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Research ArticleCharacterizing Critical Transition State forNetwork Fundamental Diagram

Rongrong Hong1 Chengchuan An1 Zhenbo Lu 1 Jingxin Xia 1

Qinghui Nie2 and Wenming Rao1

1 Intelligent Transportation Systems Research Center Southeast University Nanjing China2College of Civil Science and Engineering Yangzhou University Yangzhou China

Correspondence should be addressed to Zhenbo Lu seuitslzb126com

Received 8 June 2018 Revised 10 September 2018 Accepted 24 October 2018 Published 18 November 2018

Guest Editor Lele Zhang

Copyright copy 2018 RongrongHong et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Macroscopic Fundamental Diagram (MFD) reveals the relationship between network accumulation and flow at the macroscopiclevelThenetwork traffic flow state analysis is a fundamental problem for theMFD-based applicationsTheoretical and experimentalinvestigations have provided insights into the dynamics and characters of traffic flow states Although many empirical studies hadbeen conducted in the field of MFD few studies were dedicated to investigate the network traffic flow states with field data Thisstudy aims to develop a data-driven method based on time series analysis of MFD state points to characterize critical transitionstate (CTS) of network traffic flow using field data The proposed method was tested in a real network of Kunshan City ChinaThetest results showed that the CTS points can be well captured by the proposed methodThe identified CTS points distinguished thetraffic states between free-flow state and optimal accumulation state and the optimal accumulation state was characterized Theday-to-day pattern of CTS pointswas investigated by the GaussianMixtureModel-based clusteringmodel An extended applicationof real-time identification of CTS points was also discussedThe proposed method is helpful to understand the temporal evolutionprocess of network traffic flow and provides potentials for developing more reliable network traffic flow management strategiessuch as optimizing traffic signal plans and developing strategies for congestion tooling

1 Introduction

Macroscopic Fundamental Diagram (MFD) reveals the rela-tionship between network accumulation and flow at themacroscopic level and can be readily obtained with trafficflow data from existing detectors (eg loop detectors) Thecore property of MFD is that the network flow is maximizedwhen network traffic stays at an optimal accumulation stateThis property can be used to develop effective applicationssuch as traffic signal gating control [1ndash3] congestion pricing[4] route guidance [5] and traffic safety analysis [6] To thesuccesses of these applications one of the important basesis to understand the characteristic of network traffic flowstates in MFD including the state transitions in the temporalevolution process

To characterize the network traffic flow states with MFDanalytical methods were proposed in theoretical studiesDaganzo and Gayah [7] illustrated that bifurcations occurred

when the network accumulation came up to a certain valueIn light of their work a method based on the phase portraitsof the dynamic Equations was developed to determine theboundaries of the stable and unstable regions of states inMFD[8] Gayah and Daganzo [9] found that the clockwise hys-teresis phenomenon during the congestion recovery periodwas caused by the instability of network traffic flow Daganzoand Geroliminis [10] developed an analytical approximationmethod to derive the upper bound of the average flowconditional on average densityThe derived network flowwasinterpreted as the theoretical capacity of network Followingtheir work Geroliminis and Boyac [11] investigated theeffects of different link lengths and signal timing on networkcapacity Although these studies have revealed importantproperties of macroscopic network traffic flow states thetheoretical methods were not ready to apply in practice dueto limitations such as the overestimation of network capacity[12 13]

HindawiJournal of Advanced TransportationVolume 2018 Article ID 4839729 13 pageshttpsdoiorg10115520184839729

2 Journal of Advanced Transportation

More practically empirical studies using field data gainedmore attentions In literature most studies focused on theestimation of MFD with different sources of data includingloop detector data [14 15] probe vehicle data [15ndash17] andconnected vehicle data [18] However few studies dedicatedto develop methods for the analysis of network traffic flowstates such as the network traffic flow state identification andstate evolution pattern recognition Saberi and Mahmassani[19] conducted a comparison analysis to investigate thecapacity drop phenomena using loop detector data fromthree different freeway networks The optimal accumulationstates were typically identified by observation [20 21] Arecent study by Ampountolas and Kouvelas [22] proposeda Kalman Filter based method to identify the maximumnetwork flow according to the changes of slopes betweenconsecutive points in MFD

In contrast to the limited analysis of network traffic flowstates the analysis of link traffic flow states was extensivelydiscussed using field data in the past decadesThe equilibriumrelationship among the traffic flow density and speed onlinks of urban roads and freeways was obtained by fitting thescatter state points in link traffic flow fundamental diagram[23ndash25] Further clustering methods were applied to thescatter state points to identify the trafficflow statesThemeth-ods include nested clustering [26] fuzzy c-means clustering[27 28] online agglomerative clustering [29] K-means [30]multivariate clustering [31] and DBSCAN clustering [32]These studies intended to answer the questions that howmany traffic states could be described and what the pattern oftraffic flow is within each state However the temporal evolu-tion patterns of traffic states are not specifically addressed inthe clusteringmethodsThus in another perspective analysismethodswere developed to identify the transition state on thetime series of the trafficflow statesThekey procedure of thesestudies is to identify the breakpoints in a time continuoustraffic flow state series Banks [33] measured the significantchanges of the slopes between consecutive flow-occupancystate points and identified the transition states between afree-flow state and a synchronized flow state Vlahogianniet al [34] used the Recurrence Plot (RP) to identify thepoint where a shift or abrupt change in the traffic volumeseries taken place Kamarianakis et al [35] used a smooth-transition regression to identify an asymmetric transitionbehaviour between free-flow and congestion state Tang et al[36] and Yan et al [37] reconstructed the time series of trafficflow occupancy and speed as a network graph and identifieddifferent traffic flow states within the time series based ona complex network method Based on the analysis of thetime series of traffic flow states Blandin et al [38] proposeda modified LWR model to describe the temporal transitioncharacteristic of traffic flow states Despite the similarity ofthe problems between the link and network traffic flow stateanalysis the characteristic of traffic flow states on networkcould be very different from that on a single link

The problem addressed in this study is to identify andcharacterize the critical transition state (CTS) point in MFDusing field data The CTS point is defined as the point wheresignificant changes occurs in a MFD time series meaningthat the network traffic flow state before and after a CTS

point is significantly different First the similarity of twosubseries before and after a point in a MFD time series wasmeasured Second the CTS points were identified based onthe similarity measurement Third the day-to-day patternswere investigated based on the identified CTS points Theproposed method was designed to analyse network trafficflow states based on historical data In addition an extendedapplication of real-time identification of CTS points was alsodiscussed in the endTheproposedmethod is helpful to betterunderstand the evolution process of network traffic flow andoffers opportunities to develop more reliable and responsivenetwork traffic flow management strategies For examplewith the identification of CTS point it is helpful to determinewhen to use the peak control strategy to improve trafficflow conditions in real time Besides it also can be used todetermine to the best period to implement congestion toolingpolicy before the urban traffic flow becomes too congested

2 Methodology

21 Framework The framework of the methodology involvesthree parts as illustrated in Figure 1 The first part is tomeasure the similarity between two subseries around eachpoint in aMFD time seriesThe two subseries are extracted bya sliding window structure and the Dynamic TimeWrapping(DTW) distance [39] is used as the similarity measurementThe second part is to find the most significant DTW distancethrough a peak searching process to identify the CTS pointThe Locally Weighted Regression (LOESS) [40] is initiallyused to smooth the DTW distance series extracted in thefirst part Then the peaks of the smoothed DTW distanceseries are determined based on a root-finding method [41]The third part is to analyse the day-to-day patterns of the CTSpoints identified fromMFD time series of different days Thepatterns are recognized by aGaussianMixtureModel (GMM)clustering method [42]

22 Measuring Pattern Similarity with Sliding Windows Let119904119905 be a state point in aMFD time series for the 119905th interval in aday and 119904119905 is a two-dimensional vector defined as the networkoccupancy and network flow (119900119888119888119905 V119900119897119905) The network occu-pancy and network flow are the average occupancy over eachlink and average volume weighted by link length in a network[14]

The two subseries before and after 119904119905 are definedin two sliding windows with the length 119897 as 119878119861119905 =(119904119905minus119897 119904119905+1minus119897 119904119905minus1) = (1199041198871 119904119887119894 119904119887119897) and 119878119860119905 = (119904119905119904119905+1 119904119905+119897minus1) = (1199041198861 119904119886119895 119904119886119897) respectively The simi-larity measurement between the two subseries is notated as119889(119878119861119905 119878119860119905) and attributed to 119904119905 119878119861119905 and 119878119860119905 are normalized toensure the two subseries are comparable The Dynamic TimeWrapping (DTW) distance is used to calculate 119889(119878119861119905 119878119860119905)The reason that DTW distance is adopted as the similaritymeasurement of the two subseries rather than Euclideandistance is that the DTW distance is able to compare thetrends (eg increasing or decreasing trends) between twoseries Let 119904119887119894 and 119904119886119895 be the 119894th and 119895th element of 119878119861119905 and119878119860119905 respectively

Journal of Advanced Transportation 3

Smooth the DTW distance serieswith LOESS

Extract the two sub-seriesby sliding windows

Window 1

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wor

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Network occupancy

Measuring similarity between two sub-series

DTW distanceseries

Time of day

CTS

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Identify the CTS points by root-finding

Cluster the CTS pointsusing GMM method

Peak searching for the CTS points Investigating the day-to-day patterns of the CTS points

CTS

The set of CTSpoints from

different days

Window 2

Calculate DTWDistance

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Figure 1 Framework of proposed method

Time series alignment

4 6 8 10 122SB index

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Figure 2 An optimal correspondence on warped evolution paths for 119904119905

The schematic diagram for calculation of the DTWdistance is illustrated in Figure 2 The DTW distance iscalculated in two steps First calculate the local cost matrixEach itemof thematrixwith index (119894 119895) represents a local cost119888(119894 119895) between 119904119887119894 and 119904119886119895 Second calculate the accumulatedcost matrix or global cost matrix as shown in (1) Third findthe shortest path with the global cost matrix 120573(119894 119895) shownas the path 0 in Figure 2 The global cost 120573(119897 119897) is the DTWdistance between the two series The DTW distance can bemeasured for two series with different lengths In this study

the lengths of two series are the same as 119897 Thus the costmatrix is 119897lowast119897The 119888(119894 119895) is calculated as the Euclidean distancebetween 119904119887119894 and 119904119886119895 To effectively find the shortest pathdynamic programming algorithm is used

120573 (119894 119895)= 119888 (119894 119895)+min 120573 (119894 minus 1 119895 minus 1) 120573 (119894 minus 1 119895) 120573 (119894 119895 minus 1)

(1)

4 Journal of Advanced Transportation

where 120573(119894 119895) is the global cost from item (1 1) to item(119894 119895) By iteratively solving (1) the shortest path can be ob-tained

23 Peak Searching for Critical Transition State Point Basedon the DTW distance the CTS points could be identified asthe points with significantly large DTWdistances Intuitivelya threshold-based method could be used Thus all pointswith DTW distances larger than a predetermined thresholdwould be regarded as the CTS points However the selectionof the thresholds could be difficulty and arbitrary becausedifferent networksmay have distinct thresholds Basically theidentification of the CTS points is an extreme value anal-ysis problem over a series Therefore a more sophisticatedalgorithm was developed in this study The algorithm wasdescribed as follows

Step 1 Smooth the distance series with Locally WeightedRegression (LOESS) [40] The smoothing process is initiallyconducted in order to filter the occasionally and locallyfluctuated points in the DTW distance series Let 119910119895 be thesmoothed value of the jth item in the DTW distance serieswhich were divided into 119899 equally spaced intervals

Step 2 Determine the ranges that contain the CTS pointsThe identification of CTS is to find the extreme point (EP)of DTW series To determine the range of each extreme point(EP) the approximate extreme point (AEP) is captured firstThe AEP locates at a small scale that an extreme point wascontained Notate Δ119910119895 and 2119910119895 as the first and second orderdifference of 119910119895 The monotonicity of 119910(119895) is determined bycomparing the sign of 119910119895 Δ1199101198952119910119895 which were expressed in(2) and (3)

119910119895 = 119891 (119895) Δ119910119895 = 119891 (119895 + 1) minus 119891 (119895) 2119910119895 = Δ119910119895+1 minus Δ119910119895

(2)

Δ119910119895 = 1 Δ119910119895 gt 00 Δ119910119895 lt 0 (3)

If (2119910119895) = 0 the monotonicity of 119891(119895 + 1) changedcompared with 119891(119895) and there is an AEP around (119895 + 1)119905ℎinterval Based on the determined AEPs the ranges thatcontain EPs are determined by the centres of neighbouringAEPs

Step 3 Identify the CTS points using Brendrsquos method Theidentification for CTS (EP) point can be viewed as rooting-finding of the first derivative of 119910 within each range Brendrsquosmethod [41] is used to find the extreme point 119910119888 with localmaximum distance within each range The method is aroot-finding algorithm combining the bisection method thesecantmethod and inverse quadratic interpolationwhich hasthe advantage of fast-converging Corresponding to 119910119888 thecritical state point 119904119888 is achieved

24 Day-to-Day Pattern Recognition of the CTS Points Dueto the stochastic nature of urban traffic flow it is assumedthat the day-to-day distribution of the CTS points is subjectto a Gaussian Mixture Distribution In this study a GaussianMixture Model (GMM) is used to cluster the CTS pointsextracted from different days Compared to other clusteringmethods such as the K-means one appealing feature of theGMM clustering method is that the probability of a statepoint belonging to a cluster is available The GMM clusteringmethod is described as follows

In GMM the data is assumed to be generated froma mixture of a finite number of Gaussian distributionsEach Gaussian distribution is regarded as a component ofthe GMM and the components are linearly combined bymixing weights In fact the GMM clustering is equivalentto estimate the unknown mixing weights and parameters ofeach Gaussian distribution In this study in addition to thenetwork flow and occupancy the temporal feature (eg timeindex during a day) of the CTS points is also included to theclustering process Let 119904119888 represents the set of the identifiedCTS points from multiple days 119904119888119894 which represents for CTSpoint is a vector formed by network flow-occupancy andthe time of day for the 119894th element in 119904119888 0119896(119904119888119894 | 120579119896) is theProbability Density Function (PDF) of kth component TheGMMmodel is shown in

0 (119904119888119894 | 120579) = 119866sum119896=1

1205911198960119896 (119904119888119894 | 120579119896)

(119904119888119894 isin 1198773 119896 isin 1 2 119866 120591119896 isin (0 1) 119866sum119896

120591119896 = 1 ))(4)

where 119896 presents the kth component in the GMM 119866 is thetotal number of the components 120579119896 = (120583119896 sum119896) 120583119896 and sum119896are the mean and covariance of the kth component and 120591119896 isthe mixing weight of the kth component

As described in (5) the GMM-based clustering methodcan be viewed as a process to find the optimal parameters 120591119896and 120579119896 to maximize the likelihood function of GMM

119871 (120579 119904119888 119885) = 119873prod119894=1

119866prod119896=1

[1205911198960119896 (119904119888119894 | 120579119896)]I(119885119894=119896) (5)

120579119896 and 120591119896 were iteratively estimated using theExpectation-Maximization (EM) algorithm [42] The E-step which is expressed in (6) aims at getting the posteriorprobability (120574(119904119888119894 119896)) of 119904119888119894 belonging to component 119896 withinitialled 120579119896 and 120591119896 The M-step is to get the new estimateof 120579119896 and 120591119896 through maximizing 119871(120579 119904119888 119885) with posteriorprobability achieved in E-step The estimation of parametersis shown in (7)-(9)

120574 (119904119888119894 119896) = 1205911198960119896 (119904119888119894 | 120579119896)sum119866119896=1 1205911198960119896 (119904119888119894 | 120579119896) (6)

120583119896 = sum119873119894=1 120574 (119904119888119894 119896) 119904119888119894sum119873119894=1 120574 (119904119888119894 119896) (7)

Journal of Advanced Transportation 5

The clustering of critical transition state points based on GMMInput 119904119888 120579119905 119866Output 120583119896lowast sum119896lowast 120591119896lowast

Process(1) Giving the initial 120579119905 and 119866(2) 119864 larr997888 120579119905+1 minus 120579119905(3) When 119864 gt 120576 do(4) 120574(119904119888119894 119896) larr997888 F(120579119905) step 4 is E-step (Equation (6))(5) 120579119905+1 larr997888 argmax120579 step 5 is M-step (Equation (7)-(9))(6) replace 120579119905 with 120579119905+1(7) end for

Algorithm 1 The EM Algorithm to estimate parameters in GMM

sum119896

= sum119873119894=1 120574 (119904119888119894 119896) (119904119888119894 minus 120583119896) (119904119888119894 minus 120583119896)119879sum119873119894=1 120574 (119904119888119894 119896) (8)

120591119896 = sum119873119894=1 120574 (119904119888119894 119896)119873 (9)

where 119873 is the total number of all critical transition statepoints 119885119895 is a latent variable that determines the componentfrom which the observation originates The EM Algorithm isshown in Algorithm 1 120579119905 is the initial value for 120579 and 119905119894 is the119894th literation 119864 is the difference between two neighbouringiterations and 120576 is a small real number and 120579lowast is theoptimal value It is worth mentioning that the BIC value isused to avoid overfitting and determine the optimal modelparameters and the number of clusters [43]

3 Implementation

31 Data Collection A real network with 13 signalized inter-sections and 26 links in Kunshan City China was selectedas the test site as shown in Figure 3 All intersectionsoperated under pretimed signal control with four differentsignal plans in a day The four arterials in the network hadsignificant traffic volume during the day time Oversaturatedand spillover conditions were frequently observed duringmorning and afternoon peaks Microwave-based detectorswere installed on links except the ones shown as the dash linkin Figure 3 All detectors were located approximately at themiddle of the links and collected traffic counts and occupancyin every 5minutes Twomonths of data collected fromAugust1 2015 to September 30 2015 were used in this study Inaddition for validation purpose GPS probe vehicle data from1526 taxi cabs during the same period were also collected tocalculate the network average speed The space-mean speedof the network was simply calculated as the ratio between thetotal distance travelled and the total time spent by the taxicabs in the test site for every 5 minutes

32 Identification of the CTS Points Before calculating DTWdistance for the two subseries in the sliding windows it isimportant to select a reasonable window length since thetwo subseries are expected to reflect the traffic evolution

patterns However due to the different OD patterns andsignal operations in different networks the window lengthis more likely network-dependent In general the windowlength should be shorter than the peak duration Besidesthe window length should not be too short to filter randomfluctuations in data series and not too long to reflect trafficdynamics Basically determining a desirable length is a trade-off between reliability and accuracy

In our case the morning peak is from 600 to 900and the afternoon peak is from 1630 to 1930 Thereforethe window length should be shorter than 3 hours Basedon that the impacts of different window lengths on themeasurement of the DTW distance were investigated Anindex of AverageChange Rate (ACR)was used to describe thefluctuation in theDTWdistance seriesTheACR is calculatedas

119860119862119877 = 1119899119899sum119894=1

119889119894+1 minus 119889119894119889119894 (10)

where 119889119894 is the normalized DTWdistance associated with theith state point in a MFD time series and 119899 is the number ofstate points

Figure 4(a) shows the measurement of DTW distancecalculated from the MFD time series on August 10 2015with window length ranging from 15 minutes to 3 hours Itis observed that the curves of the DTW distance becomesmoother with the increasing of window length The curveswith window lengths of 1 hour and 15 hours are observed asthe relatively balanced cases where the trends are clear andinformative short-term changes are retained In addition theACR was calculated using data in the two months as shownin Figure 4(b) Each point in Figure 4(b) represents the ACRof the DTW distance series of a day with a particular windowlength When the window length is lower than 1 hour theACR decrease significantly from about 011 to 001 A turningpoint can be clearly spotted at the window length of 1 hourBased on the investigation the window length was selected as1 hour for our network

With the measurement of DTW distance a peak search-ing method was applied Figure 5 illustrates the identifiedpeaks on the DTW distance curve of August 11 2015The redand blue marks represent the local maximum and minimum

6 Journal of Advanced Transportation

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TongfengRoad

Qianjin Road

Kuntai Road

Link length(m)

Signalized Intersection

Link IDLink with detectorLink without detector

Figure 3 Layout of the test network and detectors in Kunshan City China

extreme points extracted by the proposed method The localmaximum extreme points are regarded as the CTS points inthis study As shown in Figure 5 the CTS points (see thepoints located at the high peaks of the curves) appear duringthe morning and afternoon peaks Although the most criticaltransition state points are successfully identified suspiciousCTS points are also found at the local peaks on the curvesSome suspicious points (shown in the blue circle in Figure 5)are obviously invalid since the corresponding DTWdistancesare so small These points could be readily excluded bya lower bound threshold In this study any state pointswith DTW distance less than 15 were considered as invalidpoints

33 Validation of the CTS Points To validate the identifiedCTS points the locations of the CTS points at the MFD timeseries the correlation between the CTS points and networkspace-mean speed were investigated For ease of illustrationthe morning and afternoon peak were separately consideredThe identified CTS points on August 10 Monday 2015and August 12 Wednesday 2015 are illustrated in Figure 6Figures 6(a) 6(d) and 6(g) show the CTS points during themorning peaks of the three days and Figures 6(b) 6(e) and

6(h) show the CTS points during the afternoon peaks of thethree days Figures 6(c) 6(f) and 6(i) illustrate the correlationbetween the CTS points and network space-mean speed forthe three days The red markers represent the identified CTSpoints The arrows indicate the time sequence within theMFD time series

For the locations at the MFD time series the identifiedCTS points approximately locate at the boundaries of the twodistinguished traffic states see Figures 6(a) 6(d) 6(g) 6(b)6(e) and 6(h) The state points on the left side of the CTSpoints show a positive correlation between network flow andoccupancy while the correlation of state points on the rightside is vague and the slopes between the consecutive statepoints are much more fluctuated than the left side A flatshape of the right side state points can be observed and thenetwork flow fluctuates within a small range from 75 to 90veh5min Based on this it is reasonable to infer that the leftand right side of the identified CTS points are correspondingto the free-flow and optimal accumulation state respectivelyIn other words the identified CTS points well captured theshifts of network traffic states between the free-flow andoptimal accumulation state

By incorporating the network speed in all these threedays the CTS points are identified at the times when the

Journal of Advanced Transportation 7

15min 30min

45min 1h

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(a)

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2 31Length of window (h)

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Figure 4 Impacts of different window lengths on the measurement of DTW distance (a) DTW distance curves with different lengths ofwindow on August 10 Tuesday 2015 (b) ACR calculation with different window lengths during August 1 2015 and September 30 2015

network congestion just appeared and nearly recovered seeFigures 6(c) 6(f) and 6(i) Moreover we can find thatonly the CTS points during the congestion formation stageare identified in the morning peaks of the three days Thereason may be that the traffic flow decreases slowly duringthe congestion recovery stage at the end of the morningpeaks and the traffic state after the morning peaks doesnot experience a significant change According to the pairedCTS points in afternoon peaks the optimal accumulationstate can be approximately estimated as (012 73veh5min) byaveraging over the three days

34 Day-to-Day Pattern Analysis The empirical distributionof the identified CTS points based on two-month weekdaydata is visualized in as shown Figure 7 A clear bimodalitycan be observed in the distribution In fact the bimodality iscaused by the different distributions of the CTS points in themorning and afternoon peaks

Considering the most congested afternoon peaks inour case day-to-day patterns of the CTS points from theafternoon peaks in 38 weekdays were investigated with theGMM clustering Two distinct clusters are found as shownin Figure 8 The cluster centre for Cluster 1 (in blue) is (1718

8 Journal of Advanced Transportation

2 4 6 8 10 12 14 16 18 20 22 240Time of day

0

10

20

DTW

dist

ance

Figure 5The DTW distance on August 11 Tuesday 2015

013 728 (veh5min)) and for Cluster 2 (in red) is (1810 010722(veh5min)) 516 and 484 of the total identified CTSpoints are in Cluster 1 and Cluster 2 respectively Figures 8(a)and 8(b) illustrate the distribution of flow-occupancy andtime in the two clusters respectively According to the timedistribution Cluster 1 could be the critical transition stateindicating a transition from the free-flow state to optimalaccumulation state while Cluster 2 is the critical transitionstate from the optimal accumulation state to free-flow stateIn terms of the flow-occupancy distribution it is found thatthe CTS points in Cluster 1 are much more scattered thanthose in Cluster 2 This indicates that the transition from theoptimal accumulation state to free-flow state is much morepredictable and the transition from the free-flow to optimalaccumulation state has higher uncertainty

35 Extended Application of Real-Time CTS Point Identifi-cation In addition to the analysis of the CTS points usinghistorical data an extended application of real-time CTSpoint identification is also discussed Based on the proposedmethod theMFD state points can be labelled as three classesthe CTS points of Cluster 1 the CTS points of Cluster2 and non-CTS points Thus a probabilistic discriminantanalysis [43] was conducted using the labelled dataset Thediscriminant analysis also known as supervised classifica-tion assumes that the observations in each class are generatedby a distribution specific to that class Let 0119896 be the probabilitydistribution of the kth class which is Gaussian distributionin this study 120591119896 be the proportion of observations of class kin the population and 119904119905 be a real-time observation of MFDstate points The posterior probability that 119904119905 belongs to thekth class is expressed in

119901119903 (119904119905 isin 119888119897119886119904119904 119896) = 1205911198960119896 (119904119905 120583119896 sum119896) (119904119905)sum119866119896=1 1205911198960119896 (119904119905 120583119896 sum119896) (119904119905) 119896 isin 1 2 3

(11)

In this study the labelled MFD state points in theafternoon peaks of 45 days are used as the training set andanother 15 days as the test set Each MFD state point was

identified as an observation of the class with the highestprobability that belongs to According to the test results theerror of the discriminant method is about 81

The discrimination results for three typical weekdays areillustrated in Figure 9Theblack red and blue dots in Figure 9represent the non-CTS points the CTS points of Cluster 1and the CTS point of Cluster 2 respectively The probabilisticdiscriminant analysis performed well in identifying the CTSpoints All the identified CTS points are located at theboundaries between the free-flow and optimal accumulationstate as shown in Figures 9(a) 9(c) and 9(e) and illustrate astrong correlation to the changes of network speed during theformation and recovery period of the network congestion

4 Conclusions

This study proposed a data-driven method to identify andcharacterize the critical transition state in MFD using fielddata The identification of the critical transition state isbased on the analysis of the MFD time series The DTWdistance is used to measure the similarity of subseries beforeand after each state point in a MFD time series and apeak searching method is developed to identify the criticaltransition state points based on the DTW distance seriesThe GMM clustering is used to characterize the day-to-daypattern of the identified critical transition state points Theprobabilistic discriminant analysis is performed to identifythe CTS points

The identification and validation of CTS points indicatedthat the proposed method well identified the network tran-sition states By analysis the identified CTS points well cap-tured the shifts of network traffic states between the free-flowand optimal accumulation state The optimal accumulationstate can be approximately estimated as (012 73veh5min)by averaging over the three days A clear bimodality can beobserved by exploring the probability density of identifiedCTS points and the bimodality is caused by different dis-tributions of the CTS points in the morning and afternoonpeaks Moreover the CTS points were classified as twocategories One cluster represents the transition from free-flow state to optimal accumulation state and the otherrepresents the transition from optimal accumulation state

Journal of Advanced Transportation 9

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)01 02 03 0400

Network occupancy

(b)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(c)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(d)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(g)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(i)

Figure 6The validation of the identified CTS points on August 10 Monday 2015 and August 12 Wednesday 2015 (a) (d) and (g)The CTSpoints during morning peaks of the three days (b) (e) and ( h) The CTS points during afternoon peaks of the three days (c) (f) and (i)Correlation between the CTS points and network speed

to free-flow state Besides the probabilistic discriminantanalysis performed well in identifying the CTS points All theidentified CTS points were located at the boundaries betweenthe free-flow and optimal accumulation state The proposed

method is helpful to better understand the evolution processof network traffic flow and offers opportunities to developmore reliable and responsive network traffic flow manage-ment strategies

10 Journal of Advanced Transportation

0

4020

6080

100

Networ

k flow

(veh

5m

in)

0000002040608

005010 015 020 025

Network occupancy

Figure 7 Visualization of the distribution of the CTS points

0

40

20

60

80

100

Net

wor

k flo

w (v

eh5

min

)

000 005 010 015 020 025Network occupancy

(a)

Cluster12

0

5

10

15C

ount

17 18 1916Time of day

(b)

Figure 8 Day-to-day pattern of the CTS points from the afternoon peaks of 38 weekdays (a) Flow-occupancy distribution in two clusters(b) Time distribution in two clusters

In future the impact factors of critical transition statesin actual network will be investigated through comparativeanalysis under certain conditions Besides it is encouragedto test the proposed method with different sources of data(eg probe vehicle or vehicle license plate reidentificationdata) Also more field validation works are needed to furtherinterpret the critical transition states considering real trafficbehaviours

Data Availability

Partial of the detector data used to support the findings ofthis study are includedwithin the supplementary informationfile

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The work was funded by the National Natural Science Foun-dation of China (no 71871055) and theKeyResearchProgramof Jiangsu Province Science and Technology Department (noBE2017027)

Supplementary Materials

The supplementary material file is partial of the detectordata used to derive the MFD for test site The data are

Journal of Advanced Transportation 11

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

(b)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(c)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60N

etw

ork

spee

d (k

mh

)

(d)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

Figure 9 The discrimination results on three typical weekdays (a) (c) and (e) The identified CTS points in the network flow-occupancyseries during afternoon peak of the three days (b) (d) and (f ) The identified CTS points in the network speed series of the three days

12 Journal of Advanced Transportation

valid and they were formed by STATIONID (the ID numberof specific link) CALENDAR KEY (the date of traffic flowdata for specific link) FIVEMINX (the time of traffic flowdata for specific date) OCCUPANCY (the occupancy5minfor specific link) and VOLUME (the flow5min for spe-cific link) from August 1 2015 to September 30 2015(Supplementary Materials)

References

[1] M Keyvan-Ekbatani M Papageorgiou and I PapamichailldquoPerimeterTraffic Control via Remote FeedbackGatingrdquo Proce-dia - Social and Behavioral Sciences vol 111 pp 645ndash653 2014

[2] Y M Du J P Wu and Y Jia H ldquoMFD-Based Regional TrafficVolume Dynamic Controlrdquo Journal of Transportation SystemsEngineering Information Technology vol 14 no 3 pp 162ndash1672014

[3] A Csikos T Tettamanti and I Varga ldquoMacroscopic modelingand control of emission in urban road traffic networksrdquo Trans-port vol 30 no 2 pp 152ndash161 2015

[4] M D Simoni A J Pel R A Waraich and S P HoogendoornldquoMarginal cost congestion pricing based on the network fun-damental diagramrdquo Transportation Research Part C EmergingTechnologies vol 56 pp 221ndash238 2015

[5] M Hajiahmadi V L Knoop B De Schutter and H Hellen-doorn ldquoOptimal dynamic route guidance A model predictiveapproach using the macroscopic fundamental diagramrdquo inProceedings of the 2013 16th International IEEE Conferenceon Intelligent Transportation Systems Intelligent TransportationSystems for All Modes ITSC 2013 pp 1022ndash1028 NetherlandsOctober 2013

[6] R Alsalhi and V Dixit ldquoThe relationship between traffic safetyand macroscopic fundamental diagram (MFD)rdquo AustralasianTransport Research Forum (ATRF) vol 37 2015

[7] C F Daganzo V V Gayah and E J Gonzales ldquoMacroscopicrelations of urban traffic variables bifurcations multivalued-ness and instabilityrdquo Transportation Research Part B Method-ological vol 45 no 1 pp 278ndash288 2011

[8] J Haddad and N Geroliminis ldquoOn the stability of trafficperimeter control in two-region urban citiesrdquo TransportationResearch Part B Methodological vol 46 no 9 pp 1159ndash11762012

[9] V V Gayah and C F Daganzo ldquoClockwise hysteresis loops inthe Macroscopic Fundamental Diagram An effect of networkinstabilityrdquo Transportation Research Part B Methodological vol45 no 4 pp 643ndash655 2011

[10] C FDaganzo andNGeroliminis ldquoAn analytical approximationfor the macroscopic fundamental diagram of urban trafficrdquoTransportation Research Part B Methodological vol 42 no 9pp 771ndash781 2008

[11] N Geroliminis and B Boyaci ldquoThe effect of variability of urbansystems characteristics in the network capacityrdquo TransportationResearch Part B Methodological vol 46 no 10 pp 1607ndash16232012

[12] E J Gonzales C Chavis Y Li and C F Daganzo Mul-timodal transport modeling for nairobi kenya insights andrecommendations with an evidence-based model insights andrecommendations with an evidence-based model kenya 2009

[13] A Mazloumian N Geroliminis and D Helbing ldquoThe spatialvariability of vehicle densities as determinant of urban networkcapacityrdquo Philosophical Transactions of the Royal Society A

Mathematical Physical amp Engineering Sciences vol 368 no1928 pp 4627ndash4647 2010

[14] N Geroliminis and C F Daganzo ldquoExistence of urban-scalemacroscopic fundamental diagrams some experimental find-ingsrdquo Transportation Research Part B Methodological vol 42no 9 pp 759ndash770 2008

[15] T Courbon and L Leclercq ldquoCross-comparison of macroscopicfundamental diagram estimation methodsrdquo in Proceedings ofthe 14thMeeting of the EuroWorking Group on Transp - In Questfor Adv Models Tools and Methods for Transp and LogistEWGT 26th Mini-EURO Conf - Intelligent Decis Making inTransp and Logist MEC 1st Eur Sci Conf on Air Transp - RHpp 417ndash426 Poland September 2011

[16] VGayah andV Dixit ldquoUsingmobile probe data and themacro-scopic fundamental diagram to estimate network densitiesrdquoTransportation Research Record no 2390 pp 76ndash86 2013

[17] J Du H Rakha and V V Gayah ldquoDeriving macroscopicfundamental diagrams from probe data Issues and proposedsolutionsrdquo Transportation Research Part C Emerging Technolo-gies vol 66 pp 136ndash149 2016

[18] Z Xu P J Jin and X Jiang ldquoMacroscopic FundamentalDiagram (MFD) Based on Vehicle-to-Infrastructure (V2I)Connected Vehicle Data Transportation Research Board 94thAnnual Meetingrdquo Macroscopic Fundamental Diagram (MFD)Based on Vehicle-to-Infrastructure (V2I) Connected Vehicle Data Transportation Research Board 94th Annual Meeting 2015

[19] M Saberi and H Mahmassani ldquoHysteresis and capacity dropphenomena in freeway networks Empirical characterizationand interpretationrdquo Transportation Research Record no 2391pp 44ndash55 2013

[20] M Keyvan-Ekbatani A Kouvelas I Papamichail and MPapageorgiou ldquoExploiting the fundamental diagram of urbannetworks for feedback-based gatingrdquo Transportation ResearchPart B Methodological vol 46 no 10 pp 1393ndash1403 2012

[21] K Aboudolas and N Geroliminis ldquoPerimeter and boundaryflow control inmulti-reservoir heterogeneous networksrdquoTrans-portation Research Part B Methodological vol 55 pp 265ndash2812013

[22] K Ampountolas and A Kouvelas ldquoReal-Time Estimation ofCritical Values of the Macroscopic Fundamental Diagramfor Maximum Network Throughputrdquo Transportation ResearchBoard 94th Annual Meeting 2015

[23] B Ponnu and B Coifman ldquoWhen adjacent lane dependenciesdominate the uncongested regime of the fundamental relation-shiprdquo Transportation Research Part B Methodological vol 104pp 602ndash615 2017

[24] X Qu J Zhang and S Wang ldquoOn the stochastic fundamentaldiagram for freeway traffic Model development analyticalproperties validation and extensive applicationsrdquo Transporta-tion Research Part B Methodological vol 104 pp 256ndash271 2017

[25] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015

[26] J Xia and M Chen ldquoA nested custering technique for freewayoperating condition classificationrdquo Computer-Aided Civil andInfrastructure Engineering vol 22 no 6 pp 430ndash437 2007

[27] H-p Lu Z-y Sun and W-c Qu ldquoBig data-driven based real-time traffic flow state identification and predictionrdquo DiscreteDynamics in Nature and Society Art ID 284906 11 pages 2015

Journal of Advanced Transportation 13

[28] M Azimi and Y Zhang ldquoCategorizing Freeway Flow Condi-tions by Using Clustering Methodsrdquo Transportation ResearchRecord vol 2173 no 1 pp 105ndash114 2018

[29] J Xia W Huang and J Guo ldquoA clustering approach to onlinefreeway traffic state identification using ITS datardquoKSCE Journalof Civil Engineering vol 16 no 3 pp 426ndash432 2012

[30] J Tang F Liu W Zhang S Zhang and Y Wang ldquoExploringdynamic property of traffic flow time series in multi-statesbased on complex networks Phase space reconstruction versusvisibility graphrdquo Physica A Statistical Mechanics and its Appli-cations vol 450 pp 635ndash648 2016

[31] H B Celikoglu and M A Silgu ldquoExtension of traffic flowpattern dynamic classification by a macroscopic model usingmultivariate clusteringrdquo Transportation Science vol 50 no 3pp 966ndash981 2016

[32] S Oh Y-J Byon and H Yeo ldquoImpact of Traffic State Transitionand Oscillation on Highway Performance with Section-BasedApproachrdquo in Proceedings of the 18th IEEE International Confer-ence on Intelligent Transportation Systems ITSC 2015 pp 2141ndash2146 Spain September 2015

[33] J H Banks ldquoInvestigation of some characteristics of congestedflowrdquo Transportation Research Record no 1678 pp 128ndash1341999

[34] E I Vlahogianni M G Karlaftis and J C Golias ldquoStatisticalmethods for detecting nonlinearity and non-stationarity in uni-variate short-term time-series of traffic volumerdquoTransportationResearch Part C Emerging Technologies vol 14 no 5 pp 351ndash367 2006

[35] Y Kamarianakis H Oliver Gao and P Prastacos ldquoCharacter-izing regimes in daily cycles of urban traffic using smooth-transition regressionsrdquo Transportation Research Part C Emerg-ing Technologies vol 18 no 5 pp 821ndash840 2010

[36] J Tang Y Wang and F Liu ldquoCharacterizing traffic timeseries based on complex network theoryrdquo Physica A StatisticalMechanics and its Applications vol 392 no 18 pp 4192ndash42012013

[37] Y Yan S Zhang J Tang and X Wang ldquoUnderstandingcharacteristics in multivariate traffic flow time series fromcomplex network structurerdquo Physica A Statistical Mechanicsand its Applications vol 477 pp 149ndash160 2017

[38] S Blandin J Argote A M Bayen and D B Work ldquoPhasetransition model of non-stationary traffic flow Definitionproperties and solution methodrdquo Transportation Research PartB Methodological vol 52 pp 31ndash55 2013

[39] D F Silva and G E Batista ldquoSpeeding up all-pairwise dynamictime warping matrix calculationrdquo in Proceedings of the 2016SIAM International Conference on Data Mining Society forIndustrial and Applied Mathematics pp 837ndash845 2016

[40] O B Linton ldquoLocal Regression Modelsrdquo inMicroeconometricsPalgrave Macmillan London UK 2010

[41] R P Brent Algorithms for Minimization without DrivativesPrentice Hall Englewood Cliffs NJ USA 2013

[42] NM Nasrabadi ldquoPattern Recognition andMachine LearningrdquoJournal of Electronic Imaging vol 16 no 4 p 049901 2007

[43] C Fraley and A E Raftery ldquoModel-based clustering discrimi-nant analysis and density estimationrdquo Journal of the AmericanStatistical Association vol 97 no 458 pp 611ndash631 2002

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2 Journal of Advanced Transportation

More practically empirical studies using field data gainedmore attentions In literature most studies focused on theestimation of MFD with different sources of data includingloop detector data [14 15] probe vehicle data [15ndash17] andconnected vehicle data [18] However few studies dedicatedto develop methods for the analysis of network traffic flowstates such as the network traffic flow state identification andstate evolution pattern recognition Saberi and Mahmassani[19] conducted a comparison analysis to investigate thecapacity drop phenomena using loop detector data fromthree different freeway networks The optimal accumulationstates were typically identified by observation [20 21] Arecent study by Ampountolas and Kouvelas [22] proposeda Kalman Filter based method to identify the maximumnetwork flow according to the changes of slopes betweenconsecutive points in MFD

In contrast to the limited analysis of network traffic flowstates the analysis of link traffic flow states was extensivelydiscussed using field data in the past decadesThe equilibriumrelationship among the traffic flow density and speed onlinks of urban roads and freeways was obtained by fitting thescatter state points in link traffic flow fundamental diagram[23ndash25] Further clustering methods were applied to thescatter state points to identify the trafficflow statesThemeth-ods include nested clustering [26] fuzzy c-means clustering[27 28] online agglomerative clustering [29] K-means [30]multivariate clustering [31] and DBSCAN clustering [32]These studies intended to answer the questions that howmany traffic states could be described and what the pattern oftraffic flow is within each state However the temporal evolu-tion patterns of traffic states are not specifically addressed inthe clusteringmethodsThus in another perspective analysismethodswere developed to identify the transition state on thetime series of the trafficflow statesThekey procedure of thesestudies is to identify the breakpoints in a time continuoustraffic flow state series Banks [33] measured the significantchanges of the slopes between consecutive flow-occupancystate points and identified the transition states between afree-flow state and a synchronized flow state Vlahogianniet al [34] used the Recurrence Plot (RP) to identify thepoint where a shift or abrupt change in the traffic volumeseries taken place Kamarianakis et al [35] used a smooth-transition regression to identify an asymmetric transitionbehaviour between free-flow and congestion state Tang et al[36] and Yan et al [37] reconstructed the time series of trafficflow occupancy and speed as a network graph and identifieddifferent traffic flow states within the time series based ona complex network method Based on the analysis of thetime series of traffic flow states Blandin et al [38] proposeda modified LWR model to describe the temporal transitioncharacteristic of traffic flow states Despite the similarity ofthe problems between the link and network traffic flow stateanalysis the characteristic of traffic flow states on networkcould be very different from that on a single link

The problem addressed in this study is to identify andcharacterize the critical transition state (CTS) point in MFDusing field data The CTS point is defined as the point wheresignificant changes occurs in a MFD time series meaningthat the network traffic flow state before and after a CTS

point is significantly different First the similarity of twosubseries before and after a point in a MFD time series wasmeasured Second the CTS points were identified based onthe similarity measurement Third the day-to-day patternswere investigated based on the identified CTS points Theproposed method was designed to analyse network trafficflow states based on historical data In addition an extendedapplication of real-time identification of CTS points was alsodiscussed in the endTheproposedmethod is helpful to betterunderstand the evolution process of network traffic flow andoffers opportunities to develop more reliable and responsivenetwork traffic flow management strategies For examplewith the identification of CTS point it is helpful to determinewhen to use the peak control strategy to improve trafficflow conditions in real time Besides it also can be used todetermine to the best period to implement congestion toolingpolicy before the urban traffic flow becomes too congested

2 Methodology

21 Framework The framework of the methodology involvesthree parts as illustrated in Figure 1 The first part is tomeasure the similarity between two subseries around eachpoint in aMFD time seriesThe two subseries are extracted bya sliding window structure and the Dynamic TimeWrapping(DTW) distance [39] is used as the similarity measurementThe second part is to find the most significant DTW distancethrough a peak searching process to identify the CTS pointThe Locally Weighted Regression (LOESS) [40] is initiallyused to smooth the DTW distance series extracted in thefirst part Then the peaks of the smoothed DTW distanceseries are determined based on a root-finding method [41]The third part is to analyse the day-to-day patterns of the CTSpoints identified fromMFD time series of different days Thepatterns are recognized by aGaussianMixtureModel (GMM)clustering method [42]

22 Measuring Pattern Similarity with Sliding Windows Let119904119905 be a state point in aMFD time series for the 119905th interval in aday and 119904119905 is a two-dimensional vector defined as the networkoccupancy and network flow (119900119888119888119905 V119900119897119905) The network occu-pancy and network flow are the average occupancy over eachlink and average volume weighted by link length in a network[14]

The two subseries before and after 119904119905 are definedin two sliding windows with the length 119897 as 119878119861119905 =(119904119905minus119897 119904119905+1minus119897 119904119905minus1) = (1199041198871 119904119887119894 119904119887119897) and 119878119860119905 = (119904119905119904119905+1 119904119905+119897minus1) = (1199041198861 119904119886119895 119904119886119897) respectively The simi-larity measurement between the two subseries is notated as119889(119878119861119905 119878119860119905) and attributed to 119904119905 119878119861119905 and 119878119860119905 are normalized toensure the two subseries are comparable The Dynamic TimeWrapping (DTW) distance is used to calculate 119889(119878119861119905 119878119860119905)The reason that DTW distance is adopted as the similaritymeasurement of the two subseries rather than Euclideandistance is that the DTW distance is able to compare thetrends (eg increasing or decreasing trends) between twoseries Let 119904119887119894 and 119904119886119895 be the 119894th and 119895th element of 119878119861119905 and119878119860119905 respectively

Journal of Advanced Transportation 3

Smooth the DTW distance serieswith LOESS

Extract the two sub-seriesby sliding windows

Window 1

Net

wor

k flo

w

Network occupancy

Measuring similarity between two sub-series

DTW distanceseries

Time of day

CTS

DTW

dist

ance

Identify the CTS points by root-finding

Cluster the CTS pointsusing GMM method

Peak searching for the CTS points Investigating the day-to-day patterns of the CTS points

CTS

The set of CTSpoints from

different days

Window 2

Calculate DTWDistance

0

40

20

60

80

100

Net

wor

k flo

w (v

eh5

min

)

000 005 010 015 020 025Network occupancy

0

4020

6080

100

Networ

k flow

(veh

5m

in)

0000002040608

005 010 015 020 025Network occupancy

Figure 1 Framework of proposed method

Time series alignment

4 6 8 10 122SB index

2

4

6

8

10

12

SA in

dex

l = 12

( )

Figure 2 An optimal correspondence on warped evolution paths for 119904119905

The schematic diagram for calculation of the DTWdistance is illustrated in Figure 2 The DTW distance iscalculated in two steps First calculate the local cost matrixEach itemof thematrixwith index (119894 119895) represents a local cost119888(119894 119895) between 119904119887119894 and 119904119886119895 Second calculate the accumulatedcost matrix or global cost matrix as shown in (1) Third findthe shortest path with the global cost matrix 120573(119894 119895) shownas the path 0 in Figure 2 The global cost 120573(119897 119897) is the DTWdistance between the two series The DTW distance can bemeasured for two series with different lengths In this study

the lengths of two series are the same as 119897 Thus the costmatrix is 119897lowast119897The 119888(119894 119895) is calculated as the Euclidean distancebetween 119904119887119894 and 119904119886119895 To effectively find the shortest pathdynamic programming algorithm is used

120573 (119894 119895)= 119888 (119894 119895)+min 120573 (119894 minus 1 119895 minus 1) 120573 (119894 minus 1 119895) 120573 (119894 119895 minus 1)

(1)

4 Journal of Advanced Transportation

where 120573(119894 119895) is the global cost from item (1 1) to item(119894 119895) By iteratively solving (1) the shortest path can be ob-tained

23 Peak Searching for Critical Transition State Point Basedon the DTW distance the CTS points could be identified asthe points with significantly large DTWdistances Intuitivelya threshold-based method could be used Thus all pointswith DTW distances larger than a predetermined thresholdwould be regarded as the CTS points However the selectionof the thresholds could be difficulty and arbitrary becausedifferent networksmay have distinct thresholds Basically theidentification of the CTS points is an extreme value anal-ysis problem over a series Therefore a more sophisticatedalgorithm was developed in this study The algorithm wasdescribed as follows

Step 1 Smooth the distance series with Locally WeightedRegression (LOESS) [40] The smoothing process is initiallyconducted in order to filter the occasionally and locallyfluctuated points in the DTW distance series Let 119910119895 be thesmoothed value of the jth item in the DTW distance serieswhich were divided into 119899 equally spaced intervals

Step 2 Determine the ranges that contain the CTS pointsThe identification of CTS is to find the extreme point (EP)of DTW series To determine the range of each extreme point(EP) the approximate extreme point (AEP) is captured firstThe AEP locates at a small scale that an extreme point wascontained Notate Δ119910119895 and 2119910119895 as the first and second orderdifference of 119910119895 The monotonicity of 119910(119895) is determined bycomparing the sign of 119910119895 Δ1199101198952119910119895 which were expressed in(2) and (3)

119910119895 = 119891 (119895) Δ119910119895 = 119891 (119895 + 1) minus 119891 (119895) 2119910119895 = Δ119910119895+1 minus Δ119910119895

(2)

Δ119910119895 = 1 Δ119910119895 gt 00 Δ119910119895 lt 0 (3)

If (2119910119895) = 0 the monotonicity of 119891(119895 + 1) changedcompared with 119891(119895) and there is an AEP around (119895 + 1)119905ℎinterval Based on the determined AEPs the ranges thatcontain EPs are determined by the centres of neighbouringAEPs

Step 3 Identify the CTS points using Brendrsquos method Theidentification for CTS (EP) point can be viewed as rooting-finding of the first derivative of 119910 within each range Brendrsquosmethod [41] is used to find the extreme point 119910119888 with localmaximum distance within each range The method is aroot-finding algorithm combining the bisection method thesecantmethod and inverse quadratic interpolationwhich hasthe advantage of fast-converging Corresponding to 119910119888 thecritical state point 119904119888 is achieved

24 Day-to-Day Pattern Recognition of the CTS Points Dueto the stochastic nature of urban traffic flow it is assumedthat the day-to-day distribution of the CTS points is subjectto a Gaussian Mixture Distribution In this study a GaussianMixture Model (GMM) is used to cluster the CTS pointsextracted from different days Compared to other clusteringmethods such as the K-means one appealing feature of theGMM clustering method is that the probability of a statepoint belonging to a cluster is available The GMM clusteringmethod is described as follows

In GMM the data is assumed to be generated froma mixture of a finite number of Gaussian distributionsEach Gaussian distribution is regarded as a component ofthe GMM and the components are linearly combined bymixing weights In fact the GMM clustering is equivalentto estimate the unknown mixing weights and parameters ofeach Gaussian distribution In this study in addition to thenetwork flow and occupancy the temporal feature (eg timeindex during a day) of the CTS points is also included to theclustering process Let 119904119888 represents the set of the identifiedCTS points from multiple days 119904119888119894 which represents for CTSpoint is a vector formed by network flow-occupancy andthe time of day for the 119894th element in 119904119888 0119896(119904119888119894 | 120579119896) is theProbability Density Function (PDF) of kth component TheGMMmodel is shown in

0 (119904119888119894 | 120579) = 119866sum119896=1

1205911198960119896 (119904119888119894 | 120579119896)

(119904119888119894 isin 1198773 119896 isin 1 2 119866 120591119896 isin (0 1) 119866sum119896

120591119896 = 1 ))(4)

where 119896 presents the kth component in the GMM 119866 is thetotal number of the components 120579119896 = (120583119896 sum119896) 120583119896 and sum119896are the mean and covariance of the kth component and 120591119896 isthe mixing weight of the kth component

As described in (5) the GMM-based clustering methodcan be viewed as a process to find the optimal parameters 120591119896and 120579119896 to maximize the likelihood function of GMM

119871 (120579 119904119888 119885) = 119873prod119894=1

119866prod119896=1

[1205911198960119896 (119904119888119894 | 120579119896)]I(119885119894=119896) (5)

120579119896 and 120591119896 were iteratively estimated using theExpectation-Maximization (EM) algorithm [42] The E-step which is expressed in (6) aims at getting the posteriorprobability (120574(119904119888119894 119896)) of 119904119888119894 belonging to component 119896 withinitialled 120579119896 and 120591119896 The M-step is to get the new estimateof 120579119896 and 120591119896 through maximizing 119871(120579 119904119888 119885) with posteriorprobability achieved in E-step The estimation of parametersis shown in (7)-(9)

120574 (119904119888119894 119896) = 1205911198960119896 (119904119888119894 | 120579119896)sum119866119896=1 1205911198960119896 (119904119888119894 | 120579119896) (6)

120583119896 = sum119873119894=1 120574 (119904119888119894 119896) 119904119888119894sum119873119894=1 120574 (119904119888119894 119896) (7)

Journal of Advanced Transportation 5

The clustering of critical transition state points based on GMMInput 119904119888 120579119905 119866Output 120583119896lowast sum119896lowast 120591119896lowast

Process(1) Giving the initial 120579119905 and 119866(2) 119864 larr997888 120579119905+1 minus 120579119905(3) When 119864 gt 120576 do(4) 120574(119904119888119894 119896) larr997888 F(120579119905) step 4 is E-step (Equation (6))(5) 120579119905+1 larr997888 argmax120579 step 5 is M-step (Equation (7)-(9))(6) replace 120579119905 with 120579119905+1(7) end for

Algorithm 1 The EM Algorithm to estimate parameters in GMM

sum119896

= sum119873119894=1 120574 (119904119888119894 119896) (119904119888119894 minus 120583119896) (119904119888119894 minus 120583119896)119879sum119873119894=1 120574 (119904119888119894 119896) (8)

120591119896 = sum119873119894=1 120574 (119904119888119894 119896)119873 (9)

where 119873 is the total number of all critical transition statepoints 119885119895 is a latent variable that determines the componentfrom which the observation originates The EM Algorithm isshown in Algorithm 1 120579119905 is the initial value for 120579 and 119905119894 is the119894th literation 119864 is the difference between two neighbouringiterations and 120576 is a small real number and 120579lowast is theoptimal value It is worth mentioning that the BIC value isused to avoid overfitting and determine the optimal modelparameters and the number of clusters [43]

3 Implementation

31 Data Collection A real network with 13 signalized inter-sections and 26 links in Kunshan City China was selectedas the test site as shown in Figure 3 All intersectionsoperated under pretimed signal control with four differentsignal plans in a day The four arterials in the network hadsignificant traffic volume during the day time Oversaturatedand spillover conditions were frequently observed duringmorning and afternoon peaks Microwave-based detectorswere installed on links except the ones shown as the dash linkin Figure 3 All detectors were located approximately at themiddle of the links and collected traffic counts and occupancyin every 5minutes Twomonths of data collected fromAugust1 2015 to September 30 2015 were used in this study Inaddition for validation purpose GPS probe vehicle data from1526 taxi cabs during the same period were also collected tocalculate the network average speed The space-mean speedof the network was simply calculated as the ratio between thetotal distance travelled and the total time spent by the taxicabs in the test site for every 5 minutes

32 Identification of the CTS Points Before calculating DTWdistance for the two subseries in the sliding windows it isimportant to select a reasonable window length since thetwo subseries are expected to reflect the traffic evolution

patterns However due to the different OD patterns andsignal operations in different networks the window lengthis more likely network-dependent In general the windowlength should be shorter than the peak duration Besidesthe window length should not be too short to filter randomfluctuations in data series and not too long to reflect trafficdynamics Basically determining a desirable length is a trade-off between reliability and accuracy

In our case the morning peak is from 600 to 900and the afternoon peak is from 1630 to 1930 Thereforethe window length should be shorter than 3 hours Basedon that the impacts of different window lengths on themeasurement of the DTW distance were investigated Anindex of AverageChange Rate (ACR)was used to describe thefluctuation in theDTWdistance seriesTheACR is calculatedas

119860119862119877 = 1119899119899sum119894=1

119889119894+1 minus 119889119894119889119894 (10)

where 119889119894 is the normalized DTWdistance associated with theith state point in a MFD time series and 119899 is the number ofstate points

Figure 4(a) shows the measurement of DTW distancecalculated from the MFD time series on August 10 2015with window length ranging from 15 minutes to 3 hours Itis observed that the curves of the DTW distance becomesmoother with the increasing of window length The curveswith window lengths of 1 hour and 15 hours are observed asthe relatively balanced cases where the trends are clear andinformative short-term changes are retained In addition theACR was calculated using data in the two months as shownin Figure 4(b) Each point in Figure 4(b) represents the ACRof the DTW distance series of a day with a particular windowlength When the window length is lower than 1 hour theACR decrease significantly from about 011 to 001 A turningpoint can be clearly spotted at the window length of 1 hourBased on the investigation the window length was selected as1 hour for our network

With the measurement of DTW distance a peak search-ing method was applied Figure 5 illustrates the identifiedpeaks on the DTW distance curve of August 11 2015The redand blue marks represent the local maximum and minimum

6 Journal of Advanced Transportation

120104

120204

106104

106204

108130

108230

108131

108231

1091

09

1092

09

1091

10

1092

10

1091

11

1092

11

106105

106205

106106

106206

108129

108229

1091

00

1092

00

1111

01

1112

01

120105

120205

737

313

303

359

589

350

120104

623

335 884

333 411

777

150

Chan

gjia

ng R

oad

HeilongjiangRoad

TongfengRoad

Qianjin Road

Kuntai Road

Link length(m)

Signalized Intersection

Link IDLink with detectorLink without detector

Figure 3 Layout of the test network and detectors in Kunshan City China

extreme points extracted by the proposed method The localmaximum extreme points are regarded as the CTS points inthis study As shown in Figure 5 the CTS points (see thepoints located at the high peaks of the curves) appear duringthe morning and afternoon peaks Although the most criticaltransition state points are successfully identified suspiciousCTS points are also found at the local peaks on the curvesSome suspicious points (shown in the blue circle in Figure 5)are obviously invalid since the corresponding DTWdistancesare so small These points could be readily excluded bya lower bound threshold In this study any state pointswith DTW distance less than 15 were considered as invalidpoints

33 Validation of the CTS Points To validate the identifiedCTS points the locations of the CTS points at the MFD timeseries the correlation between the CTS points and networkspace-mean speed were investigated For ease of illustrationthe morning and afternoon peak were separately consideredThe identified CTS points on August 10 Monday 2015and August 12 Wednesday 2015 are illustrated in Figure 6Figures 6(a) 6(d) and 6(g) show the CTS points during themorning peaks of the three days and Figures 6(b) 6(e) and

6(h) show the CTS points during the afternoon peaks of thethree days Figures 6(c) 6(f) and 6(i) illustrate the correlationbetween the CTS points and network space-mean speed forthe three days The red markers represent the identified CTSpoints The arrows indicate the time sequence within theMFD time series

For the locations at the MFD time series the identifiedCTS points approximately locate at the boundaries of the twodistinguished traffic states see Figures 6(a) 6(d) 6(g) 6(b)6(e) and 6(h) The state points on the left side of the CTSpoints show a positive correlation between network flow andoccupancy while the correlation of state points on the rightside is vague and the slopes between the consecutive statepoints are much more fluctuated than the left side A flatshape of the right side state points can be observed and thenetwork flow fluctuates within a small range from 75 to 90veh5min Based on this it is reasonable to infer that the leftand right side of the identified CTS points are correspondingto the free-flow and optimal accumulation state respectivelyIn other words the identified CTS points well captured theshifts of network traffic states between the free-flow andoptimal accumulation state

By incorporating the network speed in all these threedays the CTS points are identified at the times when the

Journal of Advanced Transportation 7

15min 30min

45min 1h

15h 2h

25h 3h

Time of day

Dyn

amic

Tim

e Wra

ppin

g di

stan

ce

0

1

2

3

0

5

10

2 4 6 8 10 12 14 16 18 20 22 240

0102030

20406080

4 6 8 10 12 14 16 18 20 222 6 8 10 12 14 16 18 204

25

50

75

100

0102030405060

4 6 8 10 12 14 16 18 20 2222 4 6 8 10 12 14 16 18 20 22 240

2 4 6 8 10 12 14 16 18 20 22 24005

101520

2 4 6 8 10 12 14 16 18 20 22 24002468

2 4 6 8 10 12 14 16 18 20 22 240

(a)

000

003

006

009

Aver

age c

hang

e rat

e of d

istan

ce

2 31Length of window (h)

(b)

Figure 4 Impacts of different window lengths on the measurement of DTW distance (a) DTW distance curves with different lengths ofwindow on August 10 Tuesday 2015 (b) ACR calculation with different window lengths during August 1 2015 and September 30 2015

network congestion just appeared and nearly recovered seeFigures 6(c) 6(f) and 6(i) Moreover we can find thatonly the CTS points during the congestion formation stageare identified in the morning peaks of the three days Thereason may be that the traffic flow decreases slowly duringthe congestion recovery stage at the end of the morningpeaks and the traffic state after the morning peaks doesnot experience a significant change According to the pairedCTS points in afternoon peaks the optimal accumulationstate can be approximately estimated as (012 73veh5min) byaveraging over the three days

34 Day-to-Day Pattern Analysis The empirical distributionof the identified CTS points based on two-month weekdaydata is visualized in as shown Figure 7 A clear bimodalitycan be observed in the distribution In fact the bimodality iscaused by the different distributions of the CTS points in themorning and afternoon peaks

Considering the most congested afternoon peaks inour case day-to-day patterns of the CTS points from theafternoon peaks in 38 weekdays were investigated with theGMM clustering Two distinct clusters are found as shownin Figure 8 The cluster centre for Cluster 1 (in blue) is (1718

8 Journal of Advanced Transportation

2 4 6 8 10 12 14 16 18 20 22 240Time of day

0

10

20

DTW

dist

ance

Figure 5The DTW distance on August 11 Tuesday 2015

013 728 (veh5min)) and for Cluster 2 (in red) is (1810 010722(veh5min)) 516 and 484 of the total identified CTSpoints are in Cluster 1 and Cluster 2 respectively Figures 8(a)and 8(b) illustrate the distribution of flow-occupancy andtime in the two clusters respectively According to the timedistribution Cluster 1 could be the critical transition stateindicating a transition from the free-flow state to optimalaccumulation state while Cluster 2 is the critical transitionstate from the optimal accumulation state to free-flow stateIn terms of the flow-occupancy distribution it is found thatthe CTS points in Cluster 1 are much more scattered thanthose in Cluster 2 This indicates that the transition from theoptimal accumulation state to free-flow state is much morepredictable and the transition from the free-flow to optimalaccumulation state has higher uncertainty

35 Extended Application of Real-Time CTS Point Identifi-cation In addition to the analysis of the CTS points usinghistorical data an extended application of real-time CTSpoint identification is also discussed Based on the proposedmethod theMFD state points can be labelled as three classesthe CTS points of Cluster 1 the CTS points of Cluster2 and non-CTS points Thus a probabilistic discriminantanalysis [43] was conducted using the labelled dataset Thediscriminant analysis also known as supervised classifica-tion assumes that the observations in each class are generatedby a distribution specific to that class Let 0119896 be the probabilitydistribution of the kth class which is Gaussian distributionin this study 120591119896 be the proportion of observations of class kin the population and 119904119905 be a real-time observation of MFDstate points The posterior probability that 119904119905 belongs to thekth class is expressed in

119901119903 (119904119905 isin 119888119897119886119904119904 119896) = 1205911198960119896 (119904119905 120583119896 sum119896) (119904119905)sum119866119896=1 1205911198960119896 (119904119905 120583119896 sum119896) (119904119905) 119896 isin 1 2 3

(11)

In this study the labelled MFD state points in theafternoon peaks of 45 days are used as the training set andanother 15 days as the test set Each MFD state point was

identified as an observation of the class with the highestprobability that belongs to According to the test results theerror of the discriminant method is about 81

The discrimination results for three typical weekdays areillustrated in Figure 9Theblack red and blue dots in Figure 9represent the non-CTS points the CTS points of Cluster 1and the CTS point of Cluster 2 respectively The probabilisticdiscriminant analysis performed well in identifying the CTSpoints All the identified CTS points are located at theboundaries between the free-flow and optimal accumulationstate as shown in Figures 9(a) 9(c) and 9(e) and illustrate astrong correlation to the changes of network speed during theformation and recovery period of the network congestion

4 Conclusions

This study proposed a data-driven method to identify andcharacterize the critical transition state in MFD using fielddata The identification of the critical transition state isbased on the analysis of the MFD time series The DTWdistance is used to measure the similarity of subseries beforeand after each state point in a MFD time series and apeak searching method is developed to identify the criticaltransition state points based on the DTW distance seriesThe GMM clustering is used to characterize the day-to-daypattern of the identified critical transition state points Theprobabilistic discriminant analysis is performed to identifythe CTS points

The identification and validation of CTS points indicatedthat the proposed method well identified the network tran-sition states By analysis the identified CTS points well cap-tured the shifts of network traffic states between the free-flowand optimal accumulation state The optimal accumulationstate can be approximately estimated as (012 73veh5min)by averaging over the three days A clear bimodality can beobserved by exploring the probability density of identifiedCTS points and the bimodality is caused by different dis-tributions of the CTS points in the morning and afternoonpeaks Moreover the CTS points were classified as twocategories One cluster represents the transition from free-flow state to optimal accumulation state and the otherrepresents the transition from optimal accumulation state

Journal of Advanced Transportation 9

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)01 02 03 0400

Network occupancy

(b)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(c)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(d)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(g)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(i)

Figure 6The validation of the identified CTS points on August 10 Monday 2015 and August 12 Wednesday 2015 (a) (d) and (g)The CTSpoints during morning peaks of the three days (b) (e) and ( h) The CTS points during afternoon peaks of the three days (c) (f) and (i)Correlation between the CTS points and network speed

to free-flow state Besides the probabilistic discriminantanalysis performed well in identifying the CTS points All theidentified CTS points were located at the boundaries betweenthe free-flow and optimal accumulation state The proposed

method is helpful to better understand the evolution processof network traffic flow and offers opportunities to developmore reliable and responsive network traffic flow manage-ment strategies

10 Journal of Advanced Transportation

0

4020

6080

100

Networ

k flow

(veh

5m

in)

0000002040608

005010 015 020 025

Network occupancy

Figure 7 Visualization of the distribution of the CTS points

0

40

20

60

80

100

Net

wor

k flo

w (v

eh5

min

)

000 005 010 015 020 025Network occupancy

(a)

Cluster12

0

5

10

15C

ount

17 18 1916Time of day

(b)

Figure 8 Day-to-day pattern of the CTS points from the afternoon peaks of 38 weekdays (a) Flow-occupancy distribution in two clusters(b) Time distribution in two clusters

In future the impact factors of critical transition statesin actual network will be investigated through comparativeanalysis under certain conditions Besides it is encouragedto test the proposed method with different sources of data(eg probe vehicle or vehicle license plate reidentificationdata) Also more field validation works are needed to furtherinterpret the critical transition states considering real trafficbehaviours

Data Availability

Partial of the detector data used to support the findings ofthis study are includedwithin the supplementary informationfile

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The work was funded by the National Natural Science Foun-dation of China (no 71871055) and theKeyResearchProgramof Jiangsu Province Science and Technology Department (noBE2017027)

Supplementary Materials

The supplementary material file is partial of the detectordata used to derive the MFD for test site The data are

Journal of Advanced Transportation 11

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

(b)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(c)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60N

etw

ork

spee

d (k

mh

)

(d)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

Figure 9 The discrimination results on three typical weekdays (a) (c) and (e) The identified CTS points in the network flow-occupancyseries during afternoon peak of the three days (b) (d) and (f ) The identified CTS points in the network speed series of the three days

12 Journal of Advanced Transportation

valid and they were formed by STATIONID (the ID numberof specific link) CALENDAR KEY (the date of traffic flowdata for specific link) FIVEMINX (the time of traffic flowdata for specific date) OCCUPANCY (the occupancy5minfor specific link) and VOLUME (the flow5min for spe-cific link) from August 1 2015 to September 30 2015(Supplementary Materials)

References

[1] M Keyvan-Ekbatani M Papageorgiou and I PapamichailldquoPerimeterTraffic Control via Remote FeedbackGatingrdquo Proce-dia - Social and Behavioral Sciences vol 111 pp 645ndash653 2014

[2] Y M Du J P Wu and Y Jia H ldquoMFD-Based Regional TrafficVolume Dynamic Controlrdquo Journal of Transportation SystemsEngineering Information Technology vol 14 no 3 pp 162ndash1672014

[3] A Csikos T Tettamanti and I Varga ldquoMacroscopic modelingand control of emission in urban road traffic networksrdquo Trans-port vol 30 no 2 pp 152ndash161 2015

[4] M D Simoni A J Pel R A Waraich and S P HoogendoornldquoMarginal cost congestion pricing based on the network fun-damental diagramrdquo Transportation Research Part C EmergingTechnologies vol 56 pp 221ndash238 2015

[5] M Hajiahmadi V L Knoop B De Schutter and H Hellen-doorn ldquoOptimal dynamic route guidance A model predictiveapproach using the macroscopic fundamental diagramrdquo inProceedings of the 2013 16th International IEEE Conferenceon Intelligent Transportation Systems Intelligent TransportationSystems for All Modes ITSC 2013 pp 1022ndash1028 NetherlandsOctober 2013

[6] R Alsalhi and V Dixit ldquoThe relationship between traffic safetyand macroscopic fundamental diagram (MFD)rdquo AustralasianTransport Research Forum (ATRF) vol 37 2015

[7] C F Daganzo V V Gayah and E J Gonzales ldquoMacroscopicrelations of urban traffic variables bifurcations multivalued-ness and instabilityrdquo Transportation Research Part B Method-ological vol 45 no 1 pp 278ndash288 2011

[8] J Haddad and N Geroliminis ldquoOn the stability of trafficperimeter control in two-region urban citiesrdquo TransportationResearch Part B Methodological vol 46 no 9 pp 1159ndash11762012

[9] V V Gayah and C F Daganzo ldquoClockwise hysteresis loops inthe Macroscopic Fundamental Diagram An effect of networkinstabilityrdquo Transportation Research Part B Methodological vol45 no 4 pp 643ndash655 2011

[10] C FDaganzo andNGeroliminis ldquoAn analytical approximationfor the macroscopic fundamental diagram of urban trafficrdquoTransportation Research Part B Methodological vol 42 no 9pp 771ndash781 2008

[11] N Geroliminis and B Boyaci ldquoThe effect of variability of urbansystems characteristics in the network capacityrdquo TransportationResearch Part B Methodological vol 46 no 10 pp 1607ndash16232012

[12] E J Gonzales C Chavis Y Li and C F Daganzo Mul-timodal transport modeling for nairobi kenya insights andrecommendations with an evidence-based model insights andrecommendations with an evidence-based model kenya 2009

[13] A Mazloumian N Geroliminis and D Helbing ldquoThe spatialvariability of vehicle densities as determinant of urban networkcapacityrdquo Philosophical Transactions of the Royal Society A

Mathematical Physical amp Engineering Sciences vol 368 no1928 pp 4627ndash4647 2010

[14] N Geroliminis and C F Daganzo ldquoExistence of urban-scalemacroscopic fundamental diagrams some experimental find-ingsrdquo Transportation Research Part B Methodological vol 42no 9 pp 759ndash770 2008

[15] T Courbon and L Leclercq ldquoCross-comparison of macroscopicfundamental diagram estimation methodsrdquo in Proceedings ofthe 14thMeeting of the EuroWorking Group on Transp - In Questfor Adv Models Tools and Methods for Transp and LogistEWGT 26th Mini-EURO Conf - Intelligent Decis Making inTransp and Logist MEC 1st Eur Sci Conf on Air Transp - RHpp 417ndash426 Poland September 2011

[16] VGayah andV Dixit ldquoUsingmobile probe data and themacro-scopic fundamental diagram to estimate network densitiesrdquoTransportation Research Record no 2390 pp 76ndash86 2013

[17] J Du H Rakha and V V Gayah ldquoDeriving macroscopicfundamental diagrams from probe data Issues and proposedsolutionsrdquo Transportation Research Part C Emerging Technolo-gies vol 66 pp 136ndash149 2016

[18] Z Xu P J Jin and X Jiang ldquoMacroscopic FundamentalDiagram (MFD) Based on Vehicle-to-Infrastructure (V2I)Connected Vehicle Data Transportation Research Board 94thAnnual Meetingrdquo Macroscopic Fundamental Diagram (MFD)Based on Vehicle-to-Infrastructure (V2I) Connected Vehicle Data Transportation Research Board 94th Annual Meeting 2015

[19] M Saberi and H Mahmassani ldquoHysteresis and capacity dropphenomena in freeway networks Empirical characterizationand interpretationrdquo Transportation Research Record no 2391pp 44ndash55 2013

[20] M Keyvan-Ekbatani A Kouvelas I Papamichail and MPapageorgiou ldquoExploiting the fundamental diagram of urbannetworks for feedback-based gatingrdquo Transportation ResearchPart B Methodological vol 46 no 10 pp 1393ndash1403 2012

[21] K Aboudolas and N Geroliminis ldquoPerimeter and boundaryflow control inmulti-reservoir heterogeneous networksrdquoTrans-portation Research Part B Methodological vol 55 pp 265ndash2812013

[22] K Ampountolas and A Kouvelas ldquoReal-Time Estimation ofCritical Values of the Macroscopic Fundamental Diagramfor Maximum Network Throughputrdquo Transportation ResearchBoard 94th Annual Meeting 2015

[23] B Ponnu and B Coifman ldquoWhen adjacent lane dependenciesdominate the uncongested regime of the fundamental relation-shiprdquo Transportation Research Part B Methodological vol 104pp 602ndash615 2017

[24] X Qu J Zhang and S Wang ldquoOn the stochastic fundamentaldiagram for freeway traffic Model development analyticalproperties validation and extensive applicationsrdquo Transporta-tion Research Part B Methodological vol 104 pp 256ndash271 2017

[25] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015

[26] J Xia and M Chen ldquoA nested custering technique for freewayoperating condition classificationrdquo Computer-Aided Civil andInfrastructure Engineering vol 22 no 6 pp 430ndash437 2007

[27] H-p Lu Z-y Sun and W-c Qu ldquoBig data-driven based real-time traffic flow state identification and predictionrdquo DiscreteDynamics in Nature and Society Art ID 284906 11 pages 2015

Journal of Advanced Transportation 13

[28] M Azimi and Y Zhang ldquoCategorizing Freeway Flow Condi-tions by Using Clustering Methodsrdquo Transportation ResearchRecord vol 2173 no 1 pp 105ndash114 2018

[29] J Xia W Huang and J Guo ldquoA clustering approach to onlinefreeway traffic state identification using ITS datardquoKSCE Journalof Civil Engineering vol 16 no 3 pp 426ndash432 2012

[30] J Tang F Liu W Zhang S Zhang and Y Wang ldquoExploringdynamic property of traffic flow time series in multi-statesbased on complex networks Phase space reconstruction versusvisibility graphrdquo Physica A Statistical Mechanics and its Appli-cations vol 450 pp 635ndash648 2016

[31] H B Celikoglu and M A Silgu ldquoExtension of traffic flowpattern dynamic classification by a macroscopic model usingmultivariate clusteringrdquo Transportation Science vol 50 no 3pp 966ndash981 2016

[32] S Oh Y-J Byon and H Yeo ldquoImpact of Traffic State Transitionand Oscillation on Highway Performance with Section-BasedApproachrdquo in Proceedings of the 18th IEEE International Confer-ence on Intelligent Transportation Systems ITSC 2015 pp 2141ndash2146 Spain September 2015

[33] J H Banks ldquoInvestigation of some characteristics of congestedflowrdquo Transportation Research Record no 1678 pp 128ndash1341999

[34] E I Vlahogianni M G Karlaftis and J C Golias ldquoStatisticalmethods for detecting nonlinearity and non-stationarity in uni-variate short-term time-series of traffic volumerdquoTransportationResearch Part C Emerging Technologies vol 14 no 5 pp 351ndash367 2006

[35] Y Kamarianakis H Oliver Gao and P Prastacos ldquoCharacter-izing regimes in daily cycles of urban traffic using smooth-transition regressionsrdquo Transportation Research Part C Emerg-ing Technologies vol 18 no 5 pp 821ndash840 2010

[36] J Tang Y Wang and F Liu ldquoCharacterizing traffic timeseries based on complex network theoryrdquo Physica A StatisticalMechanics and its Applications vol 392 no 18 pp 4192ndash42012013

[37] Y Yan S Zhang J Tang and X Wang ldquoUnderstandingcharacteristics in multivariate traffic flow time series fromcomplex network structurerdquo Physica A Statistical Mechanicsand its Applications vol 477 pp 149ndash160 2017

[38] S Blandin J Argote A M Bayen and D B Work ldquoPhasetransition model of non-stationary traffic flow Definitionproperties and solution methodrdquo Transportation Research PartB Methodological vol 52 pp 31ndash55 2013

[39] D F Silva and G E Batista ldquoSpeeding up all-pairwise dynamictime warping matrix calculationrdquo in Proceedings of the 2016SIAM International Conference on Data Mining Society forIndustrial and Applied Mathematics pp 837ndash845 2016

[40] O B Linton ldquoLocal Regression Modelsrdquo inMicroeconometricsPalgrave Macmillan London UK 2010

[41] R P Brent Algorithms for Minimization without DrivativesPrentice Hall Englewood Cliffs NJ USA 2013

[42] NM Nasrabadi ldquoPattern Recognition andMachine LearningrdquoJournal of Electronic Imaging vol 16 no 4 p 049901 2007

[43] C Fraley and A E Raftery ldquoModel-based clustering discrimi-nant analysis and density estimationrdquo Journal of the AmericanStatistical Association vol 97 no 458 pp 611ndash631 2002

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Journal of Advanced Transportation 3

Smooth the DTW distance serieswith LOESS

Extract the two sub-seriesby sliding windows

Window 1

Net

wor

k flo

w

Network occupancy

Measuring similarity between two sub-series

DTW distanceseries

Time of day

CTS

DTW

dist

ance

Identify the CTS points by root-finding

Cluster the CTS pointsusing GMM method

Peak searching for the CTS points Investigating the day-to-day patterns of the CTS points

CTS

The set of CTSpoints from

different days

Window 2

Calculate DTWDistance

0

40

20

60

80

100

Net

wor

k flo

w (v

eh5

min

)

000 005 010 015 020 025Network occupancy

0

4020

6080

100

Networ

k flow

(veh

5m

in)

0000002040608

005 010 015 020 025Network occupancy

Figure 1 Framework of proposed method

Time series alignment

4 6 8 10 122SB index

2

4

6

8

10

12

SA in

dex

l = 12

( )

Figure 2 An optimal correspondence on warped evolution paths for 119904119905

The schematic diagram for calculation of the DTWdistance is illustrated in Figure 2 The DTW distance iscalculated in two steps First calculate the local cost matrixEach itemof thematrixwith index (119894 119895) represents a local cost119888(119894 119895) between 119904119887119894 and 119904119886119895 Second calculate the accumulatedcost matrix or global cost matrix as shown in (1) Third findthe shortest path with the global cost matrix 120573(119894 119895) shownas the path 0 in Figure 2 The global cost 120573(119897 119897) is the DTWdistance between the two series The DTW distance can bemeasured for two series with different lengths In this study

the lengths of two series are the same as 119897 Thus the costmatrix is 119897lowast119897The 119888(119894 119895) is calculated as the Euclidean distancebetween 119904119887119894 and 119904119886119895 To effectively find the shortest pathdynamic programming algorithm is used

120573 (119894 119895)= 119888 (119894 119895)+min 120573 (119894 minus 1 119895 minus 1) 120573 (119894 minus 1 119895) 120573 (119894 119895 minus 1)

(1)

4 Journal of Advanced Transportation

where 120573(119894 119895) is the global cost from item (1 1) to item(119894 119895) By iteratively solving (1) the shortest path can be ob-tained

23 Peak Searching for Critical Transition State Point Basedon the DTW distance the CTS points could be identified asthe points with significantly large DTWdistances Intuitivelya threshold-based method could be used Thus all pointswith DTW distances larger than a predetermined thresholdwould be regarded as the CTS points However the selectionof the thresholds could be difficulty and arbitrary becausedifferent networksmay have distinct thresholds Basically theidentification of the CTS points is an extreme value anal-ysis problem over a series Therefore a more sophisticatedalgorithm was developed in this study The algorithm wasdescribed as follows

Step 1 Smooth the distance series with Locally WeightedRegression (LOESS) [40] The smoothing process is initiallyconducted in order to filter the occasionally and locallyfluctuated points in the DTW distance series Let 119910119895 be thesmoothed value of the jth item in the DTW distance serieswhich were divided into 119899 equally spaced intervals

Step 2 Determine the ranges that contain the CTS pointsThe identification of CTS is to find the extreme point (EP)of DTW series To determine the range of each extreme point(EP) the approximate extreme point (AEP) is captured firstThe AEP locates at a small scale that an extreme point wascontained Notate Δ119910119895 and 2119910119895 as the first and second orderdifference of 119910119895 The monotonicity of 119910(119895) is determined bycomparing the sign of 119910119895 Δ1199101198952119910119895 which were expressed in(2) and (3)

119910119895 = 119891 (119895) Δ119910119895 = 119891 (119895 + 1) minus 119891 (119895) 2119910119895 = Δ119910119895+1 minus Δ119910119895

(2)

Δ119910119895 = 1 Δ119910119895 gt 00 Δ119910119895 lt 0 (3)

If (2119910119895) = 0 the monotonicity of 119891(119895 + 1) changedcompared with 119891(119895) and there is an AEP around (119895 + 1)119905ℎinterval Based on the determined AEPs the ranges thatcontain EPs are determined by the centres of neighbouringAEPs

Step 3 Identify the CTS points using Brendrsquos method Theidentification for CTS (EP) point can be viewed as rooting-finding of the first derivative of 119910 within each range Brendrsquosmethod [41] is used to find the extreme point 119910119888 with localmaximum distance within each range The method is aroot-finding algorithm combining the bisection method thesecantmethod and inverse quadratic interpolationwhich hasthe advantage of fast-converging Corresponding to 119910119888 thecritical state point 119904119888 is achieved

24 Day-to-Day Pattern Recognition of the CTS Points Dueto the stochastic nature of urban traffic flow it is assumedthat the day-to-day distribution of the CTS points is subjectto a Gaussian Mixture Distribution In this study a GaussianMixture Model (GMM) is used to cluster the CTS pointsextracted from different days Compared to other clusteringmethods such as the K-means one appealing feature of theGMM clustering method is that the probability of a statepoint belonging to a cluster is available The GMM clusteringmethod is described as follows

In GMM the data is assumed to be generated froma mixture of a finite number of Gaussian distributionsEach Gaussian distribution is regarded as a component ofthe GMM and the components are linearly combined bymixing weights In fact the GMM clustering is equivalentto estimate the unknown mixing weights and parameters ofeach Gaussian distribution In this study in addition to thenetwork flow and occupancy the temporal feature (eg timeindex during a day) of the CTS points is also included to theclustering process Let 119904119888 represents the set of the identifiedCTS points from multiple days 119904119888119894 which represents for CTSpoint is a vector formed by network flow-occupancy andthe time of day for the 119894th element in 119904119888 0119896(119904119888119894 | 120579119896) is theProbability Density Function (PDF) of kth component TheGMMmodel is shown in

0 (119904119888119894 | 120579) = 119866sum119896=1

1205911198960119896 (119904119888119894 | 120579119896)

(119904119888119894 isin 1198773 119896 isin 1 2 119866 120591119896 isin (0 1) 119866sum119896

120591119896 = 1 ))(4)

where 119896 presents the kth component in the GMM 119866 is thetotal number of the components 120579119896 = (120583119896 sum119896) 120583119896 and sum119896are the mean and covariance of the kth component and 120591119896 isthe mixing weight of the kth component

As described in (5) the GMM-based clustering methodcan be viewed as a process to find the optimal parameters 120591119896and 120579119896 to maximize the likelihood function of GMM

119871 (120579 119904119888 119885) = 119873prod119894=1

119866prod119896=1

[1205911198960119896 (119904119888119894 | 120579119896)]I(119885119894=119896) (5)

120579119896 and 120591119896 were iteratively estimated using theExpectation-Maximization (EM) algorithm [42] The E-step which is expressed in (6) aims at getting the posteriorprobability (120574(119904119888119894 119896)) of 119904119888119894 belonging to component 119896 withinitialled 120579119896 and 120591119896 The M-step is to get the new estimateof 120579119896 and 120591119896 through maximizing 119871(120579 119904119888 119885) with posteriorprobability achieved in E-step The estimation of parametersis shown in (7)-(9)

120574 (119904119888119894 119896) = 1205911198960119896 (119904119888119894 | 120579119896)sum119866119896=1 1205911198960119896 (119904119888119894 | 120579119896) (6)

120583119896 = sum119873119894=1 120574 (119904119888119894 119896) 119904119888119894sum119873119894=1 120574 (119904119888119894 119896) (7)

Journal of Advanced Transportation 5

The clustering of critical transition state points based on GMMInput 119904119888 120579119905 119866Output 120583119896lowast sum119896lowast 120591119896lowast

Process(1) Giving the initial 120579119905 and 119866(2) 119864 larr997888 120579119905+1 minus 120579119905(3) When 119864 gt 120576 do(4) 120574(119904119888119894 119896) larr997888 F(120579119905) step 4 is E-step (Equation (6))(5) 120579119905+1 larr997888 argmax120579 step 5 is M-step (Equation (7)-(9))(6) replace 120579119905 with 120579119905+1(7) end for

Algorithm 1 The EM Algorithm to estimate parameters in GMM

sum119896

= sum119873119894=1 120574 (119904119888119894 119896) (119904119888119894 minus 120583119896) (119904119888119894 minus 120583119896)119879sum119873119894=1 120574 (119904119888119894 119896) (8)

120591119896 = sum119873119894=1 120574 (119904119888119894 119896)119873 (9)

where 119873 is the total number of all critical transition statepoints 119885119895 is a latent variable that determines the componentfrom which the observation originates The EM Algorithm isshown in Algorithm 1 120579119905 is the initial value for 120579 and 119905119894 is the119894th literation 119864 is the difference between two neighbouringiterations and 120576 is a small real number and 120579lowast is theoptimal value It is worth mentioning that the BIC value isused to avoid overfitting and determine the optimal modelparameters and the number of clusters [43]

3 Implementation

31 Data Collection A real network with 13 signalized inter-sections and 26 links in Kunshan City China was selectedas the test site as shown in Figure 3 All intersectionsoperated under pretimed signal control with four differentsignal plans in a day The four arterials in the network hadsignificant traffic volume during the day time Oversaturatedand spillover conditions were frequently observed duringmorning and afternoon peaks Microwave-based detectorswere installed on links except the ones shown as the dash linkin Figure 3 All detectors were located approximately at themiddle of the links and collected traffic counts and occupancyin every 5minutes Twomonths of data collected fromAugust1 2015 to September 30 2015 were used in this study Inaddition for validation purpose GPS probe vehicle data from1526 taxi cabs during the same period were also collected tocalculate the network average speed The space-mean speedof the network was simply calculated as the ratio between thetotal distance travelled and the total time spent by the taxicabs in the test site for every 5 minutes

32 Identification of the CTS Points Before calculating DTWdistance for the two subseries in the sliding windows it isimportant to select a reasonable window length since thetwo subseries are expected to reflect the traffic evolution

patterns However due to the different OD patterns andsignal operations in different networks the window lengthis more likely network-dependent In general the windowlength should be shorter than the peak duration Besidesthe window length should not be too short to filter randomfluctuations in data series and not too long to reflect trafficdynamics Basically determining a desirable length is a trade-off between reliability and accuracy

In our case the morning peak is from 600 to 900and the afternoon peak is from 1630 to 1930 Thereforethe window length should be shorter than 3 hours Basedon that the impacts of different window lengths on themeasurement of the DTW distance were investigated Anindex of AverageChange Rate (ACR)was used to describe thefluctuation in theDTWdistance seriesTheACR is calculatedas

119860119862119877 = 1119899119899sum119894=1

119889119894+1 minus 119889119894119889119894 (10)

where 119889119894 is the normalized DTWdistance associated with theith state point in a MFD time series and 119899 is the number ofstate points

Figure 4(a) shows the measurement of DTW distancecalculated from the MFD time series on August 10 2015with window length ranging from 15 minutes to 3 hours Itis observed that the curves of the DTW distance becomesmoother with the increasing of window length The curveswith window lengths of 1 hour and 15 hours are observed asthe relatively balanced cases where the trends are clear andinformative short-term changes are retained In addition theACR was calculated using data in the two months as shownin Figure 4(b) Each point in Figure 4(b) represents the ACRof the DTW distance series of a day with a particular windowlength When the window length is lower than 1 hour theACR decrease significantly from about 011 to 001 A turningpoint can be clearly spotted at the window length of 1 hourBased on the investigation the window length was selected as1 hour for our network

With the measurement of DTW distance a peak search-ing method was applied Figure 5 illustrates the identifiedpeaks on the DTW distance curve of August 11 2015The redand blue marks represent the local maximum and minimum

6 Journal of Advanced Transportation

120104

120204

106104

106204

108130

108230

108131

108231

1091

09

1092

09

1091

10

1092

10

1091

11

1092

11

106105

106205

106106

106206

108129

108229

1091

00

1092

00

1111

01

1112

01

120105

120205

737

313

303

359

589

350

120104

623

335 884

333 411

777

150

Chan

gjia

ng R

oad

HeilongjiangRoad

TongfengRoad

Qianjin Road

Kuntai Road

Link length(m)

Signalized Intersection

Link IDLink with detectorLink without detector

Figure 3 Layout of the test network and detectors in Kunshan City China

extreme points extracted by the proposed method The localmaximum extreme points are regarded as the CTS points inthis study As shown in Figure 5 the CTS points (see thepoints located at the high peaks of the curves) appear duringthe morning and afternoon peaks Although the most criticaltransition state points are successfully identified suspiciousCTS points are also found at the local peaks on the curvesSome suspicious points (shown in the blue circle in Figure 5)are obviously invalid since the corresponding DTWdistancesare so small These points could be readily excluded bya lower bound threshold In this study any state pointswith DTW distance less than 15 were considered as invalidpoints

33 Validation of the CTS Points To validate the identifiedCTS points the locations of the CTS points at the MFD timeseries the correlation between the CTS points and networkspace-mean speed were investigated For ease of illustrationthe morning and afternoon peak were separately consideredThe identified CTS points on August 10 Monday 2015and August 12 Wednesday 2015 are illustrated in Figure 6Figures 6(a) 6(d) and 6(g) show the CTS points during themorning peaks of the three days and Figures 6(b) 6(e) and

6(h) show the CTS points during the afternoon peaks of thethree days Figures 6(c) 6(f) and 6(i) illustrate the correlationbetween the CTS points and network space-mean speed forthe three days The red markers represent the identified CTSpoints The arrows indicate the time sequence within theMFD time series

For the locations at the MFD time series the identifiedCTS points approximately locate at the boundaries of the twodistinguished traffic states see Figures 6(a) 6(d) 6(g) 6(b)6(e) and 6(h) The state points on the left side of the CTSpoints show a positive correlation between network flow andoccupancy while the correlation of state points on the rightside is vague and the slopes between the consecutive statepoints are much more fluctuated than the left side A flatshape of the right side state points can be observed and thenetwork flow fluctuates within a small range from 75 to 90veh5min Based on this it is reasonable to infer that the leftand right side of the identified CTS points are correspondingto the free-flow and optimal accumulation state respectivelyIn other words the identified CTS points well captured theshifts of network traffic states between the free-flow andoptimal accumulation state

By incorporating the network speed in all these threedays the CTS points are identified at the times when the

Journal of Advanced Transportation 7

15min 30min

45min 1h

15h 2h

25h 3h

Time of day

Dyn

amic

Tim

e Wra

ppin

g di

stan

ce

0

1

2

3

0

5

10

2 4 6 8 10 12 14 16 18 20 22 240

0102030

20406080

4 6 8 10 12 14 16 18 20 222 6 8 10 12 14 16 18 204

25

50

75

100

0102030405060

4 6 8 10 12 14 16 18 20 2222 4 6 8 10 12 14 16 18 20 22 240

2 4 6 8 10 12 14 16 18 20 22 24005

101520

2 4 6 8 10 12 14 16 18 20 22 24002468

2 4 6 8 10 12 14 16 18 20 22 240

(a)

000

003

006

009

Aver

age c

hang

e rat

e of d

istan

ce

2 31Length of window (h)

(b)

Figure 4 Impacts of different window lengths on the measurement of DTW distance (a) DTW distance curves with different lengths ofwindow on August 10 Tuesday 2015 (b) ACR calculation with different window lengths during August 1 2015 and September 30 2015

network congestion just appeared and nearly recovered seeFigures 6(c) 6(f) and 6(i) Moreover we can find thatonly the CTS points during the congestion formation stageare identified in the morning peaks of the three days Thereason may be that the traffic flow decreases slowly duringthe congestion recovery stage at the end of the morningpeaks and the traffic state after the morning peaks doesnot experience a significant change According to the pairedCTS points in afternoon peaks the optimal accumulationstate can be approximately estimated as (012 73veh5min) byaveraging over the three days

34 Day-to-Day Pattern Analysis The empirical distributionof the identified CTS points based on two-month weekdaydata is visualized in as shown Figure 7 A clear bimodalitycan be observed in the distribution In fact the bimodality iscaused by the different distributions of the CTS points in themorning and afternoon peaks

Considering the most congested afternoon peaks inour case day-to-day patterns of the CTS points from theafternoon peaks in 38 weekdays were investigated with theGMM clustering Two distinct clusters are found as shownin Figure 8 The cluster centre for Cluster 1 (in blue) is (1718

8 Journal of Advanced Transportation

2 4 6 8 10 12 14 16 18 20 22 240Time of day

0

10

20

DTW

dist

ance

Figure 5The DTW distance on August 11 Tuesday 2015

013 728 (veh5min)) and for Cluster 2 (in red) is (1810 010722(veh5min)) 516 and 484 of the total identified CTSpoints are in Cluster 1 and Cluster 2 respectively Figures 8(a)and 8(b) illustrate the distribution of flow-occupancy andtime in the two clusters respectively According to the timedistribution Cluster 1 could be the critical transition stateindicating a transition from the free-flow state to optimalaccumulation state while Cluster 2 is the critical transitionstate from the optimal accumulation state to free-flow stateIn terms of the flow-occupancy distribution it is found thatthe CTS points in Cluster 1 are much more scattered thanthose in Cluster 2 This indicates that the transition from theoptimal accumulation state to free-flow state is much morepredictable and the transition from the free-flow to optimalaccumulation state has higher uncertainty

35 Extended Application of Real-Time CTS Point Identifi-cation In addition to the analysis of the CTS points usinghistorical data an extended application of real-time CTSpoint identification is also discussed Based on the proposedmethod theMFD state points can be labelled as three classesthe CTS points of Cluster 1 the CTS points of Cluster2 and non-CTS points Thus a probabilistic discriminantanalysis [43] was conducted using the labelled dataset Thediscriminant analysis also known as supervised classifica-tion assumes that the observations in each class are generatedby a distribution specific to that class Let 0119896 be the probabilitydistribution of the kth class which is Gaussian distributionin this study 120591119896 be the proportion of observations of class kin the population and 119904119905 be a real-time observation of MFDstate points The posterior probability that 119904119905 belongs to thekth class is expressed in

119901119903 (119904119905 isin 119888119897119886119904119904 119896) = 1205911198960119896 (119904119905 120583119896 sum119896) (119904119905)sum119866119896=1 1205911198960119896 (119904119905 120583119896 sum119896) (119904119905) 119896 isin 1 2 3

(11)

In this study the labelled MFD state points in theafternoon peaks of 45 days are used as the training set andanother 15 days as the test set Each MFD state point was

identified as an observation of the class with the highestprobability that belongs to According to the test results theerror of the discriminant method is about 81

The discrimination results for three typical weekdays areillustrated in Figure 9Theblack red and blue dots in Figure 9represent the non-CTS points the CTS points of Cluster 1and the CTS point of Cluster 2 respectively The probabilisticdiscriminant analysis performed well in identifying the CTSpoints All the identified CTS points are located at theboundaries between the free-flow and optimal accumulationstate as shown in Figures 9(a) 9(c) and 9(e) and illustrate astrong correlation to the changes of network speed during theformation and recovery period of the network congestion

4 Conclusions

This study proposed a data-driven method to identify andcharacterize the critical transition state in MFD using fielddata The identification of the critical transition state isbased on the analysis of the MFD time series The DTWdistance is used to measure the similarity of subseries beforeand after each state point in a MFD time series and apeak searching method is developed to identify the criticaltransition state points based on the DTW distance seriesThe GMM clustering is used to characterize the day-to-daypattern of the identified critical transition state points Theprobabilistic discriminant analysis is performed to identifythe CTS points

The identification and validation of CTS points indicatedthat the proposed method well identified the network tran-sition states By analysis the identified CTS points well cap-tured the shifts of network traffic states between the free-flowand optimal accumulation state The optimal accumulationstate can be approximately estimated as (012 73veh5min)by averaging over the three days A clear bimodality can beobserved by exploring the probability density of identifiedCTS points and the bimodality is caused by different dis-tributions of the CTS points in the morning and afternoonpeaks Moreover the CTS points were classified as twocategories One cluster represents the transition from free-flow state to optimal accumulation state and the otherrepresents the transition from optimal accumulation state

Journal of Advanced Transportation 9

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)01 02 03 0400

Network occupancy

(b)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(c)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(d)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(g)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(i)

Figure 6The validation of the identified CTS points on August 10 Monday 2015 and August 12 Wednesday 2015 (a) (d) and (g)The CTSpoints during morning peaks of the three days (b) (e) and ( h) The CTS points during afternoon peaks of the three days (c) (f) and (i)Correlation between the CTS points and network speed

to free-flow state Besides the probabilistic discriminantanalysis performed well in identifying the CTS points All theidentified CTS points were located at the boundaries betweenthe free-flow and optimal accumulation state The proposed

method is helpful to better understand the evolution processof network traffic flow and offers opportunities to developmore reliable and responsive network traffic flow manage-ment strategies

10 Journal of Advanced Transportation

0

4020

6080

100

Networ

k flow

(veh

5m

in)

0000002040608

005010 015 020 025

Network occupancy

Figure 7 Visualization of the distribution of the CTS points

0

40

20

60

80

100

Net

wor

k flo

w (v

eh5

min

)

000 005 010 015 020 025Network occupancy

(a)

Cluster12

0

5

10

15C

ount

17 18 1916Time of day

(b)

Figure 8 Day-to-day pattern of the CTS points from the afternoon peaks of 38 weekdays (a) Flow-occupancy distribution in two clusters(b) Time distribution in two clusters

In future the impact factors of critical transition statesin actual network will be investigated through comparativeanalysis under certain conditions Besides it is encouragedto test the proposed method with different sources of data(eg probe vehicle or vehicle license plate reidentificationdata) Also more field validation works are needed to furtherinterpret the critical transition states considering real trafficbehaviours

Data Availability

Partial of the detector data used to support the findings ofthis study are includedwithin the supplementary informationfile

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The work was funded by the National Natural Science Foun-dation of China (no 71871055) and theKeyResearchProgramof Jiangsu Province Science and Technology Department (noBE2017027)

Supplementary Materials

The supplementary material file is partial of the detectordata used to derive the MFD for test site The data are

Journal of Advanced Transportation 11

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

(b)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(c)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60N

etw

ork

spee

d (k

mh

)

(d)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

Figure 9 The discrimination results on three typical weekdays (a) (c) and (e) The identified CTS points in the network flow-occupancyseries during afternoon peak of the three days (b) (d) and (f ) The identified CTS points in the network speed series of the three days

12 Journal of Advanced Transportation

valid and they were formed by STATIONID (the ID numberof specific link) CALENDAR KEY (the date of traffic flowdata for specific link) FIVEMINX (the time of traffic flowdata for specific date) OCCUPANCY (the occupancy5minfor specific link) and VOLUME (the flow5min for spe-cific link) from August 1 2015 to September 30 2015(Supplementary Materials)

References

[1] M Keyvan-Ekbatani M Papageorgiou and I PapamichailldquoPerimeterTraffic Control via Remote FeedbackGatingrdquo Proce-dia - Social and Behavioral Sciences vol 111 pp 645ndash653 2014

[2] Y M Du J P Wu and Y Jia H ldquoMFD-Based Regional TrafficVolume Dynamic Controlrdquo Journal of Transportation SystemsEngineering Information Technology vol 14 no 3 pp 162ndash1672014

[3] A Csikos T Tettamanti and I Varga ldquoMacroscopic modelingand control of emission in urban road traffic networksrdquo Trans-port vol 30 no 2 pp 152ndash161 2015

[4] M D Simoni A J Pel R A Waraich and S P HoogendoornldquoMarginal cost congestion pricing based on the network fun-damental diagramrdquo Transportation Research Part C EmergingTechnologies vol 56 pp 221ndash238 2015

[5] M Hajiahmadi V L Knoop B De Schutter and H Hellen-doorn ldquoOptimal dynamic route guidance A model predictiveapproach using the macroscopic fundamental diagramrdquo inProceedings of the 2013 16th International IEEE Conferenceon Intelligent Transportation Systems Intelligent TransportationSystems for All Modes ITSC 2013 pp 1022ndash1028 NetherlandsOctober 2013

[6] R Alsalhi and V Dixit ldquoThe relationship between traffic safetyand macroscopic fundamental diagram (MFD)rdquo AustralasianTransport Research Forum (ATRF) vol 37 2015

[7] C F Daganzo V V Gayah and E J Gonzales ldquoMacroscopicrelations of urban traffic variables bifurcations multivalued-ness and instabilityrdquo Transportation Research Part B Method-ological vol 45 no 1 pp 278ndash288 2011

[8] J Haddad and N Geroliminis ldquoOn the stability of trafficperimeter control in two-region urban citiesrdquo TransportationResearch Part B Methodological vol 46 no 9 pp 1159ndash11762012

[9] V V Gayah and C F Daganzo ldquoClockwise hysteresis loops inthe Macroscopic Fundamental Diagram An effect of networkinstabilityrdquo Transportation Research Part B Methodological vol45 no 4 pp 643ndash655 2011

[10] C FDaganzo andNGeroliminis ldquoAn analytical approximationfor the macroscopic fundamental diagram of urban trafficrdquoTransportation Research Part B Methodological vol 42 no 9pp 771ndash781 2008

[11] N Geroliminis and B Boyaci ldquoThe effect of variability of urbansystems characteristics in the network capacityrdquo TransportationResearch Part B Methodological vol 46 no 10 pp 1607ndash16232012

[12] E J Gonzales C Chavis Y Li and C F Daganzo Mul-timodal transport modeling for nairobi kenya insights andrecommendations with an evidence-based model insights andrecommendations with an evidence-based model kenya 2009

[13] A Mazloumian N Geroliminis and D Helbing ldquoThe spatialvariability of vehicle densities as determinant of urban networkcapacityrdquo Philosophical Transactions of the Royal Society A

Mathematical Physical amp Engineering Sciences vol 368 no1928 pp 4627ndash4647 2010

[14] N Geroliminis and C F Daganzo ldquoExistence of urban-scalemacroscopic fundamental diagrams some experimental find-ingsrdquo Transportation Research Part B Methodological vol 42no 9 pp 759ndash770 2008

[15] T Courbon and L Leclercq ldquoCross-comparison of macroscopicfundamental diagram estimation methodsrdquo in Proceedings ofthe 14thMeeting of the EuroWorking Group on Transp - In Questfor Adv Models Tools and Methods for Transp and LogistEWGT 26th Mini-EURO Conf - Intelligent Decis Making inTransp and Logist MEC 1st Eur Sci Conf on Air Transp - RHpp 417ndash426 Poland September 2011

[16] VGayah andV Dixit ldquoUsingmobile probe data and themacro-scopic fundamental diagram to estimate network densitiesrdquoTransportation Research Record no 2390 pp 76ndash86 2013

[17] J Du H Rakha and V V Gayah ldquoDeriving macroscopicfundamental diagrams from probe data Issues and proposedsolutionsrdquo Transportation Research Part C Emerging Technolo-gies vol 66 pp 136ndash149 2016

[18] Z Xu P J Jin and X Jiang ldquoMacroscopic FundamentalDiagram (MFD) Based on Vehicle-to-Infrastructure (V2I)Connected Vehicle Data Transportation Research Board 94thAnnual Meetingrdquo Macroscopic Fundamental Diagram (MFD)Based on Vehicle-to-Infrastructure (V2I) Connected Vehicle Data Transportation Research Board 94th Annual Meeting 2015

[19] M Saberi and H Mahmassani ldquoHysteresis and capacity dropphenomena in freeway networks Empirical characterizationand interpretationrdquo Transportation Research Record no 2391pp 44ndash55 2013

[20] M Keyvan-Ekbatani A Kouvelas I Papamichail and MPapageorgiou ldquoExploiting the fundamental diagram of urbannetworks for feedback-based gatingrdquo Transportation ResearchPart B Methodological vol 46 no 10 pp 1393ndash1403 2012

[21] K Aboudolas and N Geroliminis ldquoPerimeter and boundaryflow control inmulti-reservoir heterogeneous networksrdquoTrans-portation Research Part B Methodological vol 55 pp 265ndash2812013

[22] K Ampountolas and A Kouvelas ldquoReal-Time Estimation ofCritical Values of the Macroscopic Fundamental Diagramfor Maximum Network Throughputrdquo Transportation ResearchBoard 94th Annual Meeting 2015

[23] B Ponnu and B Coifman ldquoWhen adjacent lane dependenciesdominate the uncongested regime of the fundamental relation-shiprdquo Transportation Research Part B Methodological vol 104pp 602ndash615 2017

[24] X Qu J Zhang and S Wang ldquoOn the stochastic fundamentaldiagram for freeway traffic Model development analyticalproperties validation and extensive applicationsrdquo Transporta-tion Research Part B Methodological vol 104 pp 256ndash271 2017

[25] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015

[26] J Xia and M Chen ldquoA nested custering technique for freewayoperating condition classificationrdquo Computer-Aided Civil andInfrastructure Engineering vol 22 no 6 pp 430ndash437 2007

[27] H-p Lu Z-y Sun and W-c Qu ldquoBig data-driven based real-time traffic flow state identification and predictionrdquo DiscreteDynamics in Nature and Society Art ID 284906 11 pages 2015

Journal of Advanced Transportation 13

[28] M Azimi and Y Zhang ldquoCategorizing Freeway Flow Condi-tions by Using Clustering Methodsrdquo Transportation ResearchRecord vol 2173 no 1 pp 105ndash114 2018

[29] J Xia W Huang and J Guo ldquoA clustering approach to onlinefreeway traffic state identification using ITS datardquoKSCE Journalof Civil Engineering vol 16 no 3 pp 426ndash432 2012

[30] J Tang F Liu W Zhang S Zhang and Y Wang ldquoExploringdynamic property of traffic flow time series in multi-statesbased on complex networks Phase space reconstruction versusvisibility graphrdquo Physica A Statistical Mechanics and its Appli-cations vol 450 pp 635ndash648 2016

[31] H B Celikoglu and M A Silgu ldquoExtension of traffic flowpattern dynamic classification by a macroscopic model usingmultivariate clusteringrdquo Transportation Science vol 50 no 3pp 966ndash981 2016

[32] S Oh Y-J Byon and H Yeo ldquoImpact of Traffic State Transitionand Oscillation on Highway Performance with Section-BasedApproachrdquo in Proceedings of the 18th IEEE International Confer-ence on Intelligent Transportation Systems ITSC 2015 pp 2141ndash2146 Spain September 2015

[33] J H Banks ldquoInvestigation of some characteristics of congestedflowrdquo Transportation Research Record no 1678 pp 128ndash1341999

[34] E I Vlahogianni M G Karlaftis and J C Golias ldquoStatisticalmethods for detecting nonlinearity and non-stationarity in uni-variate short-term time-series of traffic volumerdquoTransportationResearch Part C Emerging Technologies vol 14 no 5 pp 351ndash367 2006

[35] Y Kamarianakis H Oliver Gao and P Prastacos ldquoCharacter-izing regimes in daily cycles of urban traffic using smooth-transition regressionsrdquo Transportation Research Part C Emerg-ing Technologies vol 18 no 5 pp 821ndash840 2010

[36] J Tang Y Wang and F Liu ldquoCharacterizing traffic timeseries based on complex network theoryrdquo Physica A StatisticalMechanics and its Applications vol 392 no 18 pp 4192ndash42012013

[37] Y Yan S Zhang J Tang and X Wang ldquoUnderstandingcharacteristics in multivariate traffic flow time series fromcomplex network structurerdquo Physica A Statistical Mechanicsand its Applications vol 477 pp 149ndash160 2017

[38] S Blandin J Argote A M Bayen and D B Work ldquoPhasetransition model of non-stationary traffic flow Definitionproperties and solution methodrdquo Transportation Research PartB Methodological vol 52 pp 31ndash55 2013

[39] D F Silva and G E Batista ldquoSpeeding up all-pairwise dynamictime warping matrix calculationrdquo in Proceedings of the 2016SIAM International Conference on Data Mining Society forIndustrial and Applied Mathematics pp 837ndash845 2016

[40] O B Linton ldquoLocal Regression Modelsrdquo inMicroeconometricsPalgrave Macmillan London UK 2010

[41] R P Brent Algorithms for Minimization without DrivativesPrentice Hall Englewood Cliffs NJ USA 2013

[42] NM Nasrabadi ldquoPattern Recognition andMachine LearningrdquoJournal of Electronic Imaging vol 16 no 4 p 049901 2007

[43] C Fraley and A E Raftery ldquoModel-based clustering discrimi-nant analysis and density estimationrdquo Journal of the AmericanStatistical Association vol 97 no 458 pp 611ndash631 2002

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

4 Journal of Advanced Transportation

(2)

Δ119910119895 = 1 Δ119910119895 gt 00 Δ119910119895 lt 0 (3)

0 (119904119888119894 | 120579) = 119866sum119896=1

1205911198960119896 (119904119888119894 | 120579119896)

(119904119888119894 isin 1198773 119896 isin 1 2 119866 120591119896 isin (0 1) 119866sum119896

120591119896 = 1 ))(4)

119871 (120579 119904119888 119885) = 119873prod119894=1

119866prod119896=1

[1205911198960119896 (119904119888119894 | 120579119896)]I(119885119894=119896) (5)

Journal of Advanced Transportation 5

Algorithm 1 The EM Algorithm to estimate parameters in GMM

sum119896

120591119896 = sum119873119894=1 120574 (119904119888119894 119896)119873 (9)

3 Implementation

119860119862119877 = 1119899119899sum119894=1

119889119894+1 minus 119889119894119889119894 (10)

6 Journal of Advanced Transportation

120104

120204

106104

106204

108130

108230

108131

108231

1091

09

1092

09

1091

10

1092

10

1091

11

1092

11

106105

106205

106106

106206

108129

108229

1091

00

1092

00

1111

01

1112

01

120105

120205

737

313

303

359

589

350

120104

623

335 884

333 411

777

150

Chan

gjia

ng R

oad

HeilongjiangRoad

TongfengRoad

Qianjin Road

Kuntai Road

Link length(m)

Signalized Intersection

Link IDLink with detectorLink without detector

Figure 3 Layout of the test network and detectors in Kunshan City China

Journal of Advanced Transportation 7

15min 30min

45min 1h

15h 2h

25h 3h

Time of day

Dyn

amic

Tim

e Wra

ppin

g di

stan

ce

0

1

2

3

0

5

10

2 4 6 8 10 12 14 16 18 20 22 240

0102030

20406080

4 6 8 10 12 14 16 18 20 222 6 8 10 12 14 16 18 204

25

50

75

100

0102030405060

4 6 8 10 12 14 16 18 20 2222 4 6 8 10 12 14 16 18 20 22 240

2 4 6 8 10 12 14 16 18 20 22 24005

101520

2 4 6 8 10 12 14 16 18 20 22 24002468

2 4 6 8 10 12 14 16 18 20 22 240

(a)

000

003

006

009

Aver

age c

hang

e rat

e of d

istan

ce

2 31Length of window (h)

(b)

8 Journal of Advanced Transportation

2 4 6 8 10 12 14 16 18 20 22 240Time of day

0

10

20

DTW

dist

ance

Figure 5The DTW distance on August 11 Tuesday 2015

(11)

4 Conclusions

Journal of Advanced Transportation 9

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)01 02 03 0400

Network occupancy

(b)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(c)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(d)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(g)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(i)

10 Journal of Advanced Transportation

0

4020

6080

100

Networ

k flow

(veh

5m

in)

0000002040608

005010 015 020 025

Network occupancy

Figure 7 Visualization of the distribution of the CTS points

0

40

20

60

80

100

Net

wor

k flo

w (v

eh5

min

)

000 005 010 015 020 025Network occupancy

(a)

Cluster12

0

5

10

15C

ount

17 18 1916Time of day

(b)

Data Availability

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Supplementary Materials

Journal of Advanced Transportation 11

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

(b)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(c)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60N

etw

ork

spee

d (k

mh

)

(d)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

12 Journal of Advanced Transportation

References

Mathematical Physical amp Engineering Sciences vol 368 no1928 pp 4627ndash4647 2010

Journal of Advanced Transportation 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Journal of Advanced Transportation 5

Algorithm 1 The EM Algorithm to estimate parameters in GMM

sum119896

120591119896 = sum119873119894=1 120574 (119904119888119894 119896)119873 (9)

3 Implementation

119860119862119877 = 1119899119899sum119894=1

119889119894+1 minus 119889119894119889119894 (10)

6 Journal of Advanced Transportation

120104

120204

106104

106204

108130

108230

108131

108231

1091

09

1092

09

1091

10

1092

10

1091

11

1092

11

106105

106205

106106

106206

108129

108229

1091

00

1092

00

1111

01

1112

01

120105

120205

737

313

303

359

589

350

120104

623

335 884

333 411

777

150

Chan

gjia

ng R

oad

HeilongjiangRoad

TongfengRoad

Qianjin Road

Kuntai Road

Link length(m)

Signalized Intersection

Link IDLink with detectorLink without detector

Figure 3 Layout of the test network and detectors in Kunshan City China

Journal of Advanced Transportation 7

15min 30min

45min 1h

15h 2h

25h 3h

Time of day

Dyn

amic

Tim

e Wra

ppin

g di

stan

ce

0

1

2

3

0

5

10

2 4 6 8 10 12 14 16 18 20 22 240

0102030

20406080

4 6 8 10 12 14 16 18 20 222 6 8 10 12 14 16 18 204

25

50

75

100

0102030405060

4 6 8 10 12 14 16 18 20 2222 4 6 8 10 12 14 16 18 20 22 240

2 4 6 8 10 12 14 16 18 20 22 24005

101520

2 4 6 8 10 12 14 16 18 20 22 24002468

2 4 6 8 10 12 14 16 18 20 22 240

(a)

000

003

006

009

Aver

age c

hang

e rat

e of d

istan

ce

2 31Length of window (h)

(b)

8 Journal of Advanced Transportation

2 4 6 8 10 12 14 16 18 20 22 240Time of day

0

10

20

DTW

dist

ance

Figure 5The DTW distance on August 11 Tuesday 2015

(11)

4 Conclusions

Journal of Advanced Transportation 9

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)01 02 03 0400

Network occupancy

(b)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(c)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(d)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(g)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(i)

10 Journal of Advanced Transportation

0

4020

6080

100

Networ

k flow

(veh

5m

in)

0000002040608

005010 015 020 025

Network occupancy

Figure 7 Visualization of the distribution of the CTS points

0

40

20

60

80

100

Net

wor

k flo

w (v

eh5

min

)

000 005 010 015 020 025Network occupancy

(a)

Cluster12

0

5

10

15C

ount

17 18 1916Time of day

(b)

Data Availability

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Supplementary Materials

Journal of Advanced Transportation 11

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

(b)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(c)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60N

etw

ork

spee

d (k

mh

)

(d)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

12 Journal of Advanced Transportation

References

Mathematical Physical amp Engineering Sciences vol 368 no1928 pp 4627ndash4647 2010

Journal of Advanced Transportation 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

6 Journal of Advanced Transportation

120104

120204

106104

106204

108130

108230

108131

108231

1091

09

1092

09

1091

10

1092

10

1091

11

1092

11

106105

106205

106106

106206

108129

108229

1091

00

1092

00

1111

01

1112

01

120105

120205

737

313

303

359

589

350

120104

623

335 884

333 411

777

150

Chan

gjia

ng R

oad

HeilongjiangRoad

TongfengRoad

Qianjin Road

Kuntai Road

Link length(m)

Signalized Intersection

Link IDLink with detectorLink without detector

Figure 3 Layout of the test network and detectors in Kunshan City China

Journal of Advanced Transportation 7

15min 30min

45min 1h

15h 2h

25h 3h

Time of day

Dyn

amic

Tim

e Wra

ppin

g di

stan

ce

0

1

2

3

0

5

10

2 4 6 8 10 12 14 16 18 20 22 240

0102030

20406080

4 6 8 10 12 14 16 18 20 222 6 8 10 12 14 16 18 204

25

50

75

100

0102030405060

4 6 8 10 12 14 16 18 20 2222 4 6 8 10 12 14 16 18 20 22 240

2 4 6 8 10 12 14 16 18 20 22 24005

101520

2 4 6 8 10 12 14 16 18 20 22 24002468

2 4 6 8 10 12 14 16 18 20 22 240

(a)

000

003

006

009

Aver

age c

hang

e rat

e of d

istan

ce

2 31Length of window (h)

(b)

8 Journal of Advanced Transportation

2 4 6 8 10 12 14 16 18 20 22 240Time of day

0

10

20

DTW

dist

ance

Figure 5The DTW distance on August 11 Tuesday 2015

(11)

4 Conclusions

Journal of Advanced Transportation 9

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)01 02 03 0400

Network occupancy

(b)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(c)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(d)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(g)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(i)

10 Journal of Advanced Transportation

0

4020

6080

100

Networ

k flow

(veh

5m

in)

0000002040608

005010 015 020 025

Network occupancy

Figure 7 Visualization of the distribution of the CTS points

0

40

20

60

80

100

Net

wor

k flo

w (v

eh5

min

)

000 005 010 015 020 025Network occupancy

(a)

Cluster12

0

5

10

15C

ount

17 18 1916Time of day

(b)

Data Availability

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Supplementary Materials

Journal of Advanced Transportation 11

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

(b)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(c)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60N

etw

ork

spee

d (k

mh

)

(d)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

12 Journal of Advanced Transportation

References

Mathematical Physical amp Engineering Sciences vol 368 no1928 pp 4627ndash4647 2010

Journal of Advanced Transportation 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Journal of Advanced Transportation 7

15min 30min

45min 1h

15h 2h

25h 3h

Time of day

Dyn

amic

Tim

e Wra

ppin

g di

stan

ce

0

1

2

3

0

5

10

2 4 6 8 10 12 14 16 18 20 22 240

0102030

20406080

4 6 8 10 12 14 16 18 20 222 6 8 10 12 14 16 18 204

25

50

75

100

0102030405060

4 6 8 10 12 14 16 18 20 2222 4 6 8 10 12 14 16 18 20 22 240

2 4 6 8 10 12 14 16 18 20 22 24005

101520

2 4 6 8 10 12 14 16 18 20 22 24002468

2 4 6 8 10 12 14 16 18 20 22 240

(a)

000

003

006

009

Aver

age c

hang

e rat

e of d

istan

ce

2 31Length of window (h)

(b)

8 Journal of Advanced Transportation

2 4 6 8 10 12 14 16 18 20 22 240Time of day

0

10

20

DTW

dist

ance

Figure 5The DTW distance on August 11 Tuesday 2015

(11)

4 Conclusions

Journal of Advanced Transportation 9

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)01 02 03 0400

Network occupancy

(b)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(c)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(d)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(g)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(i)

10 Journal of Advanced Transportation

0

4020

6080

100

Networ

k flow

(veh

5m

in)

0000002040608

005010 015 020 025

Network occupancy

Figure 7 Visualization of the distribution of the CTS points

0

40

20

60

80

100

Net

wor

k flo

w (v

eh5

min

)

000 005 010 015 020 025Network occupancy

(a)

Cluster12

0

5

10

15C

ount

17 18 1916Time of day

(b)

Data Availability

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Supplementary Materials

Journal of Advanced Transportation 11

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

(b)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(c)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60N

etw

ork

spee

d (k

mh

)

(d)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

12 Journal of Advanced Transportation

References

Mathematical Physical amp Engineering Sciences vol 368 no1928 pp 4627ndash4647 2010

Journal of Advanced Transportation 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

8 Journal of Advanced Transportation

2 4 6 8 10 12 14 16 18 20 22 240Time of day

0

10

20

DTW

dist

ance

Figure 5The DTW distance on August 11 Tuesday 2015

(11)

4 Conclusions

Journal of Advanced Transportation 9

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)01 02 03 0400

Network occupancy

(b)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(c)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(d)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(g)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(i)

10 Journal of Advanced Transportation

0

4020

6080

100

Networ

k flow

(veh

5m

in)

0000002040608

005010 015 020 025

Network occupancy

Figure 7 Visualization of the distribution of the CTS points

0

40

20

60

80

100

Net

wor

k flo

w (v

eh5

min

)

000 005 010 015 020 025Network occupancy

(a)

Cluster12

0

5

10

15C

ount

17 18 1916Time of day

(b)

Data Availability

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Supplementary Materials

Journal of Advanced Transportation 11

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

(b)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(c)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60N

etw

ork

spee

d (k

mh

)

(d)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

12 Journal of Advanced Transportation

References

Mathematical Physical amp Engineering Sciences vol 368 no1928 pp 4627ndash4647 2010

Journal of Advanced Transportation 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Journal of Advanced Transportation 9

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)01 02 03 0400

Network occupancy

(b)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(c)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(d)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(g)

01 02 03 0400Network occupancy

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

(h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(i)

10 Journal of Advanced Transportation

0

4020

6080

100

Networ

k flow

(veh

5m

in)

0000002040608

005010 015 020 025

Network occupancy

Figure 7 Visualization of the distribution of the CTS points

0

40

20

60

80

100

Net

wor

k flo

w (v

eh5

min

)

000 005 010 015 020 025Network occupancy

(a)

Cluster12

0

5

10

15C

ount

17 18 1916Time of day

(b)

Data Availability

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Supplementary Materials

Journal of Advanced Transportation 11

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

(b)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(c)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60N

etw

ork

spee

d (k

mh

)

(d)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

12 Journal of Advanced Transportation

References

Mathematical Physical amp Engineering Sciences vol 368 no1928 pp 4627ndash4647 2010

Journal of Advanced Transportation 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

10 Journal of Advanced Transportation

0

4020

6080

100

Networ

k flow

(veh

5m

in)

0000002040608

005010 015 020 025

Network occupancy

Figure 7 Visualization of the distribution of the CTS points

0

40

20

60

80

100

Net

wor

k flo

w (v

eh5

min

)

000 005 010 015 020 025Network occupancy

(a)

Cluster12

0

5

10

15C

ount

17 18 1916Time of day

(b)

Data Availability

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Supplementary Materials

Journal of Advanced Transportation 11

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

(b)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(c)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60N

etw

ork

spee

d (k

mh

)

(d)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

12 Journal of Advanced Transportation

References

Mathematical Physical amp Engineering Sciences vol 368 no1928 pp 4627ndash4647 2010

Journal of Advanced Transportation 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Journal of Advanced Transportation 11

0

25

50

75

100N

etw

ork

flow

(veh

5m

in)

01 02 03 0400Network occupancy

(a)

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

(b)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(c)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60N

etw

ork

spee

d (k

mh

)

(d)

0

25

50

75

100

Net

wor

k flo

w (v

eh5

min

)

01 02 03 0400Network occupancy

(e)

2 4 6 8 10 12 14 16 18 20 22 240Time of day

20

30

40

50

60

Net

wor

k sp

eed

(km

h)

(f)

12 Journal of Advanced Transportation

References

Mathematical Physical amp Engineering Sciences vol 368 no1928 pp 4627ndash4647 2010

Journal of Advanced Transportation 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

12 Journal of Advanced Transportation

References

Mathematical Physical amp Engineering Sciences vol 368 no1928 pp 4627ndash4647 2010

Journal of Advanced Transportation 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Journal of Advanced Transportation 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

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