Forecasing and Forecas Combinaion in Airline Revenue Managemen Applicaions Chrisiane Lemke 1, Bogdan Gabrys 1 1 School of Design, Engineering & Compuing, Bournemouh Universiy, Unied Kingdom. E-mail: {clemke, bgabrys}@bournemouh.ac.uk Absrac. Predicing a variable for a fuure poin in ime helps planning for unknown fuure siuaions and is common pracice in many areas such as economics, finance, manufacuring, weaher and naural sciences. This paper invesigaes and compares approaches o forecasing and forecas combinaion ha can be applied o service indusry in general and o airline indusry in paricular. Furhermore, possibiliies o include addiionally available daa like passenger-based informaion are discussed. Keywords. Forecasing, Forecas Combinaion, Adapive Forecasing, Airline Indusry 1 Inroducion Modern revenue managemen sysems significanly increase revenues of airline companies. Airline ickes are usually sold for several booking classes differing in price and booking condiions. Passengers buying higher class ickes are willing o pay a higher price and hus conribue more o airline revenues han low fare passengers, which is why airlines would like o give prioriy o hem. However, hose higher class bookings usually arrive quie shorly before deparure, so i becomes necessary o forecas he demand for higher class ickes o be able o reserve an appropriae number of ickes. Two variables are imporan for his ask: he demand and he cancellaion rae for airline ickes. A collaboraion beween Bournemouh Universiy and Lufhansa Sysems Berlin in a previous projec showed promising resuls and significan improvemens for forecasing he demand using novel forecas combinaion approaches ([1], [2], [3]). A new projec whose firs resuls are presened in his paper aims a invesigaing if similar improvemens can be achieved for cancellaion raes. 1.1 Airline Revenue Managemen Revenue managemen ries o maximise profis by invesigaing and forecasing cusomer behaviour and drawing appropriae conclusions. In he airline indusry, he
cenral objecive of revenue managemen is deermining how many seas for each booking class should be sold prior o deparure. The risk of rejecing a booking in a low class in order o wai for a higher class booking has o be judged. Forecasing of icke demand wih corresponding no-show and cancellaion raes are a crucial componen o his. McGill and van Ryzin give an exensive review of research in airline revenue managemen [4]. One of he fundamenal developmens described is abandoning radiional single-leg approaches only considering individual flighs in favour of so called Origin and Desinaion (O&D) approaches, which work combinaions of all connecions, iineraries and booking classes. Forecasing has a long rack record in airline indusry, mainly because forecasing fuure demand has a direc influence on he booking limis for he differen fare classes [4]. Fully occupied flighs are of boh ecological and economical ineres; bu a number of seas usually say unoccupied even on sold-ou flighs due o cancellaions or so called no-show passengers. Forecasing in his conex is a crucial help for a reasonable overbooking, finding a balance beween he number of unoccupied seas and he number of denied boardings. The sae-of-he-ar mehod for forecasing demand and cancellaion raes uses a saisical model ha akes he currenly observed rae and he reference raes obained from hisorically similar flighs ino accoun. Several approaches can be pursued when rying o improve a forecas; wo have been idenified in he scope of his work: employing and improving radiional ime series mehods and exracing informaion abou cusomer behaviour. 1.2 Tradiional ime-series forecasing Forecasing and forecas combinaion is a well-researched area ([5], [6], [7]). Time series forecasing looks a sequences of daa poins, rying o idenify paerns and regulariies in heir behaviour ha migh also apply o fuure values. A large number of ime series forecasing mehods wih differen degrees of flexibiliy and complexiy are available; consequenly, here are many ways o generae forecass and one migh end up wih more han one forecas for he same problem. This leads o he quesion, wheher or no some or all of he individual forecass can be combined o obain a superior forecas. General forecasing and forecas combinaion mehods are discussed in he secions wo and hree of his paper, secion four gives an empirical evaluaion of popular and easy-o-use approaches. 1.3 Passenger-based predicing In many real world applicaions, informaion ha goes beyond ordinary ime series daa is ofen available. In services indusry, daa is ofen colleced on a cusomer basis and can be ulised wih daa mining mehods ha help modelling and undersanding various groups of cusomer behaviour. For example, daa mining is used for cusomer relaionship managemen in reail indusry ([8]), for credi card fraud deecion ([9]) or for marke baske analysis idenifying associaions beween buying choices of cusomers ([10]).
In he las years, a few publicaions sugges ha including informaion gained from so-called Passenger Name Records (PNR) in airline forecasing applicaions migh be beneficial. In [11] and [12], Neuling e al. and Lawrence e al. look a forecasing noshow raes, i.e. he rae of passengers who book a icke and fail o show up for a fligh wihou cancelling i. Boh publicaions find ha making use of PNR daa improves forecasing performance compared o pure ime series approaches. 2 Forecasing This secion describes a selecion of popular ime series forecasing echniques inspired by a book of Makridakis e al. ([5]). I provides he background for he furher secions of his paper. In he formulas used, he ime series will have he pas observaions { y 1,.., y } and he one-sep-ahead forecas o calculae will be denoed by y ˆ + 1. 2.1 Averaging and smoohing A simple approach o forecasing is aking he arihmeic mean of he k mos recen values of he ime series as shown in formula (2-1). In ha way, old and poenially inapplicable values can be discarded. 1 k yˆ + 1 = y i (2-1) i= k + 1 Exponenial smoohing mehods apply weighs ha decay exponenially wih ime and hus also rely on he assumpion ha more recen observaions are likely o be more imporan for a forecas han hose lying furher in he pas. Single exponenial smoohing as he simples represenaive of smoohing mehods is calculaed by he previous forecas adjused by he error i produced, see formula (2-2). The parameerα conrols he exen of he adjusmen. yˆ y (1 ) yˆ + 1 = α + α (2-2) Many exensions and variaions o he basic smoohing algorihm have been proposed, see for example [13]. 2.2 Regression Regression approaches express a forecas or dependen variable as a funcion of one or more independen or explanaory variables ha relae o he oucome. Simple linear regression on a single variable x can be expressed wih formula (2-3), where a is he inercep, b he slope of he line and ε he error, ha originaes from he deviaion of he linear relaionship from he acual observaion. y = a + bx + ε (2-3) In he case of ime series, x denoes he ime index. The parameers of he regression can be esimaed using sandard leas squares approaches.
2.3 Decomposiion and Thea-Model Decomposiion aims a isolaing componens of a ime series, projecing hem separaely ino he fuure and hen recombining hem o produce a final forecas. The componens are radiionally - a rend-cycle, denoing long-erm changes, - seasonaliy, reflecing shorer-erm consan-lengh changes like monhs or holiday imes and - an irregular or random error componen. Recenly, Assimakopoulos and Nikolpoulos proposed he Thea-model in [14]. I decomposes seasonally adjused series ino shor and long erm componens by applying a coefficien θ o he second order differences of he ime series as shown in formula (2-4), hus modifying he curvaure of he ime series. y '' new ( θ ) = θ * y' ' original (2-4) Thea values bigger han one dilae he series, amplifying is shor erm-behaviour, while hea values beween zero and one have he opposie effec. 2.4 ARIMA Auoregressive inegraed moving average models (ARIMA) according o [15] are a powerful and complex ool of modelling and forecasing ime series. They are described by he noaion ARIMA (p,d,q) and consis of he following hree pars: 1. AR(p) denoes an auoregressive par of order p, auoregression is a regression where he arge variable depends on ime-lagged values of iself. 2. I(d) defines he degree of differencing involved. Differencing is a mehod of removing non-saionariy by calculaing he change beween each observaion. 3. MA(q) indicaes he order of he moving average par of he model, which is given as a moving average of he error series. I can be described as a regression agains he pas error values of he series. 2.5 Nonlinear Forecasing Regime swiching models are a popular class of nonlinear forecasing mehods, combining wo or more ses of model coefficiens in one sysem. Which se o apply for a forecasing siuaion is hen deermined by looking a he regime or sae he sysem is likely o be in. A simple example for a model wih wo auoregressive regimes of order one and he parameersα i, βi, χ, he error componen ε i and he observable variable z ha governs he change of regimes can be found in formula (2-5). y y = α1 + β1 y = α + β y 2 2 1 1 + ε1, + ε, 2 z χ z > χ (2-5)
Smooh swiching of regimes in so called smooh ransiion auoregressive (STAR) models addresses issues ha arise from abruply changing regimes in he simple model. A survey of recen developmens in his area is given in [16]. Regime swiching models are a model-driven approach. Arificial neural neworks can be used for daa-driven forecasing, wih he advanage of no having o choose an appropriae model for each problem. For ime series forecasing, ime lagged observaions and ime indices can be used as inpu variables, obaining he forecas as an oupu. Neural neworks have been frequenly and successfully used for forecasing purposes, a summary of work done in his area can be found in [17]. 2.6 Discussion Time series forecasing has been exensively researched in he las 40 years and a large number of empirical sudies have been conduced o compare ou-of-sample accuracy of various mehods. Among he bigges forecasing compeiions are he hree so called M- compeiions, consising of he M-compeiion 1982, he M2-compeiion in 1993 and he M3-compeiion in 2000. All hree of hem confirmed he same general resuls, as summarised in [18]. 1. Saisically complex models like ARIMA do no necessarily ouperform less sophisicaed approaches like exponenial smoohing. 2. Forecasing performance depends on he accuracy measure used. 3. Forecasing performance depends on differen ime horizons. 4. Combinaions of forecass do on average ouperform he individual mehods. Especially conclusion number one has been subjec o fierce discussions, many of hem disagreeing on he fundamenal quesion of wheher or no empirical evaluaions are an appropriae measure for he performance of a model. In [18], Makridakis and Hibon srongly criicise he approach of building saisically complex models, disregarding all empirical evidence ha simpler ones predic he fuure jus as well or even beer in real life siuaions. In [5] i is furhermore added, ha he only advanage a sophisicaed model has compared o a simple one is he abiliy o beer fi hisorical daa, which is no guaranee for a beer ou-of-sample performance. In [19], Sandy D. Balkin criicises he choice of he M3-compeiion daa ses as originaing mosly from financial and economic ime series and probably no conaining enough complex srucures ha could favour more sophisicaed models. Keih Ord adds in he same publicaion, ha ARIMA models need a leas 50 observaions o be efficienly esimaed, which is a requiremen ha is ofen violaed. Anoher exensive empirical sudy has been carried ou by Sock and Wason in [20], using 215 U.S. macroeconomic series, comparing 49 linear and nonlinear forecasing mehods. Looking a he nonlinear mehods, hey found ha neural neworks performed well for one-period-ahead forecasing, while showing deerioraing performance wih increasing forecasing horizons. A smooh ransiion auoregression model performed generally worse han he neural neworks. No clear-cu winner could be idenified comparing nonlinear mehods o linear ones, as he forecasing accuracy differed significanly across forecas horizons and series. In [21] however, he resuls of his sudy are quesioned by saing ha nonlinear forecasing mehods should only be
considered a all if he daa shows nonlinear characerisics. They furhermore criicise carelessness in he parameerisaion of he nonlinear mehods being examined in he sudy. A re-examinaion of he performances of linear and nonlinear approaches on ime series ha rejeced he saisical es for lineariy has consequenly been carried ou using seven macroeconomic ime series. The resuls, albei carefully considering suiable model parameerisaion, do no overly favour nonlinear models. A STAR model and one of he wo examined neural neworks had a slighly beer performance han linear models, he second neural nework had no. Summarising, he resuls of he enormous amoun of empirical sudies have been mixed and someimes conradicory; no single bes general mehod could be clearly idenified. In ([19]), Keih Ord suggess a rough guideline which mehod o choose. Generally, a small number of observaions, very erraic process behaviour and no or weak seasonal paern for a given ime series are srong indicaors ha simple mehods should be used. As he number of observaions grows and he series exhibi a sable sochasic and a srong seasonal paern, saisical crieria and conexual informaion should be used o idenify an appropriae, possibly sophisicaed model. 3 Forecas Combinaion Since he publicaion of he seminal paper on forecas combinaion by Baes and Granger in 1969 ([22]), research in his area has been acive; recen reviews and summaries can be found in [6] and [7]. In general, four main reasons for he poenial benefis of forecas combinaions have been idenified: 1. I is implausible o be able o correcly model a rue daa generaion process in only one model. Single models are mos likely o be simplificaions of a much more complex realiy, so differen models migh be complemenary o each oher and be able o approximae he rue process beer. 2. Even if a single bes model is available, a lo of specialis knowledge is required in mos cases o find he righ funcions and parameers. Forecas combinaions help achieving good resuls wihou in-deph knowledge abou he applicaion and wihou ime consuming, compuaionally complex fine-uning of a single model. 3. I is no always feasible o ake all he informaion an individual forecas is based on ino accoun and creae a superior model, because informaion may be privae, unobserved or provided by a closed source. 4. Individual models may have differen speeds o adap o changes in he daa generaion process. Those changes are difficul o deec in real ime, which is why a combinaion of forecass wih differen abiliies o adap migh perform well. This secion presens forecas combinaion mehods and provides a summary and discussion of empirical evaluaions o assess heir qualiy. 3.1 Linear Forecas Combinaion c The linear combinaion of forecass calculaes a combined forecas ŷ as he weighed y,.., ˆ ˆ1 as shown in equaion (3-1). sum of m individual forecass { } y m
yˆ ω yˆ (3-1) c = m i= 1 i i Weighs can be esimaed in various ways. One easy and ofen remarkably robus example is he simple average combinaion wih equal weighs. The opimal model proposed in [22] calculaes weighs using formula (3-2), where Σ denoes he covariance marix of he m differen forecas errors and e he n 1 uni vecor. 1 Σ e ω = (3-2) 1 e' Σ e A variance based approach firs menioned by Baes and Granger in [22] and furher exended by Newbold and Granger in [23] uses he average of he sum of he pas squared forecas errors ( MSE ) over a cerain period of ime as shown in formula (3-3). 1 MSEi ω i = m 1 (3-3) j = 1MSE j Granger and Ramanahan propose he regression mehod in [24] and rea individual forecass as regressors in an ordinary leas squares regression including a consan. Anoher group of linear forecasing mehods does no esimae a covariance marix or rely on pas error values. In a rank-based approach according o Bunn ([25]), each combinaion weigh is expressed as he likelihood ha he corresponding forecas is going o ouperform he ohers, based on he number of imes where i performed bes in he pas. Gupa and Wilon addiionally consider relaive performance of oher models using a marix wih pair-wise odd raios in [26]. Each elemen of he marix represens he probabiliy ha he model of he corresponding line will ouperform he model on he corresponding column. 3.2 Nonlinear Forecas Combinaion Poenially nonlinear relaionships among forecass are no considered in linear forecas combinaion, providing he main argumen for usage of nonlinear combinaion mehods. The mos invesigaed nonlinear mehod for forecas combinaion are backpropagaion feedforward neural neworks, where individual forecass are inpu daa and he combined forecas is obained as he oupu. This mehod was firs menioned by Shi and Liu in [27] and was also used in [28] and [29]. Fuzzy sysems for forecas combinaion can be found following wo differen paradigms. Firs, fuzzy sysems can be seen as a kind of regime model similar o he one described in secion 2.5, where wo or more differen forecasing models can be acive a one ime ([30]). In an empirical evaluaion in he same paper, he resuling fuzzy sysem almos always ouperforms or draws level wih he individual forecass and linear forecas combinaion mehods invesigaed. Two more publicaions emphasise a differen aspec of fuzzy sysems - he possibiliy of modelling linguisic and subjecive knowledge ([31], [32]). Combining exper forecass wih radiional ime series forecass resuled in significan performance gains in boh publicaions. In [33], He and Xu presen a self-organizing algorihm based on he Group Mehod of Daa Handling (GMDH) mehod which was proposed by Ivakhnenko in he 1970s
([34]). Individual forecass are aken as an inpu variable for he algorihm, differen ransfer funcions, usually polynomials, hen creae inermediae model candidaes for he firs layer. Ieraively, he bes models are seleced wih an exernal crierion and used as inpu variables for he nex layer, producing more complex model candidaes unil he bes model is found. Several auhors favour he approach of pooling forecass before combining hem. By grouping similar forecass and subsequenly combining he pooled forecass, several issues like increased weigh esimaion errors because of a high number of forecass o combine can be addressed. Research in his area recenly sared wih clusering forecass based on heir recen pas s error variance in [35] and coninued wih invesigaions by Riedel and Gabrys on how o exend and modify he clusering crieria in he conex of a big pool of individual forecass ha have been diversified by differen mehods in [2] and [3]. The ree-like srucures of hese muli-level muli-sep forecas combinaions can be evolved wih geneic programming, using he qualiy of he combined predicions on he validaion daa as he finess funcion o opimize. 3.3 Adapive forecas combinaion A consanly changing environmen is a ypical characerisic for an area in which forecass are applied. Assuming ha no individual model can be a perfec model of he rue daa generaion process and considering ha each individual model has a differen speed o adap o changes, here is a reason o believe ha forecas combinaion will perform well wherever adapiviy is needed. Taking ino accoun he fac ha he performance of individual forecass changes over ime, ime varying combinaion weighs have been invesigaed. One of he iniial papers on ha maer [36] proposes modelling a bigger impac of more recen observaions and leing he combinaion weighs be a funcion of ime. One plausible mehod in he conex of regression and variance-covariance based mehods is using a moving window of fixed size o deermine he number of laes daa collecion poins o include in he calculaion, as firs hough of by Baes and Granger in [22] and Granger and Ramanahan in [24]. Srucural breaks can degrade he performance of hese approaches. In [37], Pesaran uses a varying window size following a known srucural break by minimizing he expeced mean forecas error in an ieraive procedure. Deusch e al apply regime swiching models o combine forecass for he US and UK inflaion raes in [38]. The regime he economy is in is hen deermined dependen on differen funcions of he lagged forecas error. A more compuaionally complex approach is followed in [39] and [40], where ime varying coefficiens are deermined by applying a Kalman filer using an expanding window of observaions. Two of he fuzzy sysems menioned in he previous secion include a learning mechanism: The auhors of [32] periodically adap he rule bases by calculaing he confidence of an individual forecas based on heir pas performance. In [31], membership funcions are auomaically generaed by processing incoming daa, and hus adaped if new daa arrives.
3.4 Discussion Alhough here is a vas amoun of lieraure available on linear forecas combinaion, no sraighforward mehod of choosing he righ approach has been found. The relaive performance of he models depends on he error variance of he individual forecass, he correlaion beween forecas errors and he sample size for esimaion ([41]). Rankbased approaches work well for small sample sizes, while variance-covariance and regression mehods are more suiable for bigger daa ses. Similar error variances of individual forecass indicae ha simple averages migh be a good choice. Compared o lieraure abou linear forecas combinaion, he number of publicaions abou nonlinear mehods appears small. Empirical resuls are mosly only smaller case sudies, only one comparaive sudy invesigaing neural neworks, fuzzy sysems and neuro-fuzzy sysems for forecas combinaion has been found in [42]. Comparing algorihms using four differen error measures, i is concluded ha each of he presened nonlinear mehods always ouperforms boh of he individual forecass. The neuro-fuzzy approach performs bes among he hree. Some publicaions abou forecas combinaion wih neural neworks repor a significan improvemen over individual forecasing and linear forecas combinaion models ([43], [27]), however, in hese wo publicaions only one very shor ime series is used for evaluaion. No deails or jusificaions for he archiecure of he neural nework are given. A more exensive sudy finds ha he performance of neural neworks is mixed dependen on forecasing horizons or he error measure used ([28]). Self-organising algorihms have echnical advanages compared o neural neworks: They give an explici model for each problem and he number of hidden layers and neurons does no have o be deermined in advance. However, hey seem o be a lesserknown and popular mehod in forecas combinaion wih only one idenified publicaion using i in a very small empirical sudy ([33]). Resuls repored wih ime varying parameers are ambiguous. While Deusch e al. ([44]) repor significan improvemens of he ou-of-sample forecas error on one specific ime series, more exensive empirical sudies ([39], [40]) give discouraging resuls. According o hese papers, esimaing parameers wih a recursive Kalman Filer approach seems o be he leas fruiful approach, requiring relaively high compuaional power and yielding only small improvemens. Swiching regimes seem o be more promising. Summarising, he combinaion of forecass is generally considered as beneficial in mos of he lieraure invesigaed ([18], [41]). When i comes o nonlinear forecas combinaion or combinaion wih changing weighs, he mos opimisic sudies are based on very small daa ses while more exensive sudies repor mixed resuls, which indicaes ha he nonlineariy and adapiviy invesigaed is no beneficial for every ime series. 4 Forecas and Forecas Combinaion - Experimens A number of forecas and forecas combinaion algorihms have been described in he previous secions. Many of he empirical sudies menioned have one hing in common: forecasers spen a lo of ime and apply a lo of knowledge in designing more or less
sophisicaed mehods uned for specific ime series. In some bigger pracical applicaions however, his is ofen no feasible, because ofen a grea number of forecass has o be calculaed every second. Addiionally, expers wih sufficien knowledge o analyse a ime series and choose an appropriae model for i are rare. This secion describes an empirical experimen comparing he performance of off-he-shelf forecasing and forecas combinaion mehods ha have no been heavily uned and are likely o be employed by users who are no forecasing expers. 4.1 Mehodology A daa se consising of eleven monhly empirical business ime series wih 118 o 126 observaions each has been obained from he 2006/2007 Forecasing Compeiion for Neural Neworks and Compuaional Inelligence NN3 ([45]). Eighy percen of he daa is used for raining he forecasing models, he remaining weny percen for assessing heir ou-of-sample performance. For he forecas combinaions, hose weny percen have been divided ino wo equal pars, of which he firs one is used for calculaing combinaion weighs and he second one for comparing ou-of sample performances. The comparison beween forecass and forecas combinaions can hus only ake place in he las en percen of he daa se. The mean squared error (MSE) and he mean absolue percenage error (MAPE) have been used for assessing and comparing mehods. The MSE is used as an error measure in he majoriy of empirical forecas evaluaions, while he MAPE as a relaive error measure enables comparisons beween differen series. The following forecasing mehods have been implemened in Malab: 1. The naïve las observaion mehod, where he forecas is he laes observaion. 2. The moving average, where he forecas is calculaed according o he formula (2-1). A suiable size for he ime window is found by calculaing moving averages for values from 1 o 20 and choosing he value producing he lowes insample mean squared error. 3. Single exponenial smoohing using formula (2-2), where again a grid search beween 0 and 1 is employed o find he smoohing parameer ha produces he bes in-sample MSE. 4. An exponenial smoohing approach wih dampened rend inroduced by Taylor in [46] according o he formulas (4-1), where L denoes he esimaed level of he series and R a growh rae. The smoohing parameersα and β as well as parameerφ are again deermined by a grid search. φ yˆ = L R + 1 R = β ( L / L L = αy + (1 α)( L 1 1 R φ 1 ) ) + (1 β ) R φ 1 (4-1) 5. A polynomial regression, where a polynomial of he order four is fied o he ime series by regressing ime series indices agains ime series values. 6. The hea-model according o formula (2-4) using formulas given in [47], where wo curves are used and evenually combined by a simple average. One curve
originaes fromφ = 0, yielding a line ha can be easily exrapolaed; he oher curve usesφ = 2 and predics is fuure values by single exponenial smoohing. 7. An ARIMA model following secion 2.4. Since i is difficul o idenify an appropriae ARIMA model auomaically, a pragmaic approach was used: he firs difference of he series has been aken o remove a cerain amoun of nonsaionariy before submiing i o an auomaic ARMA selecion process ([48]). 8. A Feed-forward neural nework, whose characerisics have been seleced based on findings of an exensive review of work using arificial neural neworks for forecasing purposes by Zhang e al. in [17]. According o his publicaion, a backpropagaion algorihm wih momenum is successfully used by he majoriy of empirical sudies. One hidden layer is generally considered sufficien for forecasing purposes. I is suggesed o use as many hidden neurons as inpu variables, wih inpu variables being chosen wih regard o he applicaion. The number of lagged inpu variables and hus he number of hidden neurons has been se o 12 for he experimens here in order o capure yearly seasonal effecs, if presen. A grid search of he nine possible variaions of he values 0.1, 0.5 and 0.9 for he learning rae and he value of he momenum has been carried ou, indicaing bes resuls wih a learning rae of 0.1 and a momenum of 0.5. Furhermore, he following forecas combinaion mehods have been employed: 1. The simple average, which calculaes he average of all available forecass. 2. Similar o he previous mehod, simple average wih rimming averages individual forecass, bu only he bes performing 80% of he models are aken ino accoun. Performance was assessed by he MSE of he validaion period. 3. A variance-based forecas combinaion mehod using formula (3-3). 4. The radiional ouperformance mehod of Bunn, which was menioned in secion 3.1. Pas performance is again assessed in he validaion period. 5. A variance-based pooling approach inroduced by Aiolfi and Timmermann ([35]). Forecass are grouped ino wo clusers by a k-means algorihm according o he mean squared error of he validaion period. Forecass of he hisorically bes performing cluser are hen simply averaged o obain a final forecas. 6. A second variance-based pooling mehod, which is almos idenical o he previous one. The only difference is ha hree clusers are used raher han wo. 4.2 Resuls Table 4.1 shows average ou-of-sample MSE and MAPE values for he differen forecasing mehods numbered from one o en. MSE values are given boh relaive o he naive las observaion forecas and in absolue values. I has o be noed ha he average absolue MSE values have limied explanaory power and are only given for illusraive purposes, because he scale of each ime series is differen. The mos obvious resul is he superior performance of he neural nework (mehod eigh), srongly dominaing average MAPE values, reducing he nex bes mehod s MAPE by furher 41%. Using he relaive MSE measure, he neural nework performs marginally worse han mehods hree, four and six. The ARMA mehod, which represens he mos saisically complex mehod used in his es, fails o ouperform any oher mehod apar from he regression approach. Worh menioning is Taylor s modified dampened
rend exponenial smoohing (mehod number four), which is very easy o use and ranks bes using he MSE and second bes using he MAPE error measure. Table 4.1. MSE and MAPE error values for forecasing mehods, measured on he 20% esing se and averaged for each of he 8 mehods over he 11 series. Mehod MSE (relaive o naïve forecas) MSE (absolue, x 10 7 ) MAPE 1 1.00 2.7524 55.86 2 0.91 2.6886 55.14 3 0.88 2.6988 54.06 4 0.86 2.5561 28.31 5 98.32 27.972 3473.77 6 0.88 2.6988 55.34 7 1.40 2.6793 90.80 8 0.90 1.1682 16.67 Looking a he performance of forecas combinaion mehods in able 4.2, variancebased pooling wih hree clusers (mehod number six) is he bes performing approach in erms of he average MAPE error measure. The MSE gives differen average resuls: one big oulier for one of he ime series in he daa se corrups his mehod s overall performance, he variance-based mehod number hree performs bes insead. Simple average wih rimming resuls in second bes forecass for boh error measures. Table 4.2. MSE and MAPE error values for forecas combinaion mehods, measured on he 10% esing se and averaged for each of he 6 mehods over he 11 series. Mehod MSE (relaive o naïve forecas) MSE (absolue, x 10 7 ) MAPE 1 3.53 2.7229 57.36 2 0.79 2.1934 23.36 3 0.72 2.0632 26.85 4 8.26 1.6656 21.93 5 0.87 2.2638 24.08 6 1.66 0.96169 16.68 Table 4.3 compares forecass o forecas combinaions by showing he number of he bes performing mehod according o he MSE and MAPE of he validaion period ogeher wih he corresponding MSE and MAPE values of he esing period. Forecas combinaions are clearly beneficial in his experimen for boh error measures; for four ou of he eleven ime series hey ouperform individual forecas algorihms, five imes hey perform equally. In urn, hey are only ouperformed on wo of he series. Regarding he MAPE error measure, he error values on he wo series where he combinaions performed worse are sill quie similar, while on he oher hand, improvemens of up o 12% could be achieved from using hem on oher series. This does no hold looking a he MSE, where differences are more significan using
combinaions resuls in improvemens of up o 40% (series eigh), bu also in deerioraing performance of up o 72% (series one). Table 4.3. Bes performing mehod chosen in validaion period and corresponding MSE (relaive o naïve forecas) and MAPE error measures of he esing period, lef: forecasing mehods, righ: forecas combinaion mehods. Series mehod (MSE/ MAPE) Forecasing MSE (rel) MAPE Forecas Combinaion mehod MAPE (MSE/M APE) mehod (MSE/ MAPE) 1 8/4 0.42 2.74 5/2 1.48 2.73 2 8/8 0.46 17.27 6/6 0.46 17.27 3 8/8 0.33 44.31 6/6 0.33 44.31 4 8/8 0.39 16.44 6/6 0.38 14.96 5 8/8 2.07 3.24 6/6 2.07 3.24 6 8/8 0.62 6.87 2/2 0.70 6.94 7 5/5 9.31 9.65 6/6 9.31 9.65 8 4/8 0.29 11.57 6/6 0.25 10.50 9 7/7 0.57 4.27 6/6 0.57 4.27 10 8/8 4.87 25.36 6/6 2.91 22.40 11 3/8 0.90 20.34 6/1 0.82 21.95 avg 8/8 1.13 14.13 6/6 1.66 14.06 The resuls can be summarised as follows: When nohing is known abou a ime series and an individual forecasing mehod is needed, Taylor s dampened rend exponenial smoohing is a safe opion o improve upon he naïve las observaion forecas. When only considering a MAPE loss-funcion, a neural nework is likely o achieve consisen improvemens oo. Combinaions of forecasing mehods ouperformed forecass of individual approaches for he majoriy of he series. A simple average wih rimming performed comparaively well for boh he MSE and he MAPE error measure; variance-based pooling proved beneficial when working wih MAPE error measures. 5 Conclusion and Fuure Work This paper presened and discussed forecas and forecas combinaion approaches wih a focus on heir applicaion in airline indusry. Building on a shor inroducion o airline revenue managemen, he imporance and need for effecive forecasing procedures in airline indusry has been explained and jusified. Two differen approaches o forecasing demand and cancellaion raes ha are crucial in his conex have been idenified, one involves radiional ime series forecas and forecas combinaion mehods furher described in secions wo and hree. The second approach is predicion using daa on passenger level, which is an area of research in airline indusry forecasing ha was made possible wih he change from leg-based o O&D sysems in he 1990s. Currenly, relaed work only seems o have been done in he area of forecasing no-show raes. Tesing is conribuions for demand and cancellaion raes is an ineresing and novel issue.
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