach based on the feedback of prediction errors at each time slot without the needs of externalintervention The experimental study on short-term travel speed prediction shows that the proposed method can significantlyreduce the prediction errors of the traditional dynamic linear model and outperform two state-of-the-art methods in the case ofmajor system behavior changes

1 Introduction

Accurate short-term traffic prediction plays an importantrole for successful traffic information system application suchas en-route navigation system traffic control and trafficcongestion management [1] During past decades differentprediction methods have been proposed to predict trafficstates for developing effective traveler information systemand real-time traffic management The methodology forshort-term travel timetraffic flow prediction can be classifiedinto the data-driven approach and model-driven approach[2ndash5]Themodel-driven approach consists in applying trafficflow theory to inference traffic state dynamics based onpartial observation of traffic data [6ndash10] The advantage ofthe model-driven approach is that it can obtain accuratetraffic state estimation with fewer observations Howeverthe performance of the model-driven approach can bepoor if the applied models are not well calibrated [4] Asregards the data-driven approach it relies on the spatial-temporal correlation of traffic states for which future trafficstates can be estimated based on historical time series dataAmong different data-driven approaches which are the focus

of this study machine learning methods are widely usedfor traffic characteristics prediction eg neural networks[11 12] autoregressive integrated moving average models(ARIMA) [13 14] support vector machine methods [15]nearest neighbor classification methods [16 17] ensemblelearning approach [12] and Bayesian dynamic linear models(DLM)state space models [18ndash22] among many others Oneof the main issues in short-term traffic prediction is how todynamically adapt a predicting model to the uncertainty ofsystem behavior changes in particular in case of accidentor unforeseen events With recent vehicular communicationadvances in real-time traffic data collection the developmentof adaptive short-term traffic prediction methods becomean active research area in transportation science and indeveloping applications based on vehicular communicationtechnology

In this perspective the DLM approach provides a sys-tematic approach based on Bayesrsquo theorem for system statesupdating and prediction This approach considers systemstates of interest as unknown stochastic variables to beestimatedTheprior distribution of system states is quantifiedbased on historical data By collecting new data over time the

HindawiJournal of Advanced TransportationVolume 2019 Article ID 5314520 10 pageshttpsdoiorg10115520195314520

2 Journal of Advanced Transportation

posterior distribution of system states can be estimated basedon the Bayesrsquo theorem This sequential learning frameworkprovides an adaptive learning process for handling time seriesdata prediction It has been shown that model parametersneed to be adaptive with system behavior [23] Fei et al [19]proposed a DLM for real-time short-term freeway travel timeprediction The model adjusts the variances of disturbanceunder a user-defined threshold based on the adaptive controltheory However such an adjustment mechanism is notoptimized and relies on the intervention of expert knowledgeraising issues in its generalization in different areas For thisissue Fei et al [18] incorporated a Markov switching processin the DLMbased on the three-phase traffic flow theoryTheyshowed the Markov switching DLM approach outperformsthe ARIMA method

Another adaptive modeling approach consists in devel-oping methodology to detect change points of system statesand update system parameters to catch system behaviorchanges [24ndash29] The change-point detection methods canbe designed to monitor prediction errors and detect acci-dents providing feedback to adjust model prediction andreduce prediction errors Comert and Bezuglov [25] appliedthe hidden Markov model (HMM) and the expectation-maximization (EM) algorithm as a change-point detectionmethod to update the estimation of parameters (ie processmean) used in the autoregressive integrated moving average(ARIMA)modelMoreira-Matias and Alesiani [28] proposeda change detection method based on Page-Hinkley change-point detection method [30] for triggering an accidentalarm The threshold for alarm triggering is a user-defineddeterministic parameter corresponding to a tolerable falsealarm rate

In this study a new online adaptive parameter estimationapproach is proposed under the DLM framework to achievebetter accuracy of prediction when some external eventsor system regime changes occur This method is based oncontinuously monitoring prediction errors for adaptivelyadjusting model parameters The main contribution resideson the adaptive model parameter adjustment design toimprove classical DLM approach when unpredicted systembehavior changes are detected The performance of theproposed approach is tested on a simulated road networkunder accidental scenariosThe performance of the proposedmethod is compared with classical DLM methods and thebenchmark methods ie ARIMA method [31] and Holt-Winters Exponential Smoothing method [32] Note that wedo not intend to extensively compare the proposed approachwith other methods but instead to demonstrate the effec-tiveness of the proposed method in improving the parametersetting issues of classical DLM approaches

The rest of the paper is organized as follows In Section 2we present the general DLM forecasting framework anddifferent DLM specifications for time series data analysis InSection 3 a new adaptive parameter estimation method isproposed for online parameter learning to reduce predictionerror Section 4 reports the numerical study on real-timeshort-term road travel speed prediction under accidental

scenarios A comparative study with two other state-of-the-art methods is provided Finally conclusions are drawn andfuture extensions are discussed

2 Bayesian Dynamic Linear Modelfor Traffic State Prediction

A general DLM can be described by an observation equationand a system state equation to model the process of a system[23] The state equation describes system state evolutionmapping from a priori distribution at t-1 to posterior distri-bution at time tThe observation equation describes observedmeasurements at time t in relation to system states Theevolution of system states over time is assumed to followa stochastic process with Gaussian errors A DLM can bewritten as [23]

119909119905 = 119866119905119909119905minus1 +119908119905 119908119905 sim 119873 (0119882t) (state equation) (1)

119910119905 = 119865119905119909119905 + 119907119905119907119905 sim 119873 (0 119881t) (observation equation) (2)

In (1) 119909119905 is the system state at time t 119866119905 is the evolutionmatrix of 119909119905 In (2) 119910119905 is observation at time 119905 119865119905 is thedesign matrix of 119909119905 119908119905 and 119907119905 are white noise error termsfollowing normal distribution with 0 mean and variance119882119905 and 119881119905 respectively It is assumed that 119907119905 and 119908119905 aremutually independent ie cov(119907119905119908119905) = 0 for 119905 = 1 119879The ratio 119908119905119907119905 is called the signal-to-noise ratio at time 119905It represents the ratio of system prediction errors 119908119905 andobservation errors 119907119905 In general 119882119905 and 119881119905 are unknownand need to be estimated from data The general DLM canbe represented by a quadruple 119865119905119866119905119881119905119882119905 over time119905 = 1 2 119879 The DLM provides a probabilistic linkage toupdate the posterior distribution of system states based on apriori distribution andnewly available observations over timebased on the Bayesian forecasting framework [19 23 33]TheBayesian forecasting framework in the context of traffic speedprediction on a road network is described as follows

21 Bayesian Forecasting Framework

Step 1 (initialization) Initialize system state variables 1199090 (ietravel speed on a road sectionlink) at 119905 = 0

(1199090 | 1198630) sim N (11989801198620) (3)

where 1198980 denotes the estimated means of link travel speedat 119905 = 0 1198620 is the estimated variance based on the initialinformation set 1198630 (ie historical travel speed data onnetwork) Set 119905 = 1199050Step 2 (prior distribution estimation) Estimate the priordistribution of 119909119905 as

(119909119905 | 119863119905minus1) sim N (119886119905 119877t)with 119886119905 = 119866119905119898119905minus1 and 119877119905 = 119866119905119862119905minus11198661015840119905 +119882119905 (4)

whereN(119886119905 119877t) is the normal distribution 119886119905 is the estimatedmean of system states and 119877119905 is the estimated variance of

Journal of Advanced Transportation 3

system states We can observe that increasing 119866119905 or119882119905 willamplify the variance of 119909119905

Step 3 (one-step forecast) Estimate one-step forecast for 119910119905as(119910119905 | 119863119905minus1) sim N (119891119905119876t)

with 119891119905 = 1198651015840119905119886119905 and 119876119905 = 1198651015840119905119877119905119865119905 +119881119905 (5)

where 119891119905 is the mean of prediction at 119905 and119876119905 is the varianceof prediction at 119905 We can observed that if 119865119905 (ie designmatrix) is constant 119891119905 prop 119886119905 = 119866119905119898119905minus1 and119876119905 prop 119877119905 +119881119905Step 4 (posterior distribution at 119905) Calculate the posteriordistribution (119909119905 | 119863119905) as(119909119905 | 119863119905) sim N (119898119905119862t)

with 119898119905 = 119886119905 +119860119905119890119905 and 119862119905 = 119877119905 minus1198601199051198761199051198601015840119905 (6)

where 119860119905 = 119877119905119865119905119876minus1119905 and 119890119905 = 119910119905 minus 119891119905 If 119865119905 = 1 then 119860119905 =119877119905119876119905Step 5 (iterate) Set 119905 = 119905 + 1 If 119905 = 119879 then stop otherwise goto Step 1

The proof of the one-step forecast and the posteriordistribution can be found in [23] Note that for the univariateDLM it has been shown that the covariance of systemevolution 119882119905 needs to adapt to drastic system behaviorchanges or regime shift [23] Regardless of this issuewillmakea serious prediction bias [12] However none of the existingstudies propose any adaptive parameter estimation for119882119905 toaddress this issue

We specify three DLMs ie first-orderDLM cubic splinesmoothing DLM and second-order DLM with increasingcomplexity based on the above DLM forecasting frameworkfor travel speed prediction on a road network The aim is toprovide the benchmarks to compare with the performance ofthe proposed adaptive parameter updating DLM The threeDLMs are described as follows

(a) First-Order DLM This is the basic DLM which incorpo-rates a mean level term and a Gaussian noisy term to describesystem state evolution

119910119905 = 120583119905 + 120576obs 120576119905 sim N (0 1205902obs) (7)

120583119905 = 120583119905minus1 + 120576level 120576level sim N (0 1205902level) (8)

119909119905 = 120583119905119865 = 119866 = 1119882 = 1205902level 119881119905 = 1205902obs(t)

(9)

(b) Cubic Spline Smoothing DLM This model extends thefirst-order DLM by incorporating a local linear trend Theresulting system of equations is written as follows

Equation (4) and

120583119905 = 120583119905minus1 + 120572119905minus1 + 120576level 120576level sim N (0 1205902level) (10)

120572119905 = 120572119905minus1 + 120576trend 120576trend sim N (0 1205902trend) (11)

In terms of the quadruples of DLM it is equivalent to

119909119905 = [120583119905120572119905] 119865 = [1 0] 119866 = [1 1

0 1]

119882 = [1205902level 00 1205902trend] 119881119905 = 1205902obs(t)

(12)

(c) Second-Order DLM This model extends the cubic splinesmoothing DLM by introducing a second linear trend tomodel changes of the trend level The second-order DLM isdescribed as follows

Equations (4) and (7) and

120572119905 = 120572119905minus1 + 120573119905minus1 + 120576trend1 120576trend1 sim N (0 1205902trend1) (13)

120573119905 = 120573119905minus1 + 120576trend2 120576trend2 sim N (0 1205902trend2) (14)

119909119905 = [[[120583119905120572119905120573119905]]]

119865 = [1 0 0]

119866 = [[[1 1 00 1 10 0 1

]]]

119882 = [[[[

1205902level 0 00 1205902trend1 00 0 1205902trend2

]]]]

119881119905 = 1205902obs(t)

(15)

In our travel speed prediction context 119910119905 is observed dataof average link travel speed at time t 120583119905 is the unknownaverage speed at time 119905120572119905 is the trend of variation of averages120576level and 120576trend are the corresponding error terms respectivelyNote thatmore complicatedDLMsusing a higher-order trendcomponent or combining a systematic seasonal variationcomponent and a regression component can also be specified

4 Journal of Advanced Transportation

3 Adaptive Parameter Updating for DLM

31 Adaptive DLM We propose an adaptive parameterupdating approach based on the first-order DLM We havetwo unknown parameters ie 119881119905 and 119882119905 to be estimatedThe two parameters determine the predicted system statesand influence the accuracy of prediction To estimate theunknown model parameters we can construct the likelihoodfunction based on observed data as a function of unknownparameters The maximum likelihood estimation approachis used to estimate the parameters [34] The log-likelihoodfunction is written as follows [33 35]

119871119871 (120579) = minus12119879sum119905=1

log 10038161003816100381610038161198761199051003816100381610038161003816minus 12119879sum119905=1

(119910119905 minus 119891119905)1015840119876minus1119905 (119910119905 minus 119891119905)(16)

where 120579 denotes the unknown parameters ie 120579 = (119881119905119882119905)119876119905 and 119891119905 are the variances and means of prediction at time119905 (see (5)) respectively The maximum likelihood estimates(MLE) of parameters can then be obtained by solving thefollowing optimization problem

= argmax120579

119871119871 (120579) (17)

In classical DLMs the system parameters 120579 are constantregardless system regime changes Fei et al [19] proposed anintervention approach by adjusting the model error covari-ance based on anticipated changes from additional exteriorinformation andor expertrsquos knowledge The drawback isexpertrsquos adjustment might be trivial and lack a system-widecontrol based on the feedback of prediction errors

Different with existing approach we propose a two-stagealgorithm by first estimating initial parameters 1205790 based on atraining data set and then using an online adaptive parameterupdating based on the feedback of one-step prediction errorsIt is similar to feedback control to optimize the model param-eters The proposed two-stage adaptive parameter updatingapproach is described as follows

32 Online Adaptive Parameter Updating Approach Theproposed approach estimates the model parameters (11988101198820)based on historical data and adaptively optimizes its modelparameters over time based on one-step model predictionerrors The approach is described as follows

Step 1 (initial parameter estimation)

(i) Estimate 0 and 0 given input training data setDcomputeMLE estimates of 0 and 0 Get 0 = radic0

(ii) Optimize 1198820 given 1199070 find the optimal signal-to-noise ratio (ie1199081199070) as

s = argmin119904119866(119910 | 120579) (18)

where 119866(119910 | 120579) is a loss function defined by the rootmean square error The optimal estimates of model

error covariance for the training data set can then beobtained as lowast0 = 20

Step 2 (online adaptive parameter updating) Set119881119905 = V0 and119882119905 = lowast0 and compute one-step forecast 119910119905 and predictionerror 120576119905 = |119910119905 minus 119910119905| based on (4)-(6) Given a predefinedtolerable threshold 120591 update119882119905+1 as

119882119905+1 = (s119905+1)2 119881119905 if 120576119905 ge 120591119882119905 otherwise

(19)

where s119905+1 = argmin119904119866(119910 | 119863119905 120579) 119881119905 is kept constantNote that one can obtain s119905+1 without difficulty by the goldensection search or the line search approach [36]

The online adaptive parameter updating approach isshown in Figure 1

33 Measure Metrics for Assessing Prediction Accuracy Tomeasure the accuracy of prediction two metrics are applied(1) Root Mean Square Error (RMSE) (2) Mean AbsoluteError (MAE) The first one computes the mean of squarederror terms The second one reports the mean of absoluteerrors The definitions are as follows

(a) Mean absolute error (MAE) measures the averagemagnitude of prediction errors by taking into account allobservation equally

MAE = 1119873119873sum119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816 (20)

where119873 is the total number of observation 119910119894 and 119910119894 are theobservation and prediction values of sample i respectively

(b) Root Mean Square Error (RMSE) is a second-ordermeasure for prediction errors

RMSE = [ 1119873119873sum119894=1

(119910119894 minus 119910119894)2]12

(21)

The RMSE is a kind of second-order measure of predictionerrors Note MAE and RMSE provide similar measures forquantifying the model prediction errors However RMSEweights more to large errors providing more desired propertywhen large errors are undesirable

4 Experimental Study

41 Experimental Settings and Link Speed Data We generaterealistic travel speed data by microscopic traffic simula-tions implemented by SUMO [37] a widely used micro-scopic traffic simulation The test area is selected fromLuxembourg City and its surrounding areas [38 39] Thetravel speed data is collected from the simulator on 13main road sections in Luxembourg City (Figure 2) in themorning peak-hour period from 700-900 Link averagetravel speed is aggregated in a 5-minute aggregation inter-val (ie 700 705 710 900) as is the case for most

Journal of Advanced Transportation 5

Observed historicaltime series data

y1y2 yk

Initial DLMparameter estimation

MLE estimate of observationerror variance (V0 ) and

model error covariance (W0)

Optimize model parameters W0based on the root mean square

prediction errors

Online adaptive DLMparameters estimation

Set Vt0=V0 and Wt0=Wlowast0

t=t0

One-step prediction yt+1

New observation yt+1at time t+1

If yt+1 minus yt+1 gt optimize Wt

t=t+1

DLM model specification

Figure 1 Online adaptive parameter updating approach

Figure 2 Luxembourg road network and traffic count positionsThere are totally 13 traffic counts located over main road sectionsaround Luxembourg City and the quarter of European UnionInstitutions (Kirchberg) The red car is the accident scenariooccurring on Grand Duchess Charlotte Bridge standing in way forthe direction to east during am 730 to am 800

realistic applications [40] The data is freely available athttpsgithubcompigne2019-simulations-DLM

We consider two scenarios normal traffic without acci-dents and traffic with an accident occurred during 730-800 on the Grand Duchess Charlotte Bridge (see Figure 2)

connecting European Institution quarter and LuxembourgCity center Travel demand is generated based on the realisticLUST traffic demand scenarios for Luxembourg [41] whichrepresents the daily mobility patterns of peoples work-ingliving in the study area We generate one training dataset for initial DLMparameter estimation under normal trafficsituation and one test data set under accident situationsThe aim is to test the performance of the proposed adaptiveparameter updating approach under unforeseen event Thegenerated traffic patterns are different from one day toanother due to stochastic behavior of traffic

The travel speed profiles on three road sections aroundthe accidental site (road sections 3 4 and 5) for normal andaccidental scenarios are shown in Figure 3 In the normaltraffic scenario there are some frustrations on road section3W and 4W (direction for Luxembourg City center) Whenan accident occurs (see the right part in Figure 3) traffic isheavily impacted on the road section 5 for both directionsand on the road sections 3 and 4 to its east direction Wecan find there is significant travel speed reduction on nearbyroads due to the accident event The numerical is executed byDLM Matlab Toolbox (httpsmjlainegithubiodlm) using

Journal of Advanced Transportation 3

system states We can observe that increasing 119866119905 or119882119905 willamplify the variance of 119909119905

Step 3 (one-step forecast) Estimate one-step forecast for 119910119905as(119910119905 | 119863119905minus1) sim N (119891119905119876t)

with 119891119905 = 1198651015840119905119886119905 and 119876119905 = 1198651015840119905119877119905119865119905 +119881119905 (5)

where 119891119905 is the mean of prediction at 119905 and119876119905 is the varianceof prediction at 119905 We can observed that if 119865119905 (ie designmatrix) is constant 119891119905 prop 119886119905 = 119866119905119898119905minus1 and119876119905 prop 119877119905 +119881119905Step 4 (posterior distribution at 119905) Calculate the posteriordistribution (119909119905 | 119863119905) as(119909119905 | 119863119905) sim N (119898119905119862t)

with 119898119905 = 119886119905 +119860119905119890119905 and 119862119905 = 119877119905 minus1198601199051198761199051198601015840119905 (6)

where 119860119905 = 119877119905119865119905119876minus1119905 and 119890119905 = 119910119905 minus 119891119905 If 119865119905 = 1 then 119860119905 =119877119905119876119905Step 5 (iterate) Set 119905 = 119905 + 1 If 119905 = 119879 then stop otherwise goto Step 1

The proof of the one-step forecast and the posteriordistribution can be found in [23] Note that for the univariateDLM it has been shown that the covariance of systemevolution 119882119905 needs to adapt to drastic system behaviorchanges or regime shift [23] Regardless of this issuewillmakea serious prediction bias [12] However none of the existingstudies propose any adaptive parameter estimation for119882119905 toaddress this issue

We specify three DLMs ie first-orderDLM cubic splinesmoothing DLM and second-order DLM with increasingcomplexity based on the above DLM forecasting frameworkfor travel speed prediction on a road network The aim is toprovide the benchmarks to compare with the performance ofthe proposed adaptive parameter updating DLM The threeDLMs are described as follows

(a) First-Order DLM This is the basic DLM which incorpo-rates a mean level term and a Gaussian noisy term to describesystem state evolution

119910119905 = 120583119905 + 120576obs 120576119905 sim N (0 1205902obs) (7)

120583119905 = 120583119905minus1 + 120576level 120576level sim N (0 1205902level) (8)

119909119905 = 120583119905119865 = 119866 = 1119882 = 1205902level 119881119905 = 1205902obs(t)

(9)

(b) Cubic Spline Smoothing DLM This model extends thefirst-order DLM by incorporating a local linear trend Theresulting system of equations is written as follows

Equation (4) and

120583119905 = 120583119905minus1 + 120572119905minus1 + 120576level 120576level sim N (0 1205902level) (10)

120572119905 = 120572119905minus1 + 120576trend 120576trend sim N (0 1205902trend) (11)

In terms of the quadruples of DLM it is equivalent to

119909119905 = [120583119905120572119905] 119865 = [1 0] 119866 = [1 1

0 1]

119882 = [1205902level 00 1205902trend] 119881119905 = 1205902obs(t)

(12)

(c) Second-Order DLM This model extends the cubic splinesmoothing DLM by introducing a second linear trend tomodel changes of the trend level The second-order DLM isdescribed as follows

Equations (4) and (7) and

120572119905 = 120572119905minus1 + 120573119905minus1 + 120576trend1 120576trend1 sim N (0 1205902trend1) (13)

120573119905 = 120573119905minus1 + 120576trend2 120576trend2 sim N (0 1205902trend2) (14)

119909119905 = [[[120583119905120572119905120573119905]]]

119865 = [1 0 0]

119866 = [[[1 1 00 1 10 0 1

]]]

119882 = [[[[

1205902level 0 00 1205902trend1 00 0 1205902trend2

]]]]

119881119905 = 1205902obs(t)

(15)

In our travel speed prediction context 119910119905 is observed dataof average link travel speed at time t 120583119905 is the unknownaverage speed at time 119905120572119905 is the trend of variation of averages120576level and 120576trend are the corresponding error terms respectivelyNote thatmore complicatedDLMsusing a higher-order trendcomponent or combining a systematic seasonal variationcomponent and a regression component can also be specified

4 Journal of Advanced Transportation

3 Adaptive Parameter Updating for DLM

31 Adaptive DLM We propose an adaptive parameterupdating approach based on the first-order DLM We havetwo unknown parameters ie 119881119905 and 119882119905 to be estimatedThe two parameters determine the predicted system statesand influence the accuracy of prediction To estimate theunknown model parameters we can construct the likelihoodfunction based on observed data as a function of unknownparameters The maximum likelihood estimation approachis used to estimate the parameters [34] The log-likelihoodfunction is written as follows [33 35]

119871119871 (120579) = minus12119879sum119905=1

log 10038161003816100381610038161198761199051003816100381610038161003816minus 12119879sum119905=1

(119910119905 minus 119891119905)1015840119876minus1119905 (119910119905 minus 119891119905)(16)

where 120579 denotes the unknown parameters ie 120579 = (119881119905119882119905)119876119905 and 119891119905 are the variances and means of prediction at time119905 (see (5)) respectively The maximum likelihood estimates(MLE) of parameters can then be obtained by solving thefollowing optimization problem

= argmax120579

119871119871 (120579) (17)

In classical DLMs the system parameters 120579 are constantregardless system regime changes Fei et al [19] proposed anintervention approach by adjusting the model error covari-ance based on anticipated changes from additional exteriorinformation andor expertrsquos knowledge The drawback isexpertrsquos adjustment might be trivial and lack a system-widecontrol based on the feedback of prediction errors

Different with existing approach we propose a two-stagealgorithm by first estimating initial parameters 1205790 based on atraining data set and then using an online adaptive parameterupdating based on the feedback of one-step prediction errorsIt is similar to feedback control to optimize the model param-eters The proposed two-stage adaptive parameter updatingapproach is described as follows

32 Online Adaptive Parameter Updating Approach Theproposed approach estimates the model parameters (11988101198820)based on historical data and adaptively optimizes its modelparameters over time based on one-step model predictionerrors The approach is described as follows

Step 1 (initial parameter estimation)

(i) Estimate 0 and 0 given input training data setDcomputeMLE estimates of 0 and 0 Get 0 = radic0

(ii) Optimize 1198820 given 1199070 find the optimal signal-to-noise ratio (ie1199081199070) as

s = argmin119904119866(119910 | 120579) (18)

where 119866(119910 | 120579) is a loss function defined by the rootmean square error The optimal estimates of model

error covariance for the training data set can then beobtained as lowast0 = 20

Step 2 (online adaptive parameter updating) Set119881119905 = V0 and119882119905 = lowast0 and compute one-step forecast 119910119905 and predictionerror 120576119905 = |119910119905 minus 119910119905| based on (4)-(6) Given a predefinedtolerable threshold 120591 update119882119905+1 as

119882119905+1 = (s119905+1)2 119881119905 if 120576119905 ge 120591119882119905 otherwise

(19)

where s119905+1 = argmin119904119866(119910 | 119863119905 120579) 119881119905 is kept constantNote that one can obtain s119905+1 without difficulty by the goldensection search or the line search approach [36]

The online adaptive parameter updating approach isshown in Figure 1

33 Measure Metrics for Assessing Prediction Accuracy Tomeasure the accuracy of prediction two metrics are applied(1) Root Mean Square Error (RMSE) (2) Mean AbsoluteError (MAE) The first one computes the mean of squarederror terms The second one reports the mean of absoluteerrors The definitions are as follows

(a) Mean absolute error (MAE) measures the averagemagnitude of prediction errors by taking into account allobservation equally

MAE = 1119873119873sum119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816 (20)

where119873 is the total number of observation 119910119894 and 119910119894 are theobservation and prediction values of sample i respectively

(b) Root Mean Square Error (RMSE) is a second-ordermeasure for prediction errors

RMSE = [ 1119873119873sum119894=1

(119910119894 minus 119910119894)2]12

(21)

The RMSE is a kind of second-order measure of predictionerrors Note MAE and RMSE provide similar measures forquantifying the model prediction errors However RMSEweights more to large errors providing more desired propertywhen large errors are undesirable

4 Experimental Study

41 Experimental Settings and Link Speed Data We generaterealistic travel speed data by microscopic traffic simula-tions implemented by SUMO [37] a widely used micro-scopic traffic simulation The test area is selected fromLuxembourg City and its surrounding areas [38 39] Thetravel speed data is collected from the simulator on 13main road sections in Luxembourg City (Figure 2) in themorning peak-hour period from 700-900 Link averagetravel speed is aggregated in a 5-minute aggregation inter-val (ie 700 705 710 900) as is the case for most

Journal of Advanced Transportation 5

Observed historicaltime series data

y1y2 yk

Initial DLMparameter estimation

MLE estimate of observationerror variance (V0 ) and

model error covariance (W0)

Optimize model parameters W0based on the root mean square

prediction errors

Online adaptive DLMparameters estimation

Set Vt0=V0 and Wt0=Wlowast0

t=t0

One-step prediction yt+1

New observation yt+1at time t+1

If yt+1 minus yt+1 gt optimize Wt

t=t+1

DLM model specification

Figure 1 Online adaptive parameter updating approach

Figure 2 Luxembourg road network and traffic count positionsThere are totally 13 traffic counts located over main road sectionsaround Luxembourg City and the quarter of European UnionInstitutions (Kirchberg) The red car is the accident scenariooccurring on Grand Duchess Charlotte Bridge standing in way forthe direction to east during am 730 to am 800

realistic applications [40] The data is freely available athttpsgithubcompigne2019-simulations-DLM

We consider two scenarios normal traffic without acci-dents and traffic with an accident occurred during 730-800 on the Grand Duchess Charlotte Bridge (see Figure 2)

connecting European Institution quarter and LuxembourgCity center Travel demand is generated based on the realisticLUST traffic demand scenarios for Luxembourg [41] whichrepresents the daily mobility patterns of peoples work-ingliving in the study area We generate one training dataset for initial DLMparameter estimation under normal trafficsituation and one test data set under accident situationsThe aim is to test the performance of the proposed adaptiveparameter updating approach under unforeseen event Thegenerated traffic patterns are different from one day toanother due to stochastic behavior of traffic

The travel speed profiles on three road sections aroundthe accidental site (road sections 3 4 and 5) for normal andaccidental scenarios are shown in Figure 3 In the normaltraffic scenario there are some frustrations on road section3W and 4W (direction for Luxembourg City center) Whenan accident occurs (see the right part in Figure 3) traffic isheavily impacted on the road section 5 for both directionsand on the road sections 3 and 4 to its east direction Wecan find there is significant travel speed reduction on nearbyroads due to the accident event The numerical is executed byDLM Matlab Toolbox (httpsmjlainegithubiodlm) using

4 Journal of Advanced Transportation

3 Adaptive Parameter Updating for DLM

31 Adaptive DLM We propose an adaptive parameterupdating approach based on the first-order DLM We havetwo unknown parameters ie 119881119905 and 119882119905 to be estimatedThe two parameters determine the predicted system statesand influence the accuracy of prediction To estimate theunknown model parameters we can construct the likelihoodfunction based on observed data as a function of unknownparameters The maximum likelihood estimation approachis used to estimate the parameters [34] The log-likelihoodfunction is written as follows [33 35]

119871119871 (120579) = minus12119879sum119905=1

log 10038161003816100381610038161198761199051003816100381610038161003816minus 12119879sum119905=1

(119910119905 minus 119891119905)1015840119876minus1119905 (119910119905 minus 119891119905)(16)

where 120579 denotes the unknown parameters ie 120579 = (119881119905119882119905)119876119905 and 119891119905 are the variances and means of prediction at time119905 (see (5)) respectively The maximum likelihood estimates(MLE) of parameters can then be obtained by solving thefollowing optimization problem

= argmax120579

119871119871 (120579) (17)

In classical DLMs the system parameters 120579 are constantregardless system regime changes Fei et al [19] proposed anintervention approach by adjusting the model error covari-ance based on anticipated changes from additional exteriorinformation andor expertrsquos knowledge The drawback isexpertrsquos adjustment might be trivial and lack a system-widecontrol based on the feedback of prediction errors

Different with existing approach we propose a two-stagealgorithm by first estimating initial parameters 1205790 based on atraining data set and then using an online adaptive parameterupdating based on the feedback of one-step prediction errorsIt is similar to feedback control to optimize the model param-eters The proposed two-stage adaptive parameter updatingapproach is described as follows

32 Online Adaptive Parameter Updating Approach Theproposed approach estimates the model parameters (11988101198820)based on historical data and adaptively optimizes its modelparameters over time based on one-step model predictionerrors The approach is described as follows

Step 1 (initial parameter estimation)

(i) Estimate 0 and 0 given input training data setDcomputeMLE estimates of 0 and 0 Get 0 = radic0

(ii) Optimize 1198820 given 1199070 find the optimal signal-to-noise ratio (ie1199081199070) as

s = argmin119904119866(119910 | 120579) (18)

where 119866(119910 | 120579) is a loss function defined by the rootmean square error The optimal estimates of model

error covariance for the training data set can then beobtained as lowast0 = 20

Step 2 (online adaptive parameter updating) Set119881119905 = V0 and119882119905 = lowast0 and compute one-step forecast 119910119905 and predictionerror 120576119905 = |119910119905 minus 119910119905| based on (4)-(6) Given a predefinedtolerable threshold 120591 update119882119905+1 as

119882119905+1 = (s119905+1)2 119881119905 if 120576119905 ge 120591119882119905 otherwise

(19)

where s119905+1 = argmin119904119866(119910 | 119863119905 120579) 119881119905 is kept constantNote that one can obtain s119905+1 without difficulty by the goldensection search or the line search approach [36]

The online adaptive parameter updating approach isshown in Figure 1

33 Measure Metrics for Assessing Prediction Accuracy Tomeasure the accuracy of prediction two metrics are applied(1) Root Mean Square Error (RMSE) (2) Mean AbsoluteError (MAE) The first one computes the mean of squarederror terms The second one reports the mean of absoluteerrors The definitions are as follows

(a) Mean absolute error (MAE) measures the averagemagnitude of prediction errors by taking into account allobservation equally

MAE = 1119873119873sum119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816 (20)

where119873 is the total number of observation 119910119894 and 119910119894 are theobservation and prediction values of sample i respectively

(b) Root Mean Square Error (RMSE) is a second-ordermeasure for prediction errors

RMSE = [ 1119873119873sum119894=1

(119910119894 minus 119910119894)2]12

(21)

The RMSE is a kind of second-order measure of predictionerrors Note MAE and RMSE provide similar measures forquantifying the model prediction errors However RMSEweights more to large errors providing more desired propertywhen large errors are undesirable

4 Experimental Study

41 Experimental Settings and Link Speed Data We generaterealistic travel speed data by microscopic traffic simula-tions implemented by SUMO [37] a widely used micro-scopic traffic simulation The test area is selected fromLuxembourg City and its surrounding areas [38 39] Thetravel speed data is collected from the simulator on 13main road sections in Luxembourg City (Figure 2) in themorning peak-hour period from 700-900 Link averagetravel speed is aggregated in a 5-minute aggregation inter-val (ie 700 705 710 900) as is the case for most

Journal of Advanced Transportation 5

Observed historicaltime series data

y1y2 yk

Initial DLMparameter estimation

MLE estimate of observationerror variance (V0 ) and

model error covariance (W0)

Optimize model parameters W0based on the root mean square

prediction errors

Online adaptive DLMparameters estimation

Set Vt0=V0 and Wt0=Wlowast0

t=t0

One-step prediction yt+1

New observation yt+1at time t+1

If yt+1 minus yt+1 gt optimize Wt

t=t+1

DLM model specification

Figure 1 Online adaptive parameter updating approach

Figure 2 Luxembourg road network and traffic count positionsThere are totally 13 traffic counts located over main road sectionsaround Luxembourg City and the quarter of European UnionInstitutions (Kirchberg) The red car is the accident scenariooccurring on Grand Duchess Charlotte Bridge standing in way forthe direction to east during am 730 to am 800

realistic applications [40] The data is freely available athttpsgithubcompigne2019-simulations-DLM

We consider two scenarios normal traffic without acci-dents and traffic with an accident occurred during 730-800 on the Grand Duchess Charlotte Bridge (see Figure 2)

connecting European Institution quarter and LuxembourgCity center Travel demand is generated based on the realisticLUST traffic demand scenarios for Luxembourg [41] whichrepresents the daily mobility patterns of peoples work-ingliving in the study area We generate one training dataset for initial DLMparameter estimation under normal trafficsituation and one test data set under accident situationsThe aim is to test the performance of the proposed adaptiveparameter updating approach under unforeseen event Thegenerated traffic patterns are different from one day toanother due to stochastic behavior of traffic

The travel speed profiles on three road sections aroundthe accidental site (road sections 3 4 and 5) for normal andaccidental scenarios are shown in Figure 3 In the normaltraffic scenario there are some frustrations on road section3W and 4W (direction for Luxembourg City center) Whenan accident occurs (see the right part in Figure 3) traffic isheavily impacted on the road section 5 for both directionsand on the road sections 3 and 4 to its east direction Wecan find there is significant travel speed reduction on nearbyroads due to the accident event The numerical is executed byDLM Matlab Toolbox (httpsmjlainegithubiodlm) using

Journal of Advanced Transportation 5

Observed historicaltime series data

y1y2 yk

Initial DLMparameter estimation

MLE estimate of observationerror variance (V0 ) and

model error covariance (W0)

Optimize model parameters W0based on the root mean square

prediction errors

Online adaptive DLMparameters estimation

Set Vt0=V0 and Wt0=Wlowast0

t=t0

One-step prediction yt+1

New observation yt+1at time t+1

If yt+1 minus yt+1 gt optimize Wt

t=t+1

DLM model specification

Figure 1 Online adaptive parameter updating approach

Figure 2 Luxembourg road network and traffic count positionsThere are totally 13 traffic counts located over main road sectionsaround Luxembourg City and the quarter of European UnionInstitutions (Kirchberg) The red car is the accident scenariooccurring on Grand Duchess Charlotte Bridge standing in way forthe direction to east during am 730 to am 800

realistic applications [40] The data is freely available athttpsgithubcompigne2019-simulations-DLM

We consider two scenarios normal traffic without acci-dents and traffic with an accident occurred during 730-800 on the Grand Duchess Charlotte Bridge (see Figure 2)

connecting European Institution quarter and LuxembourgCity center Travel demand is generated based on the realisticLUST traffic demand scenarios for Luxembourg [41] whichrepresents the daily mobility patterns of peoples work-ingliving in the study area We generate one training dataset for initial DLMparameter estimation under normal trafficsituation and one test data set under accident situationsThe aim is to test the performance of the proposed adaptiveparameter updating approach under unforeseen event Thegenerated traffic patterns are different from one day toanother due to stochastic behavior of traffic

The travel speed profiles on three road sections aroundthe accidental site (road sections 3 4 and 5) for normal andaccidental scenarios are shown in Figure 3 In the normaltraffic scenario there are some frustrations on road section3W and 4W (direction for Luxembourg City center) Whenan accident occurs (see the right part in Figure 3) traffic isheavily impacted on the road section 5 for both directionsand on the road sections 3 and 4 to its east direction Wecan find there is significant travel speed reduction on nearbyroads due to the accident event The numerical is executed byDLM Matlab Toolbox (httpsmjlainegithubiodlm) using

6 Journal of Advanced Transportation

700 725 750 815 840 905Time

0

5

10

15

20

25Av

erag

e spe

ed (m

sec

)

3E3W4E

4W5E5W

700 725 750 815 840 905Time

0

5

10

15

20

25

Aver

age s

peed

(ms

ec)

3E3W4E

4W5E5W

Figure 3 Examples of observed average speeds on road sections 3 4 and 5 for both directions (direction east (E) and direction west (W))Left normal traffic scenario Right accident scenario (accident on the Grand Duchess Charlotte Bridge)

1st order DLMCubic spline smoothing model2nd order DLM

1

2

3

4

5

6

7

8

9

Root

Mea

n Sq

uare

Err

or (R

MSE

)

1 2 3 4 5 6 7 8 9 100wv

Figure 4 Influence of the signal-to-noise ratio (wv) on one-step forecast accuracy of DLMs on traffic data on road section 4E

a Dell Latitude E5470 laptop with win64 OS Intel i5-6300UCPU 2 Cores and 8GB memory

42 Result

421 Initial Parameter Estimation of the Adaptive DLM Weuse the MLE method to obtain an initial estimate of 1198810and 1198820 based on the training data set ie observationsin the normal traffic scenario The optimal signal-to-noiseratio with a minimal RMSE value of the one-step forecastcan be obtained (Figure 4) The RMSE values is a functionof signal-to-noise ratio which decreases at the beginningand then increases until a stable value when increasingthe signal-to-noise ratio We estimate the optimal signal-to-noise ratios and optimal model error covariance for eachlink Figure 4 shows the first-order DLM obtains best fits

(ie lowest RMSE) compared to the second-order DLM andthe cubic spline smoothing DLM model In normal trafficscenario traffic speed presents small fluctuation for mostof the time The evolution function in the first-order DLMcaptures smooth changes of mean state traffic evolution withbest goodness-of-fit However higher-order DLMs mightoverfit local trend resulting in higher prediction error

Figure 5 reports the local trends of the DLMs withand without optimizing signal-to-noise ratio to minimizethe RMSE We found that after optimizing the signal-to-noise ratio the fitted Kalman filter smoother becomes moreadaptive to observations (on the right side of Figure 5)The DLM Kalman filter smoother has smaller variance withMLE parameters In terms of one-step forecast accuracythe prediction accuracy is improved when applying theoptimized signal-to-noise ratio in the MLE models

Journal of Advanced Transportation 7

95 confidence intervalBackground level(smoothed state mean)upperlower boundupperlower boundObservation

95 confidence intervalBackground level(smoothed state mean)upperlower boundupperlower boundObservation

0

5

10

15

20

25

30

Aver

age s

peed

(ms

ec)

730 800 830 900705Time

0

5

10

15

20

25

30

Aver

age s

peed

(ms

ec)

730 800 830 900705Time

Figure 5 First-order DLM Kalman filter smoother fitted on average travel speed on link 4E On the left parameters (11988101198820) estimated byMLE on the right after optimization

Table 1 Average prediction accuracy of the adaptive DLM in accident scenarios

Measuremetrics Road Section First-order

DLM

Cubic splinesmoothing

DLM

Second-orderDLM AR(2) HW Exp

Smoothing

One-stepshift

predictor

AdaptiveDLM

RMSE 3 4 and 5 4901 5450 5537 4653 4612 4830 4480Others 1948 2155 2530 2587 2596 2336 2020

MAE 3 4 and 5 3398 4029 4071 263 2666 2814 2595Others 1511 1655 1913 1446 1469 1741 1500

422 Online Adaptive Parameter Updating We test theperformance of the proposed approach to the traffic accidentscenario As we can see on Figure 2 when the traffic accidentoccurs its upstream and downstream road sections ie roadsections 3 4 and 5 would have significant impacts Henceit would be interesting to investigate the performance of theproposed method on these road sections

Table 1 shows the adaptiveDLMsignificantly outperformsthe other methods for the cases of major changes in traffic onroad sections 3 4 and 5 The average RMSE of the adaptiveDLM over the road sections 3 4 and 5 is 4480 comparedto the HW Exponential Smoothing method (4612) AR(2)(4653) and the three DLM approaches It outperforms thesimple one-step shift predictor (ie using observations attime t as predictors for t+1) in both accidental and normaltraffic road sections The values of the MAE measure claimthe same conclusion However on the other road sectionsthe adaptive DLM performs similar well compared with theother approachesThe average execution of the adaptiveDLMfor each road section is 01082 second

To illustrate the effectiveness of reducing predictionerrors of the proposed method in case of major changesin traffic we investigate two road sections which are sig-nificantly impacted by the accident ie 4E and 5W We

can find travel speed quickly drop at about 740 and thetraffic becomes fluid at about 805 on both road sections(see Figure 3 on the right) As shown in Figure 6 forroad section 5W the classical DLM with constant modelparameters generates a quite biased one-step forecast due tosuch a sudden change (black line) However the proposedmethod provides adaptive one-step forecasts during andafter accidents (red line) The comparison of absolute errorsobtained by the classical DLM and the adaptive DLM isshown on the right side of Figure 6 Figure 7 compares theperformance of different DLM models for the road section4E The result shows the adaptive DLM model obtains moreaccurate prediction compared to the other DLM modelsFigure 8 reports the profile of adaptive optimal signal-to-noise ratios at each time step We use the standard deviationof travel speed in the normal traffic scenario to estimate thetolerable threshold in (19)

5 Conclusions

In this study we propose an online adaptive DLM algorithmfor time series data analysis and forecasting The proposedmethod is applied for short-term travel speed forecasts inurban areas based on a microscopic traffic simulator The

8 Journal of Advanced Transportation

700 725 750 815 840 905 700 725 750 815 840 9050

5

10

15

20

25

0

5

10

15

20

25Av

erag

e spe

ed (m

sec

)

yyDLM

yadapDLM

||

DLM

adapDLM

Figure 6 Comparison of one-step forecasts and absolute errors for first-orderDLMand the adaptive DLM in accident scenarios (road section5W) Le the one-step forecasts of average speed Right absolute residuals

700 725 750 815 840 905Time

0

5

10

15

20

25

30

35

One

-ste

p av

erag

e spe

ed p

redi

ctio

n (m

sec

)

Obs1st orderCubic spline

2nd orderAdaptive

35

4

45

5

55

6

Aver

age a

bsol

ute r

esid

uals

Cubic spline smoothing 2nd order Adaptive1st orderDLM model

Figure 7 Comparison of the performance of different DLM models in unforeseen accident situation (road section 4E)

experiments show the proposed method allows adaptivelyoptimizing its model parameters to improve its predictionaccuracy in a continuous way under uncertainty The pro-posed method does not need the intervention of experts andcan adjust its model error covariance automatically based onfeedback information of its one-step prediction errors

Experimental studies show that our adaptive DLMapproach outperforms both autoregressive integratedmovingaverage (ARIMA) and Holt-Winters Exponential Smoothing(ETS) that are both considered to be the main time series

analysis methods employed on this type of problems [28]We thus consider that this comparison is a reasonable proxyto a comparison with other online models for travel speedprediction that use ARIMA or ETS

Future extensions concern an adaptive parameter updat-ing scheme design for the state space methods and formore complicated DLMs with seasonal and regression termsApplications of the proposed method on other time seriesdata would also be beneficial for assessing and improving itsperformance

Journal of Advanced Transportation 9

01234567

Sign

al-to

-noi

se ra

tio (w

v)

725 750 815 840 905700Time

Figure 8 Adaptive parameter updating of DLM model in unforeseen accident situation (road section 5W)

Notations

119905 Index of discretized time intervals 119905 = 1 2 3 119879119909119905 System state at time 119905119910119905 Observation at time 119905119907119905 Observation error at time 119905119881119905 Variance of 119907119905119865119905 Design matrix for observation equation at time 119905119866119905 Evolution matrix of system states at time 119905119908119905 Forecast error at time 119905119882119905 Variance of 119908119905

Data Availability

The data is freely available at httpsgithubcompigne2019-simulations-DLM

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors are grateful to the support of LuxembourgInstitute of Socio-Economic Research (LISER) under thevisiting scholar grant

References

[1] TMa ldquoSolving a dynamic user-optimal route guidance problembased on joint strategy fictitious playrdquo in Game 13eoreticAnalysis of Congestion Safety and Security pp 67ndash89 SpringerInternational Publishing 2015

[2] U Mori AMendiburuM Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for advanced travellerinformation systemsrdquo Transportmetrica A Transport Sciencevol 11 pp 119ndash157 2015

[3] S Oh Y-J Byon K Jang and H Yeo ldquoShort-term travel-timeprediction on highway a review of the data-driven approachrdquoTransport Reviews vol 35 pp 4ndash32 2015

[4] T Seo A M Bayen T Kusakabe and Y Asakura ldquoTrafficstate estimation on highway A comprehensive surveyrdquo AnnualReviews in Control vol 43 pp 128ndash151 2017

[5] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting Where we are and where we are goingrdquoTransportation Research Part C Emerging Technologies vol 43pp 3ndash19 2014

[6] CM J Tampere and L H Immers ldquoAn extended Kalman filterapplication for traffic state estimation using CTM with implicitmode switching and dynamic parametersrdquo in Proceedings of theIEEE Intelligent Transportation Systems Conference 2007

[7] YWangM Papageorgiou AMessmer P Coppola A Tzimitsiand A Nuzzolo ldquoAn adaptive freeway traffic state estimatorrdquoAutomatica vol 45 no 1 pp 10ndash24 2009

[8] Y Wang and M Papageorgiou ldquoReal-time freeway trafficstate estimation based on extended Kalman filter a generalapproachrdquo Transportation Research Part B Methodological vol39 no 2 pp 141ndash167 2005

[9] Y Yang Y Xu J Han E Wang W Chen and L Yue ldquoEfficienttraffic congestion estimation using multiple spatio-temporalpropertiesrdquo Neurocomputing vol 267 pp 344ndash353 2017

[10] S FanM Herty and B Seibold ldquoComparative model accuracyof a data-fitted generalized Aw-Rascle-ZhangmodelrdquoNetworksand Heterogeneous Media vol 9 no 2 pp 239ndash268 2014

[11] K Y Chan T S Dillon J Singh and E Chang ldquoNeural-network-based models for short-term traffic flow forecast-ing using a hybrid exponential smoothing and levenberg-marquardt algorithmrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 13 no 2 pp 644ndash654 2012

[12] L Chen and C L Chen ldquoEnsemble learning approach forfreeway short-term traffic flow predictionrdquo in Proceedings ofthe 2007 IEEE International Conference on System of SystemsEngineering pp 1ndash6 San Antonio Tex USA April 2007

[13] Y-S Jeong Y-J Byon M M Castro-Neto and S M EasaldquoSupervised weighting-online learning algorithm for short-term traffic flow predictionrdquo IEEE Transactions on IntelligentTransportation Systems vol 14 no 4 pp 1700ndash1707 2013

[14] B L Smith B M Williams and R K Oswald ldquoComparison ofparametric and nonparametric models for traffic flow forecast-ingrdquoTransportation Research Part C Emerging Technologies vol10 no 4 pp 303ndash321 2002

[15] M Castro-Neto Y-S Jeong M-K Jeong and L D HanldquoOnline-SVR for short-term traffic flow prediction undertypical and atypical traffic conditionsrdquo Expert Systems withApplications vol 36 no 3 pp 6164ndash6173 2009

[16] F G Habtemichael and M Cetin ldquoShort-term traffic flowrate forecasting based on identifying similar traffic patternsrdquoTransportation Research Part C Emerging Technologies vol 66pp 61ndash78 2016

[17] A Salamanis G Margaritis D D Kehagias G Matzoulasand D Tzovaras ldquoIdentifying patterns under both normal andabnormal traffic conditions for short-term traffic predictionrdquoTransportation Research Procedia vol 22 pp 665ndash674 2017

[18] X Fei Y Zhang K Liu and M Guo ldquoBayesian dynamic linearmodel with switching for real-time short-term freeway travel

10 Journal of Advanced Transportation

time prediction with license plate recognition datardquo Journal ofTransportation Engineering vol 139 no 11 pp 1058ndash1067 2013

[19] X Fei C C Lu and K Liu ldquoA bayesian dynamic linear modelapproach for real-time short-term freeway travel time predic-tionrdquo Transportation Research Part C Emerging Technologiesvol 19 no 6 pp 1306ndash1318 2011

[20] Y Kawasaki Y Hara andM Kuwahara ldquoReal-timemonitoringof dynamic traffic states by state-space modelrdquo TransportationResearch Procedia vol 21 pp 42ndash55 2017

[21] C Lu and X Zhou ldquoShort-term highway traffic state predictionusing structural state space modelsrdquo Journal of IntelligentTransportation Systems Technology Planning and Operationsvol 18 no 3 pp 309ndash322 2014

[22] A Stathopoulos and M G Karlaftis ldquoA multivariate statespace approach for urban traffic flowmodeling and predictionrdquoTransportation Research Part C Emerging Technologies vol 11no 2 pp 121ndash135 2003

[23] M West and J Harrison Bayesian Forecasting and DynamicModels Springer New York NY USA 1997

[24] L Auret and C Aldrich ldquoChange point detection in time seriesdata with random forestsrdquo Control Engineering Practice vol 18no 8 pp 990ndash1002 2010

[25] G Comert and A Bezuglov ldquoAn Online Change-Point-BasedModel for Traffic Parameter Predictionrdquo IEEE Transactions onIntelligent Transportation Systems vol 14 no 3 pp 1360ndash13692013

[26] M Daumer and M Falk ldquoOn-line change-point detection (forstate space models) using multi-process Kalman filtersrdquo LinearAlgebra and its Applications vol 284 no 1-3 pp 125ndash135 1998

[27] S Liu M Yamada N Collier and M Sugiyama ldquoChange-point detection in time-series data by relative density-ratioestimationrdquo Neural Networks vol 43 pp 72ndash83 2013

[28] L Moreira-Matias and F Alesiani ldquoDrift3Flow freeway-incident prediction using real-time learningrdquo in Proceedings ofthe IEEE 18th International Conference on Intelligent Transporta-tion Systems 571 566 pages October 2015

[29] E Ruggieri and M Antonellis ldquoAn exact approach to Bayesiansequential change point detectionrdquo Computational Statistics ampData Analysis vol 97 pp 71ndash86 2016

[30] E S Page ldquoContinuous inspection schemesrdquo Biometrika vol41 pp 100ndash114 1954

[31] G E Box G M Jenkins G C Reinsel and G M Ljung TimeSeries Analysis Forecasting and Control Wiley-Blackwell 2015

[32] C C Holt ldquoForecasting seasonals and trends by exponentiallyweightedmoving averagesrdquo International Journal of Forecastingvol 20 no 1 pp 5ndash10 2004

[33] J Durbin and S J Koopman Time Series Analysis by State SpaceMethods vol 38 Oxford University Press Oxford UK 2ndedition 2012

[34] ldquoMaximum-likelihood methodrdquo in Encyclopedia of Mathemat-ics 2001 httpswwwencyclopediaofmathorgindexphpMax-imum-likelihood method

[35] G Petris S Petrone and P Campagnoli Dynamic LinearModels with R Springer New York NY USA 2009

[36] J Kiefer ldquoSequentialMinimax Search for aMaximumrdquoProceed-ings of the American Mathematical Society vol 4 no 3 p 5021953

[37] SUMO Simulation of UrbanMobility 2018 httpswwwdlrdetsendesktopdefaultaspxtabid-988316931 read-41000

[38] Y Pigne G Danoy and P Bouvry ldquoA platform for realisticonline vehicular network management inrdquo in IEEE GlobecomWorkshops pp 595ndash599 IEEE 2010

[39] Y Pigne G Danoy and P Bouvry ldquoA vehicular mobilitymodel based on real traffic counting datardquo in CommunicationTechnologies for Vehicles vol 6596 pp 131ndash142 Springer BerlinHeidelberg Heidelberg Germany 2011

[40] Z Liang and YWakahara ldquoReal-time urban traffic amount pre-diction models for dynamic route guidance systemsrdquo EURASIPJournal on Wireless Communications and Networking vol 852014

[41] L Codeca R Frank and T Engel ldquoLuxembourg SUMO traffic(LuST) scenario 24 hours of mobility for vehicular networkingresearchrdquo in Proceedings of the IEEE Vehicular NetworkingConference VNC 2015 pp 1ndash8 Japan December 2015

Journal of Advanced Transportation 7

95 confidence intervalBackground level(smoothed state mean)upperlower boundupperlower boundObservation

95 confidence intervalBackground level(smoothed state mean)upperlower boundupperlower boundObservation

0

5

10

15

20

25

30

Aver

age s

peed

(ms

ec)

730 800 830 900705Time

0

5

10

15

20

25

30

Aver

age s

peed

(ms

ec)

730 800 830 900705Time

Figure 5 First-order DLM Kalman filter smoother fitted on average travel speed on link 4E On the left parameters (11988101198820) estimated byMLE on the right after optimization

Table 1 Average prediction accuracy of the adaptive DLM in accident scenarios

Measuremetrics Road Section First-order

DLM

Cubic splinesmoothing

DLM

Second-orderDLM AR(2) HW Exp

Smoothing

One-stepshift

predictor

AdaptiveDLM

RMSE 3 4 and 5 4901 5450 5537 4653 4612 4830 4480Others 1948 2155 2530 2587 2596 2336 2020

MAE 3 4 and 5 3398 4029 4071 263 2666 2814 2595Others 1511 1655 1913 1446 1469 1741 1500

422 Online Adaptive Parameter Updating We test theperformance of the proposed approach to the traffic accidentscenario As we can see on Figure 2 when the traffic accidentoccurs its upstream and downstream road sections ie roadsections 3 4 and 5 would have significant impacts Henceit would be interesting to investigate the performance of theproposed method on these road sections

Table 1 shows the adaptiveDLMsignificantly outperformsthe other methods for the cases of major changes in traffic onroad sections 3 4 and 5 The average RMSE of the adaptiveDLM over the road sections 3 4 and 5 is 4480 comparedto the HW Exponential Smoothing method (4612) AR(2)(4653) and the three DLM approaches It outperforms thesimple one-step shift predictor (ie using observations attime t as predictors for t+1) in both accidental and normaltraffic road sections The values of the MAE measure claimthe same conclusion However on the other road sectionsthe adaptive DLM performs similar well compared with theother approachesThe average execution of the adaptiveDLMfor each road section is 01082 second

To illustrate the effectiveness of reducing predictionerrors of the proposed method in case of major changesin traffic we investigate two road sections which are sig-nificantly impacted by the accident ie 4E and 5W We

can find travel speed quickly drop at about 740 and thetraffic becomes fluid at about 805 on both road sections(see Figure 3 on the right) As shown in Figure 6 forroad section 5W the classical DLM with constant modelparameters generates a quite biased one-step forecast due tosuch a sudden change (black line) However the proposedmethod provides adaptive one-step forecasts during andafter accidents (red line) The comparison of absolute errorsobtained by the classical DLM and the adaptive DLM isshown on the right side of Figure 6 Figure 7 compares theperformance of different DLM models for the road section4E The result shows the adaptive DLM model obtains moreaccurate prediction compared to the other DLM modelsFigure 8 reports the profile of adaptive optimal signal-to-noise ratios at each time step We use the standard deviationof travel speed in the normal traffic scenario to estimate thetolerable threshold in (19)

5 Conclusions

In this study we propose an online adaptive DLM algorithmfor time series data analysis and forecasting The proposedmethod is applied for short-term travel speed forecasts inurban areas based on a microscopic traffic simulator The

8 Journal of Advanced Transportation

700 725 750 815 840 905 700 725 750 815 840 9050

5

10

15

20

25

0

5

10

15

20

25Av

erag

e spe

ed (m

sec

)

yyDLM

yadapDLM

||

DLM

adapDLM

Figure 6 Comparison of one-step forecasts and absolute errors for first-orderDLMand the adaptive DLM in accident scenarios (road section5W) Le the one-step forecasts of average speed Right absolute residuals

700 725 750 815 840 905Time

0

5

10

15

20

25

30

35

One

-ste

p av

erag

e spe

ed p

redi

ctio

n (m

sec

)

Obs1st orderCubic spline

2nd orderAdaptive

35

4

45

5

55

6

Aver

age a

bsol

ute r

esid

uals

Cubic spline smoothing 2nd order Adaptive1st orderDLM model

Figure 7 Comparison of the performance of different DLM models in unforeseen accident situation (road section 4E)

experiments show the proposed method allows adaptivelyoptimizing its model parameters to improve its predictionaccuracy in a continuous way under uncertainty The pro-posed method does not need the intervention of experts andcan adjust its model error covariance automatically based onfeedback information of its one-step prediction errors

Experimental studies show that our adaptive DLMapproach outperforms both autoregressive integratedmovingaverage (ARIMA) and Holt-Winters Exponential Smoothing(ETS) that are both considered to be the main time series

analysis methods employed on this type of problems [28]We thus consider that this comparison is a reasonable proxyto a comparison with other online models for travel speedprediction that use ARIMA or ETS

Future extensions concern an adaptive parameter updat-ing scheme design for the state space methods and formore complicated DLMs with seasonal and regression termsApplications of the proposed method on other time seriesdata would also be beneficial for assessing and improving itsperformance

Journal of Advanced Transportation 9

01234567

Sign

al-to

-noi

se ra

tio (w

v)

725 750 815 840 905700Time

Figure 8 Adaptive parameter updating of DLM model in unforeseen accident situation (road section 5W)

Notations

119905 Index of discretized time intervals 119905 = 1 2 3 119879119909119905 System state at time 119905119910119905 Observation at time 119905119907119905 Observation error at time 119905119881119905 Variance of 119907119905119865119905 Design matrix for observation equation at time 119905119866119905 Evolution matrix of system states at time 119905119908119905 Forecast error at time 119905119882119905 Variance of 119908119905

Data Availability

The data is freely available at httpsgithubcompigne2019-simulations-DLM

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors are grateful to the support of LuxembourgInstitute of Socio-Economic Research (LISER) under thevisiting scholar grant

References

[1] TMa ldquoSolving a dynamic user-optimal route guidance problembased on joint strategy fictitious playrdquo in Game 13eoreticAnalysis of Congestion Safety and Security pp 67ndash89 SpringerInternational Publishing 2015

[2] U Mori AMendiburuM Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for advanced travellerinformation systemsrdquo Transportmetrica A Transport Sciencevol 11 pp 119ndash157 2015

[3] S Oh Y-J Byon K Jang and H Yeo ldquoShort-term travel-timeprediction on highway a review of the data-driven approachrdquoTransport Reviews vol 35 pp 4ndash32 2015

[4] T Seo A M Bayen T Kusakabe and Y Asakura ldquoTrafficstate estimation on highway A comprehensive surveyrdquo AnnualReviews in Control vol 43 pp 128ndash151 2017

[5] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting Where we are and where we are goingrdquoTransportation Research Part C Emerging Technologies vol 43pp 3ndash19 2014

[6] CM J Tampere and L H Immers ldquoAn extended Kalman filterapplication for traffic state estimation using CTM with implicitmode switching and dynamic parametersrdquo in Proceedings of theIEEE Intelligent Transportation Systems Conference 2007

[7] YWangM Papageorgiou AMessmer P Coppola A Tzimitsiand A Nuzzolo ldquoAn adaptive freeway traffic state estimatorrdquoAutomatica vol 45 no 1 pp 10ndash24 2009

[8] Y Wang and M Papageorgiou ldquoReal-time freeway trafficstate estimation based on extended Kalman filter a generalapproachrdquo Transportation Research Part B Methodological vol39 no 2 pp 141ndash167 2005

[9] Y Yang Y Xu J Han E Wang W Chen and L Yue ldquoEfficienttraffic congestion estimation using multiple spatio-temporalpropertiesrdquo Neurocomputing vol 267 pp 344ndash353 2017

[10] S FanM Herty and B Seibold ldquoComparative model accuracyof a data-fitted generalized Aw-Rascle-ZhangmodelrdquoNetworksand Heterogeneous Media vol 9 no 2 pp 239ndash268 2014

[11] K Y Chan T S Dillon J Singh and E Chang ldquoNeural-network-based models for short-term traffic flow forecast-ing using a hybrid exponential smoothing and levenberg-marquardt algorithmrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 13 no 2 pp 644ndash654 2012

[12] L Chen and C L Chen ldquoEnsemble learning approach forfreeway short-term traffic flow predictionrdquo in Proceedings ofthe 2007 IEEE International Conference on System of SystemsEngineering pp 1ndash6 San Antonio Tex USA April 2007

[13] Y-S Jeong Y-J Byon M M Castro-Neto and S M EasaldquoSupervised weighting-online learning algorithm for short-term traffic flow predictionrdquo IEEE Transactions on IntelligentTransportation Systems vol 14 no 4 pp 1700ndash1707 2013

[14] B L Smith B M Williams and R K Oswald ldquoComparison ofparametric and nonparametric models for traffic flow forecast-ingrdquoTransportation Research Part C Emerging Technologies vol10 no 4 pp 303ndash321 2002

[15] M Castro-Neto Y-S Jeong M-K Jeong and L D HanldquoOnline-SVR for short-term traffic flow prediction undertypical and atypical traffic conditionsrdquo Expert Systems withApplications vol 36 no 3 pp 6164ndash6173 2009

[16] F G Habtemichael and M Cetin ldquoShort-term traffic flowrate forecasting based on identifying similar traffic patternsrdquoTransportation Research Part C Emerging Technologies vol 66pp 61ndash78 2016

[17] A Salamanis G Margaritis D D Kehagias G Matzoulasand D Tzovaras ldquoIdentifying patterns under both normal andabnormal traffic conditions for short-term traffic predictionrdquoTransportation Research Procedia vol 22 pp 665ndash674 2017

[18] X Fei Y Zhang K Liu and M Guo ldquoBayesian dynamic linearmodel with switching for real-time short-term freeway travel

10 Journal of Advanced Transportation

time prediction with license plate recognition datardquo Journal ofTransportation Engineering vol 139 no 11 pp 1058ndash1067 2013

[19] X Fei C C Lu and K Liu ldquoA bayesian dynamic linear modelapproach for real-time short-term freeway travel time predic-tionrdquo Transportation Research Part C Emerging Technologiesvol 19 no 6 pp 1306ndash1318 2011

[20] Y Kawasaki Y Hara andM Kuwahara ldquoReal-timemonitoringof dynamic traffic states by state-space modelrdquo TransportationResearch Procedia vol 21 pp 42ndash55 2017

[21] C Lu and X Zhou ldquoShort-term highway traffic state predictionusing structural state space modelsrdquo Journal of IntelligentTransportation Systems Technology Planning and Operationsvol 18 no 3 pp 309ndash322 2014

[22] A Stathopoulos and M G Karlaftis ldquoA multivariate statespace approach for urban traffic flowmodeling and predictionrdquoTransportation Research Part C Emerging Technologies vol 11no 2 pp 121ndash135 2003

[23] M West and J Harrison Bayesian Forecasting and DynamicModels Springer New York NY USA 1997

[24] L Auret and C Aldrich ldquoChange point detection in time seriesdata with random forestsrdquo Control Engineering Practice vol 18no 8 pp 990ndash1002 2010

[25] G Comert and A Bezuglov ldquoAn Online Change-Point-BasedModel for Traffic Parameter Predictionrdquo IEEE Transactions onIntelligent Transportation Systems vol 14 no 3 pp 1360ndash13692013

[26] M Daumer and M Falk ldquoOn-line change-point detection (forstate space models) using multi-process Kalman filtersrdquo LinearAlgebra and its Applications vol 284 no 1-3 pp 125ndash135 1998

[27] S Liu M Yamada N Collier and M Sugiyama ldquoChange-point detection in time-series data by relative density-ratioestimationrdquo Neural Networks vol 43 pp 72ndash83 2013

[28] L Moreira-Matias and F Alesiani ldquoDrift3Flow freeway-incident prediction using real-time learningrdquo in Proceedings ofthe IEEE 18th International Conference on Intelligent Transporta-tion Systems 571 566 pages October 2015

[29] E Ruggieri and M Antonellis ldquoAn exact approach to Bayesiansequential change point detectionrdquo Computational Statistics ampData Analysis vol 97 pp 71ndash86 2016

[30] E S Page ldquoContinuous inspection schemesrdquo Biometrika vol41 pp 100ndash114 1954

[31] G E Box G M Jenkins G C Reinsel and G M Ljung TimeSeries Analysis Forecasting and Control Wiley-Blackwell 2015

[32] C C Holt ldquoForecasting seasonals and trends by exponentiallyweightedmoving averagesrdquo International Journal of Forecastingvol 20 no 1 pp 5ndash10 2004

[33] J Durbin and S J Koopman Time Series Analysis by State SpaceMethods vol 38 Oxford University Press Oxford UK 2ndedition 2012

[34] ldquoMaximum-likelihood methodrdquo in Encyclopedia of Mathemat-ics 2001 httpswwwencyclopediaofmathorgindexphpMax-imum-likelihood method

[35] G Petris S Petrone and P Campagnoli Dynamic LinearModels with R Springer New York NY USA 2009

[36] J Kiefer ldquoSequentialMinimax Search for aMaximumrdquoProceed-ings of the American Mathematical Society vol 4 no 3 p 5021953

[37] SUMO Simulation of UrbanMobility 2018 httpswwwdlrdetsendesktopdefaultaspxtabid-988316931 read-41000

[38] Y Pigne G Danoy and P Bouvry ldquoA platform for realisticonline vehicular network management inrdquo in IEEE GlobecomWorkshops pp 595ndash599 IEEE 2010

[39] Y Pigne G Danoy and P Bouvry ldquoA vehicular mobilitymodel based on real traffic counting datardquo in CommunicationTechnologies for Vehicles vol 6596 pp 131ndash142 Springer BerlinHeidelberg Heidelberg Germany 2011

[40] Z Liang and YWakahara ldquoReal-time urban traffic amount pre-diction models for dynamic route guidance systemsrdquo EURASIPJournal on Wireless Communications and Networking vol 852014

[41] L Codeca R Frank and T Engel ldquoLuxembourg SUMO traffic(LuST) scenario 24 hours of mobility for vehicular networkingresearchrdquo in Proceedings of the IEEE Vehicular NetworkingConference VNC 2015 pp 1ndash8 Japan December 2015

8 Journal of Advanced Transportation

700 725 750 815 840 905 700 725 750 815 840 9050

5

10

15

20

25

0

5

10

15

20

25Av

erag

e spe

ed (m

sec

)

yyDLM

yadapDLM

||

DLM

adapDLM

Figure 6 Comparison of one-step forecasts and absolute errors for first-orderDLMand the adaptive DLM in accident scenarios (road section5W) Le the one-step forecasts of average speed Right absolute residuals

700 725 750 815 840 905Time

0

5

10

15

20

25

30

35

One

-ste

p av

erag

e spe

ed p

redi

ctio

n (m

sec

)

Obs1st orderCubic spline

2nd orderAdaptive

35

4

45

5

55

6

Aver

age a

bsol

ute r

esid

uals

Cubic spline smoothing 2nd order Adaptive1st orderDLM model

Figure 7 Comparison of the performance of different DLM models in unforeseen accident situation (road section 4E)

experiments show the proposed method allows adaptivelyoptimizing its model parameters to improve its predictionaccuracy in a continuous way under uncertainty The pro-posed method does not need the intervention of experts andcan adjust its model error covariance automatically based onfeedback information of its one-step prediction errors

Experimental studies show that our adaptive DLMapproach outperforms both autoregressive integratedmovingaverage (ARIMA) and Holt-Winters Exponential Smoothing(ETS) that are both considered to be the main time series

analysis methods employed on this type of problems [28]We thus consider that this comparison is a reasonable proxyto a comparison with other online models for travel speedprediction that use ARIMA or ETS

Future extensions concern an adaptive parameter updat-ing scheme design for the state space methods and formore complicated DLMs with seasonal and regression termsApplications of the proposed method on other time seriesdata would also be beneficial for assessing and improving itsperformance

Journal of Advanced Transportation 9

01234567

Sign

al-to

-noi

se ra

tio (w

v)

725 750 815 840 905700Time

Figure 8 Adaptive parameter updating of DLM model in unforeseen accident situation (road section 5W)

Notations

119905 Index of discretized time intervals 119905 = 1 2 3 119879119909119905 System state at time 119905119910119905 Observation at time 119905119907119905 Observation error at time 119905119881119905 Variance of 119907119905119865119905 Design matrix for observation equation at time 119905119866119905 Evolution matrix of system states at time 119905119908119905 Forecast error at time 119905119882119905 Variance of 119908119905

Data Availability

The data is freely available at httpsgithubcompigne2019-simulations-DLM

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors are grateful to the support of LuxembourgInstitute of Socio-Economic Research (LISER) under thevisiting scholar grant

References

[1] TMa ldquoSolving a dynamic user-optimal route guidance problembased on joint strategy fictitious playrdquo in Game 13eoreticAnalysis of Congestion Safety and Security pp 67ndash89 SpringerInternational Publishing 2015

[2] U Mori AMendiburuM Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for advanced travellerinformation systemsrdquo Transportmetrica A Transport Sciencevol 11 pp 119ndash157 2015

[3] S Oh Y-J Byon K Jang and H Yeo ldquoShort-term travel-timeprediction on highway a review of the data-driven approachrdquoTransport Reviews vol 35 pp 4ndash32 2015

[4] T Seo A M Bayen T Kusakabe and Y Asakura ldquoTrafficstate estimation on highway A comprehensive surveyrdquo AnnualReviews in Control vol 43 pp 128ndash151 2017

[5] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting Where we are and where we are goingrdquoTransportation Research Part C Emerging Technologies vol 43pp 3ndash19 2014

[6] CM J Tampere and L H Immers ldquoAn extended Kalman filterapplication for traffic state estimation using CTM with implicitmode switching and dynamic parametersrdquo in Proceedings of theIEEE Intelligent Transportation Systems Conference 2007

[7] YWangM Papageorgiou AMessmer P Coppola A Tzimitsiand A Nuzzolo ldquoAn adaptive freeway traffic state estimatorrdquoAutomatica vol 45 no 1 pp 10ndash24 2009

[8] Y Wang and M Papageorgiou ldquoReal-time freeway trafficstate estimation based on extended Kalman filter a generalapproachrdquo Transportation Research Part B Methodological vol39 no 2 pp 141ndash167 2005

[9] Y Yang Y Xu J Han E Wang W Chen and L Yue ldquoEfficienttraffic congestion estimation using multiple spatio-temporalpropertiesrdquo Neurocomputing vol 267 pp 344ndash353 2017

[10] S FanM Herty and B Seibold ldquoComparative model accuracyof a data-fitted generalized Aw-Rascle-ZhangmodelrdquoNetworksand Heterogeneous Media vol 9 no 2 pp 239ndash268 2014

[11] K Y Chan T S Dillon J Singh and E Chang ldquoNeural-network-based models for short-term traffic flow forecast-ing using a hybrid exponential smoothing and levenberg-marquardt algorithmrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 13 no 2 pp 644ndash654 2012

[12] L Chen and C L Chen ldquoEnsemble learning approach forfreeway short-term traffic flow predictionrdquo in Proceedings ofthe 2007 IEEE International Conference on System of SystemsEngineering pp 1ndash6 San Antonio Tex USA April 2007

[13] Y-S Jeong Y-J Byon M M Castro-Neto and S M EasaldquoSupervised weighting-online learning algorithm for short-term traffic flow predictionrdquo IEEE Transactions on IntelligentTransportation Systems vol 14 no 4 pp 1700ndash1707 2013

[14] B L Smith B M Williams and R K Oswald ldquoComparison ofparametric and nonparametric models for traffic flow forecast-ingrdquoTransportation Research Part C Emerging Technologies vol10 no 4 pp 303ndash321 2002

[15] M Castro-Neto Y-S Jeong M-K Jeong and L D HanldquoOnline-SVR for short-term traffic flow prediction undertypical and atypical traffic conditionsrdquo Expert Systems withApplications vol 36 no 3 pp 6164ndash6173 2009

[16] F G Habtemichael and M Cetin ldquoShort-term traffic flowrate forecasting based on identifying similar traffic patternsrdquoTransportation Research Part C Emerging Technologies vol 66pp 61ndash78 2016

[17] A Salamanis G Margaritis D D Kehagias G Matzoulasand D Tzovaras ldquoIdentifying patterns under both normal andabnormal traffic conditions for short-term traffic predictionrdquoTransportation Research Procedia vol 22 pp 665ndash674 2017

[18] X Fei Y Zhang K Liu and M Guo ldquoBayesian dynamic linearmodel with switching for real-time short-term freeway travel

10 Journal of Advanced Transportation

time prediction with license plate recognition datardquo Journal ofTransportation Engineering vol 139 no 11 pp 1058ndash1067 2013

[19] X Fei C C Lu and K Liu ldquoA bayesian dynamic linear modelapproach for real-time short-term freeway travel time predic-tionrdquo Transportation Research Part C Emerging Technologiesvol 19 no 6 pp 1306ndash1318 2011

[20] Y Kawasaki Y Hara andM Kuwahara ldquoReal-timemonitoringof dynamic traffic states by state-space modelrdquo TransportationResearch Procedia vol 21 pp 42ndash55 2017

[21] C Lu and X Zhou ldquoShort-term highway traffic state predictionusing structural state space modelsrdquo Journal of IntelligentTransportation Systems Technology Planning and Operationsvol 18 no 3 pp 309ndash322 2014

[22] A Stathopoulos and M G Karlaftis ldquoA multivariate statespace approach for urban traffic flowmodeling and predictionrdquoTransportation Research Part C Emerging Technologies vol 11no 2 pp 121ndash135 2003

[23] M West and J Harrison Bayesian Forecasting and DynamicModels Springer New York NY USA 1997

[24] L Auret and C Aldrich ldquoChange point detection in time seriesdata with random forestsrdquo Control Engineering Practice vol 18no 8 pp 990ndash1002 2010

[25] G Comert and A Bezuglov ldquoAn Online Change-Point-BasedModel for Traffic Parameter Predictionrdquo IEEE Transactions onIntelligent Transportation Systems vol 14 no 3 pp 1360ndash13692013

[26] M Daumer and M Falk ldquoOn-line change-point detection (forstate space models) using multi-process Kalman filtersrdquo LinearAlgebra and its Applications vol 284 no 1-3 pp 125ndash135 1998

[27] S Liu M Yamada N Collier and M Sugiyama ldquoChange-point detection in time-series data by relative density-ratioestimationrdquo Neural Networks vol 43 pp 72ndash83 2013

[28] L Moreira-Matias and F Alesiani ldquoDrift3Flow freeway-incident prediction using real-time learningrdquo in Proceedings ofthe IEEE 18th International Conference on Intelligent Transporta-tion Systems 571 566 pages October 2015

[29] E Ruggieri and M Antonellis ldquoAn exact approach to Bayesiansequential change point detectionrdquo Computational Statistics ampData Analysis vol 97 pp 71ndash86 2016

[30] E S Page ldquoContinuous inspection schemesrdquo Biometrika vol41 pp 100ndash114 1954

[31] G E Box G M Jenkins G C Reinsel and G M Ljung TimeSeries Analysis Forecasting and Control Wiley-Blackwell 2015

[32] C C Holt ldquoForecasting seasonals and trends by exponentiallyweightedmoving averagesrdquo International Journal of Forecastingvol 20 no 1 pp 5ndash10 2004

[33] J Durbin and S J Koopman Time Series Analysis by State SpaceMethods vol 38 Oxford University Press Oxford UK 2ndedition 2012

[34] ldquoMaximum-likelihood methodrdquo in Encyclopedia of Mathemat-ics 2001 httpswwwencyclopediaofmathorgindexphpMax-imum-likelihood method

[35] G Petris S Petrone and P Campagnoli Dynamic LinearModels with R Springer New York NY USA 2009

[36] J Kiefer ldquoSequentialMinimax Search for aMaximumrdquoProceed-ings of the American Mathematical Society vol 4 no 3 p 5021953

[37] SUMO Simulation of UrbanMobility 2018 httpswwwdlrdetsendesktopdefaultaspxtabid-988316931 read-41000

[38] Y Pigne G Danoy and P Bouvry ldquoA platform for realisticonline vehicular network management inrdquo in IEEE GlobecomWorkshops pp 595ndash599 IEEE 2010

[39] Y Pigne G Danoy and P Bouvry ldquoA vehicular mobilitymodel based on real traffic counting datardquo in CommunicationTechnologies for Vehicles vol 6596 pp 131ndash142 Springer BerlinHeidelberg Heidelberg Germany 2011

[40] Z Liang and YWakahara ldquoReal-time urban traffic amount pre-diction models for dynamic route guidance systemsrdquo EURASIPJournal on Wireless Communications and Networking vol 852014

[41] L Codeca R Frank and T Engel ldquoLuxembourg SUMO traffic(LuST) scenario 24 hours of mobility for vehicular networkingresearchrdquo in Proceedings of the IEEE Vehicular NetworkingConference VNC 2015 pp 1ndash8 Japan December 2015

Journal of Advanced Transportation 9

01234567

Sign

al-to

-noi

se ra

tio (w

v)

725 750 815 840 905700Time

Figure 8 Adaptive parameter updating of DLM model in unforeseen accident situation (road section 5W)

Notations

119905 Index of discretized time intervals 119905 = 1 2 3 119879119909119905 System state at time 119905119910119905 Observation at time 119905119907119905 Observation error at time 119905119881119905 Variance of 119907119905119865119905 Design matrix for observation equation at time 119905119866119905 Evolution matrix of system states at time 119905119908119905 Forecast error at time 119905119882119905 Variance of 119908119905

Data Availability

The data is freely available at httpsgithubcompigne2019-simulations-DLM

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors are grateful to the support of LuxembourgInstitute of Socio-Economic Research (LISER) under thevisiting scholar grant

References

[1] TMa ldquoSolving a dynamic user-optimal route guidance problembased on joint strategy fictitious playrdquo in Game 13eoreticAnalysis of Congestion Safety and Security pp 67ndash89 SpringerInternational Publishing 2015

[2] U Mori AMendiburuM Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for advanced travellerinformation systemsrdquo Transportmetrica A Transport Sciencevol 11 pp 119ndash157 2015

[3] S Oh Y-J Byon K Jang and H Yeo ldquoShort-term travel-timeprediction on highway a review of the data-driven approachrdquoTransport Reviews vol 35 pp 4ndash32 2015

[4] T Seo A M Bayen T Kusakabe and Y Asakura ldquoTrafficstate estimation on highway A comprehensive surveyrdquo AnnualReviews in Control vol 43 pp 128ndash151 2017

[5] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting Where we are and where we are goingrdquoTransportation Research Part C Emerging Technologies vol 43pp 3ndash19 2014

[6] CM J Tampere and L H Immers ldquoAn extended Kalman filterapplication for traffic state estimation using CTM with implicitmode switching and dynamic parametersrdquo in Proceedings of theIEEE Intelligent Transportation Systems Conference 2007

[7] YWangM Papageorgiou AMessmer P Coppola A Tzimitsiand A Nuzzolo ldquoAn adaptive freeway traffic state estimatorrdquoAutomatica vol 45 no 1 pp 10ndash24 2009

[8] Y Wang and M Papageorgiou ldquoReal-time freeway trafficstate estimation based on extended Kalman filter a generalapproachrdquo Transportation Research Part B Methodological vol39 no 2 pp 141ndash167 2005

[9] Y Yang Y Xu J Han E Wang W Chen and L Yue ldquoEfficienttraffic congestion estimation using multiple spatio-temporalpropertiesrdquo Neurocomputing vol 267 pp 344ndash353 2017

[10] S FanM Herty and B Seibold ldquoComparative model accuracyof a data-fitted generalized Aw-Rascle-ZhangmodelrdquoNetworksand Heterogeneous Media vol 9 no 2 pp 239ndash268 2014

[11] K Y Chan T S Dillon J Singh and E Chang ldquoNeural-network-based models for short-term traffic flow forecast-ing using a hybrid exponential smoothing and levenberg-marquardt algorithmrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 13 no 2 pp 644ndash654 2012

[12] L Chen and C L Chen ldquoEnsemble learning approach forfreeway short-term traffic flow predictionrdquo in Proceedings ofthe 2007 IEEE International Conference on System of SystemsEngineering pp 1ndash6 San Antonio Tex USA April 2007

[13] Y-S Jeong Y-J Byon M M Castro-Neto and S M EasaldquoSupervised weighting-online learning algorithm for short-term traffic flow predictionrdquo IEEE Transactions on IntelligentTransportation Systems vol 14 no 4 pp 1700ndash1707 2013

[14] B L Smith B M Williams and R K Oswald ldquoComparison ofparametric and nonparametric models for traffic flow forecast-ingrdquoTransportation Research Part C Emerging Technologies vol10 no 4 pp 303ndash321 2002

[15] M Castro-Neto Y-S Jeong M-K Jeong and L D HanldquoOnline-SVR for short-term traffic flow prediction undertypical and atypical traffic conditionsrdquo Expert Systems withApplications vol 36 no 3 pp 6164ndash6173 2009

[16] F G Habtemichael and M Cetin ldquoShort-term traffic flowrate forecasting based on identifying similar traffic patternsrdquoTransportation Research Part C Emerging Technologies vol 66pp 61ndash78 2016

[17] A Salamanis G Margaritis D D Kehagias G Matzoulasand D Tzovaras ldquoIdentifying patterns under both normal andabnormal traffic conditions for short-term traffic predictionrdquoTransportation Research Procedia vol 22 pp 665ndash674 2017

[18] X Fei Y Zhang K Liu and M Guo ldquoBayesian dynamic linearmodel with switching for real-time short-term freeway travel

10 Journal of Advanced Transportation

time prediction with license plate recognition datardquo Journal ofTransportation Engineering vol 139 no 11 pp 1058ndash1067 2013

[19] X Fei C C Lu and K Liu ldquoA bayesian dynamic linear modelapproach for real-time short-term freeway travel time predic-tionrdquo Transportation Research Part C Emerging Technologiesvol 19 no 6 pp 1306ndash1318 2011

[20] Y Kawasaki Y Hara andM Kuwahara ldquoReal-timemonitoringof dynamic traffic states by state-space modelrdquo TransportationResearch Procedia vol 21 pp 42ndash55 2017

[21] C Lu and X Zhou ldquoShort-term highway traffic state predictionusing structural state space modelsrdquo Journal of IntelligentTransportation Systems Technology Planning and Operationsvol 18 no 3 pp 309ndash322 2014

[22] A Stathopoulos and M G Karlaftis ldquoA multivariate statespace approach for urban traffic flowmodeling and predictionrdquoTransportation Research Part C Emerging Technologies vol 11no 2 pp 121ndash135 2003

[23] M West and J Harrison Bayesian Forecasting and DynamicModels Springer New York NY USA 1997

[24] L Auret and C Aldrich ldquoChange point detection in time seriesdata with random forestsrdquo Control Engineering Practice vol 18no 8 pp 990ndash1002 2010

[25] G Comert and A Bezuglov ldquoAn Online Change-Point-BasedModel for Traffic Parameter Predictionrdquo IEEE Transactions onIntelligent Transportation Systems vol 14 no 3 pp 1360ndash13692013

[26] M Daumer and M Falk ldquoOn-line change-point detection (forstate space models) using multi-process Kalman filtersrdquo LinearAlgebra and its Applications vol 284 no 1-3 pp 125ndash135 1998

[27] S Liu M Yamada N Collier and M Sugiyama ldquoChange-point detection in time-series data by relative density-ratioestimationrdquo Neural Networks vol 43 pp 72ndash83 2013

[28] L Moreira-Matias and F Alesiani ldquoDrift3Flow freeway-incident prediction using real-time learningrdquo in Proceedings ofthe IEEE 18th International Conference on Intelligent Transporta-tion Systems 571 566 pages October 2015

[29] E Ruggieri and M Antonellis ldquoAn exact approach to Bayesiansequential change point detectionrdquo Computational Statistics ampData Analysis vol 97 pp 71ndash86 2016

[30] E S Page ldquoContinuous inspection schemesrdquo Biometrika vol41 pp 100ndash114 1954

[31] G E Box G M Jenkins G C Reinsel and G M Ljung TimeSeries Analysis Forecasting and Control Wiley-Blackwell 2015

[32] C C Holt ldquoForecasting seasonals and trends by exponentiallyweightedmoving averagesrdquo International Journal of Forecastingvol 20 no 1 pp 5ndash10 2004

[33] J Durbin and S J Koopman Time Series Analysis by State SpaceMethods vol 38 Oxford University Press Oxford UK 2ndedition 2012

[34] ldquoMaximum-likelihood methodrdquo in Encyclopedia of Mathemat-ics 2001 httpswwwencyclopediaofmathorgindexphpMax-imum-likelihood method

[35] G Petris S Petrone and P Campagnoli Dynamic LinearModels with R Springer New York NY USA 2009

[36] J Kiefer ldquoSequentialMinimax Search for aMaximumrdquoProceed-ings of the American Mathematical Society vol 4 no 3 p 5021953

[37] SUMO Simulation of UrbanMobility 2018 httpswwwdlrdetsendesktopdefaultaspxtabid-988316931 read-41000

[38] Y Pigne G Danoy and P Bouvry ldquoA platform for realisticonline vehicular network management inrdquo in IEEE GlobecomWorkshops pp 595ndash599 IEEE 2010

[39] Y Pigne G Danoy and P Bouvry ldquoA vehicular mobilitymodel based on real traffic counting datardquo in CommunicationTechnologies for Vehicles vol 6596 pp 131ndash142 Springer BerlinHeidelberg Heidelberg Germany 2011

[40] Z Liang and YWakahara ldquoReal-time urban traffic amount pre-diction models for dynamic route guidance systemsrdquo EURASIPJournal on Wireless Communications and Networking vol 852014

[41] L Codeca R Frank and T Engel ldquoLuxembourg SUMO traffic(LuST) scenario 24 hours of mobility for vehicular networkingresearchrdquo in Proceedings of the IEEE Vehicular NetworkingConference VNC 2015 pp 1ndash8 Japan December 2015

10 Journal of Advanced Transportation

time prediction with license plate recognition datardquo Journal ofTransportation Engineering vol 139 no 11 pp 1058ndash1067 2013

[19] X Fei C C Lu and K Liu ldquoA bayesian dynamic linear modelapproach for real-time short-term freeway travel time predic-tionrdquo Transportation Research Part C Emerging Technologiesvol 19 no 6 pp 1306ndash1318 2011

[20] Y Kawasaki Y Hara andM Kuwahara ldquoReal-timemonitoringof dynamic traffic states by state-space modelrdquo TransportationResearch Procedia vol 21 pp 42ndash55 2017

[21] C Lu and X Zhou ldquoShort-term highway traffic state predictionusing structural state space modelsrdquo Journal of IntelligentTransportation Systems Technology Planning and Operationsvol 18 no 3 pp 309ndash322 2014

[22] A Stathopoulos and M G Karlaftis ldquoA multivariate statespace approach for urban traffic flowmodeling and predictionrdquoTransportation Research Part C Emerging Technologies vol 11no 2 pp 121ndash135 2003

[23] M West and J Harrison Bayesian Forecasting and DynamicModels Springer New York NY USA 1997

[24] L Auret and C Aldrich ldquoChange point detection in time seriesdata with random forestsrdquo Control Engineering Practice vol 18no 8 pp 990ndash1002 2010

[25] G Comert and A Bezuglov ldquoAn Online Change-Point-BasedModel for Traffic Parameter Predictionrdquo IEEE Transactions onIntelligent Transportation Systems vol 14 no 3 pp 1360ndash13692013

[26] M Daumer and M Falk ldquoOn-line change-point detection (forstate space models) using multi-process Kalman filtersrdquo LinearAlgebra and its Applications vol 284 no 1-3 pp 125ndash135 1998

[27] S Liu M Yamada N Collier and M Sugiyama ldquoChange-point detection in time-series data by relative density-ratioestimationrdquo Neural Networks vol 43 pp 72ndash83 2013

[28] L Moreira-Matias and F Alesiani ldquoDrift3Flow freeway-incident prediction using real-time learningrdquo in Proceedings ofthe IEEE 18th International Conference on Intelligent Transporta-tion Systems 571 566 pages October 2015

[29] E Ruggieri and M Antonellis ldquoAn exact approach to Bayesiansequential change point detectionrdquo Computational Statistics ampData Analysis vol 97 pp 71ndash86 2016

[30] E S Page ldquoContinuous inspection schemesrdquo Biometrika vol41 pp 100ndash114 1954

[31] G E Box G M Jenkins G C Reinsel and G M Ljung TimeSeries Analysis Forecasting and Control Wiley-Blackwell 2015

[32] C C Holt ldquoForecasting seasonals and trends by exponentiallyweightedmoving averagesrdquo International Journal of Forecastingvol 20 no 1 pp 5ndash10 2004

[33] J Durbin and S J Koopman Time Series Analysis by State SpaceMethods vol 38 Oxford University Press Oxford UK 2ndedition 2012

[34] ldquoMaximum-likelihood methodrdquo in Encyclopedia of Mathemat-ics 2001 httpswwwencyclopediaofmathorgindexphpMax-imum-likelihood method

[35] G Petris S Petrone and P Campagnoli Dynamic LinearModels with R Springer New York NY USA 2009

[36] J Kiefer ldquoSequentialMinimax Search for aMaximumrdquoProceed-ings of the American Mathematical Society vol 4 no 3 p 5021953

[37] SUMO Simulation of UrbanMobility 2018 httpswwwdlrdetsendesktopdefaultaspxtabid-988316931 read-41000

[38] Y Pigne G Danoy and P Bouvry ldquoA platform for realisticonline vehicular network management inrdquo in IEEE GlobecomWorkshops pp 595ndash599 IEEE 2010

[39] Y Pigne G Danoy and P Bouvry ldquoA vehicular mobilitymodel based on real traffic counting datardquo in CommunicationTechnologies for Vehicles vol 6596 pp 131ndash142 Springer BerlinHeidelberg Heidelberg Germany 2011

[40] Z Liang and YWakahara ldquoReal-time urban traffic amount pre-diction models for dynamic route guidance systemsrdquo EURASIPJournal on Wireless Communications and Networking vol 852014

[41] L Codeca R Frank and T Engel ldquoLuxembourg SUMO traffic(LuST) scenario 24 hours of mobility for vehicular networkingresearchrdquo in Proceedings of the IEEE Vehicular NetworkingConference VNC 2015 pp 1ndash8 Japan December 2015

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