DynamicOrigin-DestinationMatrixEstimationBasedonUrban ...2020/03/30 · improves estimation...

Research ArticleDynamic Origin-Destination Matrix Estimation Based on UrbanRail Transit AFC Data Deep Optimization Framework withForward Passing and Backpropagation Techniques

Yuedi Yang 1 Jun Liu 1 Pan Shang 1 Xinyue Xu 2 and Xuchao Chen 3

1School of Traffic and Transportation Beijing Jiaotong University Beijing 100044 China2State Key Laboratory of Railway Traffic Control and Safety Beijing Jiaotong University Beijing 100044 China3Beijing Infrastructure Investment Co LTD Beijing 100044 China

Correspondence should be addressed to Jun Liu jliubjtueducn and Pan Shang shangpanbjtueducn

Received 30 March 2020 Revised 10 October 2020 Accepted 23 November 2020 Published 7 December 2020

Academic Editor Jose E Naranjo

Copyright copy 2020 Yuedi Yang et al is 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

At present the existing dynamic OD estimationmethods in an urban rail transit network still need to be improved in the factors ofthe time-dependent characteristics of the system and the estimation accuracy of the results is study focuses on predicting thedynamic OD demand for a time of period in the future for an urban rail transit system We propose a nonlinear programmingmodel to predict the dynamic ODmatrix based on historic automatic fare collection (AFC) datais model assigns the passengerflow to the hierarchical flow network which can be calibrated by backpropagation of the first-order gradients and reassignment ofthe passenger flow with the updated weights between different layers e proposed model can predict the time-varying ODmatrix the number of passengers departing at each time and the travel time spent by passengers of which the results are shown inthe case study Finally the results indicate that the proposed model can effectively obtain a relatively accurate estimation resulte proposed model can integrate more traffic characteristics than traditional methods and provides an effective and hierarchicalpassenger flow estimation framework is study can provide a rich set of passenger demand for advanced transit planning andmanagement applications for instance passenger flow control adaptive travel demand management and real-timetrain scheduling

1 Introduction

As an important part of passenger flow prediction in anurban rail transit system origin-destination (OD) matrixestimation plays an important role which provides basicdata for passenger flow assignment Most of the developedapproaches are usually applied to road traffic systems suchas freeway highway and road networks in which link(section) flow can usually be obtained by detectors How-ever the development of approaches for estimating dynamictraffic demand for large-scale and complex rail transitnetworks without observed link flows such as the Beijingand Shanghai subway systems in China remains a criticaland challenging problem and this issue has been attracting asignificant amount of attention from transport operationresearchers and managers [1]

Many approaches have been proposed for distributingtrips among origins and destinations over the years egravity model is a typical traditional model that can predictthe OD distribution of traffic zones [2] However con-ventional approaches based on economic population andspatial relationship data are time-consuming and are highlyexpensive

Some estimation approaches have become popular inOD estimation or updating the OD matrix from trafficcounts e scope of the literature in this field is very widee existing works are based on the entropy maximization[3] the maximum likelihood approach [4] the generalizedleast squares estimator [5 6] the Bayesian inference ap-proach [7 8] and other approaches to solving this problemHowever due to the complexity of calculating this pa-rameter mostly the developed models based on the

HindawiJournal of Advanced TransportationVolume 2020 Article ID 8846715 16 pageshttpsdoiorg10115520208846715

assignment matrix are limited to simple networks like in-tersections interchanges and freeways [9]

ese conventional methods for estimatingorigin-destination (OD) trip matrices from link trafficcounts assume that route choice proportions are givenconstants But this assumption does not hold in a networkwith realistic congestion levels [10] A bilevel programmingapproach has been used for the estimation of the OD matrixin congested networks [11 12] is approach combined thegeneralized least squares estimation model and the networkequilibrium model into one process However the bilevelapproach has certain difficulties in finding an optimal so-lution because of nonconvex and nondifferential formula-tions Sherali et al [13] constructed a linear programmingmodel with a user-equilibrium solution for synthesizing ODtables from traffic volume counts Later Toledo andKolechkina [14] presented the methods based on use oflinear approximations of the assignment matrix in the op-timization iterations Fujita et al [15] proposed an ODmodification approach formulated as a static user-equilib-rium assignment with elastic demand based on the residualdemand at the end of each period Applying the model tolarge-scale road network demonstrates that it efficientlyimproves estimation accuracy because the 24-hour timecoefficients of survey data are slightly biased and may bemodified properly Unlike the gravity model these ap-proaches are based on traffic count data which can bedetected from links by vehicle identification or locatingtechnologies such as GPS floating automatic license platerecognition (ALPR) and radio frequency identification(RFID) Tang et al [16] proposed a newmethod based on theentropy-maximizing theory to model OD distribution inHarbin city using large-scale taxi GPS trajectories e re-sults demonstrate that the entropy-maximizing model issuperior to the gravity model which can validate the fea-sibility of OD distribution from taxi GPS data in the urbansystem Rao et al [17] formulated a particle filter model forvehicle trajectory reconstruction based on ALPR data Andthe OD patterns are estimated by adding up the path flowswhich is conducted through dividing the reconstructedcomplete trajectories Liu et al [18] predicted the ODmatrixbased on the historical ALPR data Guo et al [19] developedan optimization model based on the least squares methode optimization model estimated the dynamic OD matrixby integrating the preliminary OD matrix dynamic as-signment matrix derived by RFID data and link flow de-tected by the inductive loop detectors However the routechoice and travel time delay issues are still difficult to dealwith So establishing the dynamic flow equations is the firstchallenge to estimate dynamic passenger flow demand forrail transit systems for the lack of observed information andcomplex structure of the network

In recent decades the application of the neural networkmodel expands this problem into a new field e neuralnetwork operates as a black box model-free and adaptivetool for capturing and learning significant structures in data[20] Gong [21] used the Hopfield neural network (HNN)model to estimate the urban orientation-destination (OD)distribution matrix from the link volumes of the

transportation network so as to promote the solving speedand precision Yang et al [22] proposed a dynamic modelbased on backpropagation (BP) learning for estimating ODflows from road entering and exiting counts e OD flowsin each short time interval are estimated through theminimization of the squared errors between the predictedand observed exiting counts Li et al [23] proposed a newdynamic radial basis function neural network to forecastoutbound passenger volumes and improve passenger flowcontrol Passenger flow control was considered to improvethe prediction accuracy by adding passenger flow controlcoefficients to their model However the current perceptronneural networks may not perform well in all issues due toreasons such as model nontransferability insufficient abilityto generalize and reliance on activation functions [24]

Subsequently the computational graph was proposed asa description language to represent mathematical expres-sions It is important to understand how the underlyingcomputational graph of a deep learning network combinedwith the BP algorithm can be used to describe the forwardpropagation and backward feedback processes betweendifferent levels of transportation planning and decisionmaking [25 26] Wu et al [27] proposed a multilayeredhierarchical flow network representation to structurallymodel different levels of travel demand for road networksincluding trip generation OD matrices path and link flowsand individual behavior parameters However the traveltimes were assumed to be observed in their study In otherwords their model was constructed in a static networkrather than a time-dependent dynamic network is issuehas been improved in this paper

In this paper we aim to predict the dynamic ODdemand for a time of period in the future based on his-torical observations In most research papers it is as-sumed that the OD matrix can be predicted fromhistorical data [18 28] We apply the historic AFC to traina time-dependent hierarchical flow network en we useit to predict the future OD demand with real-time AFCdata at current as input information which is the basicdata to formulate operational and organizational strate-gies Besides the programming model proposed in thispaper can also estimate a hierarchical traveling decisionprocess for passengers in an urban rail transit systemincluding the departure time choice at the origin the pathchoice and the corresponding arrival time at the desti-nation erefore the proposed method in this studyachieves a combination prediction of dynamic ODmatrixdeparture time choice route choice and travel time

A nonlinear programmingmodel is proposed to conductreal-time OD matrix estimation for an urban rail transitsystem based on historic automatic fare collection (AFC)data in this paper Forward passing in the hierarchical flownetwork of urban rail transit sequentially assigns passengersto candidate stations paths and different travel time in-tervalse network can be calibrated by backpropagation ofthe first-order gradients and reassignment of the passengerflowwith the updated weights between different layers underthe deep optimization frameworkis model can determinethe time-varying OD matrix the number of passengers

2 Journal of Advanced Transportation

departing at each time and the travel time spent by pas-sengers of which the results are shown in the case studyFinally a comparative analysis with artificial neural net-works is conducted to illustrate the effect and efficiency ofthe proposed model

e potential contributions are as follows

(1) A modeling framework using the multilayer hier-archical flow network is applied to describe thepassenger transit process in an urban rail systemBased on the flow-oriented prediction formulationthis deep learning modeling approach can simulta-neously estimate different levels of unobserved orpartially observed passenger flow variables ismodel is applicable to the estimation of the ODmatrix of passenger flow with AFC data unlike othertraditional estimation methods based on trafficcounts

(2) is modeling paradigm enables us to capture themathematical structure inside the OD matrix esti-mation problem by representing and decomposingcomplex composite functions through a graph ofcurrent states and numerical gradients is model isconstructed by the passengersrsquo trip process unlikethe black-box model ANN erefore the compu-tational graph can express more traffic characteristicsthan the ANN and provides an effective and hier-archical passenger flow estimation

(3) e layered framework provides a flexible mecha-nism for further expansion In particular theframework can easily add a new hierarchical struc-ture to achieve OD estimation when other sensordata sources can be obtained

(4) In this model the departure time and travel time areconsidered as variables and the additional depar-ture time layer and travel time layer are constructedin the network It is more reasonable to develop adynamic hierarchical flow network to estimate thetime-dependent OD passenger flow matrix

e remainder of the paper is organized as follows enext section presents the mathematical formulation of thetime-dependent hierarchical flow network estimationmodel In the following section we present the solutionframework for implementing forward and backwardpropagation In Section 4 we describe a numerical exper-iment based on the Beijing Subway and compare the resultswith the ANN method e conclusion is given in the lastsection

2 Problem Statement

21 Problem Statement and Notation is paper aims todesign a time-dependent hierarchical flow network (TDHFN)model according to historic AFC recordsemodel is based ona passenger assignment network considering time variationsere is an abundance of historic AFC records that can beapplied to train an optimalmodel An urban rail transit networkconsists of a set of stations N N 1 2 3 e departure

time is divided into equal time intervals composed of a setT T 1 2 t e travel time set Γ(Γ 1 2 τ ) can be obtained from historic AFC datae path set P (P 1 2 p 1113864 1113865) is the given informationconsisting of alternative routes for eachODerefore there are5 layers in the passenger assignment network origin stationdeparture time destination station paths and travel time epassengers are assigned from the origin station to differentdeparture time intervals assigned to different destination sta-tions then assigned to different paths and finally assigned todifferent travel times

In the passenger assignment network design problemthe following inputs should be given (1) AFC records of howmany passengers enter at each station depart at each timeinterval exit at each station and arrive at each time intervaland (2) the supply network of the paths of each OD withminimum travel time and maximum travel time

From the perspective of system-optimal passenger as-signment we can obtain (1) the number of passengersdeparting at each time (2) the number of passengers arrivingat each station (3) the number of passengers arriving at eachtime and (4) the number of passengers choosing each path

ere is an important assumption in this model thesame departure time interval of different origin stations willbe marked as different departure time interval indices aswell as the destination station indices and travel time indicesis ensures that each path for the different destinationstations in the network belongs to a different OD

A multilayer TDHFN is adopted to describe the ODmatrix estimation of the urban rail passenger flow probleme notations used in this paper are shown in Table 1

22 Physical Description Consider a simple physical urbanrail network with four nodes as shown in Figure 1 Node 1 isthe origin station where passengers enter (tap-in) the urbanrail system Nodes 2 and 4 are the destination stations wherethe passengers exit (tap-out) the system Node 3 is thetransfer station Four paths belong to two different ODs inthis network We consider a time-space passenger networkbased on the simple physical network (from Figure 1) asshown in Figure 2

ere is a very important principle in the numberingWith different departure times but equal travel times forthe same OD the destination station path and travel timevalues should be numbered with different indices Addi-tionally when different OD pairs have the same departuretimes and equal travel times the path and travel time valuesshould be numbered with different indices Similarly whendifferent OD pairs have the same departure times equaltravel times but different paths the travel time valuesshould be numbered with different indices is principleensures that the model proposed is time-dependent Inother words the passengers departing from the originstation at different times may choose different paths anddifferent travel times However in a static network pas-sengers are often considered to be homogenous such as inthe research of Wu et al [27] In this paper the time-dependent numbering principle can be used to consider the

Journal of Advanced Transportation 3

characteristics of passenger heterogeneity which is morepractical Finally a simple example of the time-dependentnumbering principle is shown in Figure 3 which is basedon Figures 1 and 2 e indices are shown above the boldhorizontal lines

e numbering of all the stations departure times pathsand travel times as well as the determination of the connectionbetween the decision variables of each level is the basis of themodel is method is a very important and complex process

especially in a large-scale complicated urban transport networksuch as the Beijing Subway

23 Mathematical Description A TDHFN representation isused as a high-level modeling abstract to formulate the ODmatrix estimation problem Let a TDHFN G G(V E) bethe collection of all the elements of the traffic demandvariables in different layers where each layer controls asubset of the demand variables and receives network flows

Table 1 Sets indices variables vectors and parameters

DefinitionsIndicesi Index of nodes (origin stations) i isin N

j Index of destination stations j isin D

t Index of departure time intervals t isin T

τ Index of travel time τ isin Γp Index of paths p isin P

SetsN Set of stations (origin stations)D Set of destination station indicesDt Set of destination station indices for the departure time tT Set of departure time indicesTi Set of departure time indices for the origin station iTij Set of departure time indices from the origin station i to the destination station jΓ Set of travel time indicesΓp Set of travel time indices for the path pP Set of path indicesPj Set of path indices for the destination station jPijt Set of path indices from origin station i to the destination station j at departure time tVariablesxi Number of passengers entering (tap-in) the system at origin station iht Number of passengers departing at time thj Number of passengers exiting (tap-out) the system at destination station jhp Number of passengers choosing the path pyτ Number of passengers spending the travel time τVectorsX e vector of the input layer X (xi|i isin N)

HT e vector of hidden layer 1 HT (ht|t isin T)

HD e vector of hidden layer 2 HD (hj|j isin D)

HP e vector of hidden layer 3 HP (hp|p isin P)

Y e vector of the output layer Y (yτ|τ isin T)

Parametersαit e proportion of passengers departing at time t to the passengers entering (tap-in) the system at node iβtj e proportion of passengers exiting (tap-out) the system at node j to the passengers departing at time tρjp e proportion of passengers choosing the path p to the passengers exiting (tap-out) the system at node jωpτ e proportion of passengers with the travel time τ to the passengers choosing the path pη Learning rategt A gradient of hidden layer 1gj A gradient of hidden layer 2gp A gradient of hidden layer 3gτ A gradient of the output layerΔα e updated value of αit

Δβ e updated value of βtj

Δρ e updated value of ρjp

Δω e updated value of ωpτα e matrix of the departure time layer proportion αit

β e matrix of the destination station layer proportion βtj

ρ e matrix of the path layer proportion ρjp

ω e matrix of the travel time layer proportion ωpτ1113954yτ Number of real passengers spending the travel time τ from AFC records


from its upper layers Let V NcupTcupDcupPcup Γ be the setsof vertexes arranged in the different layers

Definition of vertexes (V)

(1) e first layer is the origin station layer containing eachorigin station with the index i corresponding to thenumber of passengers xi entering (tap-in) the system atorigin station i

(2) e second layer is the departure time layer con-taining each departure time interval with the index t

corresponding to the number of passengers ht

departing at time t(3) e third layer is the destination station layer

containing each destination station with the in-dex j corresponding to the number of passengershj exiting (tap-out) the system at destinationstationj

(4) e fourth layer is the path layer containing eachpath with the index p corresponding to the numberof passengers hp choosing path p

(5) e five-layer travel time layer contains each pathwith the index τ corresponding to the number ofpassengers τ with travel time τ

(6) e edges in the graph are defined asE ENT cupETD cupEDP cupEPΓ to specify the connec-tions between vertexes

Definition of edges (E)

(1) ENT contains edges connecting the vertexes in N andT where each edge corresponds to the proportion ofpassengers αit departing at time t to the passengersentering (tap-in) the system at station i

(2) ETD contains edges connecting the vertexes in T

and D where each edge corresponds to the pro-portion of passengers βtj exiting (tap-out) thesystem at station j to the passengers departing attime t

(3) EDP contains edges connecting the vertexes in D andP where each edge corresponds to the proportion ofpassengers ρjp choosing the path p to the passengersexiting (tap-out) the system at station j

(4) EDP contains edges connecting the vertexes in P andΓ where each edge corresponds to the proportion ofpassengers ωpτ with the travel time τ to the pas-sengers choosing path p

HT α times X (1)

HD β times HT (2)

HP ρ times HD (3)

Y ω times Hp (4)

Equation (1) describes the process of trip productionfrom the origin station layer to the departure time layerEquation (2) maps the flow from the departure time layer tothe destination station layer Equation (3) maps the flowfrom an OD pair to the candidate routes Equation (4)aggregates the path flows to the travel time flows

3 Model and Solution

We propose a nonlinear programming model with linearconstraints for the studied passenger assignment prob-lem Forward passing in the TDHFN sequentially assignspassengers to candidate stations paths and differenttravel time windows e network can be improved bybackward propagation of the first-order gradients andreassignment of the passenger flow with the updatedweights between different layers under the deep opti-mization framework

1

2

3

4

Origin

Destination 1

Destination 2

Transfer

1

2

3

4

Path

Note

Figure 1 Illustration of the physical urban rail network

Destination 1

Origin 1

Destination 2

Transfer

Transfer

τ1τ2

t2t1

Path 1Path 2

Path 3Path 4

hellip

Figure 2 Illustration of the time-space passenger network


31 Optimization Model We propose a nonlinear pro-gramming model with linear constraints for the OD matrixestimation problem en the optimization model isreformulated in the TDHFN for the urban rail system

311 Constraints for Passenger Assignment Assuming thetotal number of passengers entering the urban rail system atstation i is xi passengers may depart at station i at each timeinterval t erefore equation (5) formulates the assignmentprocess where the passengers in the urban rail system areassigned to each departure time interval t Equation (6)assigns the passenger flow ht in departure time interval t tothe destination station j as flow hj Equation (7) assigns thepassenger flow hj from destination station j to path p as hpEquation (8) assigns the passenger flow hp from path p to thetravel time τ as yτ

Assigning the departure time intervals

ht 1113944i

αit times xi t isin T (5)

Assigning the destination stations

hj 1113944t

βtj times ht j isin D (6)

Assigning the paths

hp ρjp times hj p isin P (7)

Assigning the travel times

yτ 1113944p

ωpτ times hp τ isin Γ (8)

312 Constraints for Flow Equilibrium e passenger flowequilibrium constraints are shown in equations (9)ndash(12)

1113944tisinTi

αit 1 (9)

1113944jisinDt

βtj 1 (10)

1113944pisinPj

ρjp 1(11)

1113944τisinΓp

ωpτ 1(12)

313 Objective Function e objective function is shown inthe following equation

min Loss 1113944τisinΓ

12

1113954yτ minus yτ( 11138572 (13)

32 BP ofGradient e Lagrangian functions are as follows

L ωpτ hp λτ1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p

ωpτ times hp⎛⎝ ⎞⎠

2

+ λτ 1113944τisinΓp

ωpτ minus 1⎛⎜⎝ ⎞⎟⎠

L ρjp hj λp1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p

ωpτ times ρjp times hj⎛⎝ ⎞⎠

2

+ λp 1113944pisinPj

ρjp minus 1⎛⎜⎝ ⎞⎟⎠

L βtj ht λj1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p

ωpτ times ρjp times 1113944t

βtj times ht⎛⎝ ⎞⎠

2

+ λj 1113944jisinDt

βtj minus 1⎛⎝ ⎞⎠

L αit xi λt1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p


βtj times 1113944i

αit times xi⎛⎝ ⎞⎠

2

+ λt 1113944tisinTi

αit minus 1⎛⎝ ⎞⎠

(14)

erefore the gradient of each level based on the KKTconditions is as shown in (15)ndash(18)

gτ 1113954yτ minus yτ (15)

gp 1113944τisinΓp

ωpτ times 1113954yτ minus yτ( 1113857(16)

gj 1113944pisinPj

ρjp times 1113944τisinΓp


gt 1113944jisinDt

βtj times 1113944pisinPj



e Lagrangian multipliers λ are known as the adjointvariables To compute the gradient we simply read thegradient concerning nablaL 0


nablaλτL 1113944τisinΓ

ωpτ minus 1 0

nablaλpL 1113944

pisinPρjp minus 1 0

nablaλjL 1113944

jisinDβtj minus 1 0

nablaλtL 1113944

tisinTαit minus 1 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

33 Reformulation in the Deep Optimization FrameworkWe extend the TDHFN as a computational graph to expressthe passenger flow assignment process of an urban railtransit system In the TDHFN we implement forwardpassing and backward propagation (BP) to update the es-timation variables to approximate the objective functionalrelationship expressed by (13) As BP is an essential part ofthe procedure we use the term BP algorithm to represent theoverall procedure throughout this paper

e model is divided into five layers e first layer is theinput layer which represents the passenger flow entering theurban rail system by tapping in the card from the originstation the second layer is the first hidden layer whichrepresents the passenger flow departing at a certain time thethird layer is the second hidden layer which represents thepassenger flow exiting the system by tapping out the card atthe destination station the fourth layer is the third hiddenlayer which represents the passenger flow choosing a certainpath the fifth layer is the output layer which represents thepassenger flow arriving at a certain time e propagationprocess of the passenger flow in the network is shown inFigure 4 rough the connection relationships betweenneurons and the weight of each layer the passenger volumesof each OD within various time periods can be predictedprecisely At this point the output layer yτ represents

y(i j t p τ) In this paper to solve the problem conve-niently we proposed a numbering principle (shown inSection 22) so that the unique τ can represent (i j t p τ)

We can calculate many complex marginal values (updatevalues of weights) using the chain rule in calculus forexample

Δω η middotzLosszyτ

middotzyτ

zω (20)

where ω is a dimension vector of partial derivatives We seethat the marginal values consist of calculating a gradientproduct for each operation in the computational graphSimilarly the updated formulas for other weights are asfollows

Δρ η middotzLosszyτ

middotzyτ

zhj

middotzhj

zρ

Δβ η middotzLosszyτ

middotzyτ

zhj

middotzhj

zht

middotzht

zβ

Δα η middotzLosszyτ

middotzyτ

zhj

middotzhj

zht

middotzht

zxi

middotzxi

zα

(21)

34 Solution Framework Table 2 shows the solution algo-rithm for determining the estimation results including thefollowing three main parts

341 Forward Passing e forward passing step sequen-tially implements trip generation trip distribution estima-tion and a route-based passenger flow assignment whichcan be viewed as a process of the 3-step (from Step 21 to Step23) approach in the area of traffic planning

Origin 1node1

i1

Departure time 1

t1

Departure time 2

t2Destination 1

node 2

Destination 2 node 4


d1


d2

d3

d4

p1Path 1

Path 2p2

p3Path 3

Path 4p4

Path 1

Path 2

p5

p6

Path 3

Path 4

p7

p8

Travel time 1

Travel time 2

Travel time 3

Travel time 4

Travel time 5

Travel time 6

Travel time 7

Travel time 8Travel time 1

Travel time 2


Travel time 5

Travel time 6Travel time 7Travel time 8

τ1

τ2τ3

τ4τ5τ6

τ7τ8

τ9

τ10τ11τ12

τ13τ14

τ15τ16

Figure 3 Illustration of the time-dependent hierarchical passenger network


342 Backward Propagation e backpropagation stepinversely implements feedback control on the forwardpassing process Different layers of first-order partial de-rivatives or ldquoloss errorsrdquo are aggregated to calculate themarginal gradients (as shown in Step 24)

343 Update Update values of variables using gradientdescent (as shown in Step 25)

4 Numerical Experiments

41 Parameter Settings A partial network of the BeijingSubway system is adopted to verify the proposed predictivemodel is portion of the network contains 12 lines (in-cluding 6 two-direction lines) and 43 stations as shown inFigure 5 e research time ranges from 7 am to 9 am which

is the early peak period of the Beijing metroe AFC recorddata collected from Sep 3rd to 7th (fromMonday to Friday)in 2018 are utilized to train the model en the data of Sep10th (Monday) are adopted for testinge time intervals areset as 10min Accordingly the passenger flow for eachstation in the early peak hour is divided into 12 groups

In this paper we mainly focus on the OD passenger flownot the section passenger flow in the subway networkMoreover the congestion of the route is mainly reflected bythe passengersrsquo travel time so the passenger flow state of thesubway section is not considered erefore we only applythe AFC record of which the origin station and destinationstation both belong to the partial network of Beijing Subwayshown in Figure 5

In this paper the travel time is defined as the time rangebetween passengers entering (tap-in) and exiting (tap-out) thestation To facilitate the data statistics the travel time in this

Input layer Hidden layer 1 Output layer

Origin xi Departure time ht Destination hj Path hp Arrival time yτ

Hidden layer 2 Hidden layer 3

i1

i2

i3

t1

t2

t3

j1

j2

j3 p3

p1

p2

τ1

τ2

τ3

hellip hellip hellip hellip

helliphelliphelliphelliphellip


βtj ρjp ωpτ

yτ = sump ωpτ times hphj = sumt βtj times htht = sumi αit times xi hp = ρjp times hj

αit

Figure 4 e forward passing process of the TDHFN

Table 2 Algorithm stepsStep 1 initializationStep 2 iterative optimization processStep 21 perform the forward propagationsBased on the fixed passenger flow proportion variables in the multilayer passenger flow network assign a passenger from the originstation layer to the departure time layer from the departure time layer to the destination station layer from the destination station layerto the path layer and from the path layer to the travel time layerStep 22 calculate the subgradient informationCalculate the subgradient of the passenger flow in the output layer of the multilayer passenger flow networkStep 23 set the ldquoerrorrdquoSet the ldquoerrorrdquo of the output layer in the multilayer passenger flow networkStep 24 perform the backward error propagationsPerform the backward error propagations in the multilayer passenger flow network from the travel time layer to the path layer from thepath layer to the destination station layer from the destination station layer to the departure time layer and from the departure timelayer to the origin station layerStep 25 update the auxiliary flow proportion variablesUpdate the auxiliary passenger flow proportion variables

Step 3 terminationDetermine if all the iterations are complete if not go back to Step 2


experiment is rounded up to an integermultiple of the time interval(ie 10min) Basedon theAFC recordswe calculate the travel timeof each passenger for each OD en the travel time-frequencydistribution histogram of each OD can be obtained Two examplesof the travel time-frequency distributions of the OD fromDongzhimen toDongdan and theODfromXizhimen toXidan arelisted in Figures 6(a) and 6(b) respectively e travel time dis-tribution of each OD is relatively concentrated In particular thetravel time of more than 90 of the passengers in both of the ODsranges from10 to 20min In contrast the proportions of passengerswith travel times that are longer than30min are less than1 for thetwoODs Because the frequencies of some travel times are relativelysmall when constructing the travel time index set Γ the travel timesfor which the frequency is less than a specific threshold (eg 5)can be eliminated to reduce the network size For instance for theOD from Xizhimen to Xidan as shown in Figure 6(b) only oneindex that points to the travel times of 20min is assembled into thesetΓe threshold canbe adjustedA smaller thresholdof less than5 can be chosen if a finer resolution is needed

e difference in travel time of each path is due to the pathrsquoscongestion and individual characteristics of passengers If a logitmodel is used to describe the choice probability and behaviors ofpassengers the path choice probability is only related to the pathcost which cannot reflect the difference of pathrsquos congestionand individual characteristics of passengers erefore we re-versely deduce the possible path for passengers based on the realtravel time data from AFC and the travel time distribution ofeach path

42 Result Analysis We implement the TDHFN usingPython 361 and a part of the Beijing Subway is selected toexamine the applicability as well as the computational ef-ficiency of our proposed model e computational envi-ronment is an Intel(R) Core(TM) i5-45900 Processor CPUwith 330GHz 800GB RAM and 64 bit OS In addition toTensorFlow we can use other off-the-shelf software tools

such as eano to construct a computation graph-basedmodel

Extracted from the AFC data the origin layer has 43nodes the departure time layer has 516 nodes the desti-nation layer has 21672 nodes the path layer has 45732nodes and the travel time layer has 39396 nodes In thisexperiment we let the maximum iterations 10000 and setthe initial learning rate 000001 e iterative curve of thecase study is presented in Figure 7 which shows that theloss function can achieve convergence at the 9000thiteration

To compare the estimated OD passenger flows with theactual passenger flows we can apply some goodness-of-fitmeasures such as the mean absolute percentage error(MAPE) the mean square error (MSE) the root meansquare error (RMSE) the root mean square normalized(RMSN) [29] and R-squared Since we adopted the time-dependent prediction errors in this article this situationcannot be avoided when the value of OD passenger flowwould be zero erefore MAPE is not available because thedivisor cannot be zero RMSE and RMSN measures can beadopted because their divisors would not be zero in thisstudy But the value of RMSE is related to the value ofvariables erefore we also adopted the RMSN to compareand show the accuracy of different variables

e classical function of RMSE is presented in equation(22) Besides RMSEi in equation (23) represents the measure ofthe output nodes belonging to the network of which the originstation index is i RMSEij in equation (24) represents themeasure of the output nodes belonging to the network of whichthe origin station layer index is i and the destination stationindex is j Moreover RMSEijt in equation (25) represents themeasure of the output nodes belonging to the network of whichthe origin station layer index is i the destination station layerindex is j and the departure time layer index is t In the samevein the functions of RMSN RMSNi RMSNij and RMSNijt

are reported in equations (26)ndash(29)

Fuxingmen Xidan Dongdan Jianguomen

Chegongzhuang

Xuanwumen Chongwenmen

Guloudajie Yonghegong

Chaoyangmen

Xizhimeng

Line 1

Line 2

Pinganli Nanluoguxiang Dongsi

Line 5Line 8

Line 6

Line 4

Figure 5 Topology map of the Beijing Subway


RMSE 1

|Γ|1113944τisinΓ

yτ minus yτ1113872 1113873212

⎛⎝ ⎞⎠ (22)

RMSEi 1

Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944tisinTi

1113944jisinDt

1113944pisinPj

1113944τisinΓp

yτ minus yτ1113872 11138732⎛⎜⎝ ⎞⎟⎠

12

(23)

RMSEij 1

Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

tisinTij

1113944pisinPj

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(24)

RMSEijt 1

middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

pisinPijt

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(25)

RMSN RMSE

(1|Γ|)1113936τisinΓ1113954yτ (26)

RMSNi RMSEi

1 Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTi1113936jisinDt

1113936pisinPj1113936τisinΓp 1113954yτ

(27)

RMSNij RMSEij

1 Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pt

11138681113868111386811138681113868111386811138681113868 middot ΓP

111386811138681113868111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTij


(28)

0000

0928

00670000 0003 0003

00000100020003000400050006000700080009001000

10 20 30 40 50 60

Freq

uenc

y

Travel time (min)

(a)

0001

0981

0013 0002 0002 00010000

0200

0400

0600

0800

1000

1200

Freq

uenc

y

Travel time (min)

(b)

Figure 6 Frequency of travel times (a) Dongzhimen to Dongdan (b) Xizhimen to Xidan

100

150

200

250

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Loss

(MSE

)

Iteration

Figure 7 Iterative process


RMSNijt RMSEijt

1 middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot ΓP1113868111386811138681113868

11138681113868111386811138681113874 11138751113874 11138751113936pisinPijt1113936τisinΓp 1113954yτ

(29)

Table 3 shows the results of RMSEi and RMSNi for eachstation Several observations can be made

(1) On the whole the results for the RMSEi andRMSNi of all the stations are relatively low eaverage RMSN is below 3 which indicates thatthe proposed TDHFN can provide an effective

estimation of passenger flow for urban railsystems

(2) Some stationsrsquo RMSNi are relatively poor such asthose of Tiananmendong Tiananmenxi Dongsiand Ciqikou e reason for these results may bethat these stations are mainly located at famous

Table 3 RMSE and RMSN of the estimation results

Station index Origin station RMSEi RMSNi ()1 Andelibeijie 0669 2712 Andingmen 1890 2013 Beihaibei 0541 2574 Beijingzhan 2347 2615 Beixinqiao 0871 3076 Caishikou 0831 2787 Changchunjie 5602 2528 Chaoyangmen 2002 3249 Chegongzhuang 1532 24710 Chegongzhuangxi 1076 23611 Chongwenmen 3475 21412 Ciqikou 0747 39213 Dengshikou 0805 34214 Dongdan 0864 30515 Dongdaqiao 1057 28316 Dongsi 0849 31317 Dongsishitiao 1644 29118 Dongwuyuan 0708 28119 Dongzhimen 3988 27020 Fuchengmen 2200 27321 Fuxingmen 1323 30022 Guloudajie 1738 19423 Hepinglibeijie 1337 21924 Hepingmen 1707 19225 Jianguomen 1627 29826 Jishuitan 4913 21727 Lingjinghutong 0692 30028 Nanlishilu 1111 24629 Nanluoguxiang 0624 26630 Pinganli 0905 23631 Qianmen 2716 26232 Shishahai 0431 32333 Tiananmendong 0634 36534 Tiananmenxi 0482 39435 Wangfujing 0771 38236 Xidan 0827 28137 Xinjiekou 1000 28438 Xisi 0601 31139 Xizhimen 3687 27740 Xuanwumen 2076 20341 Yonganli 1229 31642 Yonghegong 1571 23943 Zhangzizhonglu 0719 277

Avg 1545 279Max 5602 394Min 0431 192


scenic spots and shopping mall areas rather than theplaces where residents live or work us withmorning peak data on working days the charac-teristics of the passenger flow in these types ofstations cannot be fully captured In the future all-day data can be collected to improve the estimationeffect

To explore the estimation results among the passengerODs a 3-dimensional surface map of the RMSNij matrix isshown in Figure 8(a) where the indices of the origin anddestination stations are considered as the x-axis and y-axisrespectively and the RMSNij value is considered as the z-axis Besides the contour line of the RMSNij matrix from a2-dimensional perspective is given in Figure 8(b) Note thatthe station indices in Figure 8 are the same as the indicespresented in Table 3

Furthermore we produce a 3-dimensional surface mapand a contour graph as shown in Figure 9 for the specificorigin station in Chongwenmen In Figure 9 the departuretime destination station and RMSNij values are consid-ered as the x-axis y-axis and z-axis respectively edefinitions of the departure time indices in Figure 9 aregiven in Table 4

From the contour graph in Figure 8 we can see that mostof the RMSNij values are relatively small is result indi-cates that TDHFN is effective in estimating the ODmatrix ofurban rail transit passenger flow However we can see thatthere is one point drawn in a dark red color that representsthe value of the OD from Tiananmendong to Beijingzhane passenger flow between Tiananmendong and Bei-jingzhan is quite small during the morning peak whichresults in a relatively large error

Most of the points in Figure 9 are drawn with cool colorswhich further validates the effectiveness of the proposedmethod in estimating the time-dependent OD matrix pas-senger flow ere are few points marked with warm colorsof which the destination stations include Hepingmen Bei-jingzhan etc In terms of the time dimension the time range

of these data points is mainly concentrated between 750 and810

In addition to the time-dependent OD estimationsthe time-dependent travel times for passengers can alsobe obtained based on the TDHFN method e results forpassengers from Chongwenmen to Changchunjie areillustrated in Figure 10 where the estimated and actualtime-dependent travel time distributions are presentede fluctuation trend of the estimated values is consistentwith the trend of the actual values which shows theeffectiveness of the proposed method in travel timeestimation

43 Comparative Analysis e estimation results ofTDHFN are compared with the results of an artificial neuralnetwork (ANN) For a detailed introduction of the ANNmethod we refer to the literature by Remya and Mathew[20] and Mozolin et al [24] e eigenvalues selected in thispaper are obtained from AFC data and urban rail networktopology including the daily average passenger flow of theorigin station the daily average passenger flow of the des-tination station the number of alternative paths the averagetravel time the distance (replaced by section number) thedeparture time and the average transfer times After trainingand adjusting we got a well-trained ANN model ere arethree layers in the network including the input layer theoutput layer and one hidden layere activation function isRelu and Sigmoid and the number of hidden layer nodes is5

e comparison results are illustrated in Figure 11 andTable 5 which show that the results of themodel proposed inthis paper are significantly better than those of the ANNHowever it should be noted that the source of the input datafor ANN is the same as that of the TDHFN model eperformance of the ANN method can be improved whenadditional data are collected such as commuter numberscommuter properties and land types However in a

010 20 30

40 010

2030

40

Origin station Destinatio

n statio

n

14121008060402

06

05

04

03

02

01

(a)

40

35

30

25

20

15

10

5

403530252015105

16

14

12

10

08

06

04

02

00

Origin station

Des

tinat

ion

staito

n

(b)

Figure 8 RMSN for the origin-destination matrix (a) ree-dimensional surface graph of RMSN (b) Contour graph of RMSN


Table 4 Departure time indexIndex Departure time1 7002 7103 7204 7305 7406 7507 8008 8109 82010 83011 84012 850

010

2030

40

Destinatio

n statio

n

2 4 6 8 10 12Departure time

175

150

125

100

075

050

025

000

403530252015100500

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

staito

n

42

36

30

24

18

12

06

001 2 3 4 5 6 7 8 9 10 11 12

Departure time

(b)

Figure 9 RMSN from Chongwenmen Station (a) ree-dimensional surface graph of RMSN (b) Contour graph of RMSN

(500)000

500

1000

1500

2000

2500

3000

3500

4000

20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 2010710 720 730 740 750 800 810 820 830 840 850 900

Pass

enge

r num

ber

Travel time (min)Departure time

Real travel timeEstimation travel time

Real departure timeEstimation departure time

Figure 10 Estimation results from Chongwenmen to Changchunjie


0

10

20

30

40

50

60

ndash5 5 15 25 35 45RM

SEOrigin station

TDHFNANN

Figure 11 Comparative analysis with the ANN method

Table 5 RMSE and RMSN of TDHFN compared with the ANN methodError TDHFN ANNRMSE 25749 223663RMSN () 054 466

40

35

30

25

20

15

10

5

403530252015105

Des

tinat

ion

stat

ion

Origin station

40

35

30

25

20

15

10

05

00

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion

403530252015105Origin station

200

175

150

125

100

75

50

25

0

(b)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


240

210

180

150

120

90

60

30

0

(c)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


175

150

125

100

75

50

25

0

(d)

Figure 12 Dynamic ODmatrix estimation of passenger flow (a) OD passenger volume from 700 to 730 (b) OD passenger volume from 730 to 800 (c) OD passenger volume from 800 to 830 (d) OD passenger volume from 830 to 900


practical situation a detailed and comprehensive collectionis difficult

e difference between the ANN and computationalgraph algorithm is that the former neural network is a black-boxmodel and the number of neurons activation functionsand neural network layers is not certain so this methodoften requires continuous experiments and adjustments tofind the optimal model However in TDHFN the number ofneurons the form of the activation function and thenumber of layers of the neural network are determinedvalues with practical physical significance Only the weightmatrix of each layer in the network is unknown and needs tobe determined through learning erefore the computa-tional graph can express more traffic characteristics than theANN and provides an effective and hierarchical passengerflow estimation

Finally the dynamic OD matrix estimation of passengerflow is shown in Figure 12 It shows the passenger flowchanges of each OD in different periods e dynamic ODmatrix estimation of passenger flow can provide basic datafor the passenger flow control strategy of urban rail transit

5 Conclusions

is study proposed a time-dependent hierarchical flownetwork for urban rail transit passengers e OD passengerflow matrix at each time in the subway network can beobtained by inputting the incoming passenger volume ofeach station during the morning peak to the model ismodel can be improved by backpropagation of the first-order gradients and reassignment of the passenger flow withthe updated weights between different layers under the deepoptimization framework e result analysis indicates thatthe TDHFN can provide abundant and hierarchical pas-senger flow estimation results A comparative analysis showsthat the proposed model can effectively obtain relativelyaccurate passenger flow estimation results

At present the existing OD dynamic estimation methodsof urban rail network passenger flow still need to be improvedin the factors of timeliness and accuracy e most importantcontribution of this paper is to propose a multilayer hier-archical flow network applied to urban rail with deep learningresearch is method can solve the dynamic OD matrixestimation problem is flow-oriented prediction formula-tion can simultaneously estimate different levels of unob-served or partially observed passenger flow variablesFurthermore when more data sources are available thismethod can achieve hierarchical expansion making thismethod more flexible To build a theoretically sound mod-eling framework this paper hopes to trace back to the fun-damentals or low-level representation of deep learningnetworks and construct a transportation-focused computa-tional graph as a structured modeling language is mod-eling paradigm enables us to capture the mathematicalstructure inside the OD matrix estimation problem by rep-resenting and decomposing complex composite functionsthrough a graph of current states and numerical gradients

However the model proposed in this study does notapply to all stations e model function is better when the

subway stations are mainly the distribution of the placeswhere residents live or work By only using the data of themorning peaks over a few working days we cannot deter-mine the characteristics of passenger flow through trainingIn the future more comprehensive data should be collectedsuch as GPS trajectory data [16] land-use data or the (pointof interest) POI features [30] Tang et al [31] applied touncover the characteristics of travel patterns from temporaland spatial dimensions in the metro network according tothe POI data Based on their study the stations can beclustered by node significance on the metro network or POIfeatures of stationsus the applicability of this model maybe improved

Data Availability

e numerical data used to support the findings of this studyare available from the corresponding author upon request

Disclosure

e funders had no role in the design of the study in thecollection analyses or interpretation of data in the writingof the manuscript or in the decision to publish the results

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors wish to thank Jingjia Cao Qingying Lai FeiranLiu Xu Xu and Linqi Xia for helpful discussions isresearch was funded in part by the National Natural ScienceFoundation of China under grant nos 71871012 and72001020 in part by the State Key Lab of Rail Traffic Controland Safety of China under grant no RCS2020ZT003 in partby the China Postdoctoral Science Foundation under grantno 2020M670128 and in part by the Beijing MunicipalNatural Science Foundation under grant no L181007

References

[1] X-M Yao P Zhao and D-D Yu ldquoReal-time origin-destinationmatrices estimation for urban rail transit network based onstructural state-space modelrdquo Journal of Central South Univer-sity vol 22 no 11 pp 4498ndash4506 2015

[2] J de D Ortuzar and L G Willumsen Modeling TransportJohn Wiley amp Sons New York NY USA 1994

[3] S Kikuchi and N Kronprasert ldquoConstructing a transitorigin-destination table using the uncertainty maximizationconceptrdquo Transportation Research Record Journal of theTransportation Research Board vol 2112 no 1 pp 43ndash522009

[4] M V Aerde H Rakha andH Paramahamsan ldquoEstimation oforigin-destination matrices relationship between practicaland theoretical considerationsrdquo Transportation ResearchRecord Journal of the Transportation Research Boardvol 1831 no 1 pp 122ndash130 2003

[5] L Caggiani M Ottomanelli and D Sassanelli ldquoA fixed pointapproach to origin-destination matrices estimation using


uncertain data and fuzzy programming on congested net-worksrdquo Transportation Research Part C Emerging Technol-ogies vol 28 pp 130ndash141 2013

[6] X-F Yang Y Lu and W Hao ldquoOrigin-destination esti-mation using probe vehicle trajectory and link countsrdquoJournal of Advanced Transportation vol 2017 Article ID4341532 2017

[7] L Cheng S-L Zhu Z-M Chu and J-X Cheng ldquoA bayesiannetwork model for origin-destination matrices estimationusing prior and some observed link flowsrdquo Discrete Dynamicsin Nature and Society vol 2014 p 9 Article ID 192470 2014

[8] Y-X Ji J-Z Zhao Z-M Zhang and Y-C Du ldquoEstimatingbus loads and OD flows using location-stamped farebox andWi-Fi signal datardquo Journal of Advanced Transportationvol 2017 Article ID 6374858 2017

[9] S Bera and K V K Rao ldquoEstimation of origin-destinationmatrix from traffic counts the state of the artrdquo EuropeanTransport Trasporti Europei vol 49 no 49 pp 3ndash23 2011

[10] H-P Lo and C-P Chan ldquoSimultaneous estimation of anorigin-destination matrix and link choice proportions usingtraffic countsrdquo Transportation Research Part A Policy andPractice vol 37 no 9 pp 771ndash788 2003

[11] H Yang ldquoHeuristic algorithms for the bilevel origin-destinationmatrix estimation problemrdquo Transportation Research Part BMethodological vol 29 no 4 pp 231ndash242 1995

[12] X Zhou and H S Mahmassani ldquoDynamic origin-destinationdemand estimation using automatic vehicle identificationdatardquo IEEE Transactions on Intelligent Transportation Sys-tems vol 7 no 1 pp 105ndash114 2006

[13] H D Sherali R Sivanandan and A G Hobeika ldquoA linearprogramming approach for synthesizing origin-destinationtrip tables from link traffic volumesrdquo Transportation ResearchPart B Methodological vol 28 no 3 pp 213ndash233 1994

[14] T Toledo and T Kolechkina ldquoEstimation of dynamicorigin-destination matrices using linear assignment matrixapproximationsrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 14 no 2 pp 618ndash626 2013

[15] M Fujita S Yamada and S Murakami ldquoTime coefficientestimation for hourly origin-destination demand from ob-served link flow based on semidynamic traffic assignmentrdquoJournal of Advanced Transportation vol 2017 Article ID6495861 2017

[16] J Tang S Zhang X Chen F Liu and Y Zou ldquoTaxi tripsdistribution modeling based on entropy-maximizing theory acase study in Harbin city-Chinardquo Physica A Statistical Me-chanics and Its Applications vol 493 pp 430ndash443 2018

[17] W Rao YWu J Xia J Ou and R Kluger ldquoOrigin-destinationpattern estimation based on trajectory reconstruction usingautomatic license plate recognition datardquo Transportation Re-search Part C vol 95 2017

[18] J Liu F Zheng H J Van Zuylen and J Li ldquoA dynamic ODprediction approach for urban networks based on A dynamicOD prediction approach for urban networks based on au-tomatic number plate recognition data automatic numberplate recognition datardquo Transportation Research Procediavol 47 2019

[19] J Guo Y Liu X Li W Huang J Cao and YWei ldquoEnhancedleast square based dynamic ODmatrix estimation using RadioFrequency Identification datardquo Mathematics and Computersin Simulation vol 155 pp 27ndash40 2019

[20] K P Remya and S Mathew ldquoODmatrix estimation from linkcounts using artificial neural networkrdquo International Journalof Scientific amp Engineering Research vol 4 no 5 pp 293ndash2962013

[21] Z Gong ldquoEstimating the urban o-d matrix a neural networkapproachrdquo European Journal of Operational Researchvol 106 pp 108ndash115 1998

[22] H Yang T Akiyama and T Sasaki ldquoEstimation of time-varying origin-destination flows from traffic counts a neuralnetwork approachrdquo Mathematical and Computer Modellingvol 27 no 9ndash11 pp 323ndash334 1998

[23] H-Y Li Y-T Wang X-Y Xu L-Q Qin and H-Y ZhangldquoShort-term passenger flow prediction under passenger flowcontrol using a dynamic radial basis function networkrdquoApplied Soft Computing vol 83 2019

[24] M Mozolin J-C ill and E Lynn Usery ldquoTrip distributionforecasting with multilayer perceptron neural networks acritical evaluationrdquo Transportation Research Part B Meth-odological vol 34 no 1 pp 53ndash73 2000

[25] P Shang R-M Li Z-Y Liu and K Xian ldquoTimetable syn-chronization and optimization considering time-dependentpassenger demand in an urban subway networkrdquo Trans-portation Research Record vol 34 2018

[26] P Shang R Li J Guo K Xian and X Zhou ldquoIntegratingLagrangian and Eulerian observations for passenger flow stateestimation in an urban rail transit network a space-time-statehyper network-based assignment approachrdquo TransportationResearch Part B Methodological vol 121 pp 135ndash167 2019

[27] X Wu J Guo K Xian and X Zhou ldquoHierarchical traveldemand estimation usingmultiple data sources a forward andbackward propagation algorithmic framework on a layeredcomputational graphrdquo Transportation Research Part CEmerging Technologies vol 96 pp 321ndash346 2018

[28] K Lu A Khani and B Han ldquoA trip purpose-based datadriven alighting station choice model using transit smart carddatardquo Complexity vol 2018 Article ID 3412070 2018

[29] R Frederix F Viti W Himpe and C Tampere ldquoA hierar-chical approach for dynamic origin-destination matrix esti-mation on large-scale congested networksrdquo InternationalIEEE Conference on ITS vol 53 2011

[30] Z-Y Du J-J Tang Y Qi Y-W Wang C-Y Han andY-F Yang ldquoIdentifying critical nodes in metro networkconsidering topological potential a case study in shenzhencity-Chinardquo Physica A vol 539 pp 1ndash13 2020

[31] J-J Tang X-L Wang F Zong and Z Hu ldquoUncoveringspatio-temporal travel patterns using a tensor-based modelfrom metro smart card data in Shenzhen Chinardquo Sustain-ability vol 12 no 1475 pp 1ndash16 2020


assignment matrix are limited to simple networks like in-tersections interchanges and freeways [9]

ese conventional methods for estimatingorigin-destination (OD) trip matrices from link trafficcounts assume that route choice proportions are givenconstants But this assumption does not hold in a networkwith realistic congestion levels [10] A bilevel programmingapproach has been used for the estimation of the OD matrixin congested networks [11 12] is approach combined thegeneralized least squares estimation model and the networkequilibrium model into one process However the bilevelapproach has certain difficulties in finding an optimal so-lution because of nonconvex and nondifferential formula-tions Sherali et al [13] constructed a linear programmingmodel with a user-equilibrium solution for synthesizing ODtables from traffic volume counts Later Toledo andKolechkina [14] presented the methods based on use oflinear approximations of the assignment matrix in the op-timization iterations Fujita et al [15] proposed an ODmodification approach formulated as a static user-equilib-rium assignment with elastic demand based on the residualdemand at the end of each period Applying the model tolarge-scale road network demonstrates that it efficientlyimproves estimation accuracy because the 24-hour timecoefficients of survey data are slightly biased and may bemodified properly Unlike the gravity model these ap-proaches are based on traffic count data which can bedetected from links by vehicle identification or locatingtechnologies such as GPS floating automatic license platerecognition (ALPR) and radio frequency identification(RFID) Tang et al [16] proposed a newmethod based on theentropy-maximizing theory to model OD distribution inHarbin city using large-scale taxi GPS trajectories e re-sults demonstrate that the entropy-maximizing model issuperior to the gravity model which can validate the fea-sibility of OD distribution from taxi GPS data in the urbansystem Rao et al [17] formulated a particle filter model forvehicle trajectory reconstruction based on ALPR data Andthe OD patterns are estimated by adding up the path flowswhich is conducted through dividing the reconstructedcomplete trajectories Liu et al [18] predicted the ODmatrixbased on the historical ALPR data Guo et al [19] developedan optimization model based on the least squares methode optimization model estimated the dynamic OD matrixby integrating the preliminary OD matrix dynamic as-signment matrix derived by RFID data and link flow de-tected by the inductive loop detectors However the routechoice and travel time delay issues are still difficult to dealwith So establishing the dynamic flow equations is the firstchallenge to estimate dynamic passenger flow demand forrail transit systems for the lack of observed information andcomplex structure of the network

In recent decades the application of the neural networkmodel expands this problem into a new field e neuralnetwork operates as a black box model-free and adaptivetool for capturing and learning significant structures in data[20] Gong [21] used the Hopfield neural network (HNN)model to estimate the urban orientation-destination (OD)distribution matrix from the link volumes of the

transportation network so as to promote the solving speedand precision Yang et al [22] proposed a dynamic modelbased on backpropagation (BP) learning for estimating ODflows from road entering and exiting counts e OD flowsin each short time interval are estimated through theminimization of the squared errors between the predictedand observed exiting counts Li et al [23] proposed a newdynamic radial basis function neural network to forecastoutbound passenger volumes and improve passenger flowcontrol Passenger flow control was considered to improvethe prediction accuracy by adding passenger flow controlcoefficients to their model However the current perceptronneural networks may not perform well in all issues due toreasons such as model nontransferability insufficient abilityto generalize and reliance on activation functions [24]

Subsequently the computational graph was proposed asa description language to represent mathematical expres-sions It is important to understand how the underlyingcomputational graph of a deep learning network combinedwith the BP algorithm can be used to describe the forwardpropagation and backward feedback processes betweendifferent levels of transportation planning and decisionmaking [25 26] Wu et al [27] proposed a multilayeredhierarchical flow network representation to structurallymodel different levels of travel demand for road networksincluding trip generation OD matrices path and link flowsand individual behavior parameters However the traveltimes were assumed to be observed in their study In otherwords their model was constructed in a static networkrather than a time-dependent dynamic network is issuehas been improved in this paper

In this paper we aim to predict the dynamic ODdemand for a time of period in the future based on his-torical observations In most research papers it is as-sumed that the OD matrix can be predicted fromhistorical data [18 28] We apply the historic AFC to traina time-dependent hierarchical flow network en we useit to predict the future OD demand with real-time AFCdata at current as input information which is the basicdata to formulate operational and organizational strate-gies Besides the programming model proposed in thispaper can also estimate a hierarchical traveling decisionprocess for passengers in an urban rail transit systemincluding the departure time choice at the origin the pathchoice and the corresponding arrival time at the desti-nation erefore the proposed method in this studyachieves a combination prediction of dynamic ODmatrixdeparture time choice route choice and travel time

A nonlinear programmingmodel is proposed to conductreal-time OD matrix estimation for an urban rail transitsystem based on historic automatic fare collection (AFC)data in this paper Forward passing in the hierarchical flownetwork of urban rail transit sequentially assigns passengersto candidate stations paths and different travel time in-tervalse network can be calibrated by backpropagation ofthe first-order gradients and reassignment of the passengerflowwith the updated weights between different layers underthe deep optimization frameworkis model can determinethe time-varying OD matrix the number of passengers









2 Problem Statement
















































HT α times X (1)

HD β times HT (2)

HP ρ times HD (3)

Y ω times Hp (4)




1

2

3

4

Origin

Destination 1

Destination 2

Transfer

1

2

3

4

Path

Note


Destination 1

Origin 1

Destination 2

Transfer

Transfer

τ1τ2

t2t1

Path 1Path 2

Path 3Path 4

hellip






ht 1113944i



hj 1113944t


Assigning the paths



yτ 1113944p



1113944tisinTi

αit 1 (9)

1113944jisinDt

βtj 1 (10)

1113944pisinPj

ρjp 1(11)

1113944τisinΓp

ωpτ 1(12)



12

1113954yτ minus yτ( 11138572 (13)


L ωpτ hp λτ1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p


2



L ρjp hj λp1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p


2



L βtj ht λj1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p



2



L αit xi λt1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p


βtj times 1113944i


2



(14)



gp 1113944τisinΓp


gj 1113944pisinPj



gt 1113944jisinDt







ωpτ minus 1 0

nablaλpL 1113944


nablaλjL 1113944


nablaλtL 1113944


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)






middotzyτ

zω (20)



middotzyτ

zhj

middotzhj

zρ


middotzyτ

zhj

middotzhj

zht

middotzht

zβ


middotzyτ

zhj

middotzhj

zht

middotzht

zxi

middotzxi

zα

(21)



Origin 1node1

i1

Departure time 1

t1

Departure time 2

t2Destination 1

node 2



d1


d2

d3

d4

p1Path 1

Path 2p2

p3Path 3

Path 4p4

Path 1

Path 2

p5

p6

Path 3

Path 4

p7

p8

Travel time 1

Travel time 2

Travel time 3

Travel time 4

Travel time 5

Travel time 6

Travel time 7


Travel time 2


Travel time 5


τ1

τ2τ3

τ4τ5τ6

τ7τ8

τ9

τ10τ11τ12

τ13τ14

τ15τ16













i1

i2

i3

t1

t2

t3

j1

j2

j3 p3

p1

p2

τ1

τ2

τ3




βtj ρjp ωpτ


αit














Chegongzhuang



Chaoyangmen

Xizhimeng

Line 1

Line 2


Line 5Line 8

Line 6

Line 4



RMSE 1

|Γ|1113944τisinΓ

yτ minus yτ1113872 1113873212

⎛⎝ ⎞⎠ (22)

RMSEi 1

Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944tisinTi

1113944jisinDt

1113944pisinPj

1113944τisinΓp

yτ minus yτ1113872 11138732⎛⎜⎝ ⎞⎟⎠

12

(23)

RMSEij 1

Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

tisinTij

1113944pisinPj

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(24)

RMSEijt 1

middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

pisinPijt

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(25)

RMSN RMSE

(1|Γ|)1113936τisinΓ1113954yτ (26)

RMSNi RMSEi

1 Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTi1113936jisinDt


(27)

RMSNij RMSEij

1 Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pt

11138681113868111386811138681113868111386811138681113868 middot ΓP

111386811138681113868111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTij


(28)

0000

0928

00670000 0003 0003

00000100020003000400050006000700080009001000

10 20 30 40 50 60

Freq

uenc

y

Travel time (min)

(a)

0001

0981

0013 0002 0002 00010000

0200

0400

0600

0800

1000

1200

Freq

uenc

y

Travel time (min)

(b)


100

150

200

250

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Loss

(MSE

)

Iteration



RMSNijt RMSEijt

1 middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot ΓP1113868111386811138681113868


(29)







Avg 1545 279Max 5602 394Min 0431 192











010 20 30

40 010

2030

40


n statio

n

14121008060402

06

05

04

03

02

01

(a)

40

35

30

25

20

15

10

5

403530252015105

16

14

12

10

08

06

04

02

00

Origin station

Des

tinat

ion

staito

n

(b)




010

2030

40

Destinatio

n statio

n


175

150

125

100

075

050

025

000

403530252015100500

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

staito

n

42

36

30

24

18

12

06

001 2 3 4 5 6 7 8 9 10 11 12

Departure time

(b)


(500)000

500

1000

1500

2000

2500

3000

3500

4000

20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 2010710 720 730 740 750 800 810 820 830 840 850 900

Pass

enge

r num

ber






0

10

20

30

40

50

60

ndash5 5 15 25 35 45RM

SEOrigin station

TDHFNANN



40

35

30

25

20

15

10

5

403530252015105

Des

tinat

ion

stat

ion

Origin station

40

35

30

25

20

15

10

05

00

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


200

175

150

125

100

75

50

25

0

(b)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


240

210

180

150

120

90

60

30

0

(c)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


175

150

125

100

75

50

25

0

(d)






5 Conclusions





Data Availability


Disclosure




Acknowledgments


References










































2 Problem Statement
















































HT α times X (1)

HD β times HT (2)

HP ρ times HD (3)

Y ω times Hp (4)




1

2

3

4

Origin

Destination 1

Destination 2

Transfer

1

2

3

4

Path

Note


Destination 1

Origin 1

Destination 2

Transfer

Transfer

τ1τ2

t2t1

Path 1Path 2

Path 3Path 4

hellip






ht 1113944i



hj 1113944t


Assigning the paths



yτ 1113944p



1113944tisinTi

αit 1 (9)

1113944jisinDt

βtj 1 (10)

1113944pisinPj

ρjp 1(11)

1113944τisinΓp

ωpτ 1(12)



12

1113954yτ minus yτ( 11138572 (13)


L ωpτ hp λτ1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p


2



L ρjp hj λp1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p


2



L βtj ht λj1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p



2



L αit xi λt1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p


βtj times 1113944i


2



(14)



gp 1113944τisinΓp


gj 1113944pisinPj



gt 1113944jisinDt







ωpτ minus 1 0

nablaλpL 1113944


nablaλjL 1113944


nablaλtL 1113944


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)






middotzyτ

zω (20)



middotzyτ

zhj

middotzhj

zρ


middotzyτ

zhj

middotzhj

zht

middotzht

zβ


middotzyτ

zhj

middotzhj

zht

middotzht

zxi

middotzxi

zα

(21)



Origin 1node1

i1

Departure time 1

t1

Departure time 2

t2Destination 1

node 2



d1


d2

d3

d4

p1Path 1

Path 2p2

p3Path 3

Path 4p4

Path 1

Path 2

p5

p6

Path 3

Path 4

p7

p8

Travel time 1

Travel time 2

Travel time 3

Travel time 4

Travel time 5

Travel time 6

Travel time 7


Travel time 2


Travel time 5


τ1

τ2τ3

τ4τ5τ6

τ7τ8

τ9

τ10τ11τ12

τ13τ14

τ15τ16













i1

i2

i3

t1

t2

t3

j1

j2

j3 p3

p1

p2

τ1

τ2

τ3




βtj ρjp ωpτ


αit














Chegongzhuang



Chaoyangmen

Xizhimeng

Line 1

Line 2


Line 5Line 8

Line 6

Line 4



RMSE 1

|Γ|1113944τisinΓ

yτ minus yτ1113872 1113873212

⎛⎝ ⎞⎠ (22)

RMSEi 1

Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944tisinTi

1113944jisinDt

1113944pisinPj

1113944τisinΓp

yτ minus yτ1113872 11138732⎛⎜⎝ ⎞⎟⎠

12

(23)

RMSEij 1

Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

tisinTij

1113944pisinPj

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(24)

RMSEijt 1

middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

pisinPijt

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(25)

RMSN RMSE

(1|Γ|)1113936τisinΓ1113954yτ (26)

RMSNi RMSEi

1 Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTi1113936jisinDt


(27)

RMSNij RMSEij

1 Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pt

11138681113868111386811138681113868111386811138681113868 middot ΓP

111386811138681113868111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTij


(28)

0000

0928

00670000 0003 0003

00000100020003000400050006000700080009001000

10 20 30 40 50 60

Freq

uenc

y

Travel time (min)

(a)

0001

0981

0013 0002 0002 00010000

0200

0400

0600

0800

1000

1200

Freq

uenc

y

Travel time (min)

(b)


100

150

200

250

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Loss

(MSE

)

Iteration



RMSNijt RMSEijt

1 middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot ΓP1113868111386811138681113868


(29)







Avg 1545 279Max 5602 394Min 0431 192











010 20 30

40 010

2030

40


n statio

n

14121008060402

06

05

04

03

02

01

(a)

40

35

30

25

20

15

10

5

403530252015105

16

14

12

10

08

06

04

02

00

Origin station

Des

tinat

ion

staito

n

(b)




010

2030

40

Destinatio

n statio

n


175

150

125

100

075

050

025

000

403530252015100500

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

staito

n

42

36

30

24

18

12

06

001 2 3 4 5 6 7 8 9 10 11 12

Departure time

(b)


(500)000

500

1000

1500

2000

2500

3000

3500

4000

20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 2010710 720 730 740 750 800 810 820 830 840 850 900

Pass

enge

r num

ber






0

10

20

30

40

50

60

ndash5 5 15 25 35 45RM

SEOrigin station

TDHFNANN



40

35

30

25

20

15

10

5

403530252015105

Des

tinat

ion

stat

ion

Origin station

40

35

30

25

20

15

10

05

00

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


200

175

150

125

100

75

50

25

0

(b)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


240

210

180

150

120

90

60

30

0

(c)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


175

150

125

100

75

50

25

0

(d)






5 Conclusions





Data Availability


Disclosure




Acknowledgments


References









































































HT α times X (1)

HD β times HT (2)

HP ρ times HD (3)

Y ω times Hp (4)




1

2

3

4

Origin

Destination 1

Destination 2

Transfer

1

2

3

4

Path

Note


Destination 1

Origin 1

Destination 2

Transfer

Transfer

τ1τ2

t2t1

Path 1Path 2

Path 3Path 4

hellip






ht 1113944i



hj 1113944t


Assigning the paths



yτ 1113944p



1113944tisinTi

αit 1 (9)

1113944jisinDt

βtj 1 (10)

1113944pisinPj

ρjp 1(11)

1113944τisinΓp

ωpτ 1(12)



12

1113954yτ minus yτ( 11138572 (13)


L ωpτ hp λτ1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p


2



L ρjp hj λp1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p


2



L βtj ht λj1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p



2



L αit xi λt1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p


βtj times 1113944i


2



(14)



gp 1113944τisinΓp


gj 1113944pisinPj



gt 1113944jisinDt







ωpτ minus 1 0

nablaλpL 1113944


nablaλjL 1113944


nablaλtL 1113944


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)






middotzyτ

zω (20)



middotzyτ

zhj

middotzhj

zρ


middotzyτ

zhj

middotzhj

zht

middotzht

zβ


middotzyτ

zhj

middotzhj

zht

middotzht

zxi

middotzxi

zα

(21)



Origin 1node1

i1

Departure time 1

t1

Departure time 2

t2Destination 1

node 2



d1


d2

d3

d4

p1Path 1

Path 2p2

p3Path 3

Path 4p4

Path 1

Path 2

p5

p6

Path 3

Path 4

p7

p8

Travel time 1

Travel time 2

Travel time 3

Travel time 4

Travel time 5

Travel time 6

Travel time 7


Travel time 2


Travel time 5


τ1

τ2τ3

τ4τ5τ6

τ7τ8

τ9

τ10τ11τ12

τ13τ14

τ15τ16













i1

i2

i3

t1

t2

t3

j1

j2

j3 p3

p1

p2

τ1

τ2

τ3




βtj ρjp ωpτ


αit














Chegongzhuang



Chaoyangmen

Xizhimeng

Line 1

Line 2


Line 5Line 8

Line 6

Line 4



RMSE 1

|Γ|1113944τisinΓ

yτ minus yτ1113872 1113873212

⎛⎝ ⎞⎠ (22)

RMSEi 1

Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944tisinTi

1113944jisinDt

1113944pisinPj

1113944τisinΓp

yτ minus yτ1113872 11138732⎛⎜⎝ ⎞⎟⎠

12

(23)

RMSEij 1

Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

tisinTij

1113944pisinPj

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(24)

RMSEijt 1

middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

pisinPijt

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(25)

RMSN RMSE

(1|Γ|)1113936τisinΓ1113954yτ (26)

RMSNi RMSEi

1 Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTi1113936jisinDt


(27)

RMSNij RMSEij

1 Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pt

11138681113868111386811138681113868111386811138681113868 middot ΓP

111386811138681113868111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTij


(28)

0000

0928

00670000 0003 0003

00000100020003000400050006000700080009001000

10 20 30 40 50 60

Freq

uenc

y

Travel time (min)

(a)

0001

0981

0013 0002 0002 00010000

0200

0400

0600

0800

1000

1200

Freq

uenc

y

Travel time (min)

(b)


100

150

200

250

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Loss

(MSE

)

Iteration



RMSNijt RMSEijt

1 middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot ΓP1113868111386811138681113868


(29)







Avg 1545 279Max 5602 394Min 0431 192











010 20 30

40 010

2030

40


n statio

n

14121008060402

06

05

04

03

02

01

(a)

40

35

30

25

20

15

10

5

403530252015105

16

14

12

10

08

06

04

02

00

Origin station

Des

tinat

ion

staito

n

(b)




010

2030

40

Destinatio

n statio

n


175

150

125

100

075

050

025

000

403530252015100500

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

staito

n

42

36

30

24

18

12

06

001 2 3 4 5 6 7 8 9 10 11 12

Departure time

(b)


(500)000

500

1000

1500

2000

2500

3000

3500

4000

20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 2010710 720 730 740 750 800 810 820 830 840 850 900

Pass

enge

r num

ber






0

10

20

30

40

50

60

ndash5 5 15 25 35 45RM

SEOrigin station

TDHFNANN



40

35

30

25

20

15

10

5

403530252015105

Des

tinat

ion

stat

ion

Origin station

40

35

30

25

20

15

10

05

00

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


200

175

150

125

100

75

50

25

0

(b)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


240

210

180

150

120

90

60

30

0

(c)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


175

150

125

100

75

50

25

0

(d)






5 Conclusions





Data Availability


Disclosure




Acknowledgments


References






































ht 1113944i



hj 1113944t


Assigning the paths



yτ 1113944p



1113944tisinTi

αit 1 (9)

1113944jisinDt

βtj 1 (10)

1113944pisinPj

ρjp 1(11)

1113944τisinΓp

ωpτ 1(12)



12

1113954yτ minus yτ( 11138572 (13)


L ωpτ hp λτ1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p


2



L ρjp hj λp1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p


2



L βtj ht λj1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p



2



L αit xi λt1113872 1113873 1113944τisinΓ

12

1113954yτ minus 1113944p


βtj times 1113944i


2



(14)



gp 1113944τisinΓp


gj 1113944pisinPj



gt 1113944jisinDt







ωpτ minus 1 0

nablaλpL 1113944


nablaλjL 1113944


nablaλtL 1113944


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)






middotzyτ

zω (20)



middotzyτ

zhj

middotzhj

zρ


middotzyτ

zhj

middotzhj

zht

middotzht

zβ


middotzyτ

zhj

middotzhj

zht

middotzht

zxi

middotzxi

zα

(21)



Origin 1node1

i1

Departure time 1

t1

Departure time 2

t2Destination 1

node 2



d1


d2

d3

d4

p1Path 1

Path 2p2

p3Path 3

Path 4p4

Path 1

Path 2

p5

p6

Path 3

Path 4

p7

p8

Travel time 1

Travel time 2

Travel time 3

Travel time 4

Travel time 5

Travel time 6

Travel time 7


Travel time 2


Travel time 5


τ1

τ2τ3

τ4τ5τ6

τ7τ8

τ9

τ10τ11τ12

τ13τ14

τ15τ16













i1

i2

i3

t1

t2

t3

j1

j2

j3 p3

p1

p2

τ1

τ2

τ3




βtj ρjp ωpτ


αit














Chegongzhuang



Chaoyangmen

Xizhimeng

Line 1

Line 2


Line 5Line 8

Line 6

Line 4



RMSE 1

|Γ|1113944τisinΓ

yτ minus yτ1113872 1113873212

⎛⎝ ⎞⎠ (22)

RMSEi 1

Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944tisinTi

1113944jisinDt

1113944pisinPj

1113944τisinΓp

yτ minus yτ1113872 11138732⎛⎜⎝ ⎞⎟⎠

12

(23)

RMSEij 1

Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

tisinTij

1113944pisinPj

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(24)

RMSEijt 1

middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

pisinPijt

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(25)

RMSN RMSE

(1|Γ|)1113936τisinΓ1113954yτ (26)

RMSNi RMSEi

1 Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTi1113936jisinDt


(27)

RMSNij RMSEij

1 Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pt

11138681113868111386811138681113868111386811138681113868 middot ΓP

111386811138681113868111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTij


(28)

0000

0928

00670000 0003 0003

00000100020003000400050006000700080009001000

10 20 30 40 50 60

Freq

uenc

y

Travel time (min)

(a)

0001

0981

0013 0002 0002 00010000

0200

0400

0600

0800

1000

1200

Freq

uenc

y

Travel time (min)

(b)


100

150

200

250

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Loss

(MSE

)

Iteration



RMSNijt RMSEijt

1 middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot ΓP1113868111386811138681113868


(29)







Avg 1545 279Max 5602 394Min 0431 192











010 20 30

40 010

2030

40


n statio

n

14121008060402

06

05

04

03

02

01

(a)

40

35

30

25

20

15

10

5

403530252015105

16

14

12

10

08

06

04

02

00

Origin station

Des

tinat

ion

staito

n

(b)




010

2030

40

Destinatio

n statio

n


175

150

125

100

075

050

025

000

403530252015100500

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

staito

n

42

36

30

24

18

12

06

001 2 3 4 5 6 7 8 9 10 11 12

Departure time

(b)


(500)000

500

1000

1500

2000

2500

3000

3500

4000

20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 2010710 720 730 740 750 800 810 820 830 840 850 900

Pass

enge

r num

ber






0

10

20

30

40

50

60

ndash5 5 15 25 35 45RM

SEOrigin station

TDHFNANN



40

35

30

25

20

15

10

5

403530252015105

Des

tinat

ion

stat

ion

Origin station

40

35

30

25

20

15

10

05

00

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


200

175

150

125

100

75

50

25

0

(b)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


240

210

180

150

120

90

60

30

0

(c)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


175

150

125

100

75

50

25

0

(d)






5 Conclusions





Data Availability


Disclosure




Acknowledgments


References




































ωpτ minus 1 0

nablaλpL 1113944


nablaλjL 1113944


nablaλtL 1113944


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)






middotzyτ

zω (20)



middotzyτ

zhj

middotzhj

zρ


middotzyτ

zhj

middotzhj

zht

middotzht

zβ


middotzyτ

zhj

middotzhj

zht

middotzht

zxi

middotzxi

zα

(21)



Origin 1node1

i1

Departure time 1

t1

Departure time 2

t2Destination 1

node 2



d1


d2

d3

d4

p1Path 1

Path 2p2

p3Path 3

Path 4p4

Path 1

Path 2

p5

p6

Path 3

Path 4

p7

p8

Travel time 1

Travel time 2

Travel time 3

Travel time 4

Travel time 5

Travel time 6

Travel time 7


Travel time 2


Travel time 5


τ1

τ2τ3

τ4τ5τ6

τ7τ8

τ9

τ10τ11τ12

τ13τ14

τ15τ16













i1

i2

i3

t1

t2

t3

j1

j2

j3 p3

p1

p2

τ1

τ2

τ3




βtj ρjp ωpτ


αit














Chegongzhuang



Chaoyangmen

Xizhimeng

Line 1

Line 2


Line 5Line 8

Line 6

Line 4



RMSE 1

|Γ|1113944τisinΓ

yτ minus yτ1113872 1113873212

⎛⎝ ⎞⎠ (22)

RMSEi 1

Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944tisinTi

1113944jisinDt

1113944pisinPj

1113944τisinΓp

yτ minus yτ1113872 11138732⎛⎜⎝ ⎞⎟⎠

12

(23)

RMSEij 1

Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

tisinTij

1113944pisinPj

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(24)

RMSEijt 1

middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

pisinPijt

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(25)

RMSN RMSE

(1|Γ|)1113936τisinΓ1113954yτ (26)

RMSNi RMSEi

1 Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTi1113936jisinDt


(27)

RMSNij RMSEij

1 Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pt

11138681113868111386811138681113868111386811138681113868 middot ΓP

111386811138681113868111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTij


(28)

0000

0928

00670000 0003 0003

00000100020003000400050006000700080009001000

10 20 30 40 50 60

Freq

uenc

y

Travel time (min)

(a)

0001

0981

0013 0002 0002 00010000

0200

0400

0600

0800

1000

1200

Freq

uenc

y

Travel time (min)

(b)


100

150

200

250

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Loss

(MSE

)

Iteration



RMSNijt RMSEijt

1 middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot ΓP1113868111386811138681113868


(29)







Avg 1545 279Max 5602 394Min 0431 192











010 20 30

40 010

2030

40


n statio

n

14121008060402

06

05

04

03

02

01

(a)

40

35

30

25

20

15

10

5

403530252015105

16

14

12

10

08

06

04

02

00

Origin station

Des

tinat

ion

staito

n

(b)




010

2030

40

Destinatio

n statio

n


175

150

125

100

075

050

025

000

403530252015100500

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

staito

n

42

36

30

24

18

12

06

001 2 3 4 5 6 7 8 9 10 11 12

Departure time

(b)


(500)000

500

1000

1500

2000

2500

3000

3500

4000

20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 2010710 720 730 740 750 800 810 820 830 840 850 900

Pass

enge

r num

ber






0

10

20

30

40

50

60

ndash5 5 15 25 35 45RM

SEOrigin station

TDHFNANN



40

35

30

25

20

15

10

5

403530252015105

Des

tinat

ion

stat

ion

Origin station

40

35

30

25

20

15

10

05

00

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


200

175

150

125

100

75

50

25

0

(b)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


240

210

180

150

120

90

60

30

0

(c)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


175

150

125

100

75

50

25

0

(d)






5 Conclusions





Data Availability


Disclosure




Acknowledgments


References













































i1

i2

i3

t1

t2

t3

j1

j2

j3 p3

p1

p2

τ1

τ2

τ3




βtj ρjp ωpτ


αit














Chegongzhuang



Chaoyangmen

Xizhimeng

Line 1

Line 2


Line 5Line 8

Line 6

Line 4



RMSE 1

|Γ|1113944τisinΓ

yτ minus yτ1113872 1113873212

⎛⎝ ⎞⎠ (22)

RMSEi 1

Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944tisinTi

1113944jisinDt

1113944pisinPj

1113944τisinΓp

yτ minus yτ1113872 11138732⎛⎜⎝ ⎞⎟⎠

12

(23)

RMSEij 1

Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

tisinTij

1113944pisinPj

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(24)

RMSEijt 1

middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

pisinPijt

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(25)

RMSN RMSE

(1|Γ|)1113936τisinΓ1113954yτ (26)

RMSNi RMSEi

1 Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTi1113936jisinDt


(27)

RMSNij RMSEij

1 Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pt

11138681113868111386811138681113868111386811138681113868 middot ΓP

111386811138681113868111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTij


(28)

0000

0928

00670000 0003 0003

00000100020003000400050006000700080009001000

10 20 30 40 50 60

Freq

uenc

y

Travel time (min)

(a)

0001

0981

0013 0002 0002 00010000

0200

0400

0600

0800

1000

1200

Freq

uenc

y

Travel time (min)

(b)


100

150

200

250

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Loss

(MSE

)

Iteration



RMSNijt RMSEijt

1 middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot ΓP1113868111386811138681113868


(29)







Avg 1545 279Max 5602 394Min 0431 192











010 20 30

40 010

2030

40


n statio

n

14121008060402

06

05

04

03

02

01

(a)

40

35

30

25

20

15

10

5

403530252015105

16

14

12

10

08

06

04

02

00

Origin station

Des

tinat

ion

staito

n

(b)




010

2030

40

Destinatio

n statio

n


175

150

125

100

075

050

025

000

403530252015100500

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

staito

n

42

36

30

24

18

12

06

001 2 3 4 5 6 7 8 9 10 11 12

Departure time

(b)


(500)000

500

1000

1500

2000

2500

3000

3500

4000

20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 2010710 720 730 740 750 800 810 820 830 840 850 900

Pass

enge

r num

ber






0

10

20

30

40

50

60

ndash5 5 15 25 35 45RM

SEOrigin station

TDHFNANN



40

35

30

25

20

15

10

5

403530252015105

Des

tinat

ion

stat

ion

Origin station

40

35

30

25

20

15

10

05

00

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


200

175

150

125

100

75

50

25

0

(b)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


240

210

180

150

120

90

60

30

0

(c)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


175

150

125

100

75

50

25

0

(d)






5 Conclusions





Data Availability


Disclosure




Acknowledgments


References












































Chegongzhuang



Chaoyangmen

Xizhimeng

Line 1

Line 2


Line 5Line 8

Line 6

Line 4



RMSE 1

|Γ|1113944τisinΓ

yτ minus yτ1113872 1113873212

⎛⎝ ⎞⎠ (22)

RMSEi 1

Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944tisinTi

1113944jisinDt

1113944pisinPj

1113944τisinΓp

yτ minus yτ1113872 11138732⎛⎜⎝ ⎞⎟⎠

12

(23)

RMSEij 1

Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

tisinTij

1113944pisinPj

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(24)

RMSEijt 1

middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

pisinPijt

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(25)

RMSN RMSE

(1|Γ|)1113936τisinΓ1113954yτ (26)

RMSNi RMSEi

1 Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTi1113936jisinDt


(27)

RMSNij RMSEij

1 Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pt

11138681113868111386811138681113868111386811138681113868 middot ΓP

111386811138681113868111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTij


(28)

0000

0928

00670000 0003 0003

00000100020003000400050006000700080009001000

10 20 30 40 50 60

Freq

uenc

y

Travel time (min)

(a)

0001

0981

0013 0002 0002 00010000

0200

0400

0600

0800

1000

1200

Freq

uenc

y

Travel time (min)

(b)


100

150

200

250

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Loss

(MSE

)

Iteration



RMSNijt RMSEijt

1 middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot ΓP1113868111386811138681113868


(29)







Avg 1545 279Max 5602 394Min 0431 192











010 20 30

40 010

2030

40


n statio

n

14121008060402

06

05

04

03

02

01

(a)

40

35

30

25

20

15

10

5

403530252015105

16

14

12

10

08

06

04

02

00

Origin station

Des

tinat

ion

staito

n

(b)




010

2030

40

Destinatio

n statio

n


175

150

125

100

075

050

025

000

403530252015100500

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

staito

n

42

36

30

24

18

12

06

001 2 3 4 5 6 7 8 9 10 11 12

Departure time

(b)


(500)000

500

1000

1500

2000

2500

3000

3500

4000

20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 2010710 720 730 740 750 800 810 820 830 840 850 900

Pass

enge

r num

ber






0

10

20

30

40

50

60

ndash5 5 15 25 35 45RM

SEOrigin station

TDHFNANN



40

35

30

25

20

15

10

5

403530252015105

Des

tinat

ion

stat

ion

Origin station

40

35

30

25

20

15

10

05

00

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


200

175

150

125

100

75

50

25

0

(b)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


240

210

180

150

120

90

60

30

0

(c)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


175

150

125

100

75

50

25

0

(d)






5 Conclusions





Data Availability


Disclosure




Acknowledgments


References



































RMSE 1

|Γ|1113944τisinΓ

yτ minus yτ1113872 1113873212

⎛⎝ ⎞⎠ (22)

RMSEi 1

Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944tisinTi

1113944jisinDt

1113944pisinPj

1113944τisinΓp

yτ minus yτ1113872 11138732⎛⎜⎝ ⎞⎟⎠

12

(23)

RMSEij 1

Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

tisinTij

1113944pisinPj

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(24)

RMSEijt 1

middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113944

pisinPijt

1113944τisinΓp

1113954yτ minus yτ( 11138572⎛⎜⎝ ⎞⎟⎠

12

(25)

RMSN RMSE

(1|Γ|)1113936τisinΓ1113954yτ (26)

RMSNi RMSEi

1 Ti

11138681113868111386811138681113868111386811138681113868 middot Dt

11138681113868111386811138681113868111386811138681113868 middot Pj

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Γp11138681113868111386811138681113868

111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTi1113936jisinDt


(27)

RMSNij RMSEij

1 Tij

11138681113868111386811138681113868

11138681113868111386811138681113868 middot Pt

11138681113868111386811138681113868111386811138681113868 middot ΓP

111386811138681113868111386811138681113868111386811138681113874 11138751113874 11138751113936tisinTij


(28)

0000

0928

00670000 0003 0003

00000100020003000400050006000700080009001000

10 20 30 40 50 60

Freq

uenc

y

Travel time (min)

(a)

0001

0981

0013 0002 0002 00010000

0200

0400

0600

0800

1000

1200

Freq

uenc

y

Travel time (min)

(b)


100

150

200

250

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Loss

(MSE

)

Iteration



RMSNijt RMSEijt

1 middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot ΓP1113868111386811138681113868


(29)







Avg 1545 279Max 5602 394Min 0431 192











010 20 30

40 010

2030

40


n statio

n

14121008060402

06

05

04

03

02

01

(a)

40

35

30

25

20

15

10

5

403530252015105

16

14

12

10

08

06

04

02

00

Origin station

Des

tinat

ion

staito

n

(b)




010

2030

40

Destinatio

n statio

n


175

150

125

100

075

050

025

000

403530252015100500

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

staito

n

42

36

30

24

18

12

06

001 2 3 4 5 6 7 8 9 10 11 12

Departure time

(b)


(500)000

500

1000

1500

2000

2500

3000

3500

4000

20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 2010710 720 730 740 750 800 810 820 830 840 850 900

Pass

enge

r num

ber






0

10

20

30

40

50

60

ndash5 5 15 25 35 45RM

SEOrigin station

TDHFNANN



40

35

30

25

20

15

10

5

403530252015105

Des

tinat

ion

stat

ion

Origin station

40

35

30

25

20

15

10

05

00

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


200

175

150

125

100

75

50

25

0

(b)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


240

210

180

150

120

90

60

30

0

(c)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


175

150

125

100

75

50

25

0

(d)






5 Conclusions





Data Availability


Disclosure




Acknowledgments


References



































RMSNijt RMSEijt

1 middot Pijt

11138681113868111386811138681113868

11138681113868111386811138681113868 middot ΓP1113868111386811138681113868


(29)







Avg 1545 279Max 5602 394Min 0431 192











010 20 30

40 010

2030

40


n statio

n

14121008060402

06

05

04

03

02

01

(a)

40

35

30

25

20

15

10

5

403530252015105

16

14

12

10

08

06

04

02

00

Origin station

Des

tinat

ion

staito

n

(b)




010

2030

40

Destinatio

n statio

n


175

150

125

100

075

050

025

000

403530252015100500

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

staito

n

42

36

30

24

18

12

06

001 2 3 4 5 6 7 8 9 10 11 12

Departure time

(b)


(500)000

500

1000

1500

2000

2500

3000

3500

4000

20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 2010710 720 730 740 750 800 810 820 830 840 850 900

Pass

enge

r num

ber






0

10

20

30

40

50

60

ndash5 5 15 25 35 45RM

SEOrigin station

TDHFNANN



40

35

30

25

20

15

10

5

403530252015105

Des

tinat

ion

stat

ion

Origin station

40

35

30

25

20

15

10

05

00

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


200

175

150

125

100

75

50

25

0

(b)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


240

210

180

150

120

90

60

30

0

(c)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


175

150

125

100

75

50

25

0

(d)






5 Conclusions





Data Availability


Disclosure




Acknowledgments


References












































010 20 30

40 010

2030

40


n statio

n

14121008060402

06

05

04

03

02

01

(a)

40

35

30

25

20

15

10

5

403530252015105

16

14

12

10

08

06

04

02

00

Origin station

Des

tinat

ion

staito

n

(b)




010

2030

40

Destinatio

n statio

n


175

150

125

100

075

050

025

000

403530252015100500

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

staito

n

42

36

30

24

18

12

06

001 2 3 4 5 6 7 8 9 10 11 12

Departure time

(b)


(500)000

500

1000

1500

2000

2500

3000

3500

4000

20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 2010710 720 730 740 750 800 810 820 830 840 850 900

Pass

enge

r num

ber






0

10

20

30

40

50

60

ndash5 5 15 25 35 45RM

SEOrigin station

TDHFNANN



40

35

30

25

20

15

10

5

403530252015105

Des

tinat

ion

stat

ion

Origin station

40

35

30

25

20

15

10

05

00

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


200

175

150

125

100

75

50

25

0

(b)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


240

210

180

150

120

90

60

30

0

(c)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


175

150

125

100

75

50

25

0

(d)






5 Conclusions





Data Availability


Disclosure




Acknowledgments


References




































010

2030

40

Destinatio

n statio

n


175

150

125

100

075

050

025

000

403530252015100500

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

staito

n

42

36

30

24

18

12

06

001 2 3 4 5 6 7 8 9 10 11 12

Departure time

(b)


(500)000

500

1000

1500

2000

2500

3000

3500

4000

20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 20 10 2010710 720 730 740 750 800 810 820 830 840 850 900

Pass

enge

r num

ber






0

10

20

30

40

50

60

ndash5 5 15 25 35 45RM

SEOrigin station

TDHFNANN



40

35

30

25

20

15

10

5

403530252015105

Des

tinat

ion

stat

ion

Origin station

40

35

30

25

20

15

10

05

00

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


200

175

150

125

100

75

50

25

0

(b)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


240

210

180

150

120

90

60

30

0

(c)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


175

150

125

100

75

50

25

0

(d)






5 Conclusions





Data Availability


Disclosure




Acknowledgments


References



































0

10

20

30

40

50

60

ndash5 5 15 25 35 45RM

SEOrigin station

TDHFNANN



40

35

30

25

20

15

10

5

403530252015105

Des

tinat

ion

stat

ion

Origin station

40

35

30

25

20

15

10

05

00

(a)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


200

175

150

125

100

75

50

25

0

(b)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


240

210

180

150

120

90

60

30

0

(c)

40

35

30

25

20

15

10

5

Des

tinat

ion

stat

ion


175

150

125

100

75

50

25

0

(d)






5 Conclusions





Data Availability


Disclosure




Acknowledgments


References






































5 Conclusions





Data Availability


Disclosure




Acknowledgments


References



































DynamicOrigin-DestinationMatrixEstimationBasedonUrban ...2020/03/30 · improves estimation...

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Transcript of DynamicOrigin-DestinationMatrixEstimationBasedonUrban ...2020/03/30 · improves estimation...