TRAVEL TIME ESTIMATION FOR URBAN ROAD NETWORKS USING LOW FREQUENCY PROBE VEHICLE DATA

Transcription

1 TRAVEL TIME ESTIMATIO FOR URBA ROAD ETWORKS USIG LOW FREQUECY PROBE VEHICLE DATA Erik Jeneius Corresponding author KTH Roya Institute of Technoogy Department of Transport Science Emai: Brinevägen 3, SE Stockhom, Sweden Te: Fax: Haris. Koutsopouos KTH Roya Institute of Technoogy Department of Transport Science Emai: haris.koutsopouos@abe.kth.se Submitted for pubication in Transportation Research Part B December 01 Revised February 013, March 013

2 ABSTRACT The paper presents a statistica mode for urban road network trave time estimation using vehice trajectories obtained from ow frequency GPS probes as observations, where the vehices typicay cover mutipe network inks between reports. The network mode separates trip trave times into ink trave times and intersection deays and aows correation between trave times on different network inks based on a spatia moving average (SMA) structure. The observation mode presents a way to estimate the parameters of the network mode, incuding the correation structure, through ow frequency samping of vehice traces. Link-specific effects are combined with ink attributes (speed imit, functiona cass, etc.) and trip conditions (day of week, season, weather, etc.) as expanatory variabes. The approach captures the underying factors behind spatia and tempora variations in speeds, which is usefu for traffic management, panning and forecasting. The mode is estimated using maximum ikeihood. The mode is appied in a case study for the network of Stockhom, Sweden. Link attributes and trip conditions (incuding recent snowfa) have significant effects on trave times and there is significant positive correation between segments. The case study highights the potentia of using sparse probe vehice data for monitoring the performance of the urban transport system. KEYWORDS: trave time; network; estimation; probe vehice; ow frequency samping

3 1. ITRODUCTIO Many urban road transport systems today experience increasing congestion that threatens the environment and the transport efficiency. To tacke these probems, knowedge about traffic conditions is critica at many eves of traffic management and transport poicy. Through information and personaized advice, individuas and feet management companies can pan their trips more accuratey and increase the efficiency of the system. For traffic management, speed information at the segment eve can revea probematic ocations where new or revised traffic contro schemes may be introduced to increase performance. For transport poicy, network-wide trave time information provides input for trave demand forecasting and impact assessments of poicy instruments such as congestion charges. There are a number of we-estabished technoogies for coecting speed and trave time data, incuding oop detectors, automatic vehice identification (AVI) sensors and, probe car data. Loop detectors and AVI sensors have the merit that they, once instaed, continuousy record every vehice passing the monitored road section. However, the share of segments in the network equipped with these sensors is typicay ow and not representative of the urban network as a whoe, which eaves the traffic conditions in most of the network unknown. Dedicated probe vehices, meanwhie, are used to coect trave time and other data for designated routes in the network. However, due to cost considerations the number of traffic studies with probe vehices is typicay sma and the number of vehices invoved very ow. Hence, they can ony cover a imited number of routes for a imited duration of time. Most recenty, GPS devices, aready instaed for other purposes in vehice feets (e.g., taxis, commercia vehices, service vehices, etc.) or smartphones, provide a new type of traffic sensor. These opportunistic sensors have a great potentia for provision of data for traffic management appications. Unike stationary sensors, they can coect trave time and speed data for any part of the network where equipped vehices move, without the need for traffic fow measurements. Unike designated probe cars, they can continuousy coect data for any time and day that equipped vehices are active. Liu et a. (009) investigated the feasibiity of using data from taxi dispatching systems to coect reiabe traffic information. They concuded that rea-time detection of congestion on inks is possibe, provided that an adequate database of historica traffic conditions has been constructed and that sufficienty ong road segments are considered. However, a number of imitations mean that new sophisticated methods are needed to process the data and generate usefu information, compared to traditiona sensors (Leduc, 008). The most significant chaenge is that samping (aso caed reporting, or poing) frequencies are often ow (ess than one per minute), so that vehices may have traversed significant distances between reports. For arteria networks, ow samping frequency creates difficuties in inferring the true path of the vehice between two position reports, which may invove a considerabe number of network segments (Rahmani and Koutsopouos, 013; Miwa et a., 008). Furthermore, the fraction of the reported trave time that is spent on each individua segment is not observed, which creates chaenges for network trave time estimation. The probem considered in this paper is to estimate the trave time for any route between any two points in the network under specified trip conditions; both the mean trave time and the variabiity are of interest, considering that ink trave times aong the route may be correated. The observations used for the estimation come from probe vehices that trave on the network, reporting the time and their positions at certain intervas. The samping frequency is assumed to be ow, in the sense that the distances between consecutive reports are typicay onger than the scae desired to estimate trave times. The ony information considered is the observed trave times and distances between reports; thus, data on, for exampe, instantaneous speeds, are not avaiabe. This may be a common case due to imitations in communication bandwidth. The iterature on trave time estimation and forecasting using probe vehice data has grown in recent years as the technoogy has become more avaiabe. Most papers, however, have deat with high fre- 3

4 quency probe data, (e.g., Zou et a. 005; Work et a., 008), which eiminates many of the chaenges of interest here. Further, much of the research has focused on highway traffic estimation, in particuar on the use of probe vehice data for onine estimation of macroscopic fow modes (e.g., anthawichit et a, 003; Work et a., 008; Yuan et a., 01). Most proposed methods for trave time mode estimation using ow-frequency data divide the process into two steps, performed either once or iterativey. First, each probe vehice trave time observation is decomposed into a trave time for each traversed network segment. This is sometimes known as the trave time aocation probem (Hofeitner et a., 01a; Zheng and van Zuyen, 01). Second, the trave time distributions for the network components are estimated or cacuated empiricay. Heuristic methods for the trave time aocation are proposed by Heinga et a. (008), and Zheng and van Zuyen (01). Hunter et a. (009) present a probabiistic mode of trave times in the arteria network, based on ow frequency taxi GPS probes. The mode takes into account that the path between two consecutive position reports may contain mutipe segments, and the authors formuate a maximum ikeihood probem to estimate the segment trave time distributions based on the set of observed route trave times. A simuation-based EM estimation agorithm is proposed, which iterates between trave time aocation and parameter estimation. The authors assume that the trave times on different segments are independent and report estimation resuts using norma and og-norma distributions. A deveopment of the approach in Hunter et a. (009), more ceary aimed towards trave time forecasting, is presented in Hofeitner et a. (01a). The mode assumes that each segment can be in one of two possibe states (congested or uncongested), each with its own conditiona, independent norma trave time distribution. The transitions between states among neighboring segments are modeed as a dynamic Bayesian network mode. The unobserved state and transition probabiities and the trave time distribution parameters are estimated in a simuation-based EM approach. A further deveopment using trave time distributions inspired by traffic theory, state variabes representing the number of queuing vehices on each ink, and turn fractions at intersections, is presented in Hofeitner et a. (01b). Another approach, using ow frequency GPS data from ambuances, is presented in Westgate et a. (013). In the paper, path inference and trave time estimation are performed simutaneousy using a Bayesian approach. The framework makes use of instantaneous speed information reported by the vehices. The trave times on the road inks are assumed to be independent and og-normay distributed, and the parameters are estimated using Markov chain Monte Caro methods. Athough the previous work ceary demonstrates the possibiity to extract usefu information from ow-frequency probe vehice data, the proposed methods have certain drawbacks. First, the formuation of the trave time aocation probem requires that each network segment has a trave time distribution with its own unique parameters. This means that significant amounts of data may be required for a parts of the network to obtain reiabe and significant estimates of a mode parameters. Second, the trave times on different segments are assumed to be independent conditiona on the state of the system (which is typicay hed fixed in 15-minute or onger intervas). Since trave times are often correated across inks (Bernard et a., 006; de Fabritiis et a., 008; Ramezani and Geroiminis, 01), this may ead to incorrect estimates of the trave time variabiity on onger routes. Third, the modes do not incorporate any structura information about factors having an effect on trave times, which means that they cannot be used to predict the impacts of poicy measures or other variabes infuencing trave conditions. The methodoogy proposed in this paper extends previous work on trave time estimation using probe vehice data by considering the effect of expanatory variabes on trave times. For the network components themseves this may incude attributes such as speed imit, number of anes, functiona cass, 4

5 bus stops, traffic signas, stop signs, eft turns, etc. The effects of the conditions for the trip, such as weather, time of day, weekday, time of year, and so on, are considered. The use of expanatory variabes is attractive for at east three reasons. First, it aows the identification of the underying causes that contribute to the variabiity in speeds between inks and points in time. This is important for system management and panning, where one needs to know the reationships between possibe instruments and network trave times in order to improve the mobiity and accessibiity in the transport system. Second, the approach reduces the number of parameters to estimate and trave times can be estimated even with ow frequency probe data and in areas with reativey few observations. This aspect has not been discussed much in previous work (it is handed impicity in Bayesian approaches) but proved essentia in practica appications of the probe vehice data source used in this paper. Third, the integration of trip conditions in the mode makes it possibe to extend the historica estimation to prediction of future trave times based on forecasted conditions. Furthermore, the methodoogy extends previous work by utiizing the sequences of position reports from the same vehice to incorporate and estimate the correation experienced by the vehice traversing the segments sequentiay on a trip. The network mode assumes that segment trave times have a mutivariate norma distribution according to a spatia moving average (SMA) structure. The observation mode takes into account that the correation between segments is incorporated not ony in each probe vehice trave time observation, but in the entire sequence of observations from the same vehice trajectory. The proposed methodoogy is based on a statistica mode that consists of two ayers: The network mode specifies the joint distribution of trave times at the network segment eve and expresses it in terms of a set of unknown parameters; the observation mode specifies the information contained in sequences of probe vehice reports and provides the ink from the network mode to the maximum ikeihood estimation of the unknown parameters. The appicabiity of the method and the impacts of various factors on mean and variance of trave times are demonstrated empiricay for a network in Stockhom, Sweden, using probe vehice data from a taxi feet. The case study highights the potentia in probe vehice data for estimating traffic conditions even without the avaiabiity of cassica traffic data such as fows. The probem of determining the true positions and paths of the vehices from noisy GPS data is not considered in this paper. It is assumed that the most ikey network ocation (i.e., the network segment and position aong segment) corresponding to each GPS measurement, as we as the path (i.e., the sequence of network segments) taken between each pair of consecutive measurements, have been determined by a map-matching and path inference process (Rahmani and Koutsopouos, 013; Miwa et a., 01; Hunter et a., 01). The paper is organized as foows. The network mode is presented in Section and the observation mode is presented in Section 3. Section 4 discusses some practica considerations regarding the specification and estimation of the mode. This is foowed by a description of a rea-word appication in Section 5 and a concuding discussion in Section 6.. ETWORK MODEL The trave time of a trip is assumed to consist of two components: 1. Running trave time aong inks,. Deay at intersections and traffic signas (turns). 5

6 A ink is defined to be the road section between two adjacent intersections or traffic signas (traffic signas are not aways ocated at intersections). A ink may be divided into one or more segments, with each segment part of one specificc ink. S and L denote the tota number of segments and inks in the network, respectivey. The reationship between segments and inks is described by an S L incidence matrix S, so that eement S is 1 if segment s beongs to ink and 0 otherwise. Since a segment can ony beong to a s singe ink, S s = 1 S = 1 s, ( s). for a s. The ink ( s ) of segment s is identified as the ink such that Whie the inks are argey determined by the inherent network structure, the number of segments per ink depends on the traffic characteristics of the ink. Segments are designed to capture homogeneous traffic behavior. In this mode the average speed of a vehice can vary between segments but is assumed to be constant aong each segment. The trave time on a segment s is presented as the ength of the segment, s, mutipied with the inverse speed or trave time rate X s. As described in the foowing subsections, the trave time rate may depend on observed and unobserved properties of the segment and conditions for the trip. The second trave time component of a trip is intersection and traffic signa deay. Let turn t define the movement from a ink 1 to the downstream ink, and T denote the tota number of turns in the network. A turn can thus be defined by the pair ( 1, ). Turn t is assumed to impose a trave time penaty h t. Factors that infuence the magnitude of h t may incude the type of traffic contro in the intersection, whether it invoves a eft or a right turn, time of day, etc. Conceptuay, turns can be seen as inks having zero ength, as iustrated in Figure 1. This means that the probabiity of a vehice reporting its position on a turn ink is zero. Whie one can readiy incorpo- sets of variabes for rate both types of components in a singe set of variabes, in this paper separatee ink running times and turn penaties are used for ease of exposition. Turn t Link Link 1 Segment s Figure 1: Iustration of the three types of network components: Links, segments and turns. 6

7 X s, s = 1,, S The segment trave time rates, are modeed as stochastic variabes, in genera not independent. Both mean trave times and the variabiity around the mean vaues are of interest. Assuming that the mean vaue is finite, X s = gs + ν s, where g s is the mean trave time rate and ν s is a stochastic component with E[ ν s ] = 0 capturing the variabiity around the mean. For compact notation, it is convenient to introduce the S 1 vector X with eements X s. Then X = g + ν, (1) where g is the vector of mean segment trave time rates, and ν is the vector of zero-mean stochastic terms. Conditiona on trip conditions, the mode assumes that the stochastic components of the segment trave time rates ν foow a mutivariate norma distribution according to a covariance structure defined in Section.. The turn penaties can be treated as deterministic or stochastic variabes. In reaity the deay at an intersection certainy varies according to unobserved changes in traffic fows, signa cyces etc. For estimation purposes, however, it is difficut to separate the tota trave time variabiity into variabiity in running trave times and in turn deays. Hence, in the appication in this paper turn deays are treated as deterministic trave time penaties. The turn penaties h t are represented by the T 1 vector h..1 Mean structure Vectors g and h may be further expressed as functions of a number of factors with associated parameters to be estimated from data. The parametric structure shoud refect the way different factors affect trave times, whie aso aowing for convenient and efficient estimation. The structure can aso be used to partition the segments into arger groups to ensure that a parameters can be identified through the avaiabe observations. The expanatory variabes for the mean segment trave time rates g capture segment characteristics that vary across the network, and trip conditions that vary in time. The expanatory factors reated to segment characteristics incude reguatory factors, such as speed imit and cassification, ink ength, nearby and use and fixed effects for specific segments. Segment fows are not specificay considered in the mode structure but can be incuded if avaiabe. The different attributes are represented by the S B design matrix B. The baseine segment trave time rates are then Bβ B, where β B is an B 1 parameter vector to be estimated. ote that the segments can be modeed as having fuy distinct means, without any other expanatory variabes, by setting B equa to the S S identity matrix. The observed trip conditions are assumed to act as a mutipier to the baseine segment trave time rates. Thus, a certain trip condition mutipies the baseine trave time rates Bβ B on a segments according to a certain percentage reative to some reference conditions. Reevant trip attributes coud incude tempora variations within the day, week and year, weather conditions, etc. For a given trip, the attributes are represented by an 1 design vector o. The mutipier for the trip conditions is then ( ) o 1+ oβ o, where β o is an o 1 parameter vector to be estimated. In tota, the mean segment trave time rates can be written as ( ) g = 1 + oβ Bβ. () o B 7 B

8 The mean turn penaties h have a simiar structure. The expanatory factors for the turn penaties, which may incude indicators for traffic signas, eft/right turns etc., are coected in the T E design matrix E. The trip conditions are assumed to infuence the turn penaties in the same way as the trave time rates. Thus, with the E 1 parameter vector β E, ( ) h = 1 + oβ Eβ. (3) o E The mode can be extended to aso aow the impact of trip conditions to differ between segments or turns. This woud be appropriate, e.g., if network-coded information is used about traffic incidents or construction works that do not cover the whoe period of observations, or if tempora variations are known to be more dominant in some parts of the network than others.. Variance structure The stochastic components ν in (1) represent the variabiity in segment trave time rates due to unobserved heterogeneity in traveer characteristics, traffic conditions, and oca network characteristics. For the purposes of this paper, the error terms are modeed at the ink eve; trave time rates on different segments in a ink are aowed to differ in means but are assumed to have the same variabiity around the mean, which impies that segment trave time rates are perfecty correated within the ink. This assumption can be reaxed if needed. The stochastic component of ink is denoted u. It foows that ν s = u ( s), or in vector notation, ν = Su. (4) In genera, the segment-eve covariance matrix can be obtained from the ink-eve stochastic components as Ω T T T = E νν = S E uu S. (5) An approach from spatia econometrics is used to mode the covariance between inks (LeSage and Pace, 009; Cheng et a., 01). The genera approach is to specify the structure for how independent stochastic components ε for a inks interact to determine the tota stochastic trave time rate components ν. In this paper, a spatia moving average (SMA) specification is used (Heppe, 004). In the SMA mode the stochastic component u of each ink is expressed as an independent term ε with zero mean and variance σ, pus a inear combination of the independent components of the other inks ε,. The independent error term ε captures the variabiity in trave time rates that E ε ε = 0 for. The eve of originates on the particuar ink. Independence impies that [ ] infuence from ink on ink, denoted w, is specified through the anaysis, whie the overa magnitude of the covariance is captured by a parameter ρ to be estimated. Assuming that w = 0 for a, the tota stochastic component of ink is thus, u = ε + ρ w ε, (6) or in matrix notation, = ( + ρ ) u I W ε. The L L weight matrix W with eements w ' captures the interaction among inks. This is important since empirica evidence suggests that the structure of 8

9 W must be defined with much care in order to represent the dependencies between ink trave time rates propery and aow a meaningfu estimation of the correations between inks. The SMA mode can be extended to incude mutipe weight matrices W, i = 1,,, with each matrix representing a separate dimension of spatia dependence (Heppe, 004). This more genera mode is usefu when there are mutipe factors that contribute to correations and are distributed differenty in the network. The structure for the stochastic trave time components is then ρ u = I + ρ i i. W ε (7) i=1 With the (extended) SMA mode (7), the covariance between two inks 1 and is ( ) ρ ρ ρ E u u 1 = ρ i wi,, σ 1 + w 1 i, 1. σ + ρ iρ j wi, 1, wj,, σ. i=1 i=1 j=1 {, } (8) As can be seen, there is a first-order term that arises from the direct infuences between the inks, and a second-order term that arises from infuences through common neighbors in the different dimensions. Let vector with eements σ. Simiary to the means, the trave time rate σ may be decomposed into a number of expanatory factors with associated parameters to σ denote the L 1 variances be estimated from data. The expanatory variabes are divided into two different categories: static ink characteristics and dynamic trip conditions. The ink characteristics may incude geometric properties and fixed effects for specific inks or groups of inks. The U variance components are represented by the L U design matrix U. The baseine ink variances are then Uσ U, where σ U is an U 1 parameter vector to be estimated. ote that the simpest mode formuation woud be that a inks share a singe variance parameter σ, in which case U is an L 1 vector of ones. In the other extreme, each ink may have an individua variance parameter σ, in which case U is the L L identity matrix. Further, the observed trave conditions for the trip may impact the trave time variances as we as the means. Reevant trip attributes may be simiar as for the mean trave time rates, but the impact of each attribute may be different. For a given trip, the attributes are represented by the 1 design vector p. The mutipier for the trip conditions is then ( 1+ pβ ) p, where p 1 i p ρ β is an 1 parameter vector to be estimated. ote that the parameters actuay capture the effect on the square root of the variance, i.e., the standard deviation, which is convenient since it has the same dimension as the mean. The variances of the independent stochastic components ε are thus in tota ( ) σ = 1 + pβ Uσ. (9) P U Let σ U, n be the n th ink variance component and U n the diagona matrix with the n th coumn of U aong its diagona. The covariance matrix of ε is diagona and given by p 9

10 T ( U ) P σ U, n n n=1 E εε = 1 + pβ U. (10) Combining (5), (7) and (10), the covariance matrix for the network segments becomes U ( ) ρ ρ ρ T T T T T T Ω = 1 + pβp σ U, n SUnS + ρi ( i n + n i ) + ρiρ j i n j. SW U S SU W S SW U W S (11) n=1 i=1 i=1 j=1 The SMA mode (7) does not take into account that there may be dependencies in trave times due to unobserved variations between vehices or drivers, or perhaps more importanty, between different days. However, it is straightforward to extend the mode to capture such dependencies by incuding a stochastic component at the day or vehice eves. 3. OBSERVATIO MODEL The trave time measurements considered in this paper consist of sparsey samped vehice trajectories through the network obtained from GPS devices or simiar sensor technoogies. In genera, GPS ocation measurements are associated with errors. Here it is assumed that the most ikey network ocation corresponding to each GPS measurement, as we as the path (i.e., the sequence of network segments) taken between each pair of consecutive measurements, have been determined by a map-matching and path inference process (Rahmani and Koutsopouos, 013). A basic observation then consists of 1. a vehice identification number,. a pair of time stamps τ 1, τ, 3. a path representing the trajectory of the vehice between the two time stamps, invoving a sequence of segments ( s1,..., s n) and two offsets δ 1, δ specifying the vehice ocations at times τ 1, τ in reation to the upstream nodes of the first and ast segments of the path, respectivey. The concepts are iustrated in Figure. The sequence of segments defines the corresponding sequences of inks ( ( s ), 1, ( sn )) and turns ( t1,, t m ), respectivey, according to the network mode. 10

11 δ h X 4 X 5 τ X 3 τ 1 1 δ 1 Let X 1 X h 1 Figure : Iustration of the components of a probe vehice trave time observation. d s denote the distance traversed on segment s. With s denoting the ength of segment s, δ δ1 i f n=1, s = s1, s δ1 i f n>1, s = s1, ds = s i f n>1, s = s,, sn 1, (1) δ i f n>1, s = sn, 0 o therwise. Further, et a t be equa to 1 if turn t was undertaken and 0 otherwise. The trave time observation y = τ τ can then be written as 1 S T (13) y = d X + a h. s s t t s=1 t =1 Based on this mode, a probe vehice trave time observation is a inear combination of segment trave time rates and turn deays. Furthermore, under the assumption of mutivariate norma segment trave time rates and deterministic turn deays, the observed trave time foows a norma distribution. The cosed-form distribution of the observed trave time aows the network mode parameters to be identified directy through the probe vehice observations. This attractive property is not preserved if other distributions than the mutivariate norma distribution are used. The trave time is given by y = µ + η, (14) where µ is the mean trave time given by [ ] S T (15) µ = E y = d g + a h, s s t t s=1 t =1 that is, the sum of the mean segment trave times pus the turn deays. The zero-mean stochastic term η is normay distributed and given by 11

12 S L (16) η = d ν = d u, s s s=1 =1 where d = S s s d s is the distance traveed on ink. The variance of η, and hence of observation y, is cacuated from the variances and covariances of the traversed inks as L ρ L L [ η] d σ + ρi d d 1 ( wi,, σ + wi,. σ ) Var = =1 i=1 1 =1 = ρ ρ L L ρiρ j d d 1 w i, 1, wj,, σ. i=1 j=1 =1 = + 1 {, } (17) The first term is the sum of the ink trave time variances, which woud be the ony term if the inks were independent. The second term contains the direct dependencies between the traversed inks, and the third term contains the second-order dependencies through common neighbors, on or off the traversed path. Consecutive observations from the same vehice are correated through their common inks and whenever the inks traversed in the different observations are correated. Incorporating this in the mode heps the consistent estimation of the ink dependence parameters ρ i. A trace is a contiguous sequence of observations from the same vehice as it moves through the network. Observations from the same trace are correated through the ink correation structures, whie observations from different traces are considered independent. Figure 3 iustrates the reationship between observations and traces. Traces and observations k = 1, r = 3 k = 1, r = 4 k = 1, r = k =, r = 3 k = 1, r = 1 k =, r = k =, r = 1 Figure 3: Iustration of two traces (k = 1, ) each containing four and three observations, respectivey. 1

13 Let index k = 1,, K represent a certain vehice trace, where K (upper-case K ) is the number of traces in the data. Index r = 1,, k represents a certain observation in trace k, where k is the number of observations in the trace. The tota number of observations in the data is = K R k =1 k. D k is an k S matrix where eement d rs is the distance traversed on segment s for observation r. A k is an k T matrix with eement a rt equa to 1 if turn t is made in observation r and 0 otherwise. y k is an k 1 vector with eement y r the trave time of observation r (dependent variabe). The vector version of (13) is then given by y = D X + A h. (18) k k k The trave times y k are a inear transformation of mutivariate norma stochastic variabes and are thus distributed mutivariate norma. It foows from (18) that y = µ + η, (19) k k k where µ k is a vector of mean trave times and η k is a vector of correated zero-mean stochastic terms. Trip conditions are assumed to be the same for a observations in a trace. Thus, each trace k is associated with two vectors of trip attributes o k and p k (weather conditions, weekday, season, etc.) affecting the means and variances, respectivey. The vector of mean trave times for the trace is then simpy the vector form of (15), where the mean segment trave time rates can be expressed in the mode parameters using () and (3), ( ) [ ] ( + ) ( + ) µ β, β, β = E y = 1 oβ D Bβ A Eβ. (0) k B E o k k o k B k E Trip conditions affect a segments and observations uniformy and can be moved outside the other terms. The vector of stochastic terms for the observations in trace k is given by the vector form of (16), ηk = Dkν = DkSu. (1) ' T For two different traces k and k, E ηkη k = 0 by assumption. Within the same trace k, meanwhie, (5), (11) and (1) give the k k covariance matrix ( ) T Σk β p, σu, ρ = E ηη k k U ρ ρ ρ T T T T T T T T = ( 1 + pβ k p ) σ U, n DkSUnS Dk + ρi k ( i n + n i ) k + ρiρ j k i n j k. D S W U U W S D D SW U W S D n=1 i=1 i=1 j =1 13 () 1 + pβ to a ote that the trip conditions for the trace enter the covariance as the scaar mutipier ( ) baseine covariance matrix. This means that the correation between two observations depends on ink characteristics but not on trip conditions, since the trip conditions factor enters both the covariance and the variances and cances out. P

14 4. ESTIMATIO 4.1 Maximum ikeihood formuation Equations (0) and () provide the basis for estimation of the network mode parameters, β p, σ U and ρ using the probe vehice trave time observations. The observations are mutivariate norma within each trace and assumed independent among traces. Hence, given a observed trave times y, the og-ikeihood function is anaytica and given by K LL β, β, β, β, σ, y = og y µ Σ y µ og Σ. ( K T ρ ) ( π ) ( ) 1 ( ) β B, B E o p U R k k k k k k k =1 k =1 The computation times for parameter estimation can be reduced consideraby by pre-computing factors that do not depend on the parameter vaues outside of the optimization routine. For the mean structure, we note from (0) that the mean vector for trace k can be written as T T ( + ) ( ) T µ = 1 oβ ( D B A E) β β. (4) k k o k k B E The parameters for the mean segment trave time rates and the turn deays are thus merged into a singe parameter vector. Here ( DkB AkE ) is an k ( B + E ) matrix that is independent of the mode parameters and may be computed and stored for a traces prior to the optimization. For the covariance structure, we note from () that the mode parameters are mutipiers to constant matrices that may be pre-computed and stored for every combination of parameters. Matrix inversion is a costy operation, even if techniques such as LU or Choesky factorization are used. The SMA mode has the attractive property that it is not necessary to invert the fu L L ink-eve covariance matrix to compute the ikeihood function, but ony the k k covariance matrices for the individua traces. Since the number of observations in a trace is typicay much ower than the number of inks in the network, this means that the SMA formuation requires significanty ess computationa effort to estimate. In addition to the theoretica considerations discussed in Section this is another reason why the SMA formuation is chosen for the mode. The samping of the vehice trajectories can be interpreted as a inear projection from the space of segment trave time rates and turn penaties to the space of observed trave times. The dimension of the observation space depends on the data: the dimension increases with the samping frequency and with the number of observations, assuming that the vehice trajectories are samped at random ocations. The network mode represents another projection from the network components to the parameter space. The identification of the mode parameters depends on the reative dimensions of the parameter space and the observation space: a arger number of observations aows for the estimation of a richer mode. This determines, for exampe, to what extent fixed effects for specific segments or groups of segments can be incuded in the mode expicity. β E, βo (3) 4. etwork deimitations One may often be interested in estimating trave times in a subnetwork, here referred to as the primary network, which is smaer than the network spanned by the avaiabe GPS probes. If the primary network is sma, for exampe a singe street or road, the number of observations that traverse ony segments in the primary network may be insufficient to estimate the trave times reiaby. There may aso 14

15 be a bias since short traversed distances may be over-represented. Rather, it is important to utiize a observations that to some extent traverse at east one of the segments in the primary network (referred to as primary observations). With ow frequency probe vehice data, however, the primary observations wi invove traversas of many segments and transitions aso outside the primary network. These components, which depend on the used data, constitute the secondary network. If the primary network is sma, the size of the secondary network can be many times greater. Once the secondary network has been identified, a observations that ony traverse the secondary network, that is, do not extend the number of segments and transitions in the estimation any further, are used. These observations are referred to as secondary observations, and may be many times greater in number than the primary observations. The concepts are iustrated in Figure 4. Fu network Primary observations Primary network Secondary network Secondary observations Figure 4: Iustration of the concepts of the primary and the secondary network and observations. 15

16 4.3 Spatia custering of network inks Sparse probe vehice data may not have the resoution required to identify the trave time rate on a individua segments; indeed, this is the case in the appication presented in Section 5. The use of expanatory variabes can reduce the dimensionaity of the probem. Sti, it is very ikey in practice that there are systematic differences in segment trave time rates between different parts of the network not captured by observed segment attributes. Spatia custering of segments provides a compromise between the two extremes of fixed trave time rate effects for each segment (ink) and a singe baseine trave time rate for a segments. As a first step, segments (inks) can be grouped together into casses expected to have simiar traffic performance characteristics. A inks in the same cass share the same functiona form that characterizes their performance. Such a process can be used for grouping the inks that beong to the primary network, since they are of the most interest. The secondary network ony pays an auxiiary roe supporting the estimation of the main inks. For the inks in the secondary network the foowing process is used that appies spatia criteria to automate the grouping process. The agorithm operates at the ink eve, so that a segments in the same ink aways beong to the same group. 1. Initiaization Assign each ink to its own group Seect a min, max, and target vaue for the number of inks and the number of observations in each group. The choice of these parameters shoud provide a good baance between robust estimation and mode resoution.. Iteration Seect the group with the smaest number of observations (in case of ties, an arbitrary group is chosen). o For every ink in the current group Check whether it is connected with some ink in another group through a common node. A such identified adjacent groups are added to a ist. Rank the groups in the ist in increasing number of observations associated with each group o Merge the current group with the first adjacent group for which any of two conditions hod: The tota number of inks and the tota number of observations in the two groups do not exceed the corresponding target vaues, The number of inks or the number of observations in any of the groups is ess than the minimum vaue, and the tota number of inks and the tota number of observations in the two groups do not exceed the maximum vaues. Repeat with next group 3. Termination STOP if no groups can be merged The agorithm aims at grouping inks to baance the number of observations within each group (and therefore support estimation of the parameters of the corresponding distribution of trave time rates). Any constraint can be made non-binding by setting the corresponding threshod vaue sufficienty ow or high. The output of the agorithm can be summarized as a ink-group incidence matrix C, where eement C c is equa to 1 if ink beongs to group c and zero otherwise. Since each ink can ony beong to one group we must have C = 1 c c for a. The mapping of segments to groups is then obtained as the composite projection SC. 16

17 5. CASE STUDY The methodoogy proposed in Sections Section 3 and 4 is appied in the urban network of Stockhom, Sweden. The primary network consists ists of a route aong one of the major inner city streets, streets the southern haf of Birger Jarsgatan, southbound direction, direction shown in Figure 5. The shaded area represents the primary network. The secondary ondary network extends, in some cases, even beyond the shown area in Figure 5. The main route is chosen to coincide as cosey as possibe with a pair of automatic number pate recognirecogn tion (APR) sensors mounted at each end of the route (see further Section and Kazagi and Koutsopouos, 01). Figure 5: The casee study area in Stockhom, Sweden (Birger Jarsgatan, southbound). Shaded area shows the primary network. Map data Googe 01. The main route is about 1.4 km ong and contains 8 inks, divided into 36 segments, segments 6 intersections, and 10 traffic signas; the speed imit is constant at 50 km/h. There is a busy commercia and enterente tainment center in the midde of the route with a nearby taxi station, where it is expected that the mean trave time rates are higher than in adjacent parts, in particuar for taxis. The route ends with a compicomp cated signaized intersection in the south where deays can be significant. significant On the second haf ha of the route there is a bus ane that taxis are aowed to use. Initiay, different specifications of the mode structure are considered.. The focus is not to derive the best mode specification possibe, but to demonstrate the structure of the mode and a the impact and significance of different expanatory factors on the observed trave times during a specific time interinte va (7:30-8:00 AM). The estimated trave time for the main route under different trip conditions is aso evauated. In the second part, the estimated route trave time is compared with observed trave trav times 17

18 from the APR sensors. A sensitivity anaysis regarding the fitering of the observations is aso discussed. 5.1 Data The GPS probe vehice data are obtained from the feet dispatching system of a taxi company operating in tota about 1500 vehices in the Stockhom network (Rahmani and Koutsopouos, 013). The average samping frequency is about one report per minutes and 780 meters, which is consideraby ower than in most previousy reported studies (e.g., one per minute in Hunter et a., 009; Hofeitner et a., 01a; one per 00 meters in Westgate et a., 013). The digita network representation contains information about various geometric attributes, incuding segment speed imit, functiona cass (a five-eve hierarchica cassification of the segments with 1 denoting highways and 5 denoting the most periphera side streets), traffic signas, and faciity type (ramps, tunnes, roundabouts, etc.). The network mode is aso used to identify intersections, eft and right turns, and one-way streets. Data for weekdays (Monday to Friday) and the time interva 7:30-8:00 AM between January 1, 010 and December 31, 011 were used. For this two-year period, 63,680 observations in 44,844 traces are avaiabe after fitering. Of these, 10,604 observations are primary (that is, they cover the main route to some extent) and 53,076 secondary (covering ony the surrounding network). Across the primary and the secondary networks, the observations cover in tota 1300 segments, 83 inks and 1373 turns. On average, each observation covers 14.0 segments, 8.6 inks and 7.7 turns. Figure 6 shows the histogram of the average vehice speed for each observation, cacuated as the traversed (actua) distance divided by the time between the reports. The speed distribution has mean 1.8 km/h, median 0.4 km/h and standard deviation 9.4 km/h Fraction of observations Average speed [km/h] Figure 6: Distribution of average speeds among the observations (7:30-8:00 AM). Further, historica weather data for the same period as the probe vehice data were coected from a weather station in the area, reporting every 0-30 minutes. The data incudes temperature, quaitative precipitation information (ight/heavy rain/snow etc.), and other reated information. 18

19 Observations of ink fows were not avaiabe for this study. Severa network attributes that potentiay have significant impact on trave times were aso not avaiabe, incuding the number of anes, ocations of bus stops, pedestrian crossings, on-street parking, stop signs, traffic signa cyce times, nearby and use etc. Information about the time and ocation for incidents and road works were aso not avaiabe. If avaiabe, incusion of this type of information in the mode is straightforward. 5. Mode specification Specification of ink groups The spatia and tempora coverage of the avaiabe probe vehice data is not high enough to identify and reiaby estimate a segment specific parameters (for exampe, because of the sparse nature of the GPS probes). The number of observations covering each segment is generay ower in the secondary network than in the primary network. The network inks are therefore partitioned into groups using the manua approach for the main route combined with the automated method described in Section 4.3 for the secondary network. The main route is divided into seven groups, each containing four inks and 5.1 segments on average. The number of groups is seected so that fixed effects for the mean trave time rate of the segments in each group can be identified through the data, and the boundaries between groups are seected to capture the hypothesized heterogeneity of traffic conditions aong the route. The secondary network is partitioned with the agorithm in Section 4.3 using the foowing threshod vaues which were found to produce good resuts with the given dataset. For the number of observations covering each group, the minimum vaue is set to 300 and the target and maximum vaues are set to infinity. For the number of inks in each group, the minimum, target and maximum vaues are set to 10, 15 and 38, respectivey. This produces 47 ink groups for the secondary network, each containing 17.1 inks and 6.9 segments on average. Together with the seven ink groups for the primary network, there are 54 ink groups in tota. Specification of spatia weights For the stochastic components of the trave time rates an SMA structure as described in Section. is used, with a singe weight matrix W and associated parameter ρ. The correation between the trave time rates of consecutive observations within vehice traces in the dataset was about 6% and statisticay significant, suggesting that, in genera, there shoud be positive correation between the network inks. The structure of W was chosen after extensive experimentation with aternative specifications. Each ink is assumed to be infuenced by its nearest upstream neighbors in the network as we as the nearest upstream neighbors of those inks (first and second-order neighbors, respectivey). The second-order neighbors are given a weight of 0.5 reative to the first-order neighbors to represent the fact that infuence shoud decay with distance. Finay, each row of W is normaized to add to 1. This normaization is the standard approach in spatia econometrics (LeSage and Pace, 009), and was found to improve the mode fit. In order to investigate the effect and potentia of different types of variabes, four different specifications are considered. Mode 1: Link groups In the first specification (Mode 1), the mean segment trave time rates are expained ony with fixed effects at the ink group eve (54 groups in tota). The mapping of every segment to its assigned group thus form the design matrix B = SC. Expanatory variabes for segment attributes, trip conditions or turn penaties are not incuded. For the stochastic components, for the sake of simpicity, a formuation in which the variance parameter is common for a inks is used. 19

20 Mode : Segment and turn attributes In the second specification (Mode ), expanatory variabes for segment and turn attributes, but no variabes for trip conditions, are added. This is a reevant eve of specification for strategic appications, where typica trave times across different trave conditions are of interest in modeing and forecasting. The foowing variabes are added, based on their causa effects and their significance in the estimation. Segment attributes (matrix B): Dummy variabes for every combination of segment speed imit (in km/h) and segment functiona cass (abbreviated FC) in the data. The combinations (50,1), (50, ) and (50, 3) are grouped and used as reference eve since the number of observations of inks with combinations (50,1) or (50, ) are too few to estimate the effects separatey. The other existing combinations of speed imit and functiona cass are (10, 5), (30, 3), (30, 4), (30, 5), (50, 4) and (50, 5). A dummy variabe for the existence of a taxi station on the segment. The hypothesis is that the mean trave time rates for taxis on such segments are higher due to many stopping there. Dummy variabes for the ength of the ink to which the segment beongs: ength < 50 m and 50, 00 m. The hypothesis is that ength = 00 m, with the reference eve ength [ ) vehices have ess chance to reach high speeds on short inks, since the speed is typicay ower cose to intersections and traffic signas. Hence, short inks shoud have higher mean trave time rates. Turn attributes (matrix D): Three dummy variabes for the cases that the turn invoves a signaized eft turn, right turn and straight through movement, respectivey. Two dummy variabes for the cases that the turn invoves a non-signaized eft turn and a right turn, respectivey. Through movements are used as reference eve for a turn deays. The hypothesis is that straightthrough movements in genera have priority in intersections so that eft or right turns ead to onger deays. Mode 3: Trip conditions In the third specification (Mode 3), expanatory variabes for the trip conditions infuencing trave time rates and turn deays are introduced. For the trave time means (vector o ) the foowing variabes are considered: A dummy variabe for the ate haf of the time interva, i.e., 7:45-8:00 (not Juy or pubic hoiday). This aows the estimation of trave time variations within the interva. One dummy variabe for each weekday from Tuesday to Thursday (reference eve is Monday and Friday). A dummy variabe for pubic hoidays on weekdays (Christmas, Easter, etc.) Dummy variabes for the summer (June-August) and winter (January-February) seasons. The reference eve is spring (March-May) and fa (September-December). A dummy variabe for the month of Juy, the peak vacation period in Sweden. The tota effect of Juy is obtained by adding the effect of the summer season (June-August). 0

21 A dummy variabe for the year 011 (reference eve is 010). This accounts for changes in traffic eves between years. Recent snow: umber of hours of consecutive snowfa reports preceding the trip. A dummy variabe for the taxi being free, as opposed to occupied or assigned to a customer. For the trave time rate standard deviations (vector p ) the foowing variabes are used: A dummy variabe for the ate haf of the time interva, i.e., 7:45-8:00 (not Juy or pubic hoiday). This aows the estimation of trave time variations within the time period. One dummy variabe for each weekday from Tuesday to Friday (reference eve is Monday). Dummy variabes for the summer (June-August) season. A dummy variabe for the month of Juy, the peak vacation period in Sweden. The tota effect of Juy is obtained by adding the effect of the summer season (June-August). A dummy variabe for the taxi being free, as opposed to occupied or assigned to a customer. Mode 4: Independent inks The fourth specification (Mode 4) is identica to Mode 3 except that a inks are assumed to be independent; in other words, the dependence parameter ρ is omitted. By comparing the performance of Mode 3 and Mode 4, the contribution of the correation structure in the mode can be assessed. 5.3 Estimation resuts The mode specifications were estimated using the maximum ikeihood estimation routine in the Statistics Toobox for MATLAB and a trust-region refective ewton optimization agorithm (MATLAB, 009). Gradients of the og-ikeihood function were evauated anayticay, whie the Hessian used to cacuate standard errors was inverted numericay. Estimation resuts are shown in Tabe 1. The 54 segment group fixed effects, not shown in the tabe, are a significant with p-vaues ess than in a four mode specifications. This demonstrates that the automated grouping method is capabe of handing the identification probems associated with the ow samping frequency, even in the outer parts of the secondary network with few observations per segment. The expanatory power of Mode 1, captured by the og-ikeihood and Akaike s information criterion (AIC), is consideraby better than a mode ( Mode 0 ) with a singe mean trave time rate parameter and a singe trave time rate variance parameter (the og-ikeihood for this mode is 65,745, and the AIC is 531,494). The ogarithm of the ikeihood ratio is 3763, and a ikeihood ratio test with 54 degrees of freedom rejects Mode 0 in favor of Mode 1 with a p-vaue equa to 0 within machine precision. 1