dynamic departure time estimation

8/17/2019 Dynamic Departure Time Estimation

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Available online at www.sciencedirect.com

ScienceDirect

Transportation Research Procedia 00 (2017) 000–000

www.elsevier.com/locate/pr

ocedia

World Conerence on Transport Research ! WCTR 201" #han$hai. 10!1% &'l 201"

namic epart're Time *stimation

#'datta +ohanta, -leei Podno'hova

aUniversity of California, Berkeley, CA, USA 94704

Abstract

Traic vol'mes and con$estion patterns on a networ are ver sensitive to the distri'tion o depart're times. n activit!ased

travel demand modelin$, the assi$nment o depart're times o a$ents rom vario's activit locations 's'all relies on the travel

s'rves and accessiilit he'ristics and is ver imprecise. t leads to hi$h variailit o the modeled traic vol'mes, whether in

a$ent!ased sim'lation or n'merical T- solvers, and translates into an inailit to relial identi the so'rce o errors in

demand models. This wor miti$ates this 'ncertaint inte$ratin$ several emer$in$ so'rces o travel data into a novel

n'merical optimiation ramewor or depart're time inerence. We etend an *ntrop +aimiation approach (&anson et al.,

1332) that ass'mes a ied lin 'se proailit matri and solves or depart're times distri'tions rom each one 'sin$ oserved

lin vol'me co'nts. This method is etended in two respects. 4irst, we 'se a lar$e sample o real!time moilit traces rom a

cell'lar comm'nication networ lo$s to otain lower o'nd constraints or oserved depart'res. #econd, we ass'me thatdepart're time choices are increasin$l $overned online -dvanced Traveler normation #stems and incorporate travel times

availale rom online -Ps (5oo$le, 6oia, nri) into the desi$n o the lin 'se matri. #ince this matri relies in t'rn on the

ro'tin$, we appl the conve optimiation ramewor developed W' et al. (201%) to iner ro'te lows as a s'!ro'tine in

travel time inerence. We then introd'ce the optimiation prolem orm'lations to comp'te depart're time and solve the prolem

'sin$ a primal!d'al interior!point al$orithm with a ilter line!search method proposed Wachter et al. (200"). 4inall, we

dnamicall 'pdate the estimates thro'$h atch 5radient escent al$orithm which allows the ramewor to e 'sed in -dvanced

Traic +ana$ement, in a scenario when the aove mentioned data is availale as a live stream. The ramewor is tested and

validated on a detailed a$ent!ased micro!sim'lation calirated rom the cell phone data or a simpliied reewa networ

spannin$ over nine co'nties o the #an 4rancisco a -rea.

8 2017 The -'thors. P'lished *lsevier .9.

Peer!review 'nder responsiilit o W:R; C:64*R*6C* :6 TR-6#P:RT R*#*-RC< #:C*T=.

Keywords: *ntrop +aimiation, ;in!ro'te incidence matri, ;in!'se proailit matri, namic ro'te low estimation, atch 5radient

escent

221>!2>1? 8 2017 The -'thors. P'lished *lsevier .9.

Peer!review 'nder responsiilit o W:R; C:64*R*6C* :6 TR-6#P:RT R*#*-RC< #:C*T=.

http://www.sciencedirect.com/science/journal/22107843http://www.sciencedirect.com/science/journal/22107843http://www.sciencedirect.com/science/journal/22107843


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+ohant et al./ Transportation Research Procedia 00 (2017) 000–000 2

1. Introduction

1.1. Backro!nd and "otivation

t has een well estalished that traic vol'mes and con$estion patterns on a networ are hi$hl dependent on

depart're times and ro'te choice decisions o a$ents (A -rnott et al., 1330A Care et al.,

133BA hat, 133@A


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+ohant et al./ Transportation Research Procedia 00 (2017) 000–000 B

incorrect co'plin$ etween activities. -n eample o incorrect co'plin$ was provided almer et al. (200%)

wherein a$ents ma e assi$ned depart're time rom a destination even eore their act'al arrival. To prevent s'ch

scenarios, it is essential to ollow an aent*+ased a%%roac (i.e. tracin$ each individ'alFs activit chain d'rin$

sim'lation). To develop acc'rate and realistic activit chain or each individ'al, it is important to have etremel

ine!$rained depart're choice inormation. The dnamic nat're o the c'rrent ramewor allows to modi dail

activit plans and achieve more acc'rate res'lts.

1.1.$. &yna'ic Control Strateies

Transportation networ control strate$ies to miti$ate con$estion ma incl'de sta$$ered wor ho'rs (


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> +ohant et al./ Transportation Research Procedia 00 (2017) 000–000

(Peeta et al., 2001). There is a need or a more ro'st proced're to determine the lin!'se proailit matri.

#econdl, the orm'lation ass'mes that the ro'te choice decision is made at the time o depart're and there is no

chan$e or 'pdate to that decision. While this mi$ht e tr'e or short or re'entl travelled trips, it ma not hold tr'e

or an other ind o trip. Thereore, there is a need to 'pdate the ro'te choice dnamicall. - third possile

improvement to this orm'lation is to $et a etter lower o'nd on rd 'sin$ real!time availale moilit trace data.

1.4. &yna'ic &e%art!re (i'e )sti'ation or'!lation

1.1.-. artly 2+served &e%art!res Constraints

The standard entrop maimiation prolem is modiied to incl'de real!time inormation and overcome

drawacs o the traditional orm'lation. 4irst, moilit trace data rom each one is 'tilied to $enerate partl

oserved depart'res rom each one which provide lower o'nds on rd. The n'mer o trips departin$ rom a one

r in a partic'lar time interval d is $reater than or e'al to the dierence o the n'mer o oserved individ'als in

one r at time interval (d!1) and the n'mer o oserved individ'als at time interval d (e'ation 2).

q ≥max N − − N ,0 ,∀ r∈Z , d∈ D (2)

where,

rd G n'mer o trips departin$ rom one r to all destinations in time interval d,

6rd G n'mer o oserved individ'als at one r d'rin$ time interval d rom moilit trace data

1.1.4. 3ink Co!nts and artly 2+served ay%oints constraints

6et, cell'lar data driven conve optimiation al$orithm or ro'te low estimation 'sin$ partl oserved

wapoints proposed W' et al. (201%) is applied to $et proailit or a trip ori$inatin$ rom one r to 'se ro'te rH

(e'ation B). This al$orithm tries to minimie the s'ared error etween oserved lin co'nts in each time in vd

and predicted lin co'nts 'sin$ :ri$in!estination (:!) lows rHd and the lin!ro'te incidence matri -rHd

At any d∈ D −

¿

¿

¿cr∗¿

dq

r∗¿d }

− c

r∗∈Z ∗¿¿

{∑¿¿

r∗¿d ≥0,∀ r∈Z , d∈ Dq¿ (B)

whereI

-rHd G1 if ink k ies∈r!uter∗attimed,

rHd G n'mer o trips departin$ alon$ ro'te rH in time interval d

vd G oserved vol'me on lin d'rin$ time interval d

JcrHd G1 if cepat" c ies∈r!ute r∗at time d ,

cd G oserved cellpath vol'me on cellpath c d'rin$ time interval d


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+ohant et al./ Transportation Research Procedia 00 (2017) 000–000 %

6ow, the traditional T- sol'tion to the lin!'se proailit matri can e modiied as shown in e'ations

(e'ation >) and (e'ation %). 4irst, the ro'te choice proailities prrH are calc'lated as a ratio o predicted :!

trips rHd and all trips ori$inatin$ rom that one (e'ation >).r∗#it"!ri$in r

qr∗¿

d

(>)

whereI

prrH G proailit that a trip with ori$in r is alon$ ro'te rH

rHd G n'mer o trips departin$ alon$ ro'te rH in time interval d (derived rom sol'tion to e'ation B)

6et, this inormation is comined with online travel time -Ps (5oo$le, 6oia, nri) to otaina ro'st and

dnamic estimate or the lin!'se proailit matri or a trip (e'ation %)

pr¿

kr∗¿d A¿ (%)

whereI prtd G proailit that a trip departin$ one r in time interval d 'ses lin in time interval t

prrH G proailit that a trip with ori$in r is alon$ ro'te rH

prHtd G proailit that a trip alon$ ro'te rH in time interval d 'ses lin in time interval t

-rHd G1 if ink k ies∈r!uter∗attimed,

-rHtd G1 if ink k istraversed durin$ timeinterva t f!r a trip a!n$r∗departin$at d ,

1.1.5. Co'+ined 2%ti'i/ation ra'ework

The comined optimiation ramewor incl'din$ partl oserved depart'res constraints, partl oserved

wapoints and oserved lin co'nt constraints is descried in e'ation ("). nterior point optimiation or arrier

method is proposed or this prolem. The e'alit constraint in (e'ation 1) is moved to the cost 'nction with

;a$ran$e m'ltiplier scale parameter Kt and the lower o'nd constraint is added as a lo$arithmic arrier 'nction

(e'ation "). #ince each s'mmation term in the optimiation prolem is conve in nat're, the comined

optimiation prolem is also conve.

¿{ %k

t ¿

qrd

≥0, ∀ r∈Z , d∈ D (")

whereI

rd G n'mer o trips departin$ rom one r to all destinations in time interval d,

Kt G relative cost parameter or lin constraint or lin at time interval t (depends on the amo'nt and acc'rac o

lin co'nt inormation availale as well as the acc'rac o the lin!'se proailit matri)

vt G oserved vol'me on lin d'rin$ time interval t

prtd G proailit that a trip departin$ one r in time interval d 'ses lin in time interval t (calc'lated rom

e'ation %)

L(rd) G !lo$(rd – ma(6r(d!1) – 6rd, 0))

6rd G n'mer o oserved individ'als at one r d'rin$ time interval d rom moilit trace data


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" +ohant et al./ Transportation Research Procedia 00 (2017) 000–000

The proposed al$orithm or solvin$ the optimiation prolem descried in e'ation (MreNe"O) is a primal!d'al

interior!point ilter line!search al$orithm Wachter et al. (200").

1.1.6. ncre'ental )sti'ation and rediction

With more incomin$ data, it ecomes possile to predict the n'mer o depart'res rom a $iven one moreacc'ratel. Thereore, the inal step in the dnamic depart're time estimation ramewor is to dnamicall 'pdate

the optimal sol'tion with incomin$ data. There are two proposed al$orithms or incremental estimationI

• Jsin$ the primal!d'al interior!point ilter line!search al$orithm (Wachter et al, 200") mentioned in the

previo's section.

• Jsin$ atch 5radient escent – The initial easile sol'tion or the $radient descent al$orithm can e

chosen as Nma(6r(d!1) – 6rd, 0)1O. The step sie ma e chosen empiricall or derived rom line

search al$orithm. The conver$ence criteria proposed is checin$ the norm o matri rd as shown in

e'ation (7).

√∑ ∑ {qrd(k )−qrd(k −1)}2


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45JR* 1 #impliied reewa networ and a$$re$ated T-Ds representin$ the #an 4rancisco a -rea or testin$ and validatin$ the namic

epart're Time *stimation 4ramewor

To test the ramewor, the estimated depart're times m'st e compared to the act'al depart're times in the

sim'lation. The act'al depart're times d'rin$ the sim'lation are etracted rom the )vents ile, which is $enerated

as a standard +-T#im sim'lation o'tp't, 'sin$ an Aent &e%art!re )vent8andler (Waraich, 2003). To orm'late

the optimiation ramewor descried in e'ation ("), the two re'ired pieces o inormation are lin co'nts in

small time ins d (% min'tes or this scenario) and partl oserved depart'res rom each one. The lin co'nt

inormation rom the sim'lation is determined rom the )vents ile 'sin$ a 3ink 3eave )vent8andler (Waraich,

2003). The partl oserved depart're co'nts depend on two main actors. 4irstl, it depends on the percenta$e o

pop'lation or which moilit data, or eample cell!phone Call ata Records (CRs) data, is availale. #econdl,

it depends on the n'mer o arrivals occ'rrin$ in the partic'lar one. Thereore the n'mer o oserved depart'res

rom each one in the correspondin$ time ins are sim'lated as random n'mer etween ero and SS times the act'al

n'mer o depart'res, where represents the percenta$e o pop'lation or which moilit trace data is availale.

The optimiation step is perormed 'sin$ a +-T;- etension or optimiation sotware P:PT (nterior!Point

:ptimier) which implements a primal!d'al interior!point al$orithm with a ilter line!search method proposed

Wachter et al. (200"). P:PT is an open!so'rce sotware paca$e or lar$e!scale non!linear optimiation developed

as part o C:6!:R (The Comp'tational nrastr'ct're or :perations Research) initiative. To the nowled$e o thea'thors, this is the astest open!so'rce paca$e or lar$e!scale non!linear optimiation. 4or comp'tational ease, it

was ass'med that all trips made were within 1 ho'r in d'ration. This ass'mption was E'stiied since even ater

considerin$ pea!ho'r velocities on all lins, 30 o all :ri$in!estination (:!) cominations co'ld e covered

within 1 ho'r.

The incremental 'pdate step in the ramewor was perormed 'sin$ oth primal!d'al interior!point ilter line!

search al$orithm (Wachter, 200") and atch 5radient escent. (4or implementation o the $radient descent

al$orithm 'sin$ Pthon 2.7 with appropriate n'merical liraries, see httpsI//$ith'.com/'c!smartcities/T'torials!

5eneral!no/tree/master/6etwor!-nalsis!T'torials). The step sie or $radient descent al$orithm is empiricall

chosen as a constant val'e o 0.02. The precision parameter in e'ation (7) Q is empiricall chosen as 0.01.


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@ +ohant et al./ Transportation Research Procedia 00 (2017) 000–000

The interior!point al$orithm prod'ced m'ch more acc'rate res'lts (R+#;* val'e o 0.>73 as compared to 0."@B)

't the $radient descent al$orithm conver$ed aro'nd 10 times aster (10 seconds on avera$e as compared to 2

min'tes on avera$e).

1.6. Cali+ration of kt

The irst step in the proposed dnamic depart're time estimation proced're is caliration o the relative cost parameter Kt in e'ation ("). This parameter depends on the amo'nt and acc'rac o lin!co'nt inormation

availale in the networ. To perorm caliration o the parameter, the overall root mean s'ared lo$arithmic error

(R+#;*) is minimied or varin$ val'es o Kt or %!min'te depart're co'nt predictions over one ho'r rom %!

"-+ (e'ation @).

∨ ∨r ∈Z d∈ D

og qr + − og ar + (@)

whereI

rd G estimated n'mer o trips departin$ rom one r to all destinations in time interval d

ard G act'al n'mer o trips departin$ rom one r to all destinations in time interval d

The plot or R+#;* with several val'es o lo$(Kt) is shown or %!min'te depart're co'nts in a 1 ho'r period

etween %-+!"-+. (4i$'re 2). This plot was repeated or oth mornin$ and evenin$ peas d'rin$ the da with

similar trends. This clearl shows that the overall error is minimied when Kt e'als 1.0. ho'rs is shown in i$'re (4i$'re B) or three casesI

• -ct'al n'mer o depart'res

• *stimated n'mer o depart'res with no partl oserved depart'res inormation

• *stimated n'mer o depart'res with partl oserved depart'res constraints derived rom moilit trace

inormation (s'ch as CRs)


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45JR* B Plot showin$ c'm'lative n'mer o depart'res in %!min'te intervals across all ones with time o da or 1) -ct'al n'mer o

depart'res, 2) *stimated n'mer o depart'res with no partl oserved depart'res inormation and B) *stimated n'mer o depart'res with partl

oserved depart'res constraints derived rom moilit trace inormation (s'ch as CRs)

4i$'re B $ives preliminar indication that the c'm'lative n'mer o depart'res are predicted m'ch etter with the

addition o partl oserved depart're constraints 'sin$ moilit traces (lie CRs).

The asol'te error in the c'm'lative n'mer o depart'res across all ones is displaed in i$'re (4i$'re >) or

two casesI

• With no partl oserved depart'res inormation

• With partl oserved depart'res constraints derived rom moilit trace inormation (s'ch as CRs)

1.;. )ffect of artly 2+served &e%art!res Constraints

Partl oserved depart'res thro'$h moilit trace data (lie cell phone CRs) provide lower o'nd constraints

on the val'es o rd. To test the eect o partl oserved depart'res constraints on the optimal sol'tion, the moilit

trace data covera$e was varied rom 0 to 100 in intervals o 10. Then, the n'mer o oserved depart'res or

each one UrS and time in SdS was randoml selected etween ero and the n'mer o act'al depart'res within

covera$e. 4i$'re (%) shows the plot o R+#;* val'es or varin$ cell!phone data covera$e in the pop'lation 'sin$

optimal Kt o 1. :verall, the optimal sol'tions witho't an constraints lie within B.@ times o the act'al sol'tions.

This error red'ces drasticall to 1.@ times the act'al sol'tion with 100 moilit trace data covera$e. This shows

that partl oserved depart'res provide a m'ch ti$hter o'nd on the estimated val'es o rd.


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10 +ohant et al./ Transportation Research Procedia 00 (2017) 000–000

45JR* > Plot showin$ the asol'te error in networ!wide c'm'lative depart'res with time o da or 1) *stimated depart'res with Partl

:server epart'res Constraints 'sin$ Call ata Records (CRs) and 2) *stimated depart'res witho't Partl :served epart'res

45JR* % Plot o Root +ean #'ared ;o$arithmic *rror (R+#;*) vs o Cell Phone Covera$e or n'mer o depart'res etween %!" -+

1.9. )ffect of ncre'ental )sti'ation

-n important analsis in this st'd is to see the evol'tion error in the n'mer o depart'res with more incomin$

streamin$ data rom vario's so'rces. -s more lin!co'nt and moilit trace inormation is attained with time, more

acc'rate predictions can e made ao't trips departin$ 'p to 1 ho'r prior to the inormation. This allows constant

'pdate o the rd val'es over twelve %!min'te ins (4i$'re ").


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45JR* " Plot o evol'tion o Root +ean #'ared ;o$arithmic *rror (R+#;*) in the n'mer o depart'res etween %I%%!"I00 -+ with

incomin$ inormation in % min'te intervals rom "I00!"I%% -+ or 1) Done 2@, 2) Done 2, B) Done 1% and >) -ll ones

-s hpothesied, the plots show that there is a $eneral ne$ative trend in the R+#;* val'es as we otain more

in!co'nt and moilit trace inormation over time. 0!70).

The a'thors claim that with a s'icient selection o cells, ro'te ow estimation ma e possile witho't an other

inds o sensor data. n case the data is nois, it was shown that c'ratin$ 20!%0 ro'tes (per :) was s'icient or

achievin$ a low (X1%) ro'te ow error. This proves that oth asence o data and presence o noise in data have

reasonal low eect i the de$rees o reedom are ept airl low and i there is s'icient ine $rained data rom

other so'rces lie cell paths.


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12 +ohant et al./ Transportation Research Procedia 00 (2017) 000–000

To test the eect o noise on the predicted depart'res, two scenarios were introd'ced ! a normall distri'ted

error term, with mean 0 and standard deviation 0.1 and 1 respectivel, was added to each prrH term. The comparison

o root mean s'ared lo$arithmic error (R+#;*) val'es, oth with and witho't partiall oserved depart're

constraints, is shown in tale 1.

T-;* 1 Tale comparin$ eect o error in prrH on Root +ean #'are ;o$arithmic *rror (R+#;*) with and

witho't partiall oserved depart'res constraintsartially obser!ed departures "tandard error #M"$E

=es 0 0.>73

=es 0.1 0.>3%

=es 1 0.>3"

6o 0 1.0@

6o 0.1 1.B"

6o 1 1.%0

-s shown Tale 1, there is ver low increase in R+#;* val'es etween scenarios with standard error 0.1 and

1 when partl oserved depart'res constraints are added. This 'rther corroorates the act that availailit o ine

$rained data rom other so'rces allows to eep eect o noise in prrH val'es on predicted depart'res airl low.

1.1.;. )rror in %kr73

=es 0.1 0.>@B

=es 1 0.>3B

6o 0 1.0@

6o 0.1 1.1B

6o 1 1.2@

Jnlie the case with error in prrH val'es, the R+#;* val'es etween scenarios with standard error 0.1 and 1 are

si$niicant, oth with and witho't partiall oserved depart'res constraints.


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+ohant et al./ Transportation Research Procedia 00 (2017) 000–000 1B

1.11. "!lti*'odal )tension

The proposed ramewor ass'mes that all trips are 'nimodal and there is complete lin traic co'nt and travel!

time inormation or the mode over all lins in the networ in real!time. Jnder the validit o these ass'mptions, the

proposed dnamic depart're time estimation ramewor ma e etended to a mit're o m'ltiple modes in the

networ (as show in e'ation 3).

¿{ %k

t (m)¿

qrd(m)≥0,∀ r∈Z , d∈ D

(3)

whereI

rd(m) G n'mer o trips departin$ rom one r to all destinations in time interval d mode m,

Kt(m) G relative cost parameter or lin constraint or lin at time interval t or mode m (depends on the amo'nt

and acc'rac o lin co'nt inormation availale as well as the acc'rac o the lin!'se proailit matri)

vt G oserved vol'me on lin d'rin$ time interval t mode m prtd G proailit that a trip departin$ one r in time interval d 'ses lin in time interval t or mode m (calc'lated

rom e'ation %)

L(rd(m)) G !lo$(rd(m) – ma(6r(d!1)(m) – 6rd(m), 0))

6rd(m) G n'mer o oserved individ'als at one r d'rin$ time interval d mode m rom moilit trace data

1.1$. Scala+ility of Alorit'

To test the scalailit o the dnamic depart're time al$orithm, it was tested on the 'll scale networ o the nine

co'nties in the a -rea. The networ contains %">,B"7 lins and B%2,011 nodes (4i$'re 7). The ones 'sed or this

implementation are 1>%> Traic -nalsis Dones (T-Ds) deined the +etropolitan Transportation Commission(+TC) in 2002. ;in co'nts or ever %!min'te interval are availale or 10>% o't o the %">,B"7 lins thro'$h

sensors placed on certain reewas the Caltrans Perormance +eas'rement #stem (Pe+#).

The methodolo$ emploed or dnamic depart're time estimation on the 'll!scale networ was the same as that

or the simpliied reewa networ with the ollowin$ eceptionsI

• *traction o lin!ro'te incidence matri thro'$h sim'lation!ased methods ! #ince the previo's

shortest path ass'mption is no lon$er valid, lin!ro'te incidence matri was etracted tracin$ a$ent

ro'tes d'rin$ +-T#im sim'lation.

• :ptimiation al$orithm ! The interior!point al$orithm 'sin$ +-T;- interace o optimiation tool P!

:PT (Wachter, 200") was o'nd to e too slow or dealin$ with s'ch a lar$e!scale optimiation prolem.

Thereore, the $radient descent al$orithm was emploed to solve the optimiation prolem.

Considerations were made or the lar$e sie o the lin!'se proailit matri prtd $iven memor

constraints. (The implementation o the $radient descent al$orithm 'sin$ Pthon 2.7 or the 'll scale

networ can e o'nd at httpsI//$ith'.com/'c!smartcities/T'torials!5eneral!no/tree/master/6etwor!

-nalsis!T'torials)

https://github.com/ucb-smartcities/Tutorials-General-Info/tree/master/Network-Analysis-Tutorialshttps://github.com/ucb-smartcities/Tutorials-General-Info/tree/master/Network-Analysis-Tutorialshttps://github.com/ucb-smartcities/Tutorials-General-Info/tree/master/Network-Analysis-Tutorialshttps://github.com/ucb-smartcities/Tutorials-General-Info/tree/master/Network-Analysis-Tutorials


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1> +ohant et al./ Transportation Research Procedia 00 (2017) 000–000

45JR* 7 4'll scale networ representin$ the #an

4rancisco a -rea or testin$ scalailit o namic epart're Time 4ramewor

The res'lts or the 'll!scale implementation o the dnamic depart're time estimation al$orithm can e

s'mmaried as ollowsI

• Calc'latin$ R+#* instead o R+#;* ! #ince the act'al n'mer o depart'res in %!min ins rom each

o the 1>%> ones are m'ch ewer than the previo's %> s'per!districts, the asol'te val'es o error terms

are also low. %> ones the

optimal val'e o Kt was o'nd to e 10 (4i$'re @). This maes sense eca'se there is more spatial

variation in n'mer o depart'res now which is capt'red lin co'nts. Thereore, more wei$ht sho'ld

e $iven to lin co'nt constraints.O

• 9al'e o R+#* ! The optimal R+#* val'e was aro'nd 1.%%. This means that, on avera$e, we are o

rom the act'al co'nts aro'nd 1.%%. This is eca'se we have lin co'nts or onl 10>% o't o %">,000

lins. With hi$her sensor covera$e, this error can e red'ced.

%. &onclusions

This paper proposes a data!driven al$orithm or estimatin$ depart're times dnamicall on a traic networ. t

etends previo's research &anson et al (1332) proposin$ an *ntrop +aimiation approach to iner depart're

times rom lin co'nts. t also incorporates partiall oserved depart'res thro'$h so'rces lie Call ata Records

(CRs) as lower o'nd constraints and partiall oserved wapoints to iner ro'te choice proailities (W' et al,

201%). The entire prolem is orm'lated as a conve optimiation prolem (e'ation ") which can e solved a

primal!d'al interior!point al$orithm with a ilter line!search method proposed Wachter et al (200"). 4inall, atch

$radient descent approach is adopted to solve the prolem in real!time with incomin$ data which allows the

ramewor to e adopted or -dvanced Traic +ana$ement, in a scenario when data is availale as a live stream.

The s'$$ested al$orithm is emploed to predict depart're times in a sim'lated scenario representin$ a simpliied

reewa networ or the nine co'nties o the #an 4rancisco a -rea on a$ent!ased traic sim'lation sotware

+-T#im. The parameters o the comined optimiation prolem are calirated minimiin$ the Root!+ean

#'ared ;o$arithmic *rror (R+#;*) in n'mer o depart'res across all ones or a partic'lar time interval. Res'lts

show that incl'sion o partiall oserved depart'res and hi$her data covera$e $reatl red'ces the asol'te error in

n'mer o estimated depart'res across all ones. ncremental estimation allows 's to minimie error with incomin$

data. t was shown that error in meas'rement o :ri$in!estination (:!) proailit matri can e overcome

thro'$h availailit o ine $rained data rom other data so'rces while errors in meas'rement o lin!ro'te incidence

matri can e red'ced thro'$h 'se o sim'lation!ased techni'es. ;astl, scalailit o the al$orithm was proved


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+ohant et al./ Transportation Research Procedia 00 (2017) 000–000 1%

or a 'll!scale networ representin$ the nine co'nties o the a -rea and the modiications re'ired or the same

were disc'ssed.

This al$orithm co'ld prove 'se'l in $eneratin$ dnamic demand or a$ent!ased traic sim'lations as well as

proposin$ dnamic control strate$ies or miti$atin$ con$estion on a traic networ.

Ac'no(ledgement

The a'thors wo'ld lie to acnowled$e the research assistance provided *min -raelanian, +adeline#heehan and +o$en$ =in. We wo'ld also lie to than -ndrew Campell, #id 4e$in and anin$ Dhan$ or their s'pport and comments on the research.

#e)erences

r'ce 6 &anson and 4ran #o'thworth. Y*stimatin$ depart're times rom traic co'nts 'sin$ dnamic assi$nmentZ. nI Transportation

Research Part I +ethodolo$ical 2".1 (1332), pp. B!1".Cath W' et al. MCellpathI 4'sion o cell'lar and traic sensor data or ro'te low estimation via conve optimiationZ. nI Transportation

Research Part CI *mer$in$ Technolo$ies (201%).-ndreas Wachter and ;oren T ie$l. r. Y:n the implementation o an interior!point ilter ine!search al$orithm or lar$e!scale nonlinear

pro$rammin$Z. nI +athematical pro$rammin$ 10".1 (200"), pp. 2%!%7.Chris ), pp. 2%!B".

Richard -rnott, -ndre de Palma, and Roin ;indse. Yepart're time and ro'te choice or the mornin$ comm'teZ. nI Transportation ResearchPart I +ethodolo$ical 2>.B (1330), pp. 203!22@.

+alach Care and -sho #rinivasan. Y*ternalities, avera$e and mar$inal costs, and tolls on con$ested networs with time!varin$ lowsZ. nI:perations Research >1.1 (133B), pp. 217!2B1.

Chandra R hat. Y-nalsis o travel mode and depart're time choice or 'ran shoppin$ tripsZ. nI Transportation Research Part I+ethodolo$ical B2." (133@), pp. B"1!B71.

".onald P. 5aver. Y ) 3001" ! ?. 'rlI http I / /www . sciencedirect . com / science / article / pii /01312"1%@>3001"?.

%7.-. de Palma, C. ;eere, and +. en!-iva. Y- dnamic model o pea period traic lows and delas in a corridorZ. nI Comp'ters

+athematics with -pplications 1>.B (13@7), pp. 201!22B. issnI 0@3@!1221. doiI httpI//d.doi.or$/10.101"/0@3@!1221(@7)301%B!2. 'rlIhttpI//www.sciencedirect.com/science/article/pii/0@3@1221@7301%B2.

eepa [ +erchant and 5eor$e ; 6emha'ser. Y- model and an al$orithm or the dnamic traic assi$nment prolemsZ. nI Transportationscience 12.B (137@), pp. 1@B!133.

r'ce 6 &anson. Mnamic traic assi$nment or 'ran road networsZ. nI Transportation Research Part I +ethodolo$ical 2%.2 (1331), pp.

1>B!1"1.#rinivas Peeta and -thanasios [ Diliasopo'los. 4o'ndations o dnamic traic assi$nmentI The past, the present and the 't'reZ. nI

http://dx.doi.org/10.1016/0191-2615(83)90047-4http://dx.doi.org/10.1016/0191-2615(83)90047-4http://dx.doi.org/10.1016/0191-2615(86)90045-7http://dx.doi.org/10.1016/0191-2615(86)90045-7http://www.sciencedirect.com/science/article/pii/0191261586900457http://www.sciencedirect.com/science/article/pii/0191261586900457http://dx.doi.org/10.1016/0898-1221(87)90153-2http://dx.doi.org/10.1016/0898-1221(87)90153-2http://dx.doi.org/10.1016/0191-2615(83)90047-4http://dx.doi.org/10.1016/0191-2615(86)90045-7http://www.sciencedirect.com/science/article/pii/0191261586900457http://dx.doi.org/10.1016/0898-1221(87)90153-2


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dynamic departure time estimation

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