BBayesian Particle Tracking of Traffic Flows
Nicholas PolsonBooth School of Business, University of ChicagoandVadim SokolovArgonne National LaboratoryNovember 8, 2018 a r X i v : . [ s t a t . A P ] N ov BSTRACT
We develop a Bayesian particle filter for tracking traffic flows that is capable of capturing non-linearities and discontinuities present in flow dynamics. Our model includes a hidden state vari-able that captures sudden regime shifts between traffic free flow, breakdown and recovery. Wedevelop an efficient particle learning algorithm for real time on-line inference of states and pa-rameters. This requires a two step approach, first, resampling the current particles, with a mix-ture predictive distribution and second, propagation of states using the conditional posteriordistribution. Particle learning of parameters follows from updating recursions for conditionalsufficient statistics. To illustrate our methodology, we analyze measurements of daily traffic flowfrom the Illinois interstate I-55 highway system. We demonstrate how our filter can be used toinference the change of traffic flow regime on a highway road segment based on a measurementfrom freeway single-loop detectors. Finally, we conclude with directions for future research.olson and Sokolov 2
INTRODUCTION
Modeling traffic dynamics for a transportation network with highways, arterial roads and publictransit is an important task for effectively managing traffic flow. A major goal is to providetraffic flow conditions on highways from field measurements fusing in-ground loop detectorsor GPS probes. Traffic managers make decisions based on model forecasts to regulate rampmetering, apply speed harmonization, or change road pricing as congestion mitigation strategies.The general public uses model-based predictions for assessing departure times, and travel routechoices, among other factors.We propose a dynamic state-space model which incorporates a latent switching variabletogether with a traffic flow state variable to capture the non-linearities and discontinuities intraffic patterns. Our dynamic model allows for three traffic flow regimes: free flow, breakdownand recovery. A physical interpretation of the change in the flow regime is a traffic queue withcongestion inside the queue and free flow outside. In addition to switching variable and trafficflow states, we track a state that measures the rate of degradation or recovery. The challenge is tofilter traffic flow measurements which are sparse, from both fixed and moving sensors which arelocated at a small number of locations at any given point in time. Statistical inference is furthercomplicated by noisy non-Gaussian observations; for example, Rasschaert ( ) show that the datagenerating distribution for video cameras is a mixture of Poissons.In order to achieve real-time sequential inference, we develop an efficient particle filterand learning algorithm. This allows us to track the speed of traffic flow together with the latentstates that describe regime switches, and the rate of degradation (recovery). To illustrate ourmethodology, we model a data from a network of in-ground loop detectors on Chicago’s Inter-state I-55 with measurements on speed, counts and occupancy of traffic flow. The sequentialnature of particle filtering makes frequent updating feasible, and therefore, our method providesan alternative to Markov Chain Monte Carlo (MCMC) simulation methods. The computationalcost of the latter grows linearly with the number of observations. We build on particle filters fortraffic flow problems (see Mihaylova et al. ( )) who use the evolution dynamics as a proposal dis-tribution before re-sampling, the so-called bootstrap or sampling / importance resampling (SIR)filter. We improve the filtering efficiency by developing a Rao-Blackwellized which is also flexi-ble enough to incorporate particle learning.Our approach builds upon existing statistical methods in a number of ways. Tebaldi andWest ( ) infer network route flows and Westgate et al. ( ) develop MCMC methods to estimatetravel time reliability for ambulances using noisy GPS for both path travel time and individualroad segment travel time distributions. Anacleto et al. ( ) propose a dynamic Bayesian networkto model external intervention techniques to accommodate situations with suddenly changingtraffic variables. Another class of probabilistic methods rely on using image processing tech-niques for traffic estimation, for example in Kamijo et al. ( ) authors developed an algorithmthat is based on Markov random field that allows analyzing images from intersections to detectflows and accidents.Small changes in consecutive speed measurements can be explained by sensor noise butsignificant changes require estimation of probability of a flow regime switch, which is repre-sented by a discontinuity in traffic flow speed. Previous work on estimating traffic flows useextensions of the Kalman filter and relies heavily on Gaussianity assumptions, see Gazis andKnapp, Schreiter et al., Wang and Papageorgiou, Work et al. (
7, 8, 9, 10 ). Recently, Blandinet al., Polson and Sokolov (
11, 12 ) showed that dynamic properties of traffic flow such as dis-olson and Sokolov 3continuities (or shock waves), lead to forecast distributions that are mixtures. Our non-Gaussianstate-space model will explicitly capture such a behavior.Our approach builds on the current literature in a number of ways: • Provides a hierarchical model rather than a conservation law based model. Our ap-proach avoids boundary condition estimation • A predictive likelihood particle filter provides an efficient estimation strategy. Ourfilter is less sensitive to measurement outliers and less prone to particle degeneracy • Tracks traffic flow variables, such as traffic flow speed together with additional latentvariables for regime switching and a degeneracy (recovery) rate.One class of previously considered problems involves estimating (i) turn counts on urbancontrolled intersections [ Lan and Davis, Ghods and Fu (
13, 14 ) ] , (ii) flows or travel times on net-work routes [ Kwong et al., Wu et al., Rahmani et al., Nantes et al., Fowe and Chan, Dell’Orcoet al. (
15, 16, 17, 18, 19, 20 ) ] , (iii) travel times and densities on an individual road segment [ Coif-man and Kim, Zheng and Van Zuylen, Zhan et al., Seo and Kusakabe, Seo et al., Bachmann et al.(
21, 22, 23, 24, 25, 26 ) ] , and (iv) queue position on highways or arterial roads [ Vigos et al., Banet al., Zhan et al., Lee et al. (
27, 28, 29, 30 ) ] . There are several types of algorithms, which can becategorized into the following groups (i) time series analysis, (ii) machine or deep learning, and(iii) model-based analysis in a form of state-space models. We tackle the latter, which requires aparticle filtering algorithm to perform inference.A machine-learning algorithm was provided by Sun et al. ( ), where authors proposeda Bayes network algorithm for forecasting traffic flow. The idea is to derive the conditionalprobability of a traffic state on a given road, given states on topological neighbors on a roadnetwork. The resulting joint probability distribution is a mixture of Gaussians. Bayes networksfor estimating travel times were suggested earlier by Horvitz et al. ( ). This approach even-tually became a commercial product that led to the start of Inrix, a traffic data company. Wuet al. ( ) provides a machine-learning method based on a support vector regression (SVR) toforecast travel-times and Quek et al. ( ) use a fuzzy neural-network approach to address theissue of traffic data generating processes being non-linear and used fuzzy logic to improve theinteroperability of their model. On the contrary, Rice and van Zwet ( ), argue that there is alinear relation between the future travel times and currently estimated conditions. They demon-strate that a regression model with time varying coefficients is capable of designing a travel timeprediction scheme. Another class of forecasting models relies on classical time series modeling.For example, Tan et al. ( ) studied two classes of time series methods, auto-regressive movingaverage (ARIMA) and exponential smoothing (ES). The forecasts generated by ARIMA and ESmodels are used as inputs to neural networks, which aggregates those into a single forecast. VanDer Voort et al. ( ) also proposed combing an ARIMA model with a machine learning method,the Kohonen self-organizing map, which was used as an initial classifier. Van Lint ( ) addresses aparameter estimation problem of real-time learning that can improve the quality of forecasts viaan extended Kalman filter for incorporating data in real-time into a parameter learning process.Ban et al. ( ) proposes a method for estimating queue lengths at controlled intersections, basedon the travel time data measured by GPS probes. The method relies on detecting discontinu-ities and changes of slopes in travel time data. Another data-mining based approach for queueolson and Sokolov 4state estimation was proposed in Ramezani and Geroliminis ( ), who combined the traffic flowshockwave analysis with data mining techniques.Cheng et al. ( ) proposes a threshold-based critical point extraction algorithm, with agoal to reduce the communication cost in the future real-time probe data collection application.A probabilistic approach, proposed in Hofleitner et al. ( ) uses a learning algorithm to infer thedensity of vehicles on arterial roads. A heuristic filter that allows combining data from multipletypes of sensors with different spatial and temporal resolutions was proposed in Van Lint andHoogendoorn ( ).Our main contribution is an algorithm for quick detection of the break-downs and recov-eries in the traffic flow regimes. In many cases, we can detect the breakdown new measurementarrives. We contrast the performance of our filter with some other approaches in the “NumericalExperiments” subsection. BAYESIAN MODELING OF TRAFFIC FLOW SPEED
Traffic flow speed data often relies on sparse and noisy measurements. Sparseness occurs dueto a fixed grid of sensor or a dynamically changing data source such as GPS probes. Our state-space model is designed to be applicable in both scenarios. There are discontinuities in thetraffic flow dynamics which need to be accounted for. We build a Dynamic Model [ West andHarrison, Carlin et al. (
44, 45 ) ] for the traffic state during three regimes: free flow, breakdownand recovery.To illustrate our methodology, we use data from a sensor on I-55 with id N-6041 . Thesensor is located eight miles from the Chicago downtown on I-55 north bound (near Cicero Ave),which is part of a route used by many morning commuters to travel from southwest suburbs tothe city. As shown on Figure 1, the sensor is located 840 meters downstream of an off-ramp and970 meters upstream from an on-ramp.
FIGURE 1 Location of the N-6041 sensor and the geometry of the road segment.
Figure 2 illustrates a typical day traffic flow pattern on Chicago’s I-55 highway wheresudden breakdowns are followed by a recovery to free flow regime. This traffic pattern is recur-rent and similar to one observed on other work days. We can see a breakdown in traffic flowspeed during the morning peak (region 2) period followed by speed recovery (region 3). The freeflow regimes (regions 1,4, and 5 on the plot) are usually of little interest to traffic managers. Thisdata motivates our choice of the statistical model developed. The goal is to build a model that iscapable of capturing the sudden regime changes, such as free flow to congestion at the beginningolson and Sokolov 5of the morning rush hour (regions 1 and 2) or change in speed to the recovery regime at the endof the rush hour (regions 2 and 3).
FIGURE 2 Example of one day speed profile on May 14, 2009 (Thursday). This plotillustrates the speed profile for a segment of interstate highway I-55. Different flow regimeregions were identified and labeled by the authors.
A key modeling feature is the inclusion of a switching state variable α t + ∈ {−
1, 0, 1 } toidentify different flow regimes. Trends during the break-down periods and recovery periods canthen be modeled using the first order polynomial component.Specifically, given a discretization from t to t + ∆ t , we use a state-space model of theform Observation: y t + = H x t + + γ T z t + + v α t + , v α t + ∼ N ( V α t + ) (1)Evolution: x t + = G α t + x t + ( I − G α t + ) µ + w t + , w t + ∼ N ( W α t + ) (2)Switching Evolution: α t + ∼ p ( α t + | α t , Z t ) (3)where evolution gain matrix is G α t + = (cid:130) F α t + α t + (cid:140) , and µ = (cid:18) v f (cid:19) Our hidden state variable x t = ( θ t , β t ) T , where θ t is traffic flow speed and β t is rate of change.We model the recovery and degradation in speed using an additive component α t β t . We alsomodel β t using a random walk model. Measurements are then given by y t = ( s p e e d , c o u n t , d e n s i t y ) t ,and we incorporate a switching variable with three states α t ∈ { b r eak d ow n , f r e e f l ow , r e c ov e r y } t .The observation matrix H , which could be time varying, allows for partial observationof the state vector x t . The parameter v f the free flow speed on the road segment. We allow for thepossibility of regressors, z t + , in the observation equation which effect the sensor measurementolson and Sokolov 6model, γ are regressors parameters. The switching coefficient F α t defines weather the process ismean-reverting or not. We define it by F α t = (cid:168) α t ∈ { − } F , α t =
0. ,where F < p ( α t + | α t , Z t ) of the switching evolution depends on the set of variables Z t , that explain regime shifts. Severalapproaches to model switching process p ( α t + | α t , Z t ) and choosing Z t are described in the nextsub-section.The main task of our approach is to detect the start of breakdown and recovery as soonas possible. The causes of breakdown or recovery might be different. For example a breakdownmight be caused by a recurrent demand that increases capacity or non-recurrent conditions, suchas weather or traffic accidents. Characteristic change in the traffic flow speed would be the samein either situation and thus our model is agnostic to the cause of the start or end of a congestionperiod. Modeling p ( α t + | α t , Z t ) The discrete state α t + ∈ {−
1, 0, 1 } models breakdown ( α = − α = α =
1) regimes of a traffic system. We need to specify a transition kernel for the evolution ofthis state given a set of exogenous predictor variables, denoted by Z t , and current state α t . Given Z t , the set of probabilities, p ( α t + | α t , Z t ) , will then form a 3 × x t .Figure 4 shows that break downs happen more or less the same time during morning orevening peak periods on a work day. Therefore, it is natural to introduce an exogenous variable Z t = ( p e r i od , d ay o f w e e k ) , p e r i od ∈ {
1, 2, 3 } where the three periods correspond to morn-ing, evening peak period and the rest of the day. Incorporating additional considerations beyonda period of the day and day of the week, such as weather, month, and special event leads to Z t = ( d ay o f w e e k , mon t h , w ea t h e r , mon t h , e v e n t , ac c i d e n t ) t .One way to build such a model is to identify 3 × p ( α t + | α t , Z t ) for eachcombination of the parameters in Z t based on the historic observations.Figure 3(a) shows the average speed for a fixed location on the network for the April 4 -March 3 period in 2009, weekend days identified by blue dots. We see that the average speed onweekend is very close to free flow speed, roughly 63 mi / h, which means there is no congestionon those days. On the other hand average speed on a work day is significantly lower as a resultif congestion during rush hour. We also can see an unusual average speed on April 10 of 2009(seventh observation). It was Good Friday. Thus, far less traffic was observed compared to atypical Friday.Figure 3(b) shows a scatter plot of measured traffic flow speed for all non-holiday Wednes-days in 2009. The measurements are taken every five minutes. We can see that congestion startsroughly at the same time at around six in the morning and lasts roughly for three hours. Thebreakdown in the evening happens less frequently.olson and Sokolov 7(a) Average Speed per day (April, 2009) (b) Speed on Wednesdays (2009) FIGURE 3 Recurrent traffic flows. The left panel (a) shows average speed as measured bythe sensor N-6041 for each day during the April 4 - March 3 of 2009 time period. Weekendsare marked by blue dot. The right panel (b) shows raw speed measurements from the sensorN-6041 for each five minute interval of every Wednesday in 2009.
Another effect to be modeled is traffic congestion that is a recurrent event, and is similarfrom one week to another. Figure 4 shows the recurrent traffic conditions grouped by the dayof the week. This observation can be used to choose Z t . In particular, an approach based onnon-parametric regression that uses historical traffic flow data. For example, Smith et al. ( )and Chiou et al. ( ) showed that the last three measured speed values, perform very well forpredicting traffic flows. In this case we can write Z t as Z t = ( α t , ˆ x t , ˆ x t − , ˆ x t − , t i me o f d ay ) T ,where ˆ x t is the filtered value of state vector x t . Then the transition dynamics p ( α t + | α t , Z t ) isgenerated by computing a weighted average of those points from historic database that fall withina neighborhood of Z t . To calculate the neighborhood, we calculate distance between Z t and eachof the point from the data base, and then choose k points with smallest distance. The weightsare proportional to the distance.olson and Sokolov 8 FIGURE 4 Daily Traffic Patterns, measured in 2009. Each plot shows raw speed measure-ments averaged over five minute intervals from the sensor N-6041 for a given day of theweek, with holidays and days with erroneous measurements removed from data.TRACKING TRAFFIC FLOWS DURING SPECIAL AND WEATHER EVENTS
Our model is flexible in the sense that we can handle special events or severe weather conditionsthat can upset a typical traffic patterns. To show the empirical effect of snowy weather, Figure 5compares the expected travel time (red line), which is calculated based on historical data for thelast 150 days, with the travel time on a snow day (green line), for December 11, 2013. There were1.8 inches of snow on this day, with snow starting at midnight and continuing till noon. Therewere no traffic accidents on this road segment on this day. As we can see, even a light snow in aregion, where drivers are used to driving during snow days can cause major delays. The yellowregion is the 70% confidence interval based on historical data.olson and Sokolov 9(a) North Bound (b) South Bound
FIGURE 5 Impact of light snow on travel times on I-55 near Chicago on December 11,2013. Both plots show travel time on a 27 mile stretch of highway I-55 between I-355 andI-94 in both north and south directions. The north bound direction is from southwestsuburbs to the city. The red line is a travel time averaged over previous 150 days, theyellow area show 70% confidence interval for the data and green line is the travel time onthe day of the event. Source:
Special events is another potential cause of unusual traffic conditions. Figure 6 showsimpacts of special events on travel times on interstate I-55 north bound (towards the city). Theweekday football game, which takes place at Solder Field stadium in Chicago downtown, com-bined with typical commute traffic has a very significant impact on travel times. Weekend specialevents have a relatively minor negative impact. On the other hand, the NATO summit that washeld in Chicago’s McCormick Place located slightly south of downtown on Monday, had posi-tive impact on travel times. This can be explained away by regular commuters, who knew aboutthe event, changing their departure times, using commuter rail or simply working from homeon this day.olson and Sokolov 10(a) NATO Summit on Sunday May 20, 2012 (b) NATO Summit on Monday May 21, 2012(c) New York Giants at Chicago Bears on (d) Baltimore Ravens at Chicago Bears onThursday October 10, 2013 Sunday November 10, 2013
FIGURE 6 Impact of special events on I-55 north bound travel times. All plots show traveltime on a 27 mile stretch of highway I-55 between I-355 and I-94 in both north (towards thecity) and south directions. All plots compare travel times on the day of the special event(red line) with the travel time on the same day of the week averaged over the previous 150days (blue line).
BUILDING A PARTICLE FILTER
Given our state-space model, the goal is to provide an on-line algorithm, for finding, in an on-linefashion, the set of joint filtered posterior distributions p ( x t , α t | y t ) using a two-step procedurefor both x t , α t , the traffic flow state, and switching variable at each time point. The advantageof particle filtering is that we simply re-sample from the predictive distribution of the currentposterior and then propagate, to the next set of particles, to generate the approximation to theconditional posterior update. Appendix A provides a review of particle filtering and learningmethods.From a probabilistic viewpoint, we can re-write equations (1) - (2) as a hierarchical model ( y t + | x t + ) ∼ N ( H x t + + γ T z t + , V α t + ) (4) ( x t + | x t , α t + ) ∼ N ( G α t + x t + ( I − G α t + ) µ , W α t + ) . (5)Now, suppose that we are currently at time t . We assume a particle approximation forolson and Sokolov 11the joint posterior of ( x t , α t ) is of the form p N ( x t , α t | y t ) = N (cid:88) i = w ( i ) t φ ( x t , m ( i ) t , C ( i ) t ) δ α ( i ) t ( α t ) where y t = ( y , ..., y t ) and δ α ( · ) is a Dirac measure and w ( i ) t is a set of particle weights, whichwe will provide recursive updates for. We denote the Kalman moments by s t = ( m t , C t ) whichform a set of conditional sufficient statistics for the state. We describe the recursions later. Here φ ( x , µ , C ) denotes a normal density with mean µ and covariance C φ ( x , µ , C ) = ( π ) k | C | exp (cid:18) − ( x − µ ) T C − ( x − µ ) (cid:19) .To assimilate the next measurement, we need to find an updated posterior for ( x t + , α t + ) ,with approximate weights w ( i ) t + and particles ( x t + , α t + ) of the form p N (cid:0) x t + , α t + | y t + (cid:1) = N (cid:88) i = w ( i ) t + φ (cid:128) x t + , m ( i ) t + , C ( i ) t + (cid:138) δ α ( i ) t + ( α t + ) ,Our weights will be updated using the predictive likelihood, p ( y t + | s ( i ) t ) , which are re-normalized.We aim to provide a posterior with the same number of particles N with mixture weights of theform w ( i ) t + = w ( i ) t p ( y t + | ( α t , s t ) ( i ) ) (cid:80) Ni = w ( i ) t p ( y t + | ( α t , s t ) ( i ) ) . Recursive Updating
At time zero, we set an initial state distribution p ( x | α ) as follows p ( x | α ) = N (cid:88) i = w ( i ) α φ ( x , µ ( i ) α , C ( i ) α ) We take p ( α = s ) = / s ∈ {−
1, 0, 1 } .Conditional on the hidden switching state α t , the Kalman filter recursions, imply thatfiltered posterior distribution at time t is also mixture multivariate normal, i.e p N ( x t | α t , y t ) = N (cid:88) i = w ( i ) t φ ( x t , m ( i ) t , C ( i ) t ) where ( m t , C t ) ( i ) are functions of the whole path α t , y t .The goal is to find the next filtered posterior p ( x t + | y t + ) , which is obtained from themarginal of the joint posterior p ( x t + , α t + | y t + ) .Given, that x t ∼ N ( m t , C t ) , from evolution equation (5), it follows that p ( x t + | α t + , s t ) = φ ( x t + , G α t + m t + ( I − G α t + ) µ , G α t + C t G T α t + + W α t + ) olson and Sokolov 12To implement this algorithm, first compute the predictive likelihood of the next obser-vation y t + given α t + , s t , where s t = ( m t , C t ) is the sufficient statistics. This can be done bymarginalizing out x t + from measurement equation (4). We obtain a marginal predictive distri-bution p ( y t + | α t + , s t ) = (cid:90) p ( y t + | α t + , x t + ) p ( x t + | α t + , s t ) d x t + = (cid:90) φ ( y t + , H x t + + γ T z t + , V α t + ) φ ( x t + , G α t + m t + ( I − G α t + ) µ , G α t + C t G T α t + + W α t + ) d x t + = φ (cid:128) y t + , µ yt + , V pt + (cid:138) .where µ yt + = H ( G α t + m t + ( I − G α t + ) µ ) + γ T z t + , V Pt + = V α t + + H T (cid:16) G α t + C t G T α t + + W α t + (cid:17) H and V α t + and W α t + are given variance-covariance matrices.We can further marginalize out α t + using the transition kernel p ( α t + | α t , Z t ) . This leadsto 3-component mixture predictive model of the form p ( y t + | x t , α t ) = (cid:88) α t + ∈{ − } φ ( y t + , µ yt + , V pt + ) p ( α t + | α t , Z t ) Our model, therefore allows for heavy-tails and non-Gaussianity in the traffic flow evolution.We now show how this can be used to implement a particle filter and learning algorithm andtrack the filtered posterior distributions of the hidden state p ( x t | y t ) over time as new data y t + arrives.Given ( α t + , y t + ) , we need to update s t = ( µ t , C t ) T , where we suppress the index i forclarity. These updates are given by Kalman recursion operator, (cid:75) , which is given by µ ft = G α t + µ t + ( I − G α t + ) µ , C ft = G α t + C G T α t + + W α t + µ t = µ ft + K t ( y t − H µ ft ) , C t = ( I − K t H t ) C ft with Kalman gain matrix K t = C ft H T ( H C ft H T + V ) − . Now we are in a position to find the predictive density in equation (6), namely p ( y t + | α t , s t ) ,which is a 3-component mixture of Gaussians. This will lead to an efficient Rao-Blackwellisedparticle filter Algorithm. Particle Filtering for traffic flows : Step 1 (Draw) an index k t ( i ) ∼ M u l t i ( w t + ) -distribution with weights w ( i ) t + = p (cid:128) y t + | s ( i ) t (cid:138) / N (cid:88) i = p (cid:128) y t + | s ( i ) t (cid:138) . (6)olson and Sokolov 13 Step 2 (Propagate) switching state α ( i ) t + ∼ p ( α t + | α k t ( i ) t ) Step 3 (Propagate) sufficient statistics s k t ( i ) t using assimilated data and Kalman filter recur-sion s ( i ) t + = (cid:75) ( s k t ( i ) t , α ( i ) t + , y t + ) (7)The weights are updated according to the following rule w ( i ) t + = w ( i ) t p ( α t + | α ( i ) t ) (cid:80) Ni = w ( i ) t p ( α t + | α ( i ) t ) Finally, we draw new state vector x t + from its mixture multivariate normal distribution. TRACKING TRAFFIC FLOW ON INTERSTATE I-55Dataset Description
The data was provided by the Lake Michigan Interstate Gateway Alliance( http: // / ) formally Gary-Chicago-Milwaukee Corridor (GCM). The dataare measurements from the loop-detector sensors installed on interstate highways. A loop detec-tor is a very simple presence sensor that senses when a vehicle is on top of it and generates anon / off signal. There are slightly more then 900 loop-detector sensors that cover a large portionof the Chicago metropolitan area. Every 5 minutes a report for each of the loop detector sensorsis recorded. Our data contains averaged speed , flow , and occupancy . Flow is defined as the numberof off-on switches. Occupancy is defined as percentage of time a point on a road segment wasoccupied by a vehicle, thus it varies between 0 (empty road) to 100 (complete stand still). Weassume a constant vehicle length, and treat speed derived from a single loop-detector occupancymeasurement as the true traffic flow speed. Numerical Experiments
Consider a single road segment. We use measured data for a 24-hour period taken on a weekday. The segment we consider is a part of highway I-55 north bound. This part of the highwayis heavily congested during the morning rush hour, mostly due to commuters, who travel fromthe south-west suburbs to the central business district of Chicago. There were no special eventson that day and the weather was clear, thus a very similar congestion pattern can be observedon any “usual” work day on this road segment. The measurements are made by a single loopdetector. For this example we calculate the filtering distribution for the travel speed, flow regime,and rate of change variables on this segment. The time series of measured speeds is of length N = ( × ) .The state x t = ( θ t , β t ) T ∈ R is a true travel speed and associated trend coefficient. Theparameters of the observation and evolution models are set to: H = ( ) , V α t = F = W α t = (cid:18) (cid:19) ,Given that we did not have access to manufacturer’s specifications of the loop detectors, we usea value for the measurement error within the guidelines of the specification. The error for theevolution equation was chosen using maximum a posteriori mode estimation based on the datafrom the previous 30 days.olson and Sokolov 14In our simulation study, we define a transition matrix P with equilibrium probabilities π P = π that calibrate well to the three states in practice. Thus, we construct a Markov-switchingprocess with the following probability transition matrix P α t = ,where α t ∈ { b r eak d ow n , f r e e f l ow , r e c ov e r y } . The transition probabilities were fittedusing a maximum a posteriori mode estimator. F was fitted using data from times when trafficis stationary, and W α t was fitted using the data from both the stationary and non-stationaryregimes.Figure 7 shows the filtered speed and its quantiles, along with measured data. We seethat the filtered state curve more-or-less follows the measurement curve. The evolution modelproposed in this paper is very general and allows large changes in the speed state. Thus, thejittering behavior of measurement get mostly explained by statistical model and does not getfiltered out. S peed [ m / s ] yE[x|y]q025q975 FIGURE 7 Filtered traffic speed given loop detector measurements.
Figure 8 shows the filtered probability of α for each of the values (0,1,-1). We can see thatthe algorithm accurately captures the changes in the flow regime, assigns high probability to freeflow regime before the morning peak and the shifts probability to the breakdown regime.olson and Sokolov 15 P ( a = ) P ( a = ) Time P ( a = - ) FIGURE 8 Filtered value of P ( α = i ) , i =
1, 2, 3
Figure 9 shows the filtered values for rate of change of speed during recovery and break-down regimes. The algorithm captures all of the changes in traffic flow change rates. The algo-rithm captures “fast” breakdown a little after 6am and “slow” breakdown at around 10am. It alsocaptures, for example, the recovery between 2pm and 4pm.(a) Recovery (b) Degradation
FIGURE 9 Rate of change of traffic flow olson and Sokolov 16 −0.50.00.51.01.50 5 10 15 20
Time T r a ff i c S t a t e −0.50.00.51.00 5 10 15 20 Time T r a ff i c S t a t e (a) Naive Filter 1 (d) Naive Filter 2 −0.40.00.40.80 5 10 15 20 Time T r a ff i c S t a t e Time T r a ff i c S t a t e (c) Naive Filter 3 (d) Our Filter FIGURE 10 Comparison of filtered value of α with traffic states calculated using threeother naïve filters filters Naive filter 1 (mean filter): µ i = w (cid:80) i − j = i − w y i , y fi = y i − µ i µ i Naive filter 2 (simple smoothing): y fi = y i − y i − y i − Naive filter 3 (quantile filter): Y i = { y i − w , ..., y i − } Q i : = P r ( Y i ≤ y i ) = y fi = y i − Q i Q i The results of numerical experiments illustrate the following features. The filtered speedplot follows the measurement plot, which is an expected result. We have chosen a well-behavingsensor for the study and there is no outliers in the measurements. The filtering algorithm prop-erly identifies rate of change during break down and recovery regime. We can see that breakdownolson and Sokolov 17happens faster. This is a well known fact, that it takes less time for a queue to build up than todissipate. This difference in time can be explained by driver’s behavior and the fact that vehiclesacceleration rate is lower then deceleration rate. However, the results for filtered probabilities ofthe switching variable α t are less intuitive. Our filter properly identifies free flow regime in themorning, but then gets “slightly confused” during morning rush hour by assigning very closeprobabilities to recovery and breakdown regimes. We interpret this as saying that the Markovswitching process model used to model P ( α t | α t + , Z t ) is misspecified and need to be refined. Thisis a topic for our future research. DISCUSSION
We propose a mixture state-space model together with a particle filter and learning for trackingthe state of traffic flow and other hidden variables such as flow regime and rate of change in thetraffic flow speed. The proposed method is flexible, in a sense that it does not require the state-space model to be Gaussian. Our approach does not rely on blind particle proposal for estimatingthe forecast distribution, but instead draws are taken from a smoothed distribution, which takesthe measurement into consideration. Thus, our filter is fully adapted with exact samples from thefiltering distribution are drawn. We used the sufficient statistics representation for state particleslead to a computationally efficient method. We formulate the traffic flow evolution equation asa hierarchal Bayesian model that is capable of capturing traffic flow discontinuities.Although we have focused on representation of state-space dynamics using multi-processDLM, the same approach will work for the kinematic wave theory based approach. When theevolution equation is given by the classical macroscopic traffic flow model, namely Lighthill-Whitham-Richards (LWR) partial differential equation Lighthill and Whitham, Richards (
48, 49 ).A Bayesian analysis of traffic flows using the LWR model is described in Polson and Sokolov ( ).However, the analysis based on the LWR model requires an estimation of conditions for a roadsegment, which is not always available. The statistical model described in here has the flexibilityto avoid assumptions on locations of sensors.While we demonstrated the solution to the filtering problem in this paper, the fact thatwe provided a closed form solution to the filtering density and can do exact sampling from thefiltering density allows us to use the same approach for particle learning and smoothing Carvalhoet al. ( ). It works by augmenting the particle space to { x t , s t } , where s t is the sufficient statisticfor the parameter space γ , i.e. p ( γ | x t , y t ) = p ( γ | φ t ) . In the DLM setting we can learn β , byremoving the evolution equation for this parameter, rather then learning it directly from data,without relying on any model. Further research into predictive performance of such models iswarranted. PARTICLE FILTERING AND LEARNING METHODS
Particle filtering and learning methods are designed to provide state and parameter inference viathe set of joint posterior filtering distribution obtained in an on-line fashion [ Gordon et al. ( ),Carpenter et al. ( ),Pitt and Shephard ( ), Storvik ( ), Carvalho et al. ( ) ] .Let y t denote the data, and θ t the state variable, in our context use ( x t , α t ) . Let φ denotethe unknown parameters. For the moment, we suppress the conditioning on the parameters φ .We will show how to update state variables and sufficient statistics for φ . First, we factorize theolson and Sokolov 18joint posterior distribution of the data and state variables both ways as p ( y t + , θ t + | θ t ) = p ( y t + | θ t + ) p ( θ t + | θ t )= p ( y t + | θ t ) p ( θ t + | θ t , y t + ) The goal is to obtain the new filtering distribution p ( θ t + | y t + ) from the current p ( θ t | y t ) .A particle representation of the previous filtering distribution is a random histogram of draws.It is denoted by p N ( θ t | y t ) = / N (cid:80) Ni = δ θ ( i ) t , where δ is a Dirac measure. As the number ofparticles increases N → ∞ the law of large numbers guarantees that this distribution convergesto the true filtered distribution p ( θ t | y t ) .In order to provide random draws of the next distribution, we first resample θ t ’s usingthe smoothing distribution p ( θ t | y t + ) ∝ p ( y t + | θ t ) p ( θ t | y t ) obtained by Bayes rule. Thus, we draw θ k ( i ) t by drawing the index k ( i ) from a multinomialdistribution with weights w ( i ) t = p ( y t + | θ ( i ) t ) (cid:80) Nj = p ( y t + | θ ( j ) t ) .We set θ ( i ) t = θ k ( i ) t and “propagate” to the next time period t + p ( θ t + | y t + ) = (cid:90) p ( θ t + | θ t , y t + ) p ( θ t | y t + ) d θ t .Given a particle approximation { θ ( i ) : 1 ≤ i ≤ N } to p N (cid:0) θ t | y t (cid:1) , we can use Bayes rule to write p N (cid:0) θ t + | y t + (cid:1) ∝ N (cid:88) i = p (cid:128) y t + | θ ( i ) t (cid:138) p (cid:128) θ t + | θ ( i ) t , y t + (cid:138) = N (cid:88) i = w ( i ) t p (cid:128) θ t + | θ ( i ) t , y t + (cid:138) ,where the particle weights are given by w ( i ) t = p (cid:128) y t + | θ ( i ) t (cid:138)(cid:80) Ni = p (cid:128) y t + | θ ( i ) t (cid:138) .This mixture distribution representation leads to a simple simulation approach for propagatingparticles to the next filtering distribution.The algorithm consists of two steps:Step 1. (Resample) Draw θ ( i ) t ∼ M u l t N (cid:128) w ( ) t , ..., w ( N ) t (cid:138) for i =
1, ..., N Step 2. (Propagate) Draw θ ( i ) t + ∼ p (cid:128) θ t + | θ ( i ) t , y t + (cid:138) for i =
1, ..., N .olson and Sokolov 19To implement this algorithm, we need the predictive likelihood for the next observation, y t + , given the current state variable θ t . It is defined by p (cid:0) y t + | θ t (cid:1) = (cid:90) p (cid:0) y t + | θ t + (cid:1) p (cid:0) θ t + | θ t (cid:1) d θ t + .We also need the conditional posterior for the next states θ t + given ( θ t , y t + ) . It is given by p (cid:0) θ t + | θ t , y t + (cid:1) ∝ p (cid:0) y t + | θ t + (cid:1) p (cid:0) θ t + | θ t (cid:1) .This algorithm has several practical advantages. First, it does not suffer from the prob-lem of particle degeneracy which plagues the standard sample-importance resample filtering al-gorithms. This effect is heightened when y t + is an outlier. Second, it can easily be extendedto incorporate sequential parameter learning. It is common to also require learning about otherunknown static parameters, denoted by φ . To do this, we assume that there exists a conditionalsufficient statistic s t for φ at time t , namely p ( φ | θ t , y t ) = p ( φ | s t ) where s t = s ( θ t , y t ) . Moreover, we can propagate these sufficient statistics by the deterministicrecursion s t + = S ( s t , θ t + , y t + ) , given particles ( θ t , φ , s t ) ( i ) , i =
1, . . . , N . First, we resample ( θ t , φ , s t ) k ( i ) with weights proportional to p ( y t + | ( θ t , φ ) k ( i ) ) . Then we propagate to the nextfiltering distribution p ( θ t + | y t + ) by drawing θ ( i ) t + from p ( θ t + | θ k ( i ) t , φ k ( i ) , y t + ) , i =
1, . . . , N .We next update the sufficient statistic for i =
1, . . . , N , s t + = S ( s k ( i ) t , θ ( i ) t + , y t + ) .This represents a deterministic propagation. Parameter learning is completed by drawing φ ( i ) using p ( φ | s ( i ) t + ) for i =
1, . . . , N . We now track the state, θ t , and conditional sufficient statistics, s t , which will be used to perform off-line learning for φ .The algorithm now consists of four steps:Step 1. (Resample) Draw Index k t ( i ) ∼ M u l t N (cid:128) w ( ) t , ..., w ( N ) t (cid:138) for i =
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