A Clustering-Based Hybrid Support Vector Regression Model to Predict Container Volume at Seaport Sanitary Facilities

applied sciences A icle A Clus e ing-Based Hyb id Suppo Vec o Reg ession Model o P edic Con aine Volume a Seapo Sani a y Facili ies Juan Jesús Ruiz-Aguila 1,* , JoséAn onio Moscoso-López 1, Daniel U da 2, Ja ie González-En ique 3and Ignacio Tu ias 3 1Depa men o Indus ial and Ci il Enginee ing, Poly echnic School o Enginee ing, Uni e si y o Cadiz, 11202 Algeci as, Spain; [email p o ec ed] 2G upo de In eligencia Compu acional Aplicada (GICAP), Depa amen o de Ingenie ía In o má ica, Escuela Poli é cnica Supe io , Uni e sidad de Bu gos, A . Can ab ia s/n, 09006 Bu gos, Spain; du [email p o ec ed] 3Depa men o Compu e Science Enginee ing, Poly echnic School o Enginee ing, Uni e si y o Cadiz, 11202 Algeci as, Spain; ja ie [email p o ec ed] (J.G.-E.); [email p o ec ed] (I.T.) *Co espondence: juanjesus. [email p o ec ed] Recei ed: 7 No embe 2020; Accep ed: 22 No embe 2020; Published: 24 No embe 2020   Abs ac : An accu a e p edic ion o eigh olume a he sani a y acili ies o seapo s is a key ac o o imp o e planning ope a ions and esou ce alloca ion. This s udy p oposes a hyb id app oach o o ecas con aine olume a he sani a y acili ies o a seapo . The me hodology consis s o a h ee-s ep p ocedu e, combining he s eng hs o linea and non-linea models and he capabili y o a clus e ing echnique. Fi s , a sel -o ganizing map (SOM) is used o decompose he ime se ies in o smalle clus e s easie o p edic . Second, a seasonal au o eg essi e in eg a ed mo ing a e ages (SARIMA) model is applied in each clus e in o de o ob ain p edic ed alues and esiduals o each clus e . These alues a e inally used as inpu s o a suppo ec o eg ession (SVR) model oge he wi h he his o ical da a o he clus e . The inal p edic ion esul in eg a es he p edic ion esul s o each clus e . The expe imen al esul s showed ha he p oposed model p o ided accu a e p edic ion esul s and ou pe o ms he es o he models es ed. The p oposed model can be used as an au oma ic decision-making ool by seapo managemen due o i s capaci y o plan esou ces in ad ance, a oiding conges ion and ime delays. Keywo ds: ma i ime anspo ; con aine o ecas ing; suppo ec o eg ession; sel -o ganizing maps; machine lea ning; hyb id models 1. In oduc ion O e he las decades, po s ha e played an impo an ole in in e na ional ade and mos o he o e seas shipping o p oduc s is aboa d deep-sea con aine essels [ 1 ]. The inc ease in a ic o goods and he Eu opean uni ica ion has led o conside he enhancemen o secu i y a bo de c ossings o he Eu opean Union. In his sense, he Bo de Inspec ion Pos s (BIPs) we e c ea ed in o de o gua an ee he secu i y a bo de c ossings and he quali y o he impo -expo goods. BIPs a e he app o ed acili ies whe e he checks o goods ( anspo ed wi hin con aine s by ucks o owing ehicles) a e ca ied ou be o e en e ing he Communi y e i o y. Howe e , he sus ained g ow h in he wo ldwide exchange o goods is c ea ing he need o u he inspec ions esul ing in conges ion and high load-peaks wi hin he sani a y acili ies. This causes ime delays and highe cos s in he supply chain. The BIPs a e he eby bo lenecks ha mus be necessa ily assessed by Po Au ho i ies in o de o keep he quali y le el o he po and a oid losing compe i i eness. In o de o a oid ime delays and conges ion in he sani a y acili ies, he po managemen should be able o accu a ely Appl. Sci. 2020,10, 8326; doi:10.3390/app10238326 www.mdpi.com/jou nal/applsci Appl. Sci. 2020,10, 8326 2 o 17 o ecas he numbe o con aine passing h ough hese acili ies. An accu a e p edic ion o he olume o con aine a ic h ough he BIP may become a use ul ool o imp o e human esou ces, planning ope a ions and he se ice quali y a po s. Fo ecas ing ime se ies ha e ocused a en ion o many esea che s o a long ime. Many e o s ha e been ca ied ou o imp o e he exis ing me hodologies and o achie e enhanced models o ob ain accu a e p edic ions in any o ecas ing ields ( o ins ance, ene gy consump ion, anspo a ion, en i onmen , economy, medicine and heal h). The ocus o his s udy is mainly se on ma i ime anspo due o he a ic associa ed wi h he po BIPs. The p oposed o ecas ing echniques can be di ided in o h ee ca ego ies: single me hods, combined me hods and hyb id me hods. The i s class comp ises bo h linea and nonlinea echniques. Linea echniques a e based on he assump ion ha a linea ela ionship exis s be ween he u u e alues and he cu en and pas alues o he ime se ies. One o he linea models ha has a ac ed mo e a en ion o e he pas ew decades is he well-known au o eg essi e in eg a ed mo ing a e ages (ARIMA) model and i s u he ex ended e sions. Based on he Box and Jenkins me hodology [ 2 ] he ARIMA models ha e been cons an ly applied o sol e o ecas ing asks ela ed o ma i ime anspo . ARIMA models we e success ully applied o p edic expo ope a ions in con aine e minals [ 3 ], o p edic ce ain a ic lows o goods ha pass h ough a po [4,5] and ecen ly o p edic ing con aine h oughpu olumes a po s [6]. Mo eo e , nonlinea echniques ha e become a s eng h al e na i e agains he weaknesses o linea models. This o ecas ing echnique has p o ed o be e ec i e when ime se ies show nonlinea pa e ns, o e coming he main cons ain s o linea models. Real-wo ld p oblems a e complex and mos ly gene a ed by an unde lying nonlinea p ocess [ 7 ], such could be he case o olume o con aine s demand a BIPs. In his subca ego y, wo machine lea ning echniques highligh : a i icial neu al ne wo ks (ANNs) and suppo ec o machines o eg ession (SVR). Ins ead o ANNs, SVR has a ac ed inc easing a en ion in ecen yea s due o i s inhe en abili ies o o e come some gene al limi a ions o ANNs, such as he achie emen o a global minimum. Due o i s g ea gene aliza ion abili y, SVR has been used in o ecas ing anspo asks wi h p omising esul s. These wo echniques ha e been cons an ly compa ed in he esea ch li e a u e. The second ca ego y comp ises he combined models. These models a emp o o e come he cons ain s o he single o ecas ing models when ime se ies exhibi seasonal ea u es, empo al e olu ions, co ela ions, and o he pa e ns ha d o cap u e by single models. To his aim, single me hods a e combined o seize he abili ies o each one in ce ain o ecas ing asks. One o he mos equen ly used app oaches consis s o combining a single p edic ion echnique (mainly a so compu ing echnique) wi h a clus e ing me hod. Based on he di ide-and-conque p inciple, clus e ing me hods allow o di ide he o iginal da abase in o a numbe o smalle g oups, called clus e s. The main assump ion is ha simple clus e s a e easie o sol e han add essing he whole da abase. When he clus e ing me hod has di ided he da abase in o se e al clus e s, a p edic ion echnique is hen applied in each clus e independen ly. Sel -o ganizing maps (SOMs) [ 8 ] is p obably he bes -known clus e ing me hod. This popula unsupe ised lea ning echnique has been p o ed o be as e and mo e accu a e and e icien han o he clus e ing me hods [ 9 ], imp o ing he inal p edic ion pe o mance in ime se ies [ 10 ]. A combined SOM-ANN model was in oduced by Chen e al. [ 11 ] o p edic a ic lows in anspo a ion. This wo k also compa ed he o ecas ing pe o mance o he p oposed model o he ob ained wi h a SOM-ARIMA model and a single ARIMA model. Resul s showed ha he SOM-ANN model ou pe o med he es o models. Due o he ecen eme gence o SVR in anspo a ion, he e is ha dly any esea ch ela ed o anspo combining SOM and SVR in a wo-s age p ocedu e. Ne e heless, i is a widesp ead solu ion in many o he o ecas ing ields [12]. The hi d ca ego y includes hyb id models. Single linea models ha e shown o ha e se e al cons ain s, limi a ions and disad an ages in pa icula si ua ions and ce ain applica ions ( o ins ance, when he unde lying gene a ing mechanism is nonlinea o unce ain y is p esen ed) [ 13 ]. In a simila manne , he use o nonlinea models can be o ally inapp op ia e i linea pa e ns a e p esen ed. Mo eo e , eal-wo ld ime se ies a e no comple ely linea o nonlinea , bu a he con ain bo h Appl. Sci. 2020,10, 8326 3 o 17 componen s. Thus, a me hodology using linea and nonlinea models in a hyb id way akes he capabili ies o bo h models, imp o ing he o ecas ing accu acy and educing he isk o ailu e when an unsui able single model is used. Hyb idizing linea models and machine lea ning echniques ha e been p oposed in ecen yea s o o ecas anspo a ion ime se ies. Pa icula ly, ARIMA has been he mos commonly used linea model in li e a u e o cons uc his kind o hyb id model. The i s wo ks we e p oposed conside ing ANNs as he machine lea ning echnique in [ 7 , 14 ] o o ecas ime se ies and hey concluded ha hyb id ARIMA-ANN models we e supe io o single SARIMA and ANN models. Since hen, many esea ch wo ks can be ound in he li e a u e in many a eas linked o ime se ies analysis [ 15 , 16 ]. Se e al au ho s we e also p oposed a hyb idiza ion o SARIMA and SVR o add ess se e al o ecas ing asks ou side he anspo sec o [ 17 , 18 ], al hough i is less widesp ead compa ed o SARIMA-ANN models. Rela ed o ma i ime anspo , Xie e al. [ 19 ] p oposed se e al hyb id app oaches in a compa a i e way including he SARIMA-SVR model o con aine h oughpu o ecas ing. All hese p e ious s udies coincided in poin ing ou ha a hyb id s a egy conside ing ARIMA and SVR models o e came he pe o mance o single models in hei espec i e domains. In his s udy, a combined-hyb id o ecas ing model is p oposed in such a way ha a hyb id model (SARIMA-SVR) is combined wi h a clus e ing me hod (SOM) o o ecas he daily numbe o con aine s passing h ough a BIP, hus esul ing in a SOM-SARIMA-SVR model. This me hodology uni ies he s eng hs o clus e ing me hods in decomposing he o ecas ing ask in o some ela i ely easie sub asks (using a SOM me hod) and he s eng hs o hyb id models o i linea and nonlinea componen s (using a SARIMA-SVR model). Thus, he inal aim o his s udy is wo old: i s , o demons a e ha he SOM-SARIMA-SVR ou pe o ms he es o possible hyb id o combined models in o ecas ing he daily numbe o con aine s passing h ough BIP o a ma i ime po ; and second, o es he gene aliza ion capabili ies o SVR in o ecas ing logis ic asks, especially he con aine inspec ion p ocess ha can be a bo leneck in he supply chain. To ain he new model, a h ee-s ep p ocedu e was de eloped. In he i s s ep a SOM algo i hm allows o di ide he da abase in o di e en clus e s o egions. On a second s ep, a SARIMA model is i ed o he da a o each clus e in o de o cap u e he seasonali y and he linea beha io . Finally, a SVR model is applied o e each ob ained clus e conside ing se e al hyb id app oaches (di e en inpu con igu a ions) which can conside he o iginal da abase and he ou pu s o he i s s ep ( esidual and p edic ed alues). The es o he pape is o ganized as ollows: The second sec ion gi es an o e iew o sel -o ganizing maps (SOM), seasonal au o eg essi e in eg a ed mo ing a e age (SARIMA) and suppo ec o machines o eg ession (SVR). The empi ical da a, he combined-hyb id me hodology, and he expe imen al p ocedu e a e p esen ed in Sec ion 3. The ou h sec ion discusses he esul s ob ained. Finally, he las sec ion summa izes he impo an conclusions o his wo k. 2. Me hods The p oposed model consis s o hyb idizing a seasonal au o eg essi e in eg a ing mo ing a e age (SARIMA) model as a linea model, and a non-linea model such as a suppo ec o machine o eg ession (SVR). These me hods a e wo o he mos impo an o ecas ing models conce ning hei espec i e domains. Addi ionally, a clus e ing me hod, Kohonen sel -o ganizing map (SOM), is combined wi h his hyb id (SARIMA-SVR) model. The me hodology and basic concep s a e nex desc ibed. 2.1. Sel -O ganizing Maps (SOM) Wi hin he unsupe ised lea ning ield, a SOM is a kind o neu al ne wo k ha has gained special a en ion du ing he pas wo decades. Fi s p oposed by Kohonen [ 8 , 20 ], a SOM is a classi ica ion echnique ha g oups objec s o he sys ems in o egions called clus e s. This classi ica ion is based on he simila i y o nea ness o hese objec s wi hou ex e nal ac o s in luencing hei pe o mance. In he p ocess, he neu ons o he model o ganize hemsel es conside ing only hose ha play a simila Appl. Sci. 2020,10, 8326 4 o 17 ole, o ming a clus e . The ep esen a i e poin o his clus e is called cen oid, being he cen al poin , which can be used as he cen e poin o a classi ie based on minimum dis ance. The opology o a SOM model consis s o se e al neu ons dis ibu ed in o wo laye s: he inpu and he ou pu laye . The i s one is o med by kneu ons. Each neu on co esponds wi h one inpu . The ou pu laye , called he compe i ion laye , can consis o di e en opologies (2-D g id o his case). The p e-p ocessing is pe o med in his laye . In he p ocess, all he neu ons jo he ou pu laye a e connec ed by weigh s (w i,j ) wi h all neu ons io he inpu laye . A weigh ec o , wj=wj,1,wj,2,. . . ,wj,k , is he eby associa ed o each ou pu neu on jo he compe i i e laye (kis he eby he inpu ec o dimension). The aining o a SOM model is based on a compe i i e p ocess whe e ne wo ks a e ained i e a i ely. Di e en inpu ec o s xi= [x1,x2,. . . ,xk]T a e p esen ed o he ne wo k a each i e a ion. Du ing he ne wo k aining, he Euclidean dis ance be ween xand all he weigh ec o s a e compu ed as ollows:   x−wb  =min jnkx( )−wj( )ko,j=1, 2, . . . ,l, (1) whe e lis he numbe o ou pu neu ons. Acco ding o Equa ion (1), w b is conside ed he winning neu on, i.e., he neu on ha has he weigh ec o closes o x. In addi ion, he weigh o he winning neu on is upda ed a ime +1, acco ding o Equa ion (2). The same occu s wi h i s associa ed neighbo neu ons: wi( +1) = βi·δb,i( )·[x( )−wi( )]+wi( )(2) whe e δb,i ( ) is he neighbo hood unc ion (usually a Gaussian unc ion) associa ed o he neu on i, and β ( ) is he exponen ial decay lea ning ac o . Bo h pa ame e s, β ( ) and δb,i ( ), decay wi h ime. The aining algo i hm s ops when he maximum numbe o epochs (a ep esen a ion o all inpu s pa e ns) is achie ed, o when he pe o mance is minimized o he a ge . 2.2. Au o-Reg essi e In eg a ed Mo ing A e ages (ARIMA) ARIMA models we e in oduced by Box and Jenkins [ 2 ]. ARIMA has been a widely used o ecas ing linea model du ing se e al decades. In hese kind o models, he u u e alue is accep ed o be a linea unc ion o se e al pas obse a ion and an e o e m. Th ee p edic ion e ms compose his linea unc ion: he au o eg essi e e m (AR), he mo ing a e age e m (MA), and he in eg a ion e m (I). A SARIMA model can be ob ained by ex ending he ARIMA model o include seasonal ea u es. In his way, he model is speci ied as SARIMA(p,d,q)(P,D,Q) S , whe e q ep esen he o de o he mo ing a e age e ms, pdeno es he o de o he au o eg essi e e ms, and dis he deg ee o di e encing. (P,D,Q) deals wi h he seasonal pa and he capi al le e s co esponds o hei coun e pa s o he seasonal models wi h he seasonal o de s and he seasonali y o he model is ep esen ed by he pa ame e s. Equa ion (3) depic s a ypical exp ession o he SARIMA model: ϕp(L)ΦP(Bs)∇d∇D sy =θq(B)ΘQ(Bs)a , (3) whe e y is he obse ed alue, ∇d and ∇D s a e he egula and seasonal di e encing ope a o s, espec i ely, pand Pa e he numbe o non-seasonal and seasonal au o eg essi e e ms, qand Q a e he numbe o non-seasonal and seasonal mo ing a e age e ms, dand Da e he numbe o egula and seasonal di e ences, ϕ and Φ depic he alue weigh s o he non-seasonal and seasonal au o eg essi e e m, θ and Θ ep esen he weigh s o he non-seasonal and seasonal mo ing a e age e m, he seasonali y is ep esen ed by S, and a is he noise e m. The i s s ep is o iden i y he SARIMA s uc u e assis ed by he obse a ion o he simple and pa ial au oco ela ion unc ion (ACF and PACF) o he ime se ies. In addi ion, da a mus be s a iona i y in mean, using powe ans o ma ions, and s a iona i y in a iance, employing di e encing o he ime se ies. Second, he pa ame e s o he model a e es ima ed. Thi d, he es ima ed esiduals mus be checked. The equi emen s o a whi e noise p ocess should be sa is ied by he esiduals and hey a e assumed o be independen . Se e al Appl. Sci. 2020,10, 8326 5 o 17 s a is ic es s and plo s o he esiduals a e used o his pu poses. Finally, an es ima ion o he u u e alue o he olume o con aine s is ob ained. In his wo k, a alue ange o he model pa ame e s a e i e a i ely es ed o iden i y he mos sui able model. 2.3. Suppo Vec o Machines o Reg ession (SVR) Models In con as o ANN models, suppo ec o machines (SVM) is a kind o machine lea ning echnique ocused on he s uc u al isk minimiza ion ins ead o he empi ical isk minimiza ion p inciple. The main objec i e o his me hod is maximizing he ma gin dis ance [ 21 ]. The model can be o mula ed as he ollowing equa ion: y(x) = wTφ(x) + b, (4) whe e bdeno es he bias e m and wis he ec o o weigh s. φ (x) ep esen s he ke nel unc ion used o deal wi h he nonlinea p oblem, mapping he inpu da a in o a highe dimensional ( ea u e) space whe e da a can be linea . Fi s in oduced o classi ica ion p oblems, he ε -insensi i e loss unc ion, p esen ed in Equa ion (5), has enabled i s use in eg ession p oblems: Lε(y)(0, i  (x)−y≤ε  (x)−y−ε, o he wise , (5) In Equa ion (5), ε deno es he a ea o ε -insensi i e. The p ocess is he ollowing: i s , he inpu da a a e mapped in o a new space o highe dimensional ea u es, called ea u e space, by a non-linea mapping a p io i using a ke nel ans o ma ion. The aim o his ea u e space is o de ec a linea eg ession unc ion ha can be i he ou pu da a wi h he inpu da a. This linea eg ession co esponds o he nonlinea eg ession model in he o iginal space. Equa ion (6) ep esen s he p oblem ha should be op imized: min w,b,ξ 1 2||w||2+C N P i=1 (ξ+ i+ξ− i) subjec o : w·xi+b−yi≤ε+ξ+ i yi−w·xi−bi≤ε+ξ− i ξ+ i,ξ− i≥0 (6) wi h i= 1, . . . , l . ξi− and ξi+ a e he slack a iables ha deal wi h he aining e o on he op and he bo om, espec i ely. These a iables ake ze o alues wi hin he { −ε , ε } a ea, and non-ze o alues ou side i . 1 2||w||2 is he s uc u e isk conce ning he la ness o he model and he pa ame e Cis a co ec ion ac o ha deals wi h he ade-o be ween he la ness and he e o . In his wo k, he Gaussian ke nel was chosen as ke nel unc ion. The dual op imiza ion p oblem can be sol ed wi h he Lag angian mul iplie me hod [ 21 ]. Finally, he me hod ob ains he suppo ec o s as he inal decision, which a e he obse a ions wi h non-ze o coe icien s o he Lag angian mul iplie s. 3. Fo ecas ing App oach In his s udy, a h ee-s ep p ocedu e based on a SOM-SARIMA-SVR model is p oposed in o de o p edic he daily numbe o con aine s passing h ough a sani a y acili y o a seapo . A u he objec i e is o compa e he p edic ion esul s o he p oposed h ee-s ep p ocedu e wi h he ob ained using o he possible me hodologies, such as he SVR models in a single way, he hyb id SARIMA-SVR model and he combined SOM-SVR model. The p oposed app oach combines he capabili y o SARIMA models in cap u ing he linea pa e ns and he abili y o SVR in modelling nonlinea pa e ns, oge he wi h he ad an ages o he SOM clus e ing algo i hm. The expe imen al da abase comes om he BIP o he Po o Algeci as Bay, loca ed in he Sou h o Spain. The BIP o Algeci as is he app o ed acili y whe e impo goods a e inspec ed and checked Appl. Sci. 2020,10, 8326 6 o 17 be o e en e ing he Communi y e i o y o he Eu opean Union. The Po o Algeci as Bay was he i s po in he Medi e anean Sea and he ou h po in he Eu opean con inen ela ed o he o al h oughpu in 2019 (109.4 million ons) only su passed by he po s o Ro e dam (The Ne he lands), An we p (Belgium) and Hambu g (Ge many). The da abase was p o ided by he Po Au ho i y and con ains daily eco ds o he numbe o con aine s a he Algeci as BIP om 2010 o 2014. 3.1. The P oposed Hyb id Me hodology A SOM algo i hm (s ep I) was i s ly used o di ide he whole da abase in o se e al disjoin clus e s wi h simila s a is ical p ope ies. Then, he seasonali y and he linea pa e ns we e cap u ed using a SARIMA model in each clus e (s ep II). As a esul , a se o p edic ed alues and esiduals om each clus e was ob ained. Finally, a SVR model was hen implemen ed in he hi d s ep o e each clus e o gene alize he non-linea ela ionship, ob aining he inal p edic ion (s ep III). Inpu s o he SVR model we e he o iginal and p edic ed da a om he second (SARIMA) s ep. Two hyb id app oaches we e es ed in he las s ep conside ing he con igu a ion and he numbe o inpu s. The o ecas ing pe o mance was assessed using di e en p edic ion ho izons: o he SARIMA s ep, only one-day (ph =1) p edic ion ho izon was conside ed; and o he inal (SVR) s ep, wo p edic ion ho izons we e es ed, one-day (ph =1) and se en-day (ph =7) ahead. The p edic ion is one-s ep ahead (y +ph ) in bo h cases. In he ph =1 case, he p edic ion is y +1 , and o he ph =7 case he p edic ion is y +7 . The es ima ion can be he eby modelled as a nonlinea unc ion o he np eceding alues o he ime se ies and an e o e m. This is called he au o eg essi e window (n) and i s design is p esen ed in Figu e 1. The au o eg essi e window o ph =1 (s eps II and III) is p esen ed on he op o he igu e and o ph =7 (s ep III) is showed on he bo om o he igu e. Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 o 17 The expe imen al da abase comes om he BIP o he Po o Algeci as Bay, loca ed in he Sou h o Spain. The BIP o Algeci as is he app o ed acili y whe e impo goods a e inspec ed and checked be o e en e ing he Communi y e i o y o he Eu opean Union. The Po o Algeci as Bay was he i s po in he Medi e anean Sea and he ou h po in he Eu opean con inen ela ed o he o al h oughpu in 2019 (109.4 million ons) only su passed by he po s o Ro e dam (The Ne he lands), An we p (Belgium) and Hambu g (Ge many). The da abase was p o ided by he Po Au ho i y and con ains daily eco ds o he numbe o con aine s a he Algeci as BIP om 2010 o 2014. 3.1. The P oposed Hyb id Me hodology A SOM algo i hm (s ep I) was i s ly used o di ide he whole da abase in o se e al disjoin clus e s wi h simila s a is ical p ope ies. Then, he seasonali y and he linea pa e ns we e cap u ed using a SARIMA model in each clus e (s ep II). As a esul , a se o p edic ed alues and esiduals om each clus e was ob ained. Finally, a SVR model was hen implemen ed in he hi d s ep o e each clus e o gene alize he non-linea ela ionship, ob aining he inal p edic ion (s ep III). Inpu s o he SVR model we e he o iginal and p edic ed da a om he second (SARIMA) s ep. Two hyb id app oaches we e es ed in he las s ep conside ing he con igu a ion and he numbe o inpu s. The o ecas ing pe o mance was assessed using di e en p edic ion ho izons: o he SARIMA s ep, only one-day (ph = 1) p edic ion ho izon was conside ed; and o he inal (SVR) s ep, wo p edic ion ho izons we e es ed, one-day (ph = 1) and se en-day (ph = 7) ahead. The p edic ion is one- s ep ahead (y +ph) in bo h cases. In he ph = 1 case, he p edic ion is y +1, and o he ph = 7 case he p edic ion is y +7. The es ima ion can be he eby modelled as a nonlinea unc ion o he n p eceding alues o he ime se ies and an e o e m. This is called he au o eg essi e window (n) and i s design is p esen ed in Figu e 1. The au o eg essi e window o ph = 1 (s eps II and III) is p esen ed on he op o he igu e and o ph = 7 (s ep III) is showed on he bo om o he igu e. Figu e 1. Au o eg essi e window sizes in S eps II and III and hei p edic ion ho izons (ph): one-day p edic ion ho izon (below he imeline) and se en-day p edic ion ho izon (allow he imeline). n is he size o he au o eg essi e window. Fo he ph = 7 case, he au o eg essi e window is composed o alues o he con aine se ies pe iodically sampled e e y se en days in he pas . This is due o he weekly seasonali y ound in he analysis o he au oco ela ion unc ion o he ime se ies. The main assump ion he e is ha bes p edic ion is ob ained when using as inpu s se e al samples om he pas o he same day o he week (i.e., using se e al successi e Mondays in he pas o p edic a u u e Monday). The p oposed SOM-SARIMA-SVR p ocedu e is shown g aphically in Figu e 2. ph = 7 … n = 1 nn = 2 …… … y y -7 y -14 y -n·ph y y +7 ph = 1 … n y -1 y -2 y -3 y -4 y -5 y -6 y -7 y -n·ph n = 1n = 2 n = 3 y +1 y +7 … Pas Fu u e P esen Figu e 1. Au o eg essi e window sizes in S eps II and III and hei p edic ion ho izons (ph): one-day p edic ion ho izon (below he imeline) and se en-day p edic ion ho izon (allow he imeline). nis he size o he au o eg essi e window. Fo he ph =7 case, he au o eg essi e window is composed o alues o he con aine se ies pe iodically sampled e e y se en days in he pas . This is due o he weekly seasonali y ound in he analysis o he au oco ela ion unc ion o he ime se ies. The main assump ion he e is ha bes p edic ion is ob ained when using as inpu s se e al samples om he pas o he same day o he week (i.e., using se e al successi e Mondays in he pas o p edic a u u e Monday). The p oposed SOM-SARIMA-SVR p ocedu e is shown g aphically in Figu e 2. Appl. Sci. 2020,10, 8326 7 o 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 o 17 Con aine olume da a Y Clus e 2Clus e 1 Clus e n SOM y1y2yn SARIMA 1 SARIMA 2 SARIMA n e1, p1e2, p2en, pn SVR 2 SVR n Hyb id app oach 1 Hyb id app oach 2 1 1 1 1 (, , )y pye 11() ee 1 1 1 ype n y 2 y 1 y Final Ou comes 12 , ,...,   n Yy y y SVR 1 Hyb id app oach 1 Hyb id app oach 2 1 1 1 1 (, , )y pye 11() ee 1 1 1 ype Hyb id app oach 1 Hyb id app oach 2 1 1 1 1 (, , )y pye 11() ee 1 1 1 ype Figu e 2. The o e all p ocess scheme o he SOM-SARIMA-SVR app oach. 3.1.1. S ep I: SOM A sel -o ganizing ea u e map (SOM) is i s applied o he da a in o de o spli he da abase in o se e al disjoin g oups, called clus e s, wi h simila s a is ical dis ibu ion. The assump ion is ha , in he nex s ep, a o ecas ing echnique can p edic mo e accu a e each clus e ins ead o he whole da abase. Each clus e wo ks independen ly in he second and hi d s ep. In such cases, a single SARIMA and SVR models a e applied independen ly a e decomposing he he e ogeneous da a in o smalle homogeneous egions. A p ede ined numbe o clus e s, c, a e selec ed, be o e s a ing he SOM algo i hm, o assess and compa e he pe o mance o di e en solu ions. An expe imen al amewo k was de eloped in o de o selec he op imal numbe o pas alues (nc) o he inpu ec o ( he numbe o inpu neu ons). Thus, a da abase o inpu s was designed as in Equa ion (7):   12 , , ,..., , T i k k nc ph x y y y y        , (7) whe e is each sample in he daily ime se ies om 2010 o 2014 and k means he sample pe iod explained g aphically in Figu e 1 (k = 1 o k = 7 days). Each ow is a anged ecu si ely using di e en lagged e ms (as in an au o eg essi e window). A la ge numbe o di e en expe imen s we e pe o med modi ying he inpu ec o size and he numbe o i e a ions. In o de o conside he inhe en andomness o he p ocess, 20 epe i ions o each expe imen we e ca ied ou . Figu e 2. The o e all p ocess scheme o he SOM-SARIMA-SVR app oach. 3.1.1. S ep I: SOM A sel -o ganizing ea u e map (SOM) is i s applied o he da a in o de o spli he da abase in o se e al disjoin g oups, called clus e s, wi h simila s a is ical dis ibu ion. The assump ion is ha , in he nex s ep, a o ecas ing echnique can p edic mo e accu a e each clus e ins ead o he whole da abase. Each clus e wo ks independen ly in he second and hi d s ep. In such cases, a single SARIMA and SVR models a e applied independen ly a e decomposing he he e ogeneous da a in o smalle homogeneous egions. A p ede ined numbe o clus e s, c, a e selec ed, be o e s a ing he SOM algo i hm, o assess and compa e he pe o mance o di e en solu ions. An expe imen al amewo k was de eloped in o de o selec he op imal numbe o pas alues (nc) o he inpu ec o ( he numbe o inpu neu ons). Thus, a da abase o inpu s was designed as in Equa ion (7): xi=hy ,y −1·k,y −2·k,. . . ,y −nc·ph,iT, (7) whe e is each sample in he daily ime se ies om 2010 o 2014 and kmeans he sample pe iod explained g aphically in Figu e 1(k=1 o k=7 days). Each ow is a anged ecu si ely using di e en lagged e ms (as in an au o eg essi e window). A la ge numbe o di e en expe imen s we e pe o med modi ying he inpu ec o size and he numbe o i e a ions. In o de o conside he inhe en andomness o he p ocess, 20 epe i ions o each expe imen we e ca ied ou . Appl. Sci. 2020,10, 8326 8 o 17 3.1.2. S ep II: SARIMA A SARIMA model is i ed o each clus e gene a ed by he SOM in S ep I, ob aining di e en p edic ed and esidual alues o hese clus e s. The main eason o selec SARIMA as a o ecas ing model is i s abili y o cap u e linea pa e ns. In o de o assess he o ecas ing pe o mance, and due o he de e minis ic na u e o linea models, a hold-ou alida ion echnique was applied du ing he p ocess. The da a (o each clus e ) was di ided in o wo g oups: he aining se con aining wo- hi ds o he da ase , and he es se comp ising he es o he samples. The pa ame e s o he model we e adjus ed using he aining se and he es se was used o alida e he model. Di e en alues o he model pa ame e s we e es ed using a ial-and-e o p ocedu e. Only he ph =1 p edic ion ho izon was es ed. The alues o he pa ame e s es ed wi hin each clus e a e, o he non-seasonal pa : p=0, 1, 2, 3, 4; d=0, 1, 2 and q=1, 2, 3, 4; and o he seasonal pa : s=2, 5, 7; P=0, 1, 2; D=0, 1, 2 and Q=0, 1, 2, 3. 3.1.3. S ep III: SVR A his s age, a SVR model is applied o each clus e . The h ee di e en adjus able pa ame e s, which go e ns he SVR model, we e de e mined by an i e a i e p ocess ( ial-and-e o ). The pa ame e s o he di e en SVR models es ed a e shown in Table 1. No e ha he op imum numbe o clus e s was wo. This issue is u he explained below. Table 1. Pa ame e anges used o he SVR models in he hi d s ep. The subsc ip indica es he clus e o which he pa ame e belongs o (1 o 2). Pa ame e s Clus e Values Hyb id con igu a ion: SOM-SARIMA-SVR-1 SOM-SARIMA- SVR-2 Numbe o a iables: Inpu laye 1 e1y1,p1,e1 2e2y2,p2,e2 Ou pu laye 1 e1y1 2e2y2 Au o eg essi e window size (s ep: 1): Residual (e): ne 1, 2 1:20 none, 1:20 O iginal da a (y): ny 1, 2 - none, 1:10 P edic ed alues (p): np 1, 2 - none,1:20 Pa ame e s: ε(s ep: 1): 1, 2 2(−12:−2) 2(−12:−2) γ(s ep: 1): 1, 2 2(−12:−2) 2(−12:−2) C: 1, 2 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 50, 100:1000 (s ep: 100) P edic ion ho izons 1, 2 1, 7 days Fo each clus e , he inpu s o he SVR model a e o med by he clus e ed da a o he o iginal ime se ies, y i , and hei o ecas ed alues, p i , and esiduals, e i , ob ained in he second (SARIMA) s ep (ideno es he clus e hey belong o). The p esence o absence o hese a iables in he inpu s de e mine he hyb id con igu a ions. Each a iable is so ed ecu si ely as an au o eg essi e window. The sizes o he o iginal da a, p edic ed alues and esiduals om he SARIMA s ep a e deno ed as ny, np and ne, espec i ely. Due o he educ ion o a ailable da a in he clus e s, a andomized esampling s a egy was implemen ed du ing he SVR p ocess o ensu e he independence o he esul s. Fo his wo k, a wo old c oss- alida ion (2-CV) echnique was used. This alida ion s a egy was epea ed 20 imes and he inal alue o he p edic ion pe o mances was he a e age o hese epe i ions. The 20 epe i ions we e implemen ed o each combina ion o pa ame e alues o he ange shown in Table 1. Once all he ange alues o he pa ame e s we e es ed, he pa ame e combina ion ha achie es he bes pe o mance Appl. Sci. 2020,10, 8326 9 o 17 index alues is selec ed. The o al p edic ed ime se ies is o med by adding he p edic ions o each clus e s. No e ha , as in he SARIMA model, he bes SVR model may be di e en on each clus e . Two hyb id app oaches we e p oposed and assessed. The p edic ion esul s we e ob ained o wo p edic ion ho izons, ph =1 and ph =7. SOM-SARIMA-SVR-1 model (hyb id app oach 1). This classical app oach conside s ha he ela ionship be ween he linea componen L and he nonlinea componen NL o a ime se ies is addi i e. The e o e, he ime se ies can be decomposed in hese wo independen e ms. Then, a linea o ecas ing model such as SARIMA can be applied in o de o model he linea componen and he eby o ob ain he p edic ed alues, ˆ p , and he esidual, e . Subsequen ly, a SVR model is applied o e he esiduals in o de o i he nonlinea componen NL . The main assump ion o his adi ional app oach is ha he nonlinea ela ionship o he ime se ies can be ound, i any, in he esiduals o he linea model. As a consequence, NL is a p edic ed e o and i is a unc ion o he esiduals ob ained om he linea model: NL +ph = (e ,e −ph,. . . ,e −n·ph) + ε =ˆ e +ph, (8) whe e ê is he p edic ed esidual, is he nonlinea unc ion ob ained by he SVR model, nis he size o he au o eg essi e window, ph is he p edic ion ho izon, and ε is he e o e m. Finally, he p edic ion is ob ained by adding he wo single componen s: Y +ph =ˆ L +ph +NL +ph. (9) O he p oposals in he amewo k o hese kinds o hyb id models ha e eme ged in ecen yea s. Pa icula ly, inco po a ing he p edic ed alues om a p e ious s ep and he o iginal da a wi h he esiduals as inpu s o he non-linea o ecas ing model has gained special a en ion. The ime se ies is conside ed as a unc ion o se e al a iables, such as he o iginal da a, he esiduals, and he p edic ed alues om he linea model. Some examples o his wo-s ep me hodology a e he wo ks o Khasei and Bija i [ 15 , 21 ]. Based on hese las wo ks, he ollowing hyb id app oach 2 was de eloped (a e applying he SOM me hod) conside ing wo o h ee ypes o a iables as inpu s. SOM-SARIMA-SVR-2 model (hyb id app oach 2). The ime se ies is he e conside ed a nonlinea unc ion o he o iginal da a and he esiduals and he p edic ed alues om he i s s ep: ˆ Y +ph = (y,e,ˆ p) + ε +ph, (10) whe e is he nonlinea unc ion ob ained wi h he SVR model, y is he o iginal da a, e is he esidual ob ained in he SARIMA model, and ˆ p is he p edic ed alue om he SARIMA model a ime . These a iables a e p esen ed in an au o eg essi e o m, as i is exp essed in he ollowing equa ion: ˆ Y +ph = (y ,y −1·ph,y −2·ph . . . ,y −ny·ph,e +ph,e ,e −1·ph,. . . ,e −ne·ph,ˆ p +ph,ˆ p ,ˆ p −1·ph,. . . ,ˆ p −np·ph) + ε +ph (11) whe e ne,ny and np ep esen au o eg essi e window sizes o e,yand ˆ p a iables, espec i ely. Conside ing wo o h ee a iables, and hei au o eg essi e window sizes, a la ge numbe o possible unc ions can be es ed. These wo hyb id app oaches a e used in each clus e s o med in he second s ep. Conside ing wo clus e s, he pa ame e anges used a e also p esen ed in Table 1. No e ha in cases which a iables eand pa e in ol ed, hei i s alue used as inpu is he p edic ed alue a +ph ime. The same canno be said o he o iginal da a a iable y, due o he y alue a ime +ph is he inal p edic ion pu sued. In all ins ances, he SARIMA model is necessa y o ob ain he equi ed da a used as inpu s in he hi d s ep, oge he wi h he o iginal pa e ns o he con aine se ies. Appl. Sci. 2020,10, 8326 16 o 17 he combina ion o models and he hyb idiza ion o hem. In his s udy, a combined-hyb id SOM-SARIMA-SVR o ecas ing model has been p oposed based on a h ee-s ep p ocedu e o p edic he daily numbe o con aine s passing h ough a Bo de Inspec ion Pos o a ma i ime po . To educe he complexi y o he p oblem, a clus e ing SOM is i s applied o ob ain smalle egions wi h simila s a is ical ea u es which may be easie o p edic . A SARIMA model is hen i ed wi hin each clus e o ob ain p edic ed alues and esiduals o he clus e ed da abase. Finally, a SVR model is used o o ecas each clus e independen ly using he a iables ob ained om he second s ep oge he wi h he o iginal da a as inpu s. The combina ion o each clus e esul s in he whole p edic ed ime se ies. The abo e me hodology in ol es he ad an ages o combining a o ecas ing model wi h a clus e ing echnique and he s eng hs o he hyb id models in cap u e linea and nonlinea pa e ns. The p oposed SOM-SARIMA-SVR model has been de eloped and compa ed o o he possible me hodologies implied in he p ocess (SVR, SOM-SVR and SARIMA-SVR). The esul s showed ha he SOM-SARIMA-SVR model was he mos compe i i e model, imp o ing he o ecas ing pe o mance o he es o he models conce ning he p edic ion o he con aine demand, ou pe o ming hese me hodologies. Pa icula ly, conside ing he SOM-SARIMA-SVR model, wo hyb id app oaches we e assessed: he classical addi i e app oach, whe e he e o e m (e) o he linea model is he inpu o he nonlinea model; and he p oposed app oach, whe e he p edic ion is a unc ion o he o iginal da a (y) and he e o e m (e), and he p edic ion o he linea model (p). Mos accu a e esul s we e yielded by he second app oach in all cases es ed. In addi ion, he e a e no signi ican di e ences be ween he p edic ion pe o mance using bo h p edic ion ho izons. The same occu s wi h he in oduc ion o ce ain a iables as inpu s. These ou comes highligh he obus ness o he model. This in es iga ion sugges s he eby ha he p oposed hyb id app oach achie es be e ou comes ge ing highe o ecas ing pe o mance han he classical addi i e hyb id app oach. To conclude, his s udy is he i s one in using he SOM-SARIMA-SVR model o o ecas he olume o con aine s passing h ough a BIP o po s in pa icula , and o ime se ies o ecas ing in gene al wi h p omising esul s. Due o i s abili y o seize he s eng hs o linea and nonlinea models and he eby o cap u e linea and nonlinea pa e ns, he p oposed model could be applied in o he ime se ies whe e hese pa e ns a e join ly p esen ed. Pa icula ly, he use o a clus e ing echnique allows educing he complexi y o he ime se ies, inc easing he accu acy o he inal p edic ion. Ob iously, some limi a ions a e ound in he model. As wi h any da a-d i en model, when comple ely di e en inpu s come in o he model, he p edic ion accu acy could wo se. Thus, a e ained model and a eadjus men o he pa ame e s migh be equi ed o e ime. Fu u e wo ks will ocus on he applica ion o o he la es non-linea echniques, such as Deep Lea ning, in o de o assess any imp o emen s in he p edic ion. Knowing he daily con aine demand in ad ance allows de ec ing wo kload peaks in a po acili y. This gua an ees he co ec planning and o ganiza ion o a ailable human and ma e ial esou ces. The p oposed me hodology can p o ide an au oma ic ool o p edic wo kloads a sani a y acili ies a oiding conges ion and delays. The e o e, i can be used as a decision-making ool by po manage s due o i s capaci y o plan esou ces in ad ance. Au ho Con ibu ions: Concep ualiza ion, J.J.R.-A. and I.T.; me hodology, J.J.R.-A.; so wa e, J.J.R.-A., J.A.M.-L., D.U.; alida ion, J.J.R.-A., D.U. and J.G.-E.; o mal analysis, J.J.R.-A.; in es iga ion, J.J.R.-A. and J.A.M.-L.; esou ces, D.U. and J.G.-E.; da a cu a ion, J.J.R.-A. and J.A.M.-L.; w i ing—o iginal d a p epa a ion, J.J.R.-A. and J.A.M.-L.; w i ing— e iew and edi ing, D.U., J.G.-E. and I.T.; isualiza ion, J.J.R.-A.; supe ision, J.J.R.-A. and I.T.; p ojec adminis a ion, I.T.; unding acquisi ion, I.T. All au ho s ha e ead and ag eed o he published e sion o he manusc ip . Funding: This esea ch was unded by MICINN (Minis e io de Ciencia e Inno aci ó n-Spain), g an numbe RTI2018-098160-B-I00. Acknowledgmen s: The da abase was kindly p o ided by he Po o Algeci as Bay Au ho i y. Con lic s o In e es : The au ho s decla e no con lic o in e es . Appl. Sci. 2020,10, 8326 17 o 17 Re e ences 1. Xie, G.; Wang, S.; Zhao, Y.; Lai, K.K. Hyb id app oaches based on LSSVR model o con aine h oughpu o ecas ing: A compa a i e s udy. Appl. So Compu . 2013,13, 2232–2241. [C ossRe ] 2. Box, G.E.P.; Jenkins, G.M. Time Se ies Analysis: Fo ecas ing and Con ol; Holden-Day: Oakland, CA, USA, 1976. 3. Gamba della, L.M.; Bon empi, G.; Tailla d, E.; Romanengo, D.; Raso, G.; Pie ma i, P. Simula ion and o ecas ing in in e modal con aine e minal. In P oceedings o he 8 h Eu opean Simula ion Symposium; SCS In e na ional: Genoa, I aly, 1976; pp. 626–630. 4. Babcock, M.W.; Lu, X. Fo ecas ing inland wa e way g ain a ic. T ansp. Res. Pa E Logis . T ansp. Re . 2002 , 38, 65–74. [C ossRe ] 5. Klein, A. Fo ecas ing he An we p ma i ime a ic lows using ans o ma ions and in e en ion models. J. Fo ecas . 1998,15, 395–412. [C ossRe ] 6. Peng, W.-Y.; Chu, C.-W. A compa ison o uni a ia e me hods o o ecas ing con aine h oughpu olumes. Ma h. Compu . Model. 2009,50, 1045–1057. [C ossRe ] 7. Zhang, G.P. Time se ies o ecas ing using a hyb id ARIMA and neu al ne wo k model. Neu ocompu ing 2003 , 50, 159–175. [C ossRe ] 8. Kohonen, T. Sel O ganising Maps; Sp inge : Be lin, Ge many, 1995. 9. De Bod , E.; Co ell, M.; Le emy, P.; Ve leysen, M. On he use o sel -o ganizing maps o accele a e ec o quan iza ion. Neu ocompu ing 2004,56, 187–203. [C ossRe ] 10. Vlahogianni, E.I.; Ka la is, M.G.; Golias, J.C. Tempo al E olu ion o Sho -Te m U ban T a ic Flow: A Nonlinea Dynamics App oach. Compu . Ci . In as uc . Eng. 2008,23, 536–548. [C ossRe ] 11. Chen, H.; G an -Mulle , S.; Mussone, L.; Mon gome y, F. A s udy o hyb id neu al ne wo k app oaches and he e ec s o missing da a on a ic o ecas ing. Neu al Compu . Appl. 2001,10, 277–286. [C ossRe ] 12. Ismail, S.; Shab i, A.; Samsudin, R. A hyb id model o sel -o ganizing maps (SOM) and leas squa e suppo ec o machine (LSSVM) o ime-se ies o ecas ing. Expe Sys . Appl. 2011,38, 10574–10578. [C ossRe ] 13. Ma, Z.; Xing, J.; Mesbah, M.; Fe ei a, L. P edic ing sho - e m bus passenge demand using a pa e n hyb id app oach. T ansp. Res. Pa C Eme g. Technol. 2014,39, 148–163. [C ossRe ] 14. Tseng, F.-M.; Yu, H.-C.; Tzeng, G.-H. Combining neu al ne wo k model wi h seasonal ime se ies ARIMA model. Technol. Fo ecas . Soc. Chang. 2002,69, 71–87. [C ossRe ] 15. Khashei, M.; Bija i, M. A no el hyb idiza ion o a i icial neu al ne wo ks and ARIMA models o ime se ies o ecas ing. Appl. So Compu . 2011,11, 2664–2675. [C ossRe ] 16. Wang, J.-J.; Wang, J.-Z.; Zhang, Z.-G.; Guo, S.-P. S ock index o ecas ing based on a hyb id model. Omega 2012,40, 758–766. [C ossRe ] 17. Chen, K.-Y.; Wang, C.-H. A hyb id SARIMA and suppo ec o machines in o ecas ing he p oduc ion alues o he machine y indus y in Taiwan. Expe Sys . Appl. 2007,32, 254–264. [C ossRe ] 18. Zhu, B.; Wei, Y. Ca bon p ice o ecas ing wi h a no el hyb id ARIMA and leas squa es suppo ec o machines me hodology. Omega 2013,41, 517–524. [C ossRe ] 19. Kohonen, T. Sel -o ganized o ma ion o opologically co ec ea u e maps. Biol. Cybe n. 1982 ,43, 59–69. [C ossRe ] 20. Vapnik, V.N. S a is ical Lea ning Theo y; John Wiley and Sons: New Yo k, NY, USA, 1998. 21. Khashei, M.; Bija i, M. An a i icial neu al ne wo k (p, d, q) model o imese ies o ecas ing. Expe Sys . Appl. 2010,37, 479–489. [C ossRe ] Publishe ’s No e: MDPI s ays neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a ilia ions. © 2020 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h p://c ea i ecommons.o g/licenses/by/4.0/).