Resea ch A icle
T a ic Managemen as a Se ice: The T a ic Flow Pa e n
Classi ica ion P oblem
Ca los T. Cala a e,1Da id Sole ,2Juan-Ca los Cano,1and Pie o Manzoni1
1Depa men o Compu e Enginee ing (DISCA), Uni e si a Poli `
ecnica de Val`
encia, 46022 Valencia, Spain
2Ins i u e o Pu e and Applied Ma hema ics (IUMPA), Uni e si a Poli `
ecnica de Val`
encia, 46022 Valencia, Spain
Co espondence should be add essed o Ca los T. Cala a e; cala[email p o ec ed] .es
Recei ed 30 July 2015; Re ised 22 Sep embe 2015; Accep ed 27 Sep embe 2015
Academic Edi o : Shengbo Eben Li
Copy igh © 2015 Ca los T. Cala a e e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion
License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly
ci ed.
In elligen T anspo a ion Sys em (ITS) echnologies can be implemen ed o educe bo h uel consump ion and he associa ed
emission o g eenhouse gases. Howe e , such sys ems equi e in elligen and e ec i e ou e planning solu ions o educe a el ime
and p omo e s able a eling speeds. To achie e such goal hese sys ems should accoun o bo h es ima ed and eal- ime a ic
conges ion s a es, bu ob aining eliable a ic conges ion es ima ions o all he s ee s/a enues in a ci y o he di e en imes o he
day, o e e y day in a yea , is a complex ask. Modeling such a emendous amoun o da a can be ime-consuming and, addi ionally,
cen alized compu a ion o op imal ou es based on such ime-dependencies has e y high da a p ocessing equi emen s. In his
pape we app oach his p oblem h ough a heu is ic o conside ably educe he modeling e o while main aining he bene i s o
ime-dependen a ic conges ion modeling. In pa icula , we p opose g ouping s ee s by aking in o accoun eal aces desc ibing
he daily a ic pa e n. The e ec i eness o his heu is ic is assessed o he ci y o Valencia, Spain, and he esul s ob ained show
ha i is possible o educe he equi ed numbe o daily a ic low pa e ns by a ac o o 4210 while main aining he essence o
ime-dependen modeling equi emen s.
1. In oduc ion
In densely popula ed u ban a eas, a ic- ela ed p oblems,
such as ai quali y, noise, ib a ion, and acciden s, a e c i ical
issues o managemen au ho i ies. In e ms o solu ions o
make a ic low mo e e icien o o educe i , especially in
down owns, au ho i ies de elop ini ia i es o p omo e he use
o public anspo a ion, o bid access o he mos pollu ing
ehicles, al e na e he days o down own access acco ding o
he ehicles’ pla e numbe , cha ge d i e s o access, and so
o h. In addi ion o hese ini ia i es, a ic enginee s analyze
he a ic low in ou ci ies aking in o accoun impo an
ac o s like he adequa e s ee di ec ions o minimize a el
imes, in luence o a ic ligh s synch oniza ion and place-
men in a ic conges ion, uel consump ion and CO2emis-
sions, a ic noise modeling [1–6], and so o h.
Pa icula ly, in he ield o uel consump ion and exhaus
pollu an , In elligen T anspo a ion Sys ems (ITS) ha e
ecen ly eme ged as a powe ul ally in o de o imp o e a ic
lows [7]. Mo eo e , he massi e adop ion o sma phones
and he e e inc easing e o s o achie e sma phone- ehicle
in eg a ion [8, 9] pa e he way owa ds no el a ic manage-
men solu ions whe e eal- ime in e ac ion be ween d i e s
and a ic managemen au ho i ies becomes possible. Such
in e ac ion p o ides mu ual bene i s since a ic au ho i ies
a eable oha e eal- ime eedbackabou a icconges ion
s a es a di e en pa s o a ci y, while d i e s a e also able o
ha e mo e in o ma ion, aiding hem in he decision p ocess
o inding he op imal ou e.
In his pape we p esen a no el pla o m o cen alized
a ic managemen in u ban en i onmen s which a emp s
o a oid known p oblems associa ed wi h cu en ou e
planning solu ions based on ixed pa h cos s. The p oposed
solu ion akes in o accoun he his o ical da a abou a ic
pa e ns in o de o p o ide ime-dependen ou e ecom-
menda ions o d i e s a eling h ough dense a ic a eas.
As a i s app oach o deploy his solu ion, we p opose using
exis ing a ic measu emen s based on induc ion loop de ec-
ions [10] in o de o ob ain all he equi ed ime-dependen
a ic lowmodels.We ocuson hespeci iccaseo heci y
Hindawi Publishing Co po a ion
Ma hema ical P oblems in Enginee ing
Volume 2015, A icle ID 716598, 14 pages
h p://dx.doi.o g/10.1155/2015/716598
2Ma hema ical P oblems in Enginee ing
o Valencia,Spain, ogain u he insigh in o hep oblem.
Based on he esul s ob ained, we p opose a heu is ic o
add ess he p oblem e icien ly by g ouping elemen s wi h
a simila beha io , and we assess he e ec i eness o he
p oposed heu is ic in e ms o he numbe o in e pola ion
unc ions equi ed. We show ha i is possible o educe he
equi ed numbe o in e pola ions unc ions desc ibing daily
a ic pa e ns by a ac o o 4210, which signi ican ly educes
he p oblem complexi y.
The pape is o ganized as ollows: in he nex sec ion we
in oduce some ela ed wo ks. In Sec ion 3 we p esen he
p oposed a ic managemen pla o m. Sec ion 4 desc ibes
he ime-dependen a ic analysis p oblem and p o ides an
o e iew o he a ic pa e ns o he ci y o Valencia, Spain.
Sec ion 5 desc ibes he selec ed heu is ic o he modeling
p oblem, along wi h he esul s achie ed. Sec ion 6 hen p e-
sen s he o e all agg ega ion gains, de ailing he o igin o
hose gains. Finally, in Sec ion 7 we conclude he pape .
2. Rela ed Wo ks
A e se e al decades o esea ch, he exis ing a ic engi-
nee ing li e a u e is qui e b oad and ex ensi e. Recen ly, some
solu ions ha e eme ged ha ely on mobile de ices o mon-
i o he a ic in eal ime, o example, he Mobile Millen-
nium [11] p ojec . Such in o ma ion can be used o admin-
is a i e pu poses, o example, o isually analyze he a ic
condi ions, bu , in addi ion, i can also be use ul o op imize
he ou es aken by ehicles, as shown analy ically by Kim
e al. [12].
Among hese p oposals we can ind T a icView [13],
which de ines a amewo k o ga he and dissemina e in o -
ma ion abou he ehicles on he oad. Wi h such a sys em,
d i e s will be p o ided wi h oad a ic in o ma ion ha
helps d i ing in ad e se si ua ions such as oggy wea he o
inding an op imal ou e in a long ip. Wo k and Bayen [14]
highligh he po en ial o mobile de ices o p o ide eal- ime
a ic in o ma ion o he en i e anspo a ion ne wo k, p o-
iding some case s udies. Claudel e al. [15] emphasize how
mobile de ices may allow ob aining mo e eliable es ima ions
abou he ime equi ed o a e se speci ic ou es. Leon iadis
e al. [16] p opose an oppo unis ic a ic managemen sys-
em whe e ehicles sha e a ic in o ma ion in an ad hoc
manne , allowing hem o dynamically e ou e based on
indi idually collec ed a ic in o ma ion. Recen ly, solu ions
such as EcoT ec [2] in oduced a VANET-based eco iendly
ou ing algo i hm o ehicula a ic which conside s oad
cha ac e is ics and a ic condi ions o imp o e he uel
sa ings o ehicles, he eby educing gas emissions.
Mo eo e , when a emp ing o sol e he ehicle ou e
planning p oblem in he mos accu a e way, we mus ake
in o accoun he a ic a iabili y h oughou he day, as well
as o he si ua ions ha ake place in eal li e when d i ing
a ehicle [17, 18]. Fo ins ance, i is qui e clea ha , on la ge
me opoli an a eas, he cos o a e sing ce ain a e ies,
especially la ge a enues, hea ily depends on he ime o day,
being c i ical a peak a ic hou s [19]. Howe e , i has been
p o ed ha in eg a ing ime-dependencies in ou e op imiza-
ion algo i hms signi ican ly inc eases hei complexi y [20,
21].
To ackle his inc ease o complexi y, we p esen in his
pape anapp oach osigni ican ly educe heamoun o
da a ha ou pla o m will need o ind he ime-dependen
sho es ou es. Speci ically, we de ail how o agg ega e la ge
amoun s o his o ical a ic low da a in o he mos mean-
ing ul se o in o ma ion o p ope ly desc ibe a ic low
a ia ions h oughou he day on he di e en s ee s and
a enues o a ci y.
To his aim, we will use a clus e ing echnique. Clus e
analysis is an unsupe ised lea ning echnique used o he
classi ica ion o da a. Da a elemen s a e pa i ioned in o
g oups called clus e s ha ep esen p oxima e collec ions o
da a elemen s based on a dis ance o dissimila i y unc ion.
The e exis wo main clus e ing me hods. The hie a chical
me hodsbasicallys a wi heachmembe o hese ina
clus e o i s own and use nea es clus e s un il he e a e 𝑘
emaining. The pa i ioning me hods s a by building a se
o 𝑘 ep esen a i e objec s and clus e a ound hose, i e a ing
un il (locally) op imal clus e ing is ound. See, o example,
heclassicalbookbyKau manandRousseeuw[22]andXu
and Wunsch II [23].
Clus e ing echniques ha e been al eady used in he las
yea s as pa o ITS solu ions in o de o p o ide eal insigh s
in o a ic managemen policies. Fo b ie ness, we only e e
o some o hese wo ks. We ecommend consul ing Gua diola
e al. [24] o u he in o ma ion on he opic.
Fo example, Wang e al. [25] p esen a dynamic a ic
p edic ion model ha deals wi h a ic low da a o con e
hem in o a ic s a us. In his model, wo da a mining ech-
niques, he clus e ing analysis and he classi ica ion analysis,
a e applied o his o ical a ic low da a. Cace es e al. [26]
p esen a me hodology o es ima ing a ic lows using oad
ea u es as clus e ing a iables, so ha i can be applied o
any oad sec ion, e en wi hou de ec o da a. Mo e ecen ly,
Yildi imoglu and Ge oliminis [27] pa i ion he his o ical
da a se om loop de ec o s on Cali o nian eeways in clus-
e s wi h simila cha ac e is ics based on he a ic pa e ns
obse ed on he oadway. The building block o hei me hod-
ology is he de elopmen o s ochas ic conges ion maps,
which iden i y he p obabili y ha a space- ime domain is
conges ed. Finally, Gua diola e al. [24] p esen a new me ho-
dology o analyzing he daily a ic low p o ile using
Func ional Da a Analysis. They claim ha hei me hodology
allows a maximum exploi a ion o he eco ded his o ical
da a and esul s in he de ec ion o changes in he low pa -
e n, which would o he wise be di icul o de ec ia classical
s a is ical me hods.
3. T a ic Managemen as a Se ice
Cu en ehicle na iga ion sys ems a e ypically based on
locally s o ed s a ic in o ma ion om which ou es a e cal-
cula ed. Among such sys ems we can ind comme cial appli-
ca ions like TomTom (h p://www. om om.com/) o Ga min
(h p://www.ga min.com/). The e a e also ee ools, like
Ma hema ical P oblems in Enginee ing 3
Google Maps Na iga o and OsmAnd (h p://osmand.ne /)
ha ope a e in a simila manne . The main d awbacks o na -
iga ionsys emsbasedons a icin o ma iona e heinabili y
o adap o a ic conges ion s a es o unexpec ed e en s, like
acciden s o o he p oblems on he oad, which cause a el
imes obemuchhighe hanexpec ed.
Mo e sophis ica ed ou e na iga ion solu ions upda e
ou e in o ma ion in eal ime, based on epo ed a ic con-
di ions. As an example, he TomTom na iga ion so wa e has
been enhanced o suppo clien -se e in e ac ion in o de
o in o m clien s abou al e na i e ou es when a ypical a ic
delays a e de ec ed.
In his pape we will add ess he speci ic p oblem o a ic
conges ion in u ban en i onmen s. Ins ead o acciden s and
o he condi ions causing a ypical delays, we will ocus on
p edic ing daily a ic low pa e ns o a speci ic u ban
en i onmen , de ailing how i is possible o educe a el
imes based on his o ical in o ma ion abou he a ic densi y
dis ibu ion h oughou he day.
The p oposed a ic managemen pla o m is named
ABATIS: Au oma ic Balancing o T a ic h ough he In e-
g a ion o Sma phones wi h ehicles.Themainno el yo
ABATIS as a ou e planning sys em is p o iding ime-
dependen ou e ecommenda ions based on a ic conges-
ion his o y. Speci ically, i o e s clien -se e in e ac ion,
whe e he ou e selec ion p ocess is pe o med a he ou e
se e (see Figu e 1) based on eal- ime in o ma ion s o ed in
he ou e da abase and his o ical da a. The a ic analysis and
isualiza ion se e allows making a ic conges ion o ecas s
based on his o ical da a while also allowing a ic manage-
men au ho i ies o check he a ic condi ions in eal ime.
Clien s con ibu e o imp o ing he ou e da abase in o -
ma ion by p o iding eal- ime eedback abou a ic conges-
ion condi ions, which allows main aining bo h a eal- ime
mapo a ic luidi yinaci yandaccu a ehis o icalda a
o a icbeha io .Thisapp oachsuppo sglobal a icload
balancing and e en -based managemen (e.g., educing a ic
conges ion in he ou e o an ambulance).
This s a egy, al hough o e ing signi ican ly be e ou es,
has a highe cos since he es ima ed ime o a e sing each
pa h segmen will no longe be a ixed alue based on segmen
leng h and speed limi , bu ins ead i will a y dynamically
along he day. In o de o achie e ime-dependen cos s o
he di e en s ee s and a enues in a ci y, ABATIS will use
exis ing his o ical da a abou a ic logs in a ci y o es ima e
a el imes. Since such logs p o ide pe -hou conges ion
measu emen s o all induc ion loop de ec o s in a ci y
o a whole yea , hey mus be p ope ly summa ized and
syn hesized by he a ic analysis se e o allow seamlessly
in eg a ing such in o ma ion in he ou e se e . Thus, in he
emainde o he pape , we will ocus on he a ic analysis
componen , p oposing a heu is ic able o educe he complex-
i y o he p oblem by con e ing huge amoun s o his o ical
da a abou a ic in ensi y in o a small bu ep esen a i e
se o daily pa e ns able o desc ibe he expec able a ic
beha io in he ci y along he day.
Rou e
se e
T a ic analysis and
isualiza ion se e
Da abase
Figu e 1: ABATIS a ic managemen a chi ec u e.
4. Flow Pa e n Classi ica ion P oblem
A emp ing o model he daily a ic low pa e n o hund eds
o s ee s/a enues o e e y day o he yea would lead o
hund eds o housands o in e pola ion unc ions able o p o-
ide a smoo h desc ip ion o pe -s ee a ic low a ia ions
h oughou he day, based on se e al million inpu alues
(assuming a pe -hou g anula i y). Such modeling e o o
a single ci y can be conside ed excessi e and, in addi ion,
causes ou e ecommenda ion asks a he se e o ha e
an ex emely high compu a ional cos . Ne e heless, when
a emp ing o p o ide an accu a e cha ac e iza ion o pa h
segmen cos s in a speci ic u ban en i onmen , i quickly
becomes clea ha (i), om a yea ly pe spec i e, seasonal di -
e ences a e expec able as, o example, mo e people use
hei ehicles du ing cold wea he seasons han du ing he
wa m and ho seasons whe e, o example, bicycles o public
anspo can become a mo e a ac i e al e na i e; (ii), om
a weekly pe spec i e, labo days a e cha ac e ized by mobili y
pa e ns and a ic conges ion s a es ha d as ically di e
om he beha io du ing weekends and holidays; (iii),
om an hou ly pe spec i e, di e en hou s o he day a e
associa ed wi h di e en conges ion le els (e.g., day e sus
nigh ); and inally (i ), om a spa ial pe spec i e, di e en
s ee s/a enues ha e di e en a ic le els a any ime o he
day, equi ing independen modeling.
Taking he a o emen ioned ac o s in o conside a ion,
in his sec ion we will ake an in-dep h look in o a ic
beha io when ocusing on a medium-size Eu opean ci y like
Valencia, Spain, which is he hi d la ges me opoli an a ea
in Spain wi h abou 1.77 million inhabi an s. De ailed ace
iles con aining he amoun o a ic lowing in each o he
s ee s/a enues each hou o a ull yea (2013) we e p o ided
o us by Valencia’s Ci y Hall T a ic Depa men , in pa icula ,
da a conce ning he 421 mos ele an s ee s/a enues ( hose
moni o ed by a ic se ices h ough induc ion loop de ec-
o s).
Ou goal is o ob ain insigh in o he a ic low, de ec ing
a ic pa e ns acco ding o he day o he week, hou , and
ype o s ee . Based on he a ic pa e ns de ec ed, we will
p oposeaheu is icino de osimpli y henumbe o models
equi ed while main aining mos o he ime-dependen
modeling e ec i eness. Al hough we use he ci y o Valencia
as he a ge o ou analysis, he modeling me hodology ol-
lowed is qui e gene al, being applicable o o he ci ies as well.
4Ma hema ical P oblems in Enginee ing
Janua y
Feb ua y
Ma ch
Ap il
May
June
July
Augus
Sep embe
Oc obe
No embe
Decembe
8000 200 400 600 12001000
A e age a ic olume ( housands o ehicles)
Figu e 2: A e age a ic olume in Valencia pe mon h.
A e age a ic olume
F iday
Sunday
Tuesday
Monday
Sa u day
Thu sday
Wednesday
Day o he week
0
5
10
15
20
25
30
35
40
45
( housands o ehicles)
Figu e 3: A e age a ic olume in Valencia o he di e en days
o he week.
We s a by analyzing he mon hly a ic, assessing
whe he we can de ec signi ican seasonal di e ences. As
shown in Figu e 2, he e a e mino luc ua ions in e ms o
o e all a ic on a mon hly basis. I quickly becomes e iden
ha holiday pe iods, like Augus and also Eas e (in Ap il),
ha e a clea and expec able impac on he o e all a ic
olume. Fo he emaining mon hs o he yea he alues can
be conside ed ela i ely simila , ha ing a mean alue o abou
1 million ehicles.
Fo he analysis ha ollows we picked a mon h wi h
an a e age o e all a ic olume close o he mean; speci -
ically, we selec ed No embe , which has no holiday pe iods.
Focusing on he a ic pa e n a ia ion h oughou he week,
Figu e 3 shows ha he e a e e y signi ican di e ences be -
ween he days o he week, especially be ween he weekend
and weekdays. Also, we can obse e an o e all inc easing
end om Monday o F iday, wi h F iday being he weekday
wi h highe a ic olume.
In addi ion o he di e ences in e ms o daily a ic ol-
ume, he e a e also clea di e ences in e ms o he daily
a ic pa e n i sel . Fo ins ance, Figu e 4 shows ha on
Mondays he a ic ollows a ypical pa e n whe e he peak
hou is be ween 8 and 9 a.m., when mos people go o wo k.
Ano he peak occu s be ween 2 and 3 p.m., which deno es
mobili y om people wo king in he a e noon. Finally, a
las a ic peak is de ec ed be ween 6 and 8 p.m., when
wo ke s e u n o hei homes. O he weekdays ollow a
simila pa e n.
A o ally di e en pa e n is de ec ed, o example, on
a Sunday. Compa ed o weekdays we ind ha (i) wo k-
ela ed a ic peaks a e no longe p esen ; (ii) he o al a ic
olume is signi ican ly lowe ; and (iii) he peak hou s di e .
In pa icula , peak hou s a e now ela ed o mobili y owa ds
ood cou s a lunch ime (be ween 1 and 2 p.m.) and mobili y
om elax a eas o homes (be ween 6 and 8 p.m.).
When ocusing on he a ic dis ibu ion h oughou a
ci y, i is well known ha main s ee s and a enues will
expe ience a much highe a ic load han seconda y and
isola ed ones. Disc imina ing be ween hem is a ele an issue
since some s ee s ba ely expe ience any a ic load inc ease
du ing peak hou s, meaning ha a el imes a e no a ec ed
byconges ionin hesamewayas hemaina e ieso heci y.
To be able o disc imina e be ween he s ee s o Valencia
based on a ic low, we i s ob ained he peak a ic
in ensi y pe s ee du ing No embe , and we hen ob ained
he cumula i e dis ibu ion o hese alues (see Figu e 5).
We obse e ha 30.3% o all s ee s ha e a a ic in ensi y
lowe han 690 ehicles/hou du ing peak hou s, which
acco ding o [28] means ha hese low a ic in ensi y s ee s
will no expe ience a ic conges ion e en a peak hou s, and
so hey can be disca ded om ou ime-dependen mod-
eling e o s. Addi ionally, we obse e ha he numbe o
s ee s/a enues wi h e y high a ic olumes (mo e han
10.000 ehicles du ing he peak hou ) is a he limi ed (abou
10%). Thus, he majo i y o he s ee s in a ci y will expe ience
mode a e a ic olumes, and he global peak hou beha io
will no cause any no iceable e ec on hese s ee s. To
con i m his obse a ion, Figu e 6 shows he a ic load pe
hou in wo di e en s ee s o he same day. No ice ha
al hough bo h sha e qui e simila alues o peak a ic in en-
si y, he daily a ic pa e ns signi ican ly di e ha he peaks
in one pa e n o en ma ch alleys in he o he pa e n.
Obse ing he daily a ic pa e n in Figu e 6(a), we ind
ha i closely ma ches he a ic pa e n o a ypical Monday,
as shown in Figu e 4(a); on he con a y, Figu e 6(b) shows a
qui e di e en a ic pa e n. Hence, i becomes necessa y o
disc imina e be ween he di e en s ee s based on hei daily
a ic pa e n. To achie e his goal, we will apply a clus e ing
echnique in o de o au oma ically classi y s ee s acco ding
o hei daily a ic pa e n.
5. Clus e ing Heu is ic
In his sec ion we p opose a heu is ic o simpli y a ic mod-
eling o heci yo Valenciaby akingin oconside a ion he
esul s p esen ed in he p e ious sec ion.
The p oposed heu is ic agg ega es in o a single pa e n all
hose daily a ic pa e ns ha ing a common beha io . This
is made possible by making he ob ained ime-dependen
Ma hema ical P oblems in Enginee ing 5
0
500
1000
1500
2000
2500
3000
A e age a ic in ensi y ( ehicles/hou )
1:00
2:00
3:00
4:00
5:00
6:00
7:00
8:00
9:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
19:00
20:00
21:00
22:00
23:00
0:00
Time o day
(a) Monday
0
500
1000
1500
2000
2500
3000
A e age a ic in ensi y ( ehicles/hou )
1:00
2:00
3:00
4:00
5:00
6:00
7:00
8:00
9:00
0:00
11:00
12:00
13:00
14:00
15:00
16:00
22:00
18:00
19:00
20:00
21:00
23:00
10:00
17:00
Time o day
(b) Sunday
Figu e 4: A e age daily beha io o di e en days o he week.
0
0.1
0.2
0.3
0.4
0.5
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1
P(X < x)
100 1000 10000 10000010
T a ic in ensi y ( ehicles/hou )
Figu e 5: Cumula i e dis ibu ion o a ic in ensi y using he
mon hly peak hou s.
models independen o he ac ual numbe o ehicles in each
s ee h ough no maliza ion using he mean daily alue.
To his aim, we use Ma hema ica 9.0.1 [29], which is a
widely ecognized ool o sol e ma hema ical p oblems,
especially in enginee ing. This ool p o ides unc ion Find-
Clus e s, which e u ns he numbe o clus e s as well as he
elemen s on each clus e . This unc ion has se e al op ions
and subop ions. In ac , we can choose be ween a hie a chical
me hod o a pa i ioning me hod. The pa i ioning me hod
i usesisbasedon hePa i ioningA oundMedoids(PAM)
algo i hm [22], which seeks o ind 𝑘 ep esen a i e objec s
called medoids om he da a se such ha he sum o he dis-
simila i ies wi hin a clus e a e minimized. A medoid can be
de ined as ha objec o a clus e whose a e age dissimila i y
o all he objec s in he clus e is minimal. A e inding he
se o medoids, each objec o he da a se is assigned o he
nea es medoid.
We ha e chosen he pa i ioning me hod o FindClus e s
o wo easons. The i s one is ha his me hod is he de aul
op ion, and he second and mos impo an one is ha he
PAM algo i hm is he one used by e e ence au ho s on he
opicsuchasGua diolae al.(see[24]),whoclaim ha he
choice o PAM is due in pa o he la ge numbe o s a is ics
i p o ides o ho ough analysis o he esul an clus e s.
A his poin , we wan o s ess he ac ha while [24]
(and also [27]) y o clus e di e en days co esponding o
hesamesec iono a eeway, heaimo ou p ocedu eis
qui e di e en ; pa icula ly, we a emp o clus e di e en
s ee s co esponding o he same day. Mo eo e , as a as we
know, he clus e ing dis ance ha we will use he e has no
been used in any p e ious pape on ITS.
Finally, no e ha al hough we ha e no made use o hem,
unc ion FindClus e s has subop ions in o de , o ins ance,
o ine- une he numbe o clus e s. P obably he bes known
subop ion o do his is he silhoue e s a is ic [22], bu acco d-
ing o [23] he e is no c i e ion p o iding e idence abou i s
supe io i y compa ed o o he s in he gene al case o adjus-
ing he numbe o clus e s. In addi ion, no ice ha wo
p ope ies ha de ine a good heu is ic and ha we ha e aken
in o accoun o ou aim a e low ime o e head and simplici y
o i s s eps.
Belowwedesc ibe he i es eps ollowed o educe he
numbe o independen daily pa e ns o be modeled: (i)
selec he app op ia e clus e ing me ic, (ii) ind he op imal
numbe o clus e s pe day o he week, (iii) de e mine how
ep esen a i e mean days a e, (i ) g oup days o he week wi h
simila cha ac e is ics, and ( ) g oup clus e s wi h simila
daily pa e ns.
5.1. Selec ion o a Clus e ing Me ic o Pe -Hou S ee Beha -
io . I o each s ee (o s ee segmen ) we ha e he numbe
o ca s ha a e se i e e y hou , we can ep esen each s ee
by a poin 𝑥=(𝑥1,𝑥2,...,𝑥24)in R24,whe e𝑥𝑖is he numbe
o ca s a e sing he s ee a hou 𝑖.Supposeweha e wo
s ee s 𝑥=(𝑥
1,𝑥2,...,𝑥24)and 𝑦=(𝑦
1,𝑦2,...,𝑦24).By
de aul , he dis ance used o o m clus e s is he Euclidean

6Ma hema ical P oblems in Enginee ing
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(a) S ee ollowing he expec ed pa e n
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T a ic in ensi y ( ehicles/hou )
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(b) S ee no ollowing he expec ed pa e n
Figu e 6: Daily a ic in ensi y pa e n o s ee s wi h di e en cha ac e is ics.
dis ance, √∑24
𝑖=1(𝑥𝑖−𝑦𝑖)2. I he Euclidean dis ance be ween
wo poin s is ela i ely small, bo h s ee s will belong o he
same clus e . Howe e , i we a emp o classi y s ee s aking
in o accoun he a ic a iabili y as a unc ion o he ime o
day, we belie e ha his dis ance is no adequa e. Le us ake
a small illus a i e example in his ega d. Suppose ha we
only conside six consecu i e hou s o ou di e en s ee s
and ha hei espec i e poin s a e 𝑎 = (12,9,10,9,8,11),
𝑏 = (24,20,22,20,17,21),𝑐 = (6,16,20,25,17,7),and𝑑=
(15,35,44,48,34,17).
S ee s 𝑎and 𝑏ha e a simila beha io : he ela i e
numbe o ehicles a e sing hem e e y hou is mo e o less
he same, wi hin ce ain bounds. Al hough he ac ual numbe
o ehicles di e s g ea ly om one s ee o ano he , bo h
s ee s should be in he same g oup encompassing all hose
s ee s whe e he e is li le a ic a iabili y, whe e ehicle
speeds can be conside ed mos ly cons an o e he conside ed
pe iod.
Wi h espec o s ee s 𝑐and 𝑑, cen al hou s a e peak
pe iodswhe eweha eabou h ee imes he a ic olume
compa ed o edge alues. Al hough he numbe o ehicles
di e s g ea ly om one s ee o ano he , hey should belong
o he same g oup cha ac e ized by a single peak co espond-
ing o hou s in he mid- ange and wi h much lowe alues on
he edges.
Howe e , i we classi y he ou s ee s using he Euclidean
dis ance, he esul is qui e p edic able: {𝑎,𝑐} and {𝑏,𝑑}.In
his example he Euclidean dis ance has c ea ed wo clus e s
g ouping he wo s ee s wi h less a ic and he wo s ee s
wi h high a ic olume. To add ess his p oblem, we belie e
ha he dis ance me ic ha bes i s ou objec i e is he
co ela ion dis ance, de ined as 1−|𝑟
𝑥𝑦|,whe e𝑟𝑥𝑦 is he
co ela ion coe icien :
𝑟𝑥𝑦 =∑24
𝑖=1 (𝑥𝑖−𝑥)⋅(𝑦𝑖−𝑦)
√∑24
𝑖=1 (𝑥𝑖−𝑥)2⋅∑24
𝑖=1 (𝑦𝑖−𝑦)2.(1)
Recall ha |𝑟𝑥𝑦|is always less han o equal o 1 and ha
alues close o 1 indica e ha a iables 𝑥and 𝑦ha e a di ec
linea ela ionship, meaning ha he g aphical ep esen a ion
o he 24 poin s (𝑥𝑖,𝑦𝑖)is app oxima ely a s aigh line.
The e o e, he highe he co ela ion be ween poin s 𝑥and 𝑦
is, he close o ze o 1−|𝑟𝑥𝑦|becomes, and so he p obabili y
o belonging o he same clus e will inc ease. I we classi y
he ou s ee s acco ding o co ela ion dis ance, he esul
ob ained is he desi ed one: {𝑎,𝑏}and {𝑐,𝑑}.
On he o he hand, i is easy o see ha he co ela ion
dis ance is he same i we wo k wi h he coo dina es (𝑥𝑖,𝑦𝑖)
o wi h coo dina es (𝑥𝑖/∑24
𝑗=1 𝑥𝑗,𝑦𝑖/∑24
𝑗=1 𝑦𝑗), akingin o
accoun ha , o compa e s ee s conside ing a ic a iabili y
h oughou heday,i alsoseemsuse ul ocompa e he
pe cen age o he daily a ic passing on e e y s ee o
each hou . This way, i does no ma e whe he we compa e
bo h s ee s conside ing he numbe o ca s pe hou o he
pe cen age o a ic pe hou : he classi ica ion using he
co ela ion dis ance will gene a e he same clus e s. This is
ob iously no ue when adop ing Euclidean dis ances.
5.2. Finding he Op imal Numbe o Clus e s o Each Day o
he Week. Using he co ela ion dis ance de ined p e iously,
in his sec ion we will de e mine he op imal numbe o
clus e s o he 292 s ee s in Valencia conside ed by he Ci y
Hall as ep esen a i e in e ms o a ic low o e e y day
o he week. Subsequen ly, o educe he o e all numbe o
clus e s, we will a emp o join he di e en days in a week
whene e he same numbe o clus e s a e de ec ed.
The e o e, o ou analysis, we apply he FindClus e s
unc ion o each o he 28 days o No embe s udied enabling
he co ela ion dis ance op ion. Fo each day, he unc ion
will clus e he 292 poin s in R24 co esponding o he s ee s
aken o ou s udy.
In he analysis ha ollows we wo k wi h he pe cen age
o ehicles a e sing each s ee e e y hou wi h espec o
he o e all daily alue. As e e ed in he p e ious sec ion,
heac ualnumbe o ehiclespe se is no ele an o
Ma hema ical P oblems in Enginee ing 7
Table 1: Numbe o clus e s ob ained and associa ed s a is ics.
Mo Tu We Th F Sa Su
A: Week 1 3 3 1 2 1 2 3
B: Week 2 1 2 4 1 3 2 4
C: Week 3 5 3 1 4 2 2 3
D: Week 4 3 1 1 3 3 2 1
E: mean(A, B, C, D) 3 2.25 1.75 2.5 2.25 2 2.75
F:median(A,B,C,D) 3 2.5 1 2.5 2.5 2 3
G: a e age day 4 2 2 4 3 2 2
H: ound(E) == G False T ue T ue False False T ue False
I: mean(E, F, G) 3.3(3)— — 3 2.58(3)—2.58(3)
Numbe o clus e s 3223 3 2 3
Table 2: Pe cen ages o ma ching o he di e en clus e s compa ed o he a e age day clus e s.
Mo Tu We Th F Sa Su
Numbe o clus e s 3223323
Week 1
83.11 92.31 84.42 30.86 70.15 91.98 73.72
66.67 84.56 62.32 59.32 68.75 71.43 66.67
81.33 56.99 90.43 35.82
Week 2
60.14 89.74 80.52 81.48 70.15 96.26 74.36
55.07 58.09 59.42 43.22 60.64 81.90 69.56
80.00 58.06 51.56 73.13
Week 3
62.84 84.62 80.52 58.02 70.15 88.77 51.28
69.57 32.35 91.30 75.42 74.47 84.76 47.83
84.00 31.18 35.94 89.55
Week 4
81.76 96.15 74.68 62.96 86.57 97.87 82.05
88.41 84.56 82.61 74.58 65.96 53.30 59.42
76.00 65.59 56.25 58.21
A e age 73.63 78.68 77.05 58.56 69.18 86.21 66.70
ou pu poses, and he co ela ion dis ance me ic adop ed
p o ides he same ou pu on bo h cases.
Sinceou s udype iodencompasses4weeks,wec ea ean
“a e age day” o each day o he week, which is calcula ed o
each s ee by a e aging he numbe o ehicles a e sing i
each hou . Such “a e age day” a emp s o il e ou he peculi-
a i ies o a speci ic day, ob aining a ep esen a i e end
ins ead.
Table1shows he esul sob ained,whe e helas ow
shows he clus e alloca ion o each day o he week. To a ain
hose alues, we i s apply unc ion FindClus e s o di e en
weeks (A–D) and o he “a e age days” (G). In addi ion, we
calcula e he mean (E) and he median (F) o he clus e
g oups co esponding o he di e en weeks. I his mean
alue (E) is ounded o a numbe ha ma ches he numbe o
clus e s o he a e age day (G), hen we de ine such alue as
he numbe o clus e s o ha day o he week. O he wise, we
ob ain hea e ageo hemean(E),median(F),anda e age
day (G) o ob ain a alue (I) ha when ounded de ines he
numbe o clus e s o be used. We ind ha he p oposed
numbe o clus e s ma ches he ounded mean (E) excep o
a mino change in one day.
5.3. De e mining Clus e Ma ching on a Pe -Day Basis. Once
henumbe o clus e s o eachdayo heweekwasde ined,
he nex s ep was o alida e ha clus e elemen s o each day
o he week esembled he clus e elemen s ob ained o he
a e age day. I a good deg ee o ma ching is ob ained, hen he
conclusions associa ed wi h s ee s in ha clus e a e alid;
o he wise, we could be conside ing ha s ee s belong o a
g oup wi h a speci ic beha io , when in ac hei beha io
signi ican ly di e s.
Fo ou endea o we apply he FindClus e s unc ion o
he 35 days (28 eal days plus 7 a e age days), bu his ime
ixing he numbe o clus e s de ined a p io i, as ob ained in
he p e ious sec ion. A e wa ds, o each o he ou weeks
unde analysis, we compa e he clus e s ob ained agains
he a e age day o he week, de e mining he pe cen age o
s ee s ha bo h clus e s ha e in common. These esul s a e
p esen ed in Table 2.
8Ma hema ical P oblems in Enginee ing
Table 3: Pe cen ages o clus e ma ching o a e age days o he
week wi h same numbe o assigned clus e s. Valid combina ions a e
showninbold ace.
Combina ions Deg ee o
ma ching (%) A e age ma ching (%)
Monday-Thu sday
57.43
48.29
8.69
66.67
Monday-F iday
77.70
68.84
72.46
48.00
Monday-Sunday
58.78
43.49
27.54
28.00
Thu sday-F iday
37.04
59.25
71.19
63.44
Thu sday-Sunday
30.86
41.78
55.93
33.33
F iday-Sunday
61.94
51.37
36.17
51.56
Tuesday-Wednesday 91.67 91.78
91.91
Tuesday-Sa u day 71.15 58.56
44.11
Wednesday-Sa u day 69.48 56.51
42.03
We ind ha he a e age deg ee o ma ching o all he
days o he week is 72.71%. Globally, we ind ha his alue
is qui e accep able and ha di e ences appea ing on speci ic
days a e expec able since a ic pa e ns may su e some
changes depending on wea he , special e en s, o o he condi-
ions.
5.4. G ouping Days o he Week wi h Simila Clus e Cha -
ac e is ics. The nex s ep o ou clus e ing p ocedu e was o
assess he easibili y o g ouping hose days o he week ha ing
he same numbe o clus e s. Wi h his pu pose we es ed
all combina ions and calcula ed he pe cen age o clus e
ma ching o eachpai o meandayso heweek.The esul s
a e shown in Table 3.
All combina ions show an a e age deg ee o ma ching
below 70%, excep o he Tuesday-Wednesday combina ion
which is close o 92%. Thus, we ag ee ha hese wo weekdays
can be combined as i hey we e a single day since simila
pa e ns a e ob ained in e ms o a ic a iabili y h oughou
he day. Da a shown ea lie in Figu e 3 also emphasize his
simili ude.
To con i m ha he g ouping did no ha e a nega i e
impac on he e o associa ed wi h speci ic days, we now
p oceed o compa e he deg ee o ma ching o he di e en
clus e s agains he a e age day, he c ossed a e age day, and
hep oposeduniono bo hdays.These esul sa eshownin
Table 4.
We ind ha he di e ences be ween he h ee cases a e
qui e low. Speci ically, he impac o g ouping hese wo days
in o one is o only 1.6%, which is qui e accep able. The esul s
using c oss a e ages also s eng hen he poin o uni ying
hese wo days. As a esul , by accoun ing o he numbe
o clus e s o each a e age day and by me ging Tuesday and
Wednesday in o a single day, we ob ain a o al o 16 di e en
a ic pa e ns.
5.5. G ouping Clus e s wi h Simila Daily Pa e ns. In his sec-
ion we p esen he no malized a ic pa e ns co esponding
o he 16 clus e s c ea ed: 3 o Monday, 2 o Tuesday/
Wednesday, 3 o Thu sday, 3 o F iday, 2 o Sa u day, and 3
o Sunday.
As shown in Figu e 7, he e a e some pa e n simili-
udes be ween he i s weekdays (Monday e sus Tuesday/
Wednesday), be ween he las weekdays (Thu sday e sus F i-
day), and be ween weekend days (Sa u day e sus Sunday).
Howe e , his ini ial insigh ob ained isually mus be con-
i med h ough s a is ical e idence. Wi h his pu pose we
picked he clus e s o hose days which isually show some
simili ude and calcula ed he co ela ion be ween he daily
pa e ns associa ed wi h each clus e o ele an ime anges.
The esul s o hese analyses a e p esen ed in Table 5.
When compa ing he daily pa e n o he clus e s o
Monday agains Tuesday/Wednesday (see Table 5(a)), we ind
ha he e is a high co ela ion (>92%) be ween he pa e ns
co esponding o he i s 2 clus e s o each o hese days.
Thus, a single model will su ice when a emp ing o ep esen
he daily pa e n o hese clus e s ha only a di e en model
is equi ed o Monday’s Clus e numbe 3.
When compa ing Thu sday agains F iday, we ind ha
onlyClus e numbe 2 o Thu sdayandClus e numbe 1
o F idayp esen ahighco ela ion(∼94%).
Finally,whencompa ingSa u dayagains Sunday,we ind
ha Clus e numbe 1andClus e numbe 3p esen agood
deg ee o ma ching (∼94%), and hese wo clus e s can also
be ep esen ed h ough same daily pa e n.
6. Gene aliza ion and Bene i s o
he P oposed Model
In his sec ion we assess he bene i s o ou model in e ms
o he minimum numbe o pa e ns equi ed o adequa ely
desc ibe a ic in ensi y h oughou he day o he ci y o
Valencia. Then, we de ail how hese di e en models ob ained
can be in eg a ed in ou a ic managemen pla o m o
p edic ou e cos s. Finally we summa ize ou p oposal by
p esen ing he p oposed heu is ic in pseudocode o ma o
allow gene alizing he p oposed p ocedu e o any a ge ci y.
6.1. Agg ega ion Gains Achie ed. Below we discuss he di e -
en agg ega ion echniques ha in eg a e ou heu is ic and
he p e ious analysis.
Ma hema ical P oblems in Enginee ing 9
Table 4: Pe cen ages o ma ching o he di e en clus e s agains he a e age day, he c ossed a e age day, and he p oposed union o bo h
days.
O iginal a e age days C ossed a e age days Union o a e age days
Tu We Tu We Tu We
Week 1 92.31 84.42 90.26 86.54 86.83 86.23
84.56 62.32 81.16 65.44 84.00 69.60
Week 2 89.74 80.52 89.61 78.21 85.63 77.25
58.09 59.42 57.25 58.09 56.80 60.00
Week 3 84.62 80.52 86.36 78.85 82.04 72.46
32.35 91.30 34.06 90.44 30.40 88.00
Week 4 96.15 74.68 95.45 73.08 92.22 68.26
84.56 82.61 82.61 81.62 86.40 80.00
A e age 78.68 77.05 77.82 76.71 77.14 75.34
Table 5: Co ela ion be ween clus e s (pe iod be ween 7 a.m. and 9 p.m.).
(a) Monday and Tuesday/Wednesday
Tuesday/Wednesday
Clus e numbe 1 Clus e numbe 2
Monday
Clus e numbe 1 0.9221668 0.578729
Clus e numbe 2 0.6229643 0.9422671
Clus e numbe 3 0.5900097 0.7910942
(b) Thu sday and F iday
F iday
Clus e numbe 1 Clus e numbe 2 Clus e numbe 3
Thu sday
Clus e numbe 1 0.6741969 0.2552095 0.7292981
Clus e numbe 2 0.9393144 0.6691599 0.6666628
Clus e numbe 3 0.7247197 0.7841533 0.8645128
(c) Sa u day and Sunday
Sunday
Clus e numbe 1 Clus e numbe 2 Clus e numbe 3
Sa u day Clus e numbe 1 0.8859214 0.8585393 0.9368844
Clus e numbe 2 0.8948805 0.8840648 0.7977545
Yea ly Analysis. The mon hly beha io esul s shown be o e
allow assuming ha a ic olumes h oughou he yea a e
mos ly cons an , excep o aca ion pe iods like summe and
es i i ies las ing o long pe iods (e.g., Eas e ), meaning ha
pa i ioning weeks in o h ee g oups ( ypical week, ele an
holiday pe iod, and summe holidays) seems app op ia e.
Mon hly Analysis. Resul s ha e shown ha , o he same
ype o pe iod, da a is consis en ac oss weeks, which allows
clus e ing he di e en days o a mon h in a single a e age
ep esen a i e week.
T a ic In ensi y Analysis. Conce ning a ic conges ion o
he di e en s ee s and a enues o a ci y, ou heu is ic
assumes ha only a subse o hese s ee s/a enues ac ually
ace signi ican conges ion p oblems dese ing ime-depend-
en modeling, while o he es , he use o adi ional ixed-
cos app oaches su ices. Based on he h esholds de ined in
[28] o class IV (u ban) a e ial ypes, we conside ha only
hose s ee s wi h a peak a ic alue su passing 690 ehicles
pe hou a e ac ually expe iencing conges ion- ela ed a ic
delays. This way, he a ge numbe o s ee s/a enues can be
educed om 421 ( o al numbe o s ee s being moni o ed
by a ic se ices) o 292 (numbe o s ee s wi h a ele an
a ic load).
Clus e ing Analysis. Focusing on he s ee /a enue subse
signi ican ly a ec ed by conges ion, he clus e ing analysis
showed ha a small numbe o g oups can be c ea ed, whe e
o each g oup all s ee s/a enues ollow e y simila a ic
conges ion pa e ns. Thus, he a ge numbe o models
equi ed can be educed om 292 pe 7 days in a week o
a o al o 18, and his alue can be u he educed o 16 by
no icing he simila i y be ween Tuesday and Wednesday.
Daily Pa e n Analysis.Ananalysiso hedailypa e ns
associa ed wi h he di e en clus e s de ined o he di e en
days o he week has shown ha some o hese clus e s ha e