Insiu de Recerca en Economia Aplicada Regional i Pública Research Insiue of Applied Economics Documen de Treball 2013/21, 23 pàg. Working Paper 2013/21, 23 pag. Grup de Recerca Anàlisi Quaniaiva Regional Regional Quaniaive Analysis Research Group Documen de Treball 2013/13 23 pàg. Working Paper 2013/13, 23 pag. Tourism demand forecasing wih differen neural neworks models Oscar Claveria, Enric Mone and Salvador Torra
Research Insiue of Applied Economics Working Paper 2013/21, pàg. 2 Regional Quaniaive Analysis Research Group Working Paper 2013/13, pag. 2 WEBSITE: www.ub-irea.com CONTACT: irea@ub.edu WEBSITE: www.ub.edu/aqr/ CONTACT: aqr@ub.edu Universia de Barcelona Av. Diagonal, 690 08034 Barcelona The Research Insiue of Applied Economics (IREA) in Barcelona was founded in 2005, as a research insiue in applied economics. Three consolidaed research groups make up he insiue: AQR, RISK and GiM, and a large number of members are involved in he Insiue. IREA focuses on four prioriy lines of invesigaion: (i) he quaniaive sudy of regional and urban economic aciviy and analysis of regional and local economic policies, (ii) sudy of public economic aciviy in markes, paricularly in he fields of empirical evaluaion of privaizaion, he regulaion and compeiion in he markes of public services using sae of indusrial economy, (iii) risk analysis in finance and insurance, and (iv) he developmen of micro and macro economerics applied for he analysis of economic aciviy, paricularly for quaniaive evaluaion of public policies. IREA Working Papers ofen represen preliminary work and are circulaed o encourage discussion. Ciaion of such a paper should accoun for is provisional characer. For ha reason, IREA Working Papers may no be reproduced or disribued wihou he wrien consen of he auhor. A revised version may be available direcly from he auhor. Any opinions expressed here are hose of he auhor(s) and no hose of IREA. Research published in his series may include views on policy, bu he insiue iself akes no insiuional policy posiions. 2
Research Insiue of Applied Economics Working Paper 2013/21, pàg. 3 Regional Quaniaive Analysis Research Group Working Paper 2013/13, pag. 3 Absrac This paper aims o compare he performance of differen Arificial Neural Neworks echniques for ouris demand forecasing. We es he forecasing accuracy of hree differen ypes of archiecures: a muli-layer percepron, a radial basis funcion and an Elman nework. We also evaluae he effec of he memory by repeaing he experimen assuming differen opologies regarding he number of lags inroduced. We used ouris arrivals from all he differen counries of origin o Caalonia from 2001 o 2012. We find ha muli-layer percepron and radial basis funcion models ouperform Elman neworks, being he radial basis funcion archiecure he one providing he bes forecass when no addiional lags are incorporaed. These resuls indicae he poenial exisence of insabiliies when using dynamic neworks for forecasing purposes. We also find ha for higher memories, he forecasing performance obained for longer horizons improves, suggesing he imporance of increasing he dimensionaliy for long erm forecasing. Keywords: ourism demand; forecasing; arificial neural neworks; muli-layer percepron; radial basis funcion; Elman neworks; Caalonia JEL classificaion: L83; C53; C45; R11 Oscar Claveria. AQR Research Group-IREA. Deparmen of Economerics. Universiy of Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain. E-mail: oclaveria@ub.edu Enric Mone. Deparmen of Signal Theory and Communicaions, Polyechnic Universiy of Caalunya (UPC). Salvador Torra. AQR Research Group-IREA. Deparmen of Economerics. Universiy of Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain. E-mail: sorra@ub.edu Acknowledgemens We wish o hank Núria Caballé a he Observaori de Turisme de Caalunya for providing us wih he daa used in he sudy. 3
1. Inroducion There has been a growing ineres in ourism demand forecasing over he pas decades. Some of he reasons for his increase are he consan growh of world ourism, he availabiliy of more advanced forecasing echniques and he requiremen for more accurae forecass of ourism demand a he desinaion level. Caalonia (Spain) is one of he world s maor ouris desinaions. More han 15 million foreign visiors came o Caalonia in 2012, a 3.7% rise wih respec o he previous year. Tourism accouns for 12% of GDP and provides employmen for 15% of he working populaion in Caalonia. Therefore, accurae forecass of ourism volume a he desinaion level play a maor role in ourism planning as hey enable desinaions o predic infrasrucure developmen needs. The las couple of decades have seen many sudies of inernaional ourism demand forecasing, bu due o he insufficien daabases available few sudies have been underaken a a regional level. Despie he consensus on he need o develop more accurae forecass and he recogniion of heir corresponding benefis, here is no one model ha sands ou in erms of forecasing accuracy (Song and Li, 2008; Wi and Wi, 1995). Following Coshall and Charlesworh (2010), sudies of ourism demand forecasing can be divided ino causal economeric models and non-causal ime series models. Neverheless, here has been an increasing ineres in Arificial Neural Neworks (ANN) due o conroversial issues relaed o how o model he seasonal and rend componens in ime series and he limiaions of linear mehods. ANN have been applied in he many fields, bu only recenly o ourism demand forecasing (Kon and Turner, 2005; Palmer e al, 2006; Chen, 2011, Teixeira and Fernandes, 2012). Neural neworks can be divided ino hree ypes regarding heir learning sraegies: supervised learning, non-supervised learning and associaive learning. The neuronal nework archiecure mos widely used in ourism demand forecasing is he muli-layer percepron (MLP) mehod based on supervised learning (Paie and Snyder, 1996; Fernando e al, 1999; Uysal and El Roubi, 1999; Law, 1998, 2000, 2001; Law and Au, 1999, Burger e al, 2001; Tsaur e al, 2002; Claveria e al, 2014). MLP neural neworks consis of differen layers of neurons (linear combiners followed by a sigmoid non lineariy) wih a layered conneciviy. 2
An alernaive approach is he radial basis funcion (RBF) archiecure. RBF neworks consis of a linear combinaion of radial basis funcions such as kernels cenred a a se of cenroids wih a given spread. Lin e al (2013) have recenly compared he forecas accuracy of RBF neworks o ha of MLP and Suppor Vecor Regression (SVR) neworks. In MLP and RBF neworks informaion abou he conex is inroduced ino he inpu vecor by he concaenaion of several observaion vecors. In his sudy he conex is composed of pas values of he ime series. Whils MLP neural neworks are increasingly used wih forecasing purposes, oher more compuaionally expensive archiecures such as he Elman neural nework have been scarcely used in ourism demand forecasing. Elman neworks are a special archiecure of he class of recurren neural neworks (RNN). The opology of Elman neworks follows ha of a MLP nework wih feedback from he hidden layer neuron s acivaion. The Elman archiecure akes ino accoun he emporal srucure of he ime series by means of a feedback of he acivaions of he hidden layer. Cho (2003) used he Elman archiecure o predic he number of arrivals from differen counries o Hong Kong. As i can be seen, in spie of he increasing ineres in machine learning mehods for ime series forecasing, very few sudies compare he accuracy of differen neural neworks archiecures for ourism demand forecasing. Addiionally, he scarce informaion available a a regional level, resuls in a very limied number of published aricles which make use of such daa. This led us o compare he forecasing performance of hree differen arificial neural neworks archiecures (MLP, RBF and Elman) o predic inbound inernaional ourism demand o Caalonia. We used pre-processed official saisical daa of arrivals o Caalonia from he differen counries of origin. Several measures of forecas accuracy and he Diebold- Mariano es for significan differences beween each wo compeing series are compued for differen forecas horizons (1, 3 and 6 monhs) in order o assess he value of he differen models. We repeaed he experimen assuming differen opologies regarding he memory values so as o evaluae he effec of he memory on he forecasing resuls. The memory denoes he number of lags used for concaenaion when running he models. The srucure of he paper is as follows. Secion 2 briefly describes each ype of neworks used in he analysis. The daa se is described in Secion 3. In Secion 4 resuls of he forecasing compeiion are discussed. Concluding remarks are given in Secion 5. 3
2. Mehodology The use of Arificial Neural Neworks for ime series forecasing has aroused grea ineres in he pas wo decades. One of he feaures for which neural-based forecasing is increasingly applied is ha ANN are universal funcion approximaors capable of mapping any linear or nonlinear funcion under cerain condiions. As opposed o ime series linear models, and due o heir flexibiliy, ANN models lack a sandard sysemaic procedure for model building. The specificaion of he model is based on he knowledge of he problem a hand. Obaining a reliable neural model involves selecing a large number of parameers experimenally and require cross-validaion echniques (Bishop, 1995). Zhang e al (1998) reviewed he main ANN modelling issues: he nework archiecure (deermining he number of inpu nodes, hidden layers, hidden nodes and oupu nodes), he acivaion funcion, he raining algorihm, he raining sample and he es sample, as well as he performance measures. ANN models have hree learning mehods: supervised learning, non-supervised learning and associaive learning. Depending on he way in which he differen layers are linked, neworks can also be classified as: feed forward, cascade forward, radial and recurren. The neuronal nework model mos widely used in ime series forecasing is he muli-layer percepron mehod, which is based on supervised learning. To a lesser exen, radial basis funcion and Elman neural neworks are increasingly used for forecasing purposes. In his secion we presen he hree neural neworks archiecures used in he sudy: he muli-layer percepron nework, he radial basis funcion nework and he Elman nework. 2.1. Muli-layer percepron (MLP) neural nework The muli-layer percepron archiecure is he neuronal nework model mos frequenly used in ime series forecasing. The MLP is a supervised neural nework ha uses as a building block a simple percepron model. The opology consiss of layers of parallel perceprons, wih connecions beween layers ha include opimal connecions ha eiher skip a layer or inroduce a cerain kind of feedback. As described in Cybenko (1989), a nework wih one hidden layer can approximae a wide class of funcions as 4
long as i is given he adequae weighs. The number of neurons in he hidden layer deermines he MLP nework s capaciy o approximae a given funcion. In order o solve he problem of overfiing, he number of neurons ha bes performs on unseen daa can be esimaed eiher by regularizaion or by cross validaion (Masers, 1993). In his work we used he MLP specificaion suggesed by Bishop (1995): y x i 1, x 1, x 2,, x, i 1,, p, 1,, q i, 1,, q 0 q g 1 p i1 i x i p 0, i 1,, p (1) Where y is he oupu vecor of he MLP a ime ; g is he nonlinear funcion of he neurons in he hidden layer; x is he inpu value a ime i i where i sands for he memory (he number of lags ha are used o inroduce he conex of he acual observaion.); q is he number of neurons in he hidden layer; i are he weighs of neuron connecing he inpu wih he hidden layer; and are he weighs connecing he oupu of he neuron a he hidden layer wih he oupu neuron. Noe ha he oupu y in our sudy is he esimae of he value of he ime series a ime 1, while he inpu vecor o he neural nework will have a dimensionaliy of p 1. Once he opology of he neural nework is decided (i.e. he number of layers, he form of he nonlineariies, ec.), he parameers of he nework ( i and ) are esimaed. The esimaion can be done by means of differen algorihms, which are eiher based on gradien search, line search or quasi Newon search. A summary of he differen algorihms can be found in Bishop (1995). Anoher aspec o be aken ino accoun, is he fac ha he raining is done by ieraively esimaing he value of he parameers by local improvemens of he cos funcion. Therefore, here is he possibiliy ha he search for he opimum value of he parameers finishes in a local minimum. In order o parially solve his problem, he mulisaring echnique, iniializes he neural nework several imes for differen iniial random values, and reurns he bes of he resuls. We have considered a MLP p; q archiecure ha represens he possible nonlinear relaionship beween he inpu vecor x i and he oupu vecor y. 2.2. Radial basis funcion (RBF) neural nework 5
The radial basis funcion neural nework was firs formulaed by Broomhead and Lowe (1988). RBF neworks consis of a linear combinaion of radial basis funcions such as kernels cenred a a se of cenroids wih a given spread ha conrols he volume of he inpu space represened by a neuron (Bishop, 1995; Haykin, 1999). RBF neworks ypically include hree layers: an inpu layer; a hidden layer, which consiss of a se of neurons, each of hem compuing a symmeric radial funcion; and an oupu layer ha consiss of a se of neurons, one for each given oupu, linearly combining he oupus of he hidden layer. The inpu can be modelled as a feaure vecor of real numbers, and he hidden layer is formed by a se of radial funcions cenred each a a cenroid. The oupu of he nework is a scalar funcion of he oupu vecor of he hidden layer. The equaions ha describe he inpu/oupu relaionship of he RBF are: y 0 q 1 g x i g x x i 1, x, x, 1,, q i exp 1 2 p 1 x i 2,, x 2 p 2, i 1,, p (2) Where y is he oupu vecor of he RBF a ime ; are he weighs connecing he oupu of he neuron a he hidden layer wih he oupu neuron; q is he number of neurons in he hidden layer; shape; x is he inpu value a ime i g is he acivaion funcion, which usually has a Gaussian i where i sands for he memory (he number of lags ha are used o inroduce he conex of he acual observaion); vecor for neuron ; and he spread is he cenroid is a scalar ha measures he widh over he inpu space of he Gaussian funcion and i can be defined as he area of influence of neuron in he space of he inpus. Noe ha he oupu y in our sudy is he esimae of he value of he ime series a ime 1, while he inpu vecor o he neural nework will have a dimensionaliy of p 1. In order o assure a correc performance, before he raining phase he number of cenroids and he spread of each cenroids have o be seleced. The selecion of he number of hidden nodes mus ake ino accoun he rade-off beween he error in he 6
raining se and he generalizaion capaciy, which indicaes he performance over samples no used in he raining phase. In he limi case, assigning a cenroid o each inpu vecor and using a spread of high value would yield a look up able ha would have a low performance on unseen daa. Tha means ha he value of he exponenial is such ha he oupu of he neuron is high for a disance beween he observaion and he cenroid equal o zero, and he oupu is zero when he disance is differen from zero. Therefore, he cenroids and he spread of each neuron should be seleced so ha he performance on unseen daa is accepable. There are differen mehods for he esimaion of he number of cenroids and he spread of he nework. A complee summary can be found in Haykin (1999). In his sudy he raining was done by adding he cenroids ieraively wih he spread parameer fixed. Then a regularized linear regression was esimaed o compue he connecions beween he hidden and he oupu layer. Finally, he performance of he nework was compued on he validaion daa se. This process was repeaed unil he performance on he validaion daabase ceased o decrease. The spread is a hyperparameer, in he sense ha i is seleced before deermining he opology of he nework, and i is uned ouside he raining phase. Alhough a differen value of could be seleced for each neuron, usually a common value is used for all he neurons. 2.3. Elman neural nework The Elman nework, which is a special archiecure of he class of recurren neural neworks, i was firs proposed by Elman (1990). The archiecure is based on a hreelayer nework wih he addiion of a se of conex unis ha allow feedback on he inernal acivaion of he nework. There are connecions from he hidden layer o hese conex unis fixed wih a weigh of one. A each ime sep, he inpu is propagaed in a sandard feed-forward fashion. The fixed back connecions resul in he conex unis always mainaining a copy of he previous values of he hidden unis. Thus he nework can mainain a sor of sae of he pas decisions made by he hidden unis, allowing i o perform such asks as sequence-predicion ha are beyond he power of a sandard mulilayer percepron. The Elman archiecure is a ype of recurren neural nework. The oupu of he nework is hen a scalar funcion of he oupu vecor of he hidden layer: 7
y 0 q 1 z, z x, i g 1, x 1, x 2,, x, i 1,, p, 1,, q i, i 1,, p, 1,, q x, 1,, q i Where p i1 i i 0 p i z, 1, i 1,, p y is he oupu vecor of he Elman nework a ime ; (3) z, is he oupu of he hidden layer neuron a he momen ; g is he nonlinear funcion of he neurons in he hidden layer; x is he inpu value a ime i i where i sands for he memory (he number of lags ha are used o inroduce he conex of he acual observaion); i are he weighs of neuron connecing he inpu wih he hidden layer; q is he number of neurons in he hidden layer; are he weighs of neuron ha link he hidden layer wih he oupu; and i are he weighs ha correspond o he oupu layer and connec he acivaion a momen. Noe ha he oupu y in our sudy is he esimae of he value of he ime series a ime 1, while he inpu vecor o he neural nework will have a dimensionaliy of p 1. The parameers of he Elman neural nework are esimaed by minimizing an error cos funcion. In order o minimize oal error, gradien descen is used o change each weigh in proporion o is derivaive wih respec o he error, provided he nonlinear acivaion funcions are differeniable. A maor problem wih gradien descen for sandard RNN archiecures is ha error gradiens vanish exponenially quickly wih he size of he ime lag beween imporan evens. RNN may behave chaoically, due o he fac ha here is a feedback, followed by a se of sigmoid nonlineariies, and as he sign of he feedback loop is conrolled by he learning algorihm, he algorihm may develop an oscillaing behaviour, unrelaed o he obecive funcion, and he value of he weighs may diverge. In such cases, dynamical sysems heory may be used for analysis. Mos RNN presen scaling issues. In paricular, RNN canno be easily rained for large numbers of neuron unis nor for large numbers of inpus unis. Successful raining has been mosly in ime series wih few inpus. There are differen sraegies for esimaing he parameers of he Elman neural nework. Various mehods for doing so were developed by Werbos (1988), Pearlmuer 8
(1989) and Schmidhuber (1989). The sandard mehod is called backpropagaion hrough ime, and is a generalizaion of back-propagaion for feed-forward neworks. A more compuaionally expensive online varian is called real-ime recurren learning, which is an insance of auomaic differeniaion in he forward accumulaion mode wih sacked angen vecors. In his paper, he raining of he nework was done by backpropagaion hrough ime. 3. Daa Monhly daa of ouris arrivals over he ime period 2001:01 o 2012:07 were provided by he Direcció General de Turisme de Caalunya and he Saisical Insiue of Caalonia (IDESCAT). We have compued some of he mos commonly used mehods o es he uni roo hypohesis: he augmened Dickey-Fuller (ADF) es and he Kwiakowski-Phillips-Schmid-Shin (KPSS) es. While he ADF ess he null hypohesis of a uni roo in x and in he firs-differenced values of x, he KPSS saisic ess he null hypohesis of saionariy in boh x and x. Table 1 Uni roo ess on he rend-cycle series of ouris arrivals and he year-on-year raes Counry Tes for I(0) Tes for I(1) Tes for I(2) ADF KPSS ADF KPSS ADF KPSS France -2.39 0.60-3.19 0.64-5.11 0.12 Unied Kingdom -1.63 0.38-2.98 0.51-18.92 0.12 Belgium and he NL -3.56 0.24-2.49 0.21-8.43 0.02 Germany -1.93 0.50-3.54 0.33-8.76 0.15 Ialy -1.58 0.71-3.55 0.52-5.47 0.26 US and Japan 2.08 1.19-4.77 0.39-6.92 0.02 Norhern counries -1.14 1.24-3.88 0.06-11.41 0.03 Swizerland -3.26 0.38-6.14 0.07-6.20 0.16 Russia 1.80 1.06-3.62 0.65-8.37 0.04 Oher counries -1.33 1.30-4.53 0.07-9.88 0.02 Toal -1.98 0.87-2.97 0.29-12.51 0.06 1. Esimaion period 2001:01-2012:07. 2. Tess for uni roos. ADF Augmened Dickey and Fuller (1979) es, he 5% criical value is -2.88; KPSS Kwiakowski, Phillips, Schmid and Shin (1992) es, he 5% criical value is 0.46. As i can be seen in Table 1, in mos counries we canno reec he null hypohesis of a uni roo a he 5% level. Similar resuls are obained for he KPSS es, where he null hypohesis of saionariy is reeced in mos cases. When he ess were applied o he 9
firs difference of individual ime series, he null of non-saionariy is srongly reeced in mos cases. In he case of he KPSS es, we canno reec he null hypohesis of saionariy a he 5% level in any counry. These resuls imply ha differencing is required in mos cases and prove he imporance of deseasonalizing and derending ourism demand daa before modelling and forecasing. In order o eliminae boh linear rends as well as seasonaliy we used he year-onyear raes of he rend-cycle componen of he series. These series were obained using Seas/Tramo. Table 2 shows a descripive analysis of year-on-year raes of he rendcycle series beween January 2002 and July 2012. During his period, Russia and he Norhern counries experienced he highes growh in ouris arrivals. Russia is also he counry ha presens he highes dispersion in growh raes, while France shows he highes levels of Skewness and Kurosis. Table 2 Descripive analysis of he year-on-year raes of he rend-cycle series Counry Touris arrivals Mean SD Skew. Kur. France 5.06 13.69 2.13 8.93 Unied Kingdom 1.94 15.00 0.70 3.51 Belgium and NL 1.85 8.50 0.76 3.13 Germany 0.45 7.85 0.14 3.13 Ialy 5.48 14.58 0.88 3.39 US and Japan 4.77 11.14-0.08 2.64 Norhern counries 8.24 16.97 0.25 2.70 Swizerland -0.21 9.86 0.28 4.93 Russia 16.06 32.12-0.35 2.69 Oher counries 6.90 10.02-0.15 2.48 Toal 3.75 7.04-0.75 3.04 1. SD Sandard Deviaion, Skew. Skewness, Kur. Kurosis 4. Resuls In his secion we compared he forecasing performance of hree differen arificial neural neworks archiecures (muli-layer percepron, radial basis funcion and Elman recursive neural neworks) o predic arrivals o Caalonia from he differen visior counries. Following Bishop (1995) and Ripley (1996), we divided he colleced daa ino hree ses: raining, validaion and es ses. This division is done in order o asses he performance of he nework on unseen daa. The assessmen is underaken during 10
he raining process by means of he validaion se, which is used in order o deermine he epocs and he opology of he nework. The iniial size of he raining se was deermined o cover a five-year span in order o accuraely rain he neworks and o capure he differen behaviour of he ime series in relaion o he economic cycle. Afer each forecas, a reraining was done by increasing he size of he se by one period and sliding he validaion se by anoher period. This ieraive process is repeaed unil he es se consised of he las sample of he ime series. Based on hese consideraions, he firs sixy monhly observaions (from January 2001 o January 2006) were seleced as he iniial raining se, he nex hiry-six (from January 2007 o January 2009) as he validaion se and he las 20% as he es se. Noe ha he ses consis of consecuive subsamples, and he resuling validaion and es ses a he beginning of he experimen correspond o differen phases of he economic cycle. All neural neworks were implemened using Malab and is Neural Neworks oolbox. Due o he large number of possible neworks configuraions, he validaion se was used for deermining he following aspecs of he neural neworks: a. The opology of he neworks. b. The number of epocs for he raining of he MLP neural neworks. The ieraions in he gradien search are sopped when he error on he validaion se increases. c. The number of neurons in he hidden layer for he RBF. The sequenial increase in he number of neurons a he hidden layer is sopped when he error on he validaion increases. d- The value of he spread in he radial basis, which is a hyper parameer. Noe ha here are ineracions beween he differen parameers in a RBF neural nework. If he value of he spread increases, in order o cover he inpu space a much higher number of cenroids are needed. To make he sysem robus o local minima, we applied he mulisarings echnique, which consiss on repeaing each raining phase several imes. In our case, he mulisarings facor was hree and i was deermined by a compromise beween he improvemen obained by raining repeiion and he compuing ime needed for he experimen. By repeaing he raining hree imes, usually a good minimum of he performance error was obained. The selecion crierion for he opology and he parameers was he performance on he validaion se. The Elman neworks parameers and opology had o be opimized aking ino accoun ha i could yield an unsable 11
soluion such as divergen raining due o he fac ha during he raining he weighs of he feedback loop could give rise o an unsable nework. Using as a crierion he performance on he validaion se, he resuls ha are presened correspond o he selecion of he bes opology, he bes spread in he case of he RBF neural neworks, and he bes raining sraegy in he case of he Elman neural neworks. Forecass for 1,3 and 6 monhs ahead were compued in a recursive way. Tha is, afer each raining phase, we sared wih he firs es sample, hen added he sample o he validaion se, incorporaing he firs value of he validaion se o he raining se, which increases by one period. This procedure was repeaed up o he las elemen of he es se in a recursive way. This way he forecasing performance is analyzed by using a raining se ha increases as new daa are esed while leaving a consan validaion se. A poenial drawback of his recursive process is ha he fracion of daa assigned o he validaion se wih respec o he raining se is no consan. Addiionally, here are no clear crieria for deciding he size of he validaion se. In our sudy we adaped he incremenal disribuion of he daa beween raining, validaion and es ses so as o avoid he memory window o be shared beween he es se and he validaion or raining ses. In order o summarise his informaion, wo measures of forecas accuracy were compued o rank he mehods according o heir values for differen forecas horizons (1, 3 and 6 monhs): he Roo Mean Squared Error (RMSE) and he Mean Absolue Percenage Error (MAPE). The resuls of our forecasing compeiion are shown in Table 3 and Table 4. We also used he Diebold-Mariano es (Table 5) for significan differences beween each wo compeing series for each forecas horizons in order o assess he value of he differen models. We repeaed he experimen assuming differen opologies regarding he memory values. These values represen he number of lags inroduced when running he models, denoing he number of previous monhs used for concaenaion. The number of lags used in he differen experimens ranged from one o hree monhs for all he neworks archiecures. Therefore, when he memory is zero, he forecas is done using only he curren value of he ime series, wihou any addiional emporal conex. In Table 3, 4 and 5 we presen he resuls obained for he wo exreme cases: a memory of zero and memory of hree lags. 12
Table 3 MAPE (2010:04-2012:02) Memory (0) no addiional lags Memory (3) 3 addiional lags ANN models ANN models France MLP RBF Elman MLP RBF Elman 1 monh 0.33 0.34 9.02 0.06* 0.09 7.85 3 monhs 5.36 1.39 10.96 1.11 1.30 8.39 6 monhs 5.72 2.22 6.91 2.64 3.24 5.63 Unied Kingdom 1 monh 0.34 0.57 2.55 1.59 1.32 2.00 3 monhs 4.92 2.81 3.31 1.22 2.22 2.06 6 monhs 8.72 3.15 2.16 3.52 2.21 12.04 Belgium and he NL 1 monh 1.12 0.83 3.77 1.39 1.50 2.74 3 monhs 1.20 0.79 2.02 1.37 1.58 2.79 6 monhs 2.99 0.97 2.07 3.99 0.95 2.44 Germany 1 monh 5.57 4.95 12.47 6.43 6.37 16.42 3 monhs 2.01 1.83 5.92 5.72 6.66 13.76 6 monhs 2.14 3.30 4.74 7.66 8.34 16.04 Ialy 1 monh 1.32 1.84 17.63 0.77 2.18 20.35 3 monhs 9.74 10.42 24.83 8.51 5.92 23.81 6 monhs 11.76 13.45 11.52 22.76 13.56 20.39 US and Japan 1 monh 0.90 0.80 1.52 0.49 0.48 2.31 3 monhs 1.85 1.70 4.16 1.05 1.56 2.67 6 monhs 1.01 0.94 3.93 1.94 1.68 1.85 Norhern counries 1 monh 0.42 0.41 2.82 0.38 0.28 1.59 3 monhs 1.49 1.13 2.19 0.52 1.11 2.05 6 monhs 1.39 1.17 3.52 0.92 1.02 2.83 Swizerland 1 monh 1.33 1.25 2.39 1.63 1.32 1.15 3 monhs 0.83 0.65 1.47 1.60 1.12 1.74 6 monhs 0.76 0.50 2.35 0.95 0.57 1.37 Russia 1 monh 0.57 0.53 0.74 0.49 0.52 0.69 3 monhs 0.62 0.54 0.72 0.42 0.46 0.62 6 monhs 0.65 0.66 0.88 0.61 0.76 1.01 Oher counries 1 monh 0.41 0.35 1.30 0.54 0.60 1.78 3 monhs 0.92 0.64 1.91 0.50 0.51 1.81 6 monhs 1.01 0.68 1.96 0.67 0.61 3.05 Toal 1 monh 0.64 0.65 3.55 0.60 0.57 2.64 3 monhs 2.02 0.73 3.14 1.29 0.85 2.85 6 monhs 3.25 0.77 2.75 1.70 2.20 2.64 1. Ialics: bes model for each counry 2. * Bes model 13
Table 4 RMSE (2010:04-2012:02) Memory (0) no addiional lags Memory (3) 3 addiional lags ANN models ANN models France MLP RBF Elman MLP RBF Elman 1 monh 0.49 0.48 24.38 0.12* 0.31 18.71 3 monhs 6.93 1.85 20.33 2.15 1.71 18.51 6 monhs 10.28 3.71 17.14 6.63 5.48 13.41 Unied Kingdom 1 monh 3.35 7.81 20.53 5.02 6.10 13.08 3 monhs 15.27 8.85 21.03 8.11 9.54 12.07 6 monhs 23.84 9.58 17.45 12.25 14.17 19.60 Belgium and he NL 1 monh 9.63 6.35 19.58 8.73 8.50 17.25 3 monhs 7.31 3.90 19.69 7.38 8.33 14.04 6 monhs 15.30 5.07 15.87 20.06 5.46 12.67 Germany 1 monh 9.04 8.52 18.33 10.47 9.50 17.99 3 monhs 6.81 5.13 22.39 8.82 8.70 13.65 6 monhs 11.00 4.78 11.56 10.02 8.05 17.74 Ialy 1 monh 1.85 1.93 12.37 1.20 1.93 14.14 3 monhs 4.78 5.29 16.79 7.08 4.56 15.64 6 monhs 10.82 10.47 16.43 14.18 10.74 14.90 US and Japan 1 monh 6.00 4.96 15.26 5.94 5.84 18.87 3 monhs 11.15 9.88 24.53 8.86 11.13 20.51 6 monhs 12.73 15.08 20.31 11.28 10.95 13.28 Norhern counries 1 monh 5.34 5.27 22.77 3.56 3.80 20.48 3 monhs 11.71 11.25 20.04 5.15 7.65 16.87 6 monhs 16.19 15.10 26.69 15.19 12.09 28.67 Swizerland 1 monh 12.13 10.86 26.52 14.63 12.03 12.26 3 monhs 7.90 5.92 16.65 15.71 11.26 19.08 6 monhs 11.14 5.95 26.31 11.84 7.37 15.29 Russia 1 monh 33.38 28.64 38.66 25.91 28.46 36.93 3 monhs 39.13 32.53 35.19 25.99 28.93 34.12 6 monhs 39.64 37.38 56.48 37.11 41.42 59.06 Oher counries 1 monh 3.22 2.90 13.70 2.94 3.06 14.45 3 monhs 7.61 6.38 15.79 3.54 2.89 16.89 6 monhs 9.48 8.87 15.88 7.11 6.52 20.22 Toal 1 monh 3.94 3.90 17.25 4.14 4.23 15.75 3 monhs 11.40 4.83 17.72 7.28 5.28 15.32 6 monhs 21.84 4.27 13.86 14.05 13.28 12.89 1. Ialics: bes model for each counry 2. * Bes model 14
Table 5 Diebold-Mariano loss-differenial es saisic for predicive accuracy (2.028 criical value) Memory (0) no addiional lags Memory (3) 3 addiional lags MLP vs. RBF MLP vs. Elman RBF vs. Elman MLP vs. RBF MLP vs. Elman RBF vs. Elman France 1 monh 0.88-6.12* -6.08* -2.23* -6.19* -6.14* 3 monhs 1.38-4.37* -5.05* 0.33-10.12* -12.05* 6 monhs 1.36-1.95-3.64* 0.33-3.11* -3.92* Unied Kingdom 1 monh -1.62-7.10* -4.68* -1.13-4.20* -3.17* 3 monhs 0.46-1.65-2.58* -1.42-1.70-1.11 6 monhs 2.01 0.65-2.40* -1.24-1.69-0.88 Belgium and he NL 1 monh 2.50* -2.38* -3.26* 0.36-3.19* -3.28* 3 monhs 2.27* -2.62* -3.59* -0.09-2.91* -2.49* 6 monhs 2.19* -0.47-2.91* 1.67 0.61-2.54* Germany 1 monh 2.58* -3.51* -3.85* 1.64-1.99-2.38* 3 monhs 1.86-3.62* -3.92* -0.34-1.79-1.84 6 monhs 0.79-1.72-3.11* 0.82-1.20-1.75 Ialy 1 monh -1.33-5.83* -5.74* -2.89* -9.01* -8.81* 3 monhs -0.38-5.10* -4.78* 1.57-4.41* -6.29* 6 monhs -0.57-2.53* -1.77 1.03-0.25-1.85 US and Japan 1 monh 4.49* -4.98* -5.98* -0.62-4.77* -4.90* 3 monhs 0.64-4.63* -5.46* -2.73* -6.09* -3.56* 6 monhs 0.14-0.94-0.93-0.54-0.89-0.07 Norhern counries 1 monh 1.11-5.55* -5.54* -0.12-4.00* -3.83* 3 monhs 1.44-2.90* -3.06* -3.32* -7.12* -4.68* 6 monhs 0.77-2.65* -2.81* 0.62-6.64* -5.81* Swizerland 1 monh 1.52-2.76* -3.02* 2.29* 2.68* 0.50 3 monhs 1.96-1.61-1.96 2.85* 0.01-2.22* 6 monhs 1.33-5.08* -8.10* 1.33-1.50-2.65* Russia 1 monh 1.40-1.97-3.07* -2.01-3.47* -2.69* 3 monhs 1.40 0.48-1.33-1.65-1.74-0.88 6 monhs 0.91-3.18* -3.10* -1.62-4.54* -2.99* Oher counries 1 monh 0.80-5.48* -6.34* -0.25-5.69* -5.64* 3 monhs 2.94* -3.07* -3.91* 0.72-6.56* -7.17* 6 monhs 1.36-2.01-2.69* 0.03-3.75* -4.69* Toal 1 monh -0.50-6.55* -6.96* 0.45-4.01* -4.00* 3 monhs 1.02-2.92* -4.23* 0.38-7.46* -5.79* 6 monhs 2.21* 0.91-3.66* -0.45-0.75-0.55 1. Diebold-Mariano es saisic wih NW esimaor. Null hypohesis: he difference beween he wo compeing series is non-significan. A negaive sign of he saisic implies ha he second model has bigger forecasing errors. 2. * Significan a he 5% level. 15
When analysing he forecas accuracy for ouris arrivals, MLP and RBF neworks show lower RMSE and MAPE values han Elman neworks, specially for shorer horizons. RBF neworks display he lowes RMSE and MAPE values in mos counries when he memory is zero. When he forecass are obained incorporaing addiional lags of he ime series, he forecasing performance of MLP neworks improves. The lowes RMSE and MAPE value is obained wih he MLP nework for France (for 1 monh ahead) when using a memory of hree lags. When esing for significan differences beween each wo compeing series (Table 5), we find ha MLP and RBF neworks significanly ouperform Elman neworks in all counries and for all forecasing horizons. A possible explanaion for his resul is he lengh of he ime series used in he analysis. The fac ha he number of raining epocs had o be low in order o mainain he sabiliy of he nework suggess ha his nework archiecure requires longer ime series. For long raining phases, he gradien someimes diverged. The worse forecasing performance of he Elman neural neworks compared o ha of MLP and RBF archiecures for opologies wih no memory indicaes ha he feedback opology of he Elman nework could no capure he specificiies of he ime series. When comparing he forecasing performance beween MLP and RBF neworks, we find ha he RBF archiecure produces he bes forecass when he memory of he nework is se o zero, while he MLP archiecure improves is forecasing performance when a larger number of lags is incorporaed in he neworks. This resul can be explained because in his case he RBF operaes as a look up able, while he MLP ries o find a funcional relaionship lacking a conex ha migh give a hin of he slope of he ime series. As he number of lags increases, MLP neworks obain significanly beer forecass fore some counries (France, Ialy, Norhern counries and US and Japan). This resul can be explained by he fac ha as he hidden neurons linearly combine he inpu before applying he nonlineariy, and hus addiional lags can be used in a beer way o esimae he differen slopes and he fuure evoluion of he series. This evidence indicaes ha he number of previous monhs used for concaenaion condiions he forecasing performance of he differen neworks. The differences beween counries can be parly explained by differen paerns of consumer behaviour, bu hey are also relaed o he variabiliy due o he size of he sample, being France he mos imporan visior marke. When comparing he resuls for 16
differen predicion horizons, as i could be expeced he forecasing performance improves for shorer forecasing horizons. Neverheless, we find ha here is an ineracion beween he memory and he forecasing horizon. As i can be seen in Table 3 and Table 4, as he number of lags used in he neworks increases, he forecasing performance obained for longer horizons (3 and 6 monhs) improves. 5. Conclusions and discussion The obecive of he paper was o compare he forecasing performance of differen arificial neural neworks models, exending o ouris demand forecasing he resuls of previous research on economics. Wih his aim, we have carried ou a forecasing comparison beween hree archiecures of arificial neural neworks: he muli-layer percepron neural nework, he radial basis funcion neural nework and he Elman recursive neural nework. Using hese hree differen ses of models we obained forecass for he number of ouriss from all visior markes o Caalonia. When comparing he forecasing accuracy of he differen echniques, we find ha muli-layer percepron and radial basis funcion neural neworks ouperform Elman neural neworks. These resul sugges ha issues relaed wih he divergence of he Elman neural nework may arise when using dynamic neworks wih forecasing purposes. The comparison of he forecasing performance beween muli-layer percepron and radial basis funcion neural neworks permi o conclude ha he RBF neworks significanly ouperform he MLP neworks when no addiional lags are inroduced in he neworks. On he conrary, when he inpu has a conex of he pas, MLP neworks show a beer forecasing performance. We also find ha as he amoun of previous monhs used for concaenaion increases, he forecass obained for longer horizons improve, suggesing he imporance of increasing he dimensionaliy of he inpu o neworks for long erm forecasing. An inpu ha akes ino accoun a longer conex, migh capure no only he rend of he curren value, bu also possible cycles ha influence he forecas. These resuls show ha he number of lags inroduced in he neworks plays a fundamenal role on he forecasing performance of he differen archiecures. Summarising, he forecasing compeiion reveals he suiabiliy of applying mulilayer percepron and radial basis funcion neural neworks models o ourism demand forecasing. A quesion o be considered in furher research is wheher he 17
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