Traffic management as a service: the traffic flow pattern classification problem

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 0.6 0.7 0.8 0.9 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 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 0 2000 4000 6000 8000 10000 12000 14000 16000 T a ic in ensi y ( ehicles/hou ) (a) S ee ollowing he expec ed pa e n 0 500 1000 1500 2000 2500 3000 T 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 (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